EFFECT OF PARTIAL DEBONDING OF PRESTRESSING STRANDS ON BEAM END
CRACKING

By
Melissa Donoso Vidales

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Civil and Environmental Engineering
2011

ABSTRACT
EFFECT OF PARTIAL DEBONDING OF PRESTRESSING STRANDS ON BEAM END
CRACKING

By
Melissa Donoso Vidales

The use of shielded strands is an effective technique to reduce high tensile stresses in the
end region of prestressed concrete members. However, damage in the ends of prestressed beams
with shielded strands has been increasingly experienced during production. Guidelines on the use
of debonded strands, such as the AASHTO Bridge Design Specifications are mainly to address
concerns regarding the shear strength of pre-tensioned concrete elements. The bond behavior of
shielded 0.6 in. (15.2 mm) diameter strand was experimentally evaluated by conducting studies
on unstressed (pull-out tests) and stressed (pretensioned beams) concrete elements. The strands
were debonded using flexible sheathing. Experimentally calibrated finite element (FE) models
were developed to simulate bond behavior of shielded strand and assessed the induced stress
state in the elements. The effect of partial strand debonding on the anchorage zone of bridge
girders was studied through FE simulations within the context of a skew U-beam case study for
which evidence of damage exists. Pull-out test results showed no evidence of residual bond on
singular or doubly sheathed strand. However, experimentally calibrated numerical simulations
showed that strand debonded with flexible-slit sheathing leads to high radial stresses along the
debonded length in the concrete due to strand dilation from Poisson’s effect. The use of rigid
oversized sheathing was shown to decrease stresses in the transfer region and along the
unbonded length. A staggered distribution of the debonded strands can reduce the longitudinal
shear stresses created by the uneven distribution of prestress forces in the beam-cross section.

DEDICATION

To God, my parents, and my husband

iii

ACKNOWLEDGEMENT

I give special thanks to my advisor, Dr. Rigoberto Burgueño, for giving me the
opportunity of working with him, all the advice and encouragement, to keep doing my very best
even when I felt that things were not going the way that I wanted. I never imagined being a
research assistant and it was a great experience that I will not ever forget; thank you Dr.
Burgueño.
I thank Yi Sun and Mr. Sia Ravanbaskh for all the technical support. Yi, I will never
forget the countless hours of fun work at the CIL. That was an extraordinary experience. I also
thank all of the students and staff at the Department of Civil and Environmental Engineering at
Michigan State University for all their support. I thank Dr. Robert J. Frosch (Purdue University)
for sharing valuable information with us, which was of great contribution to the work presented
herein.
I want to deeply thank my great-uncle Guillermo Murillo for his support during my
studies in the U.S. Without his helped this would have not been possible. I am grateful for the
support that my parents have given me, all the moments that I thought this was tough, they
always will be there for me. I am grateful to my husband Darren for his moral support and
understanding, for being there for me in good and not so good times. And the most important
thank you, to God, who keeps me alive; helping me to do my very best every day of my life.
Never give up your dreams; there are the motor of our lives.

iv

Table of Contents

LIST OF TABLES

viii

LIST OF FIGURES

x

CHAPTER 1 INTRODUCTION: MOTIVATION AND DESCRIPTION OF THE PROBLEM 1
1.1 Introduction ...........................................................................................................................1
1.2 Background ...........................................................................................................................1
1.3 Problem Description ..............................................................................................................4
1.4 Hypotheses and Objective .....................................................................................................9
1.5 Project Phases ......................................................................................................................10
1.6 Scope ...................................................................................................................................11
1.7 Organization of the Report ..................................................................................................12

CHAPTER 2
BACKGROUND
14
2.1 Introduction ..........................................................................................................................14
2.2 Bond Phenomena in Pretensioned Concrete Members ........................................................14
2.2.1 Bond Mechanisms .........................................................................................................15
2.2.3 Bond Strength Determination ........................................................................................17
2.3 Transmission of Prestresing Force .......................................................................................21
2.4 End Cracking in Pretensioned Concrete Members ..............................................................24
2.4.1 Studies on Beam-end Cracking upon Detensioning ......................................................26
2.5 Use of Debonded/Shielded Strands in Prestressed Concrete Girders ..................................28
2.5.1 Code Guidelines: Provisions and Limitations ...............................................................30
2.5.2 Literature Review on Experimental and Numerical Studies on the use of Shielded
Strands in Pretensioned Concrete Beams ...............................................................................32
2.5.3 Literature Review on Numerical Models on Beam-end Cracking on Beams with
Shielded Strands .....................................................................................................................37

CHAPTER 3 EXPERIMENTAL ASSESSMENT ON BOND BEHAVIOR OF BONDED AND
SHIELDED STRAND
39
3.1 Introduction ..........................................................................................................................39
3.2 Description of the Experimental Program ............................................................................39
3.3 Assessment of Strand Quality ..............................................................................................40
3.4 Bond Evaluation of Unstressed Strand through Cylinder Pullout Tests .............................45
3.4.1 Test plan, description, and construction ........................................................................45
3.4.2 Test Procedure ...............................................................................................................47
3.4.3 Cylinder Pull-out Test Results.......................................................................................48
3.4.4 Bond Evaluation of Unstressed Strand through Cylinder Pullout Tests .......................53
v

3.5 Evaluation of Hydraulic Head Effect on Bond through Wall-like Blocks Pull-out Tests ..53
3.5.1 Test plan description and construction ..........................................................................54
3.5.2 Wall-like Block Test Procedure ....................................................................................56
3.5.3 Wall-like Blocks Pull-out Test Results .........................................................................58
3.5.3 Hydraulic Head Effect: Wall-like Blocks......................................................................62
3.6 Prestress Bond Transfer in Beams.......................................................................................63
3.6.1 Geometry, Reinforcing, and Prestressing Details..........................................................64
3.6.1 Test Procedure: Transfer of Prestressing Force into the Concrete Member .................69
3.6.2 Results of Bond Transfer in Beams ...............................................................................71
3.7 Findings and Discussion ......................................................................................................80

CHAPTER 4 COMPUTATIONAL ASSESSMENT
82
4.1 Introduction ..........................................................................................................................82
4.2 Concrete Damaged Plasticity Model ....................................................................................82
4.2.1 Parameters and Assumptions on the CDP Model..........................................................89
4.3 Description of the Contact Interaction Friction Model in ABAQUS ...................................90
4.4 Modeling of Prestress Bond Transfer in 2-D and 3-D Models ............................................92
4.5 Numerical Modeling of Laboratory-Scale Beams ...............................................................95
4.5.1 Description of the FE Beam Model ...............................................................................96
4.5.2 Results of Simulations on Bond Transfer on Prestressed Concrete Beams ................101
4.6 Influence of Confinement Reinforcement on Transfer Zone Stress State..........................119
4.6.1 Numerical Evaluation of Confinement in Prestressed Concrete Beams .....................120
4.6.2 Results .........................................................................................................................122
a) NCFBC FE Beam Model
122
4.7 Linear-Elastic vs. Concrete Damaged Plasticity FE Models .............................................128
4.7.1 Results: NCFBU Linear-elastic vs. NCFBU Inelastic FE Beam Models ...................128
4.8 Findings ..............................................................................................................................130

CHAPTER 5 NUMERICAL SIMULATIONS: STRESS ANALYSES OF ANCHORAGE
REGIONS
132
5.1 Introduction ........................................................................................................................132
5.2 Case Study: Skew U-Beam Unit ........................................................................................132
5.3 Simulation Case Studies .....................................................................................................135
5.4 As Built Condition: Case 1.................................................................................................137
5.4.1 Model Geometry, Mesh, Material Properties and Contact Definition ........................137
5.4.2 Results: Case 1 ............................................................................................................142
5.5 Case 2: Tight Fitting with Staggered Debonding Distribution ..........................................145
5.5.1 Specific Modeling Details ...........................................................................................146
5.5.2 Results .........................................................................................................................147
5.6 Case 3: Oversized Debonding with Original Unbonded Strand Configuration .................149
5.6.1 Model, Geometry, Mesh, and Generalities ..................................................................150
5.6.2 Results .........................................................................................................................152
5.6.3 Effects on Symmetry Assumptions .............................................................................155
5.7 Case 4: Oversized Debonding with Staggered Unbonded Strand Distribution .................160
vi

5.7.1 Generalities of the Model ............................................................................................161
5.7.2 Results .........................................................................................................................162
5.8 Comparison of Modeling Results .......................................................................................164
5.8.1 Case 1 vs. Case 2 .........................................................................................................165
5.8.2 Four Skew-symmetry Beam Models ...........................................................................170
5.8.3 Discussion....................................................................................................................180
5.9 Findings ..............................................................................................................................182

CHAPTER 6 RECOMMENDATIONS AND CONCLUSION
185
6.1 Relevant Findings ...............................................................................................................185
6.2 CONCLUSION ..................................................................................................................186

APPENDICES
188
APPENDIX A – Materials Testing and Concrete Mix Designs ..............................................189
APPENDIX B – Results from Experimental Program: ...........................................................191
APPENDIX C – Results from numerical small-scale beam models ........................................194
APPENDIX D – Construction Drawings and Results from Numerical Case Study: U-beam
Models ......................................................................................................................................213

REFERENCES

224

vii

LIST OF TABLES

Table 3-1 Amount of pull-out cylinders cast per batch ................................................................ 47
Table 3-2 Summary of maximum pull-out force on cylinders ..................................................... 50
Table 3-3 Maximum pull-out forces and bond strength values obtained from pull-out tests on
cylinders ........................................................................................................................................ 52
Table 3-4 Summary of average maximum pull-out forces: wall-like blocks ............................... 59
Table 3-5 Times at which final DEMEC were taken after strand release .................................... 71
Table 3-6 Transfer lengths for NCC beams ................................................................................. 75
Table 3-7 Transfer lengths for SCC beams................................................................................... 76
Table 3-8 Time normalized transfer lengths ................................................................................. 77
Table 3-9 Theoretical and experimental transferred strains ......................................................... 78
Table 4-1 Concrete damaged plasticity model parameters for NCC FE models ........................ 100
Table 4-2 Concrete damaged plasticity model parameters for SCC FE models......................... 101
Table 4-3 Summary of coefficient of friction used in calibration of models.............................. 103
Table 4-4 Normalized coefficients of friction based on f’ci= 8 ksi and on fsi = 202.5 ksi.......... 105
Table 4-5 Coefficients of friction normalized on fse = 189.5 ksi and f’ci=8 ksi ......................... 105
Table 5-1 Debonded strand length in U beam [8]....................................................................... 137
Table 5-2 Concrete damaged plasticity parameters for U-beam model ..................................... 141
Table 5-3 Interaction constraints in U-beam models .................................................................. 142
Table A-1 Seven-wire 0.6-in. (15.2 mm) diameter strand specifications ................................... 189
Table A-2Concrete Mix Design – Large Block .......................................................................... 189
Table A-3 Concrete mix design - NCC....................................................................................... 190
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Table A-4 Self-consolidating concrete mix design ..................................................................... 190

ix

LIST OF FIGURES
Figure 1-1 Debonded Strand using Flexible Sheathing .................................................................. 2
Figure 1-2 Photograph of a Shielded Strand using Rigid Debonding Material .............................. 2
Figure 1-3 Skew Box-beam damage observed during production [31] .......................................... 5
Figure 1-4 End-cracking seen on U-beam during production [23] ................................................. 5
Figure 1-5 Box-girder - cross section details. Adapted from [31] ................................................. 7
Figure 1-6 U-beam cross section details. Adapted from [8] [23] ................................................... 8
Figure 2-1 Schematic of bond stresses and wedging effect. Adapted from [16] .......................... 17
Figure 2-2 Bond stresses distribution at the anchorage-zone. Adapted from [16] ....................... 17
Figure 2-3 Radial Stresses at the interface of concrete and strand upon releasing the prestressing
force .............................................................................................................................................. 19
Figure 2-4 Dilation of the prestressing strand............................................................................... 19
Figure 2-5 Schematic of Simple Pull-out Test. Adapted from [16] .............................................. 21
Figure 2-6 Bursting zone and appearance of spalling and splitting cracks upon detensioning.
Adapted from [21]......................................................................................................................... 23
Figure 2-7 Bond stresses distribution in the end-region. Adapted from [16] ............................... 23
Figure 2-8 Cross section of a Type-I girder and typical cracks observed in the anchorage region.
Adapted from [9]........................................................................................................................... 25
Figure 3-1 Reinforcing details of large block: Front and side elevation ...................................... 41
Figure 3-2 Reinforcing details of the pull-out block .................................................................... 42
Figure 3-3 Large-block pull-out test set up................................................................................... 43
Figure 3-4 Overview of the large block pull-out test set up ......................................................... 43
Figure 3-5 Large block pull-out test results ................................................................................. 44
Figure 3-6 Reinforcement and geometry details of pull-out tests on cylinders ............................ 46
x

Figure 3-7 Test matrix and naming and matrix convention for cylinder pull-out tests ................ 46
Figure 3-8 Photograph of the cylinder pull-out test setup ............................................................ 48
Figure 3-9 Pull-out cylinder with longitudinal cracking.............................................................. 48
Figure 3-10 Typical force-slip response from pull-out test .......................................................... 49
Figure 3-11 Pull-out response for strand in NCC cylinders ........................................................ 50
Figure 3-12 Pull-out response for strand in SCC cylinders .......................................................... 51
Figure 3-13 Normalized peak bond strength values with respect to

f 'ci ................................. 52

Figure 3-14 Test matrix and naming convention for wall-like block tests ................................... 55
Figure 3-15 Wall-like block geometry details: side and front view ............................................. 56
Figure 3-16 Wall-like block reinforcing details............................................................................ 57
Figure 3-17 Overview of the wall-like block pull-out test ......................................................... 57
Figure 3-18 Pull-out curve response of the 5th row strand on wall-like blocks ............................ 59
Figure 3-19 Top-strand effect on wall-like blocks ....................................................................... 60
Figure 3-20 Bond stresses: Fully bonded strands in wall-like blocks .......................................... 61
Figure 3-21 Geometry of beam with fully-bond strand and with confinement ............................ 65
Figure 3-22 Photograph of end of fully bonded beam with confinement ..................................... 65
Figure 3-23 Test matrix and naming convention for beam tests .................................................. 66
Figure 3-24 Schematic beams layout ............................................................................................ 67
Figure 3-25 Overview photograph of the beams layout ............................................................... 67
Figure 3-26 Beam side prepared for strain measurement with DEMEC system .......................... 68
Figure 3-27 Overview of beam test setup and instrumentation at beam ends .............................. 69
Figure 3-28 Smoothing method for concrete surface strain profiles. Adapted from [32] ............ 69
Figure 3-29 Photograph of the three points where the strand was released.................................. 70
xi

Figure 3-30 Concrete surface strain profile: smoothed vs. unsmoothed curves ........................... 72
Figure 3-31 Concrete surface strains profiles for NCD1C and NCD1U beams ........................... 73
Figure 3-32 Concrete surface strain profiles for NCD2C and NCD2U beams............................. 74
Figure 3-33 Concrete surface strain profiles for SCFBC and SCFBU beams .............................. 74
Figure 3-34 Concrete surface strain profiles: SCC beams containing partially debonded strand
without confinement reinforcement .............................................................................................. 75
Figure 3-35 Theoretical and experimental strain level comparison – NCC beams ...................... 79
Figure 3-36 Theoretical and experimental strain level comparison – SCC beams ....................... 80
Figure 4-1 Failure surface on the deviatoric plane. Adapted from [1] ........................................ 86
Figure 4-2 Proposed yield surface in the stress space by Lubliner (1989). Adapted from [15] ... 86
Figure 4-3 Response of the concrete under uniaxial loading. Adapted from [1].......................... 87
Figure 4-4 Schematic of the basic Coulomb friction model and radial pressure due to dilation of
strand upon release ........................................................................................................................ 90
Figure 4-5 Sticking and slipping friction. Adapted from [1] ........................................................ 92
Figure 4-6 Prestress bond transfer in 2D ...................................................................................... 93
Figure 4-7 3D Modeling of prestress bond transfer using contact interaction definitions .......... 94
Figure 4-8 Geometry of the numerical beam model ..................................................................... 98
Figure 4-9 End region of the FE beam model............................................................................... 98
Figure 4-10 Fully bonded beam model ......................................................................................... 99
Figure 4-11 Typical debonded beam model ............................................................................... 100
Figure 4-12 Differential element subjected to normal and shear stresses .................................. 102
Figure 4-13 Longitudinal path along the FE beam models......................................................... 106
Figure 4-14 NCFBC beam model vs. experimental data ............................................................ 107
Figure 4-15 NCD1U model vs. experimental data ..................................................................... 107
xii

Figure 4-16 SCFBU beam model vs. experimental data ............................................................ 108
Figure 4-17 SCD2U beam model vs. experimental data ............................................................ 109
Figure 4-18 Maximum principal strain levels along cross section - NCC beams ...................... 110
Figure 4-19 Axial strain contour plots ........................................................................................ 112
Figure 4-20 Transverse strains contour plots: front and side views ........................................... 113
Figure 4-21 Maximum principal strain contour plots: side and front views............................... 114
Figure 4-22 Transverse stresses: side view and cross section at release end.............................. 116
Figure 4-23 Maximum principal stresses contour plots: front and side views .......................... 117
Figure 4-24 Schematic illustrating of high stresses along debonded zone ................................. 119
Figure 4-25 NCFBC beam model: close view of the end-zone .................................................. 121
Figure 4-26 NCD1C beam model with circular stirrups along debonded zone .......................... 122
Figure 4-27 Maximum principal stress: NCFBC beam model - cross-sectional path ............... 123
Figure 4-28 Axial strains: NCFBC model .................................................................................. 124
Figure 4-29 Transverse strains: NCFBC beam model – path along cross section...................... 125
Figure 4-30 Maximum principal stresses: NCFBC beam model – path along top beam center 125
Figure 4-31 NCD1C model: Maximum principal stress along top-beam path ........................... 127
Figure 4-32 Equivalent plastic strain in uniaxial tension (PEEQT) along cross section (at 4 in.
[100 mm] from the releasing end) .............................................................................................. 127
Figure 4-33 Transverse stresses – elastic vs. inelastic concrete behavior .................................. 129
Figure 4-34 Maximum principal stress: cross sectional path...................................................... 129
Figure 4-35 Maximum principal strain along front cross-section .............................................. 130
Figure 5-1 Damaged observed on skew U-beam during production at beam end [8][23] ......... 133
Figure 5-2 U-beam cross section. Adapted from [8] [23]........................................................... 134
Figure 5-3 View of the region of interest and debonding strand numbering: Cases 1 and 3...... 136
xiii

Figure 5-4 Partially debonding strand numbering: Cases 2 and 4 .............................................. 136
Figure 5-5 Overview of the U-beam girder as built condition model......................................... 138
Figure 5-6 End region of the U-beam as built condition model ................................................. 139
Figure 5-7 Damage seen during production of U-girder unit vs. maximum principal plastic
strains: Case 1 ............................................................................................................................ 143
Figure 5-8 Maximum principal stresses: U-beam as-built condition (Case 1) model ................ 144
Figure 5-9 Active Yielding of the concrete at beam cross section at release end: Case 1.......... 145
Figure 5-10 U-beam Bottom Flange for Case 2: Half-region along X-axis ............................... 147
Figure 5-11 Maximum principal plastic strain vs. damage observed during production: Case 2
..................................................................................................................................................... 148
Figure 5-12 Case 2: Maximum principal strain vs. damage observed during production .......... 149
Figure 5-13 Skew U-beam model with oversized hole and original debonded distribution: Case 3
..................................................................................................................................................... 151
Figure 5-14 End-zone view: U-beam model – Case 3 (Half-model along X-axis) .................... 152
Figure 5-15 Case 3: Maximum principal plastic strain at release end ........................................ 153
Figure 5-16 Case 3: Equivalent plastic strain in uniaxial tension ............................................... 154
Figure 5-17 Top view of the end of the bottom flange of the beam model with symmetry along
X-axis .......................................................................................................................................... 154
Figure 5-18 Maximum principal plastic strain: Case 1 vs. Skew- Symmetry Case 1 ................ 156
Figure 5-19 Effects on skew-symmetry boundary condition: Actively yielding........................ 157
Figure 5-20 Case 1 vs. Skew-symmetry conditions: Equivalent plastic strain in uniaxial tension
at beam end cross section ............................................................................................................ 158
Figure 5-21 Equivalent plastic strain in uniaxial tension at 12 in. from release end .................. 159
Figure 5-22 Maximum Principal Stresses: Case 1 vs. Skew-Symmetry .................................... 160
Figure 5-23 Overview of the model for Case 4 .......................................................................... 161
Figure 5-24 Results for Case 4: Maximum principal plastic strain ............................................ 163
xiv

Figure 5-25 Equivalent plastic strain in uniaxial tension: Case 4............................................... 164
Figure 5-26 Maximum principal stresses: Case 1 vs. Case 2 ..................................................... 165
Figure 5-27 Equivalent plastic strain in uniaxial tension: Case 1 vs. Case2 .............................. 167
Figure 5-28 Yielding in concrete: Case 1 vs. Case 2 .................................................................. 168
Figure 5-29 Shear Stresses on XZ Plane: Case 1 vs. Case 2 ...................................................... 169
Figure 5-30 Partially debonded strands selected for response traces ......................................... 170
Figure 5-31 Actively yielding: Four skew U-beam models........................................................ 171
Figure 5-32 Skew U-beam models: Maximum principal plastic strain at 12 in. (300 mm) from
release end ................................................................................................................................... 172
Figure 5-33 Maximum principal stresses at 12 in. (300 mm) from the release end ................... 173
Figure 5-35 Longitudinal path along partially debonded strand 1: Maximum principal stresses –
Skew-symmetry tight model vs. rigid and original configuration .............................................. 176
Figure 5-36 Response traces along partially debonded strand No.9: Maximum plastic strains –
Skew-symmetry U-beam models tight vs. rigid with original configuration ............................. 176
Figure 5-37 Longitudinal path along partially debonded strand No.11: Equivalent plastic strain in
uniaxial tension – Skew-symmetry tight vs. rigid with original unbonded pattern .................... 177
Figure 5-38 Maximum principal stresses: response trace along interface of partially debonded
strand No.1 – skew-symmetry model tight vs. rigid with staggered pattern .............................. 179
Figure 5-39 Maximum principal plastic strains along longitudinal path at the interface of
partially debonded strand No.9 ................................................................................................... 179
Figure 5-40 Equivalent plastic strain in uniaxial tension along longitudinal path at the interface
of partially debonded strand No.11 ............................................................................................. 180
Figure 5-41 Discussion of un-balanced forces due to strand debonding distribution................. 182
Figure A-1 Normalized average peak bond strength on pull-out cylinders ............................... 191
Figure A-2 Response of debonded strands in wall-like block cast with NCC – 5th row ............ 192
Figure A-3 Least square fit fully bonded strand in wall-like block cast with NC ...................... 192

xv

Figure A-4 Least square fit fully bonded strand: top-strand effect on wall-like block cast with
SCC ............................................................................................................................................. 193
Figure A-5 SCFBC numerical model vs. experimental data ..................................................... 194
Figure A-6 SCD1C beam model vs. experimental results .......................................................... 194
Figure A-7 NCD2C Beam Model vs. Experimental Data .......................................................... 195
Figure A-8 NCD2U Beam model vs. experimental curve .......................................................... 195
Figure A-9 Axial stresses along cross-section of beam at releasing end - NCC set ................... 196
Figure A-10 Transverse stresses along cross-sectional Path: NCC set of beams ....................... 196
Figure A-11 NCFBC Beam Model: Transverse Strains ............................................................. 197
Figure A-12 NCFBC Beam Model: Maximum principal plastic strains .................................... 197
Figure A-13 NCFBC Beam Model: Equivalent plastic strain in uniaxial tension...................... 198
Figure A-14 NCFBC Beam Model: Axial stresses ..................................................................... 198
Figure A-15 NCFBU Beam Model: Transverse strain ............................................................... 199
Figure A-16 NCFBU Beam Model: Maximum principal plastic strains .................................... 199
Figure A-17 NCD1C Beam Model: Transverse strain .............................................................. 200
Figure A-18 NCD1C Beam Model: Maximum principal stresses ............................................ 200
Figure A-19 NCD1U Beam Model: Axial stresses................................................................... 201
Figure A-20 NCD1U Beam Model: Equivalent plastic strain in uniaxial tension (PEEQT) .... 201
Figure A-21 NCD2C Beam Model: Axial stresses .................................................................... 202
Figure A-22 NCD2C Beam Model: Maximum principal strain ................................................. 202
Figure A-23 NCD2U Beam Model: Maximum principal plastic strain .................................... 203
Figure A-24 NCD2U Beam Model: Transverse stresses ........................................................... 203
Figure A-25 NCD2U Beam Model: PEEQT ............................................................................. 204
Figure A-26 SCFBC Beam Model: Maximum principal strains ............................................... 204
xvi

Figure A-27 SCFBC Beam Model: Transverse stresses ............................................................ 205
Figure A-28 SCFBC Beam Model: Maximum principal stresses ............................................. 205
Figure A-29 SCD1C Beam Model: Equivalent plastic strain in uniaxial tension (PEEQT) ...... 206
Figure A-30 SCD1C Beam Model: Axial stresses ..................................................................... 206
Figure A-31 SCD1U Beam Model: Axial strains ....................................................................... 207
Figure A-32 SCD1U Beam Model: Transverse strains .............................................................. 207
Figure A-33 SCD2U Beam Model: Maximum principal stresses .............................................. 208
Figure A-34 SCD2U Beam Model: Maximum principal plastic strains..................................... 208
Figure A-35 SCD2U Beam Model: PEEQT ............................................................................... 209
Figure A-36 PEEQT: NCFBC beam model – path along the cross section of the beam at 4 in.
(200 mm) from releasing end ...................................................................................................... 209
Figure A-37 Maximum Principal Strain – NCFBC Beam Model along cross section (4 in. [100
mm] from the releasing end) ....................................................................................................... 210
Figure A-38 NCFBC beam model – Transverse stresses along cross section (at 4 in. [100 mm]
from releasing end) ..................................................................................................................... 210
Figure A-39 NCD1C beam model – Maximum Principal Strain along cross section (at 4 in. [100
mm] from releasing end) ............................................................................................................. 211
Figure A-40 Maximum principal stress along cross section – NCD1C beam model ................. 211
Figure A-41 NCD1C beam model – Transverse Strain along cross section............................... 212
Figure A-42 Case 3: Maximum principal stresses at beam end.................................................. 213
Figure A-43 Results for Case 4: Yielding of concrete elements at beam cross section ............. 213
Figure A-44 PE Max Principal: Case 1 vs. Case 2 at beam end ................................................. 214
Figure A-45 Response traces along partially debonded strand No.9: Axial strains – Skewsymmetry U-beam models tight vs. rigid with original configuration........................................ 215
Figure A-46 Response traces along partially debonded strand No.9: Maximum principal strains –
Skew-symmetry U-beam models tight vs. rigid with original configuration ............................. 215
xvii

Figure A-47 Response traces along partially debonded strand No.1: Maximum principal plastic
strains – Skew-symmetry U-beam models tight vs. rigid with original configuration ............... 216
Figure A-48 Response traces along partially debonded strand No.9: PEEQT – Skew-symmetry
U-beam models tight vs. rigid with original configuration......................................................... 216
Figure A-49 Response traces along partially debonded strand No.11: Maximum principal plastic
strains – Skew-symmetry U-beam models tight vs. rigid with original configuration ............... 217
Figure A-50 Response traces along partially debonded strand No.9: PEEQT– Skew-symmetry Ubeam models tight vs. rigid with staggered configuration .......................................................... 217
Figure A-51 Response traces along partially debonded strand No.9: Maximum principal strains –
Skew-symmetry U-beam models tight vs. rigid with staggered configuration .......................... 218
Figure A-52 Response traces along partially debonded strand No.11: Equivalent Axial stresses –
Skew-symmetry U-beam models tight vs. rigid with original configuration ............................ 218
Figure A-53 Response traces along partially debonded strand No.11: Maximum principal plastic
strain – Skew-symmetry U-beam models tight vs. rigid with staggered configuration.............. 219
Figure A-54 Maximum principal stresses: Four skew U-beam models ..................................... 220
Figure A-55 Equivalent plastic strain in uniaxial tension at release end: Four skew U-beam
models ......................................................................................................................................... 221
Figure A-56 PEEQT: Four skew-symmetry U-beam models at 12 in. from release end ........... 222
Figure A-57 Axial stresses: Four skew-beam models ................................................................ 223

xviii

CHAPTER 1 INTRODUCTION: MOTIVATION AND
DESCRIPTION OF THE PROBLEM

1.1 Introduction
This chapter presents the motivation of this research and preliminary background
information. In addition, this chapter presents the description of the problem for which this
research was undertaken and examples that provide a better description of the issue. Moreover,
the research hypotheses, research objectives, the project phases, and the organization of the
report are also presented.

1.2 Background
Prestressed concrete members are extensively used as bridge girders. The concrete is
precompressed when the tensile force in steel strands is released into the concrete member.
Prestressed concrete members are made of high strength materials. However, when the
prestressing force is released into the concrete member the anchorage zone is highly stressed,
which can result in beam-end cracking.
The use of debonded strands in the precast/prestressed industry, especially for bridge
girders, has been beneficial as it permits the reduction of the high tensile stresses close to the free
end of the members. In addition, by debonding some strands the free length of the strand is
increased by reducing the restraining effect at the beam ends [19].
Partially debonded strands are considered a better solution than the use of draped strands
because of its easier implementation. Debonding is achieved by placing plastic tubing around the
1

strand along the desired length in order to prevent bonding between the steel and the surrounding
concrete [27], as shown in Figure 1-1 and Figure 1-2. However, cracking in the anchorage zones
of prestressed concrete beams containing unbonded strands have been observed during the
production process, as shown in Figure 1-3 and Figure 1-4. There are many factors that can
contribute to beam-end-cracking in prestressed concrete beams with blanketed strands, such as
the strand diameter, the compressive strength, the strand release rate and sequence, the restraint
effect due to free lengths strand, etc. Thus, detailed evaluation of this complex problem has been
elusive.

Figure 1-1 Debonded Strand using Flexible Sheathing
(For interpretation of the references to color in this and all other figures, the reader is referred
to the electronic version of this thesis)

Figure 1-2 Photograph of a Shielded Strand using Rigid Debonding Material

There have not been many studies conducted regarding the causes for end cracking in
prestressed beams. One of the most recognized works is the one conducted by Mirza and Tawfik
[19]. They conducted experimental studies and developed a one-dimensional numerical model to
2

sutdy the restraint effect in beam cracking. Mirza and Tawfik found that the short free strand
length can cause beam-end cracking and recommended the free strand length to be 5 percent of
the casting bed. Another representative study is that of Kannel et al. [13] who conducted
research to study the effects of strand release sequence on beam-end cracking. Kannel et al. [13]
used experimentally validated three-dimensional continuum-type finite element models with the
purpose of simulating the end region of a prestressed concrete beam. This study also verified
that unbonded strands can minimize stress levels in beam-end regions by increasing the free
strand length and thus minimizing the axial restraint effect.
Since the problem of beam-end cracking is concentrated in the anchorage zone and the
involved parameters are many, a full experimental assessment is cost-prohibitive. Numerical
simulations are thus more feasible; but they need to be realistic enough to represent geometry,
component interaction, and nonlinear material response.
Even though precast concrete fabricators follow the lastest design guidelines when
dealing with debonded strands, end-cracking is still being observed during the production of
some units. This is an important aspect that needs to be resolved in order to obtain high quality
products and maximize potential serviceability or durability performance. It shall be recognized
that the guidelines for the use of debonded strands provided by codes such as the AASHTO
Bridge Design Specifications [2] and the AASHTO Bridge Design Construction Specifications
[2] are mainly related to the shear performance of girders under service loads. Thus, these
recommendations are not aimed at avoiding possible end-cracking in girders with blanketed
strands during production.
While strand debonding has been in use with relative success for some time, the cracking
of beam-ends continues to be a problem. Furthermore, the idealized assumption that an
3

“unbonded” strand does not transfer any forces to the concrete beam has been proven wrong
through ad-hoc tests [6]. Thus, a detailed evaluation on the bond behavior of debonded strand,
and its effect on the transfer of prestressing forces at the ends of beams including cracking
potential is needed.

1.3 Problem Description
Blanketing of prestressing strands has become an effective technique to alleviate the high
tensile stresses in the anchorage region of pre-tensioned concrete beams during the strand release
procedure. However, end-cracking has been observed in some prestressed concrete beams
containing debonded strands. Two examples made accessible for this study are the damage in the
end region of skew bridge beams with U- and Box-type cross section [8][23][31] as shown in
Figure 1-3 and Figure 1-4. Figure 1-3 shows the end-cracking on the bottom of a skew-box
girder that occurred upon release of the prestressing strands. The skew angle of the box girder
was 40 degrees. Debonding of the strands was accomplished with double slit flexible sheathing
around the strand. In this study, the slit flexible sheathing will be also referred as soft debonding
material. The cross section of the box-girder is shown in Figure 1-5. The reinforcing details and
location of debonded and fully bonded strands are also shown in this figure.
The damage to a skewed U-beam containing partially debonded strand is shown in Figure
1-4. The skew angle was 18 degrees. The strands were also debonded using a soft sheathing
material. The cross section of the U-beam, the location of the fully bonded strands, and the
strand debonding pattern are shown in Figure 1-6. The U-girder was taken as case study for the
numerical simulations presented in Chapter 5 to investigate the effect of debonded strands on
end-cracking in full-size beams.
4

Figure 1-3 Skew Box-beam damage observed during production [31]

Figure 1-4 End-cracking seen on U-beam during production [23]

5

The box beam had an overall debonding percentage of 37 percent. The bottom row of
strands (Figure 1-5), 24 percent were debonded, and row 5 had 67 percent of its strands
debonded.

The AASHTO Bridge Design Specifications [2] limit the number of debonded

strands to 25 percent in total and to no more than 40 percent in a given single row.
Soft (slit) sheathing material offers a tight fit against between the strand and the
surrounding concrete, which could lead to high radial pressure at the interface of the two
materials as the strand tries to recover its original diameter upon release of the pre-tensioning
force. In some cases, it may be advisable to use a rigid (un-split) and oversized debonding
material to eliminate any degree of prestressing force that could be transferred along the
debonded zone and to ensure that radial stresses would not develop as the strand dilates.
Pavelchak [23] and Sun [31] have recommended the use of un-split (rigid) debonding material
when effectiveness of debonding becomes essential.

6

Figure 1-5 Box-girder - cross section details. Adapted from [31]
7

Figure 1-6 U-beam cross section details. Adapted from [8][23]
8

1.4 Hypotheses and Objective
Even though the use of debonded strands decreases the high tensile stresses in the endregion of prestressed concrete members, cracking in this zone has been observed during
production. The research hypotheses behind this damage are:
1) Beam-end cracking could occur because the debonding mechanism used does not
effectively break the bond between the strand and the surrounding concrete, thus
allowing some transfer of the prestressing force where it was not accounted for.
2) The radial stresses caused by the increased dilation of the partially debonded strands
can be quite significant in the end region and the cause for cracking at the beam end.
3) The strand debonding pattern could lead to unbalanced forces resulting in high
longitudinal shear stresses if all the blanketed strands are grouped together, which can
induce shear cracking at the beam end or a high-stress region in the anchorage zone
that can compromise the shear strength.
The research objective of this study was to obtain a better understanding on the bond
behavior of partially debonded strands and its effect on beam-end cracking upon release of the
prestressing force into the concrete member. Moreover, it was of importance to investigate the
influence of concrete, concrete fluidity and confinement reinforcement.

Thus, the specific

research aims were:
• To characterize the basic bond behavior of bonded and debonded 0.6-in.-diameter
(15.2 mm) Grade 270 (1860 MPa) low relaxation prestressing strand.
• To study the effect of concrete fluidity using normal consolidated concrete and selfconsolidating concrete with bonded and debonded 0.6-in.-diameter (15.2 mm) strand.

9

• To study the effect of confinement from steel reinforcement on the bond behavior of
bonded and debonded 0.6-in.-diameter (15.2 mm) strand.
• To develop high-fidelity finite element models that can emulate strand bond behavior
to further investigate the mechanisms of bond stress transfer and potential sources of
damage.
• To study the effects of strand debonding on beam end cracking in the anchorage zones
of small and full size beams through numerical simulations.
The research hypotheses were evaluated after analyzing the experimental and numerical
simulation results as presented in Chapters 3, 4, and 5. The objectives were achieved by
separating the study into three tasks as presented next.

1.5 Project Research Tasks
The research project was subdivided into the following three tasks: (1) strand bond
behavior assessment, (2) strand bond behavior simulation, and (3) simulation studies on
anchorage zones.
Task 1 focused on determining the fundamental bond behavior of 0.6-in.-diameter (15.2
mm) Grade 270 (1860 MPa) low relaxation prestressing strand under fully bonded and partially
unbonded conditions. The objective of this first task was to generate data to calibrate threedimensional finite element models for strand bond behavior simulation in Task 2. The
parameters considered for the experimental phase (Task 1) were: (1) concrete hydraulic head, (2)
concrete fluidity (SCC), (3) confining reinforcement, and (4) single vs. double layered slit
sheathing for debonding of the strand. Basic bond behavior was assessed through pull-out tests
on 6” (150 mm) diameter, 24” (610 mm) long cylinders with concentric 0.6-in.-diameter (15.2
10

mm) strand. The effects of concrete fluidity were studied conducting pull-out tests on wall-type
blocks containing 0.6-in.-diameter (15.2 mm) unstressed strand and built with normally
consolidated (NCC) and self-consolidating concrete (SCC) Finally, the stress transfer behavior of
fully bonded and debonded 0.6-in.-diameter (15.2 mm) strand was evaluated through laboratoryscale pre-tensioned concrete beams.
In Task 2, 3-D nonlinear FE beam models were developed for the laboratory-scale beams
of Task 1 using the commercial finite element program ABAQUS [1] and calibrated with
experimental data. The models used a concentric fully bonded or partially debonded strand with
tight fit. The strand was simulated as a cylindrical rod and bond was simulated by the definition
of normal and tangential surface interaction properties between the steel and concrete parts of the
model. The coefficient of friction used in the tangential surface contact interaction definitions
was calibrated with experimental data. The calibrated models were used to study the effect of
confining reinforcement along the transfer zone and debonded length.
Task 3 focused on the stress analysis of the anchorage region of a large-scale beam models.
A U-type cross section beam (seen Figure 1-6) was selected as case study based on evidence of
damage during production (Figure 1-4). Three-dimensional nonlinear numerical models were
developed using ABAQUS [1]. Four U-beam FE models were created and analyzed to study the
effect of sheathing material and the arrangement of debonded strands.

1.6 Scope
The effectiveness of debonding using a single layer or double layer of flexible-slit
sheathing was evaluated by conducting experimental studies on unstressed 0.6-in. (15.2 mm)
11

diameter fully bonded and debonded strand. Pull-out tests on unstressed strands were conducted
to evaluate basic bond behavior associated with chemical adhesion and mechanical interlock.
The effects of concrete fluidity and hydraulic head on the possible paste infiltration of slit
sheathed strand were evaluated through pull-out tests on wall-like blocks. Finally, stress-transfer
behavior of fully-bonded and debonded strands was studied with laboratory-scale pretensioned
concrete beams.
Additional investigations were carried out through high-fidelity numerical simulations.
Models of the laboratory-scale beam units were established and calibrated with the experimental
data to determine effective parameters for the simulation of bond-slip response between strand
and concrete. Stress analyses on the anchorage zone of U-beam were then used to evaluate the
effect of flexible tight fitting debonding vs. rigid debonding material as well as the influence of
strand debonding pattern on beam-end damage.
It was the goal of this study to provide recommendations to the precast/prestressed
industry on the effects of the type of sheathing material, slit-flexible or rigid (oversized),
arrangement of strand debonding distribution, and reinforcing confinement in the end zone of
pre-tensioned concrete members during production.

1.7 Organization of the Report
The present report has been organized into six chapters. Chapter 2 provides a literature
review on the fundamental concepts of bond phenomena, end-cracking on prestressed concrete
beams, top-strand effect, bond behavior on units cast with self-consolidating concrete, and a
summary of the most relevant studies related to the above mentioned topics. Chapter 3 reports on
the experimental program: tests units details, procedures, results, findings, and discussion of the
12

results. Chapter 4 presents the numerical simulation of strand bond behavior on small-scale beam
models calibrated using experimental data. Results from the finite element models, relevant
findings, and a discussion of the results are provided. This chapter also presents theoretical
concepts behind the numerical models, such as the modeling of inelastic concrete behavior and
the modeling of friction contact interaction to simulate bond. Chapter 5 provides numerical
studies on anchorage zones of skew U-girders, including the evaluation of “the as-built”
condition, parametric studies, and discussion of relevant findings. The conclusions and
recommendations of this study are presented in Chapter 6. The Appendix section provides
supplementary documentation of the results obtained in the experimental and numerical tasks.

13

CHAPTER 2 BACKGROUND

2.1 Introduction
One of the goal of this investigation was to study the bond behavior of “unbonded”
strands since it may be possible that the use of flexible tight fitting sheathing could lead to high
radial stresses in the anchorage region can cause end-cracking. First, it is needed to study bond
behavior in fully bonded strand to assess the different bond mechanisms. Thus, a literature
review on this topic and how this has been studied in the past is presented in this chapter. In
addition, a literature review on beam-end cracking in prestressed concrete beams upon release
with and without debonded strands is also presented here. Moreover, code recommendations on
the use of unbonded strands in pretensioned concrete members are also summarized. Finally, an
overview of some of the possible causes behind the end-cracking in beams with debonded
strands is presented. The chapter is organized as follows:
•

Bond behavior in pretensioned concrete member including bond mechanisms,

•

Transmission of prestresing force,

•

End-cracking in pretensioned concrete beams, and

•

Use of shielded strands in prestressed concrete girders.

2.2 Bond Phenomena in Pretensioned Concrete Members
When the steel is prestressed, bond is needed to hold the steel under tension and
consequently precompress the concrete [27]. In prestressed concrete applications, the tensile
stresses of the pretensioned strand are fully transmitted into the concrete member by bond. The
14

bond transfer varies from zero at the beam end and increases until the tensile force in the strand
is constant, reaching the effective prestressing force fse. The region along which the prestressing
force is transferred to the concrete is often called the transmission or transfer zone. If cracking
takes place in the concrete member, the bond between the strand and the surrounding concrete
plays a significant role in governing flexural response and shear [12]. Thus, it is important to
ensure a satisfactory bond quality to prevent bond-slip and thus undesired failure mode in a
member. The bond mechanisms in pretensioned concrete members are presented in the following
to better understand the components of bond between strand and the surrounding concrete.

2.2.1 Bond Mechanisms
The main factors that contribute to the bond between strand and the surrounding concrete
are [12] adhesion, friction, and mechanical interlock. The bond between strand and concrete is
created by the chemical adhesion of the cement paste to the strand. Mechanical interlock
resistance follows from the geometry of the helical strand, which provides shear resistance
against the surrounding concrete. When the steel is prestressed, the diameter is reduced, and as
the prestressing force is released into the concrete member the strand expands due to its
Poisson’s ratio and creates a wedging effect in the anchorage region [12]. In addition, the
increase in the strand diameter causes a radial pressure at the interface with the surrounding
concrete. As result, the longitudinal frictional forces preventing strand slip are increased. The
frictional forces depend on the radial pressure exerted on the surrounding concrete due to the
dilation of the strand and on the coefficient of friction between the two surfaces in contact. The
value of the coefficient of friction depends on the surface conditions of the strand and it also
depends on the character of the cement paste [12].
15

However, adhesion plays a minimal role because once there is some relative slip of the
strand the chemical adhesion mechanism is broken. On the other hand, friction has a significant
contribution in the development of transmission of shear or bond stresses because it prevents
strand-slip along the transfer zone. The strand upon release from its pretensioned state at the
abutments locally recovers its original diameter causing a wedging effect in the end-region (see
Figure 2-1) known as Hoyer’s effect. To develop frictional bond stresses, radial compressive
stresses are needed. These radial compressive stresses have been attributed to the Hoyer effect
because the reduced diameter of the steel causes radial stresses as the strand tries to recover its
original diameter upon release. The Poisson’s expansion causes compression perpendicular to the
steel-concrete interface as shown in Figure 2-1. As a result, the concrete is precompressed.
Moreover, shrinkage of the concrete can also contribute to the radial compressive stresses that
cause frictional resistance [4]. Figure 2-1 shows an schematic of the Hoyer effect upon
detensioning and the radial stresses at the interface of the strand and the surrounding concrete.
Figure 2-2 shows a schematic of the transverse tension from the pretensioned strand to the wire
and transverse compression toward the strand. High bond stresses thus develop in the zone of the
radial compressive forces due to the Hoyer effect, which is the main contributor to bond between
the strand and the surrounding concrete.

16

Figure 2-1 Schematic of bond stresses and wedging effect. Adapted from [16]

Figure 2-2 Bond stresses distribution at the anchorage-zone. Adapted from [16]

2.2.3 Bond Strength Determination
The bond between steel and the surrounding concrete can be determined by means of
analytical, experimental or numerical approaches. A summary of some of the most relevant
studies on bond strength determination are presented next.

17

Several researchers have developed expressions to estimate bond stresses. The equations
have been based on different assumptions according the type of units built and tested for the
purpose of determination of bond. In this section, some of the theoretical expressions most
commonly cited in the literature are presented.

Janney, 1954
Janney [12] conducted one of the earliest studies on bond behavior. In addition to
conducting experimental tests on bond transfer and flexural bond, he also presented theoretical
considerations to determine bond stresses. If the strand is free to expand, a decrease in the
pretensioning force from its initial value will result in an increase of the strand diameter. If the
concrete section is large compared to the radius of the prestressed steel the elastic theory of thick
walled cylinders gives a relation to find the increment of the steel using Equation 2-1.

∆r = rσ r

1 + υc
Ec

Equation 2-1

In Equation 2-1, σr is the radial stress in the steel (see Figure 2-3); υc is the Poisson’s
ratio of the concrete; Ec is the elastic modulus of the concrete; and, r represents the strand
effective radius. The radial stress is equal to the contact pressure between the steel and the
surrounding concrete. Figure 2-4 shows the change in diameter of the strand at each stage:
unstressed, prestressed, and after detensioning. As the prestressing force is released, the strand
expands creating radial pressure at the interface between the strand and concrete as shown in
Figure 2-3.

18

Figure 2-3 Radial Stresses at the interface of concrete and strand upon releasing the
prestressing force

Figure 2-4 Dilation of the prestressing strand

σr =

( f se − f s )υ s
E
1 + (1 + υ c ) s
Ec

Equation 2-2

Thus, the magnitude of the radial stress can be found using Equation 2-2, where 2-2, fse
represents the prestressing force in the steel; fs is the tensile stress in the steel at any point; υs is
the Poisson’s ratio of the steel; and Es is the Young’s modulus of the steel. Prestress bond
stresses at any location along the transmission zone are equal to the slope of the stress transfer
curve multiplied by r/2 [12] as represented in Eq.2-3.
19

u=

dfs r
dl 2

Equation 2-3

If the bond between strand and the surrounding concrete is entirely due to the friction
(φ) between the strand and the surrounding concrete, bond stresses can be related to radial
stresses (σr) using Equation 2-4 as proposed by Janney [12]. However, it is well known that
while friction is one of the main bond mechanisms, there are other mechanisms involved in
prestress bond transfer.

u = φσ r

Equation 2-4

Bond can be experimentally measured by conducting different types of bond assessment.
One of the simplest ways to measure bond strength is by conducting pull-out tests on unstressed
strand. A schematic of a simple pull-out test is shown in Figure 2-5. Most pull-out tests are
conducted on unstressed strands; thus, the bond contribution from Hoyer and Poisson’s effects
cannot be accounted for in this type of test. The bond strength by means of this test is
determined using Equation 2-5. Where, Fmax is the peak pull-out force (lbs. [N]), db is the
nominal diameter of the strand (in. [mm]), and Le is the embedment length (in. [mm]).

u=

Fmax
(4/3)πd b L e

20

Equation 2-5

Figure 2-5 Schematic of Simple Pull-out Test. Adapted from [16]

2.3 Transmission of Prestresing Force
The pre-tension force in the stressed strand is released when the concrete has attained
enough compressive strength. Once the strand is released, it tends to shorten and it dilates. The
pretensioning force in the strands is purely transmitted into the concrete member by bond, and as
a result the concrete is precompressed. According to Moon et al. [20], both sudden (flamecuting) and gradual (hydraulic jacking) cause a very rapid strain change of the stress state at the
end of the concrete beam. Some of the primary factors affecting the structural integrity of a
prestressed concrete member are the prestress release method and the bond integrity between the
strand and concrete. Cracks may thus develop in the end zone of a pretensioned concrete member
due to an undesirable tensile strain change during the detensioning procedure [20].

21

Prestress bond transfer takes place from the end of the concrete member, release end, to a
point in which the stress in the strand remains constant. This constant level of stresses in the
strand is termed the effective prestress, fse. The length over which the entire prestressing level is
transmitted into the concrete members is known as transfer length. The transfer length mainly
depends on the degree of prestressing force, surface condition of the prestressing strand, concrete
compressive strength at release, release method [9], quality of the bond, radial pressure, and
transverse reinforcement [16]. Beyond the transfer length, the stress in the steel remains constant
as shown in Figure 2-7. The determination of transfer lengths has been extensively studied since
it

is

an

important

design

parameter

in

pretensioned

concrete

structures

[4][5][12][20][22][23][25][26][30]. Numerous experimental studies on the evaluation of transfer
length have been conducted since this is a parameter that can be easily measured and used for
calibration of numerical models (as was case of this investigation). Results have then been
commonly used for the development of empirical expressions to determine transmission length.
However, this is not the scope of this investigation.
If the transmission length is short, it may lead to high compressive stresses that can
cause spalling or bursting cracks in the end region of the concrete member (see Figure 2-6). On
the other hand, long transfer lengths could reduce the shear capacity of the beam and longer
development lengths will be required, which may affect flexural behavior [5]. Previous studies
[4][22][27][31] have shown that sudden release of the prestressing force results in larger transfer
lengths. This phenomenon is mainly related to the dynamic effects associated with the transfer
energy from the strand to the concrete member [4].

22

Figure 2-6 Bursting zone and appearance of spalling and splitting cracks upon
detensioning. Adapted from [21]

Figure 2-7 Bond stresses distribution in the end-region. Adapted from [16]

23

Transfer lengths can be experimentally measured using different approaches. The most
common is to measure the development of compressive strains in the concrete along the transfer
region. This can be done by taking measurements on the surface of the concrete element or by
using strain gages on the prestressing strand or an instrumented rod embedded in the concrete.
Other simple methods, such as the measurement of the strand end-slip, or “suck-in,” into the
beam have also been used. Only the first method, that of using surface strain measurements is
reviewed here since it is the approach used in this study.
Concrete surface strains can be measured using mechanical gages metal disks that are
installed along a surface path of interest. The distance between target points is measured before
and after release of the pretensioning force. The difference between the final and initial reading
upon the initial reading gives the strain over certain gage length. Strain values are plotted along
the beam length in the form of a profile. A common method to obtain the transfer length from the
compressive strain profile is the 95 percent average maximum strain (AMS) method [26]. The
AMS is obtained from the strain values near the plateau of the trend once the prestressing force
has become uniform. The 95% AMS value is plotted on the strain profile and the point at which
the 95% AMS line intersects the curve defines the transfer length. The 95% AMS method was
proposed by Russell et al. [26] and it has been well accepted by other researchers. This method
was used for the experimental program of this investigation (see Section 3.6.1)

2.4 End Cracking in Pretensioned Concrete Members
Prestressed concrete I-girders often experience cracking at their ends upon strand
detensioning. Such initial cracking accelerates the deterioration of the anchorage region of the
24

girder with the presence of moisture principally by accelerating chloride ingres and corrosion
initiation of shear reinforcement and prestressing steel [14]. The probability of end-cracking in a
prestressed member upon release can increase when the free strand length is short since the
restraining force increases [19]. Moreover, the restraint force will be larger when the strand is
not simultaneously released at the two free ends [19].
The formation of cracks in the end-zone of prestensioned concrete girders is primarily
caused by the high concentration of stresses in the end-zone upon release of the prestressing
strands. In the case of Type-I girders, cracks are usually horizontal and take place near the joint
of the bottom flange and the web [33] as shown in Figure 2-8. According to Tuan et al. [33] the
distribution of high stresses in the anchorage zone of prestressed concrete members is dependent
of the location of strands, the degree of the prestressing force in the strands, the amount of
concrete surrounding the strands, the amount of draped strands in the beam-end zone, the
geometry of the girder, and the concrete characteristics.

Figure 2-8 Cross section of a Type-I girder and typical cracks observed in the anchorage
region. Adapted from [9].

25

In accordance with preliminary studies, Zia [19] concluded that one possible reason for
the end-cracking problem was the restraining effect caused by unreleased strands. If the distance
between abutments and casting bed is relatively small, this restraining effect can be significant.
The debonding of strands increases the strand free length, which could reduce the high stresses
induced in the concrete element due to the releasing of the prestressing force.

2.4.1 Studies on Beam-end Cracking upon Detensioning
The amount of studies conducted regarding beam-end cracking upon release is few. The
most recognized studies in this topic are those by Mirza and Tawfik [19] and Kannel et al. [13].
A recent study by Sun [31] has also addressed the problem and it is a companion to this study.
However, the research by Sun included the use of debonded strands and a summary of Sun’s
investigation and main findings are presented in Section 2.5.2 and Section 2.5.3. An overview of
the first two works is presented next.

Mirza and Tawfik, 1978

Mirza and Tawfik [19] conducted a detailed investigation to understand the causes behind
the appearance of vertical cracks at the ends of prestressed concrete beams. Their study included
experimental and analytical components on the development of a model to estimate tensile
stresses from the restraint effect from free strand lengths. The experimental phase of their
research was conducted on Type- III AASHTO girders of approximately 73 ft. (22.3 m) in length
to study beam-end cracking during production. The main goal of the experimental investigation
was to study the restraining effect from the short length of unreleased strands in beam-end
cracking upon strand detensioning. The experimental study was performed on small-scale beams
26

with 3/8-in. (9.5 mm) diameter strands. The study’s principal criterion was to cause enough
cracking in the concrete due to the strand restraining effect. Prestressing force was transferred
into the concrete by simultaneous release at the beam ends at three intermediate points (two
beams in a line). Mirza and Tawfik found that the release restraint effect can be significant if the
strand free length is short.
The analytical model resulting from this work was a one-dimensional representation of
the casting bed, concrete beams and prestressing strands. Beam members and each length of free
strand were represented using individual springs. The mathematical model was based on the
following assumptions: a) the materials were modeled using linear-elastic properties, b) no
friction was provided between beams and casting bed, c) dynamic effects were neglected, and d)
the stress calculation was based on beam theory. The model provided useful information in
respect to the effect of the free strand length in the casting bed and its distribution when casting
multiple beams in a single line. The numerical model did not attempt to replicate the threedimensional stress distribution in the end the regions of the prestressed concrete member, and
thus the regions subjected to high stress were not identified and bond behavior was not replicated
in the model. Satisfactory agreement between the model and the experimental results was
observed after half of the strands were released. The effects of un-released strand and free strand
length were successfully investigated with the model. Mirza and Tawfik recommended that the
free strand length shall be equal or greater to 5% of the length of the casting bed.

Kannel, French, and Stolarski (1997)

Kannel, French, and Stolarski [13] studied the cracking at beam ends within the context
of the sequence of strand release during detensioning, the debonding of strands to increase free
27

length, and the pre-cutting of some strands. They conducted experimental and numerical studies
and their approach was to utilize experimentally calibrated finite element models to account for
the stress components that take place in the transfer of prestress forces into the concrete member.
Their finite element model was developed using the program ABAQUS [1] and calibrated with
experimental data. The strands were modeled using discrete truss elements, thus, the Poisson’s
and Hoyer effects due to the dilation of the strand upon detensioning was not replicated. The
concrete was modeled using 3-D continuum elements (along the region of interest) and linearelastic material properties. The experimental studies were conducted on 54 in. (1370 mm) deep
prestressed concrete beams. The main objective was to study release patterns that could prevent
cracking, thus the researchers did not consider important to model the inelastic behavior of the
concrete and the model was considered to be truthful until the first crack developed. They
observed that the faster the stress concentration at the base of the web increased in proportion to
the rate upon which the compression is transmitted into the concrete member. Thus, they
recommended combining pre-releasing of the strands with debonding of some strands to prevent
shifting the vertical cracking further into the concrete member when using debonded strands.
They also found that altering the release sequence and using debonded strands could help to
reduce the negative restraint effect that can cause end cracking. The model was judged
satisfactory, showing general trends, such as the ones related to altering the strand release order.

2.5 Use of Debonded/Shielded Strands in Prestressed Concrete Girders
Strand debonding consists in the elimination of the strand-concrete bond over a portion of
the strand in a prestressed concrete element, and it is an option to regulate the high compressive
and tensile stresses in the prestressed concrete members [27]. Debonding practice has been
28

established based on constrained empirical data and engineering judgment. Strand debonding is
an option to the more traditional method to reduce high stresses in the end zone of prestressed
concrete members by draping the strands upward, which modifies the center of gravity of the
strands and thus efficiently decreases concrete stresses in the anchorage region [27].
Strand debonding, or sheathing, is done by placing plastic tubing around the strand to
separate it from the concrete and thus eliminate bond resistance. The sheathing material is of two
general types: soft polyethylene slit tubing used in either a single or double layer with opposing
locations for the slit opening (see Figure 1-1), or a more rigid oversized plastic tubing (see
Figure 1-2). In this study, two types of debonding material were considered: soft (tight fitting)
and rigid (loose fitting) as shown in Figure 1-1 and Figure 1-2, respectively. Soft slit sheathing
fits closely to the strand and it is easily “squeezed” for a tight fit by the fresh concrete; thus
creating geometric contact between the strand and the concrete, albeit separated by the polymer.
By contrast, the oversized polymer tube is stiff enough that it does not collapse with the fresh
concrete and thus leaves a gap between the strand and concrete.
The use of partially debonded strands reduces the axial tension because the effective free
length of the strand is increased. This effect can be debated. Yet, with the use of debonded
strands, the axial restraint effect decreases, shear stress distribution is improved, and the shear
stresses are reduced. Moreover, the stress concentration at the base of the web, as in the case of
girders, is also decreased after full release of the strands due to the decrease in compressive force
transferred at the end of the concrete member [13]. Hence, the use of debonded strands has been
beneficial in reducing the occurrence of cracking in the beam-end regions.
As introduced before, strand debonding has been generally successful in minimizing
beam-end cracking. However, concerns have recently emerged on the effect that partially
29

debonded strand may have on the damage in beam ends (see Figure 1-3 and Figure 1-4)
[8][23][31]. Unfortunately, only limited research has been done to evaluate the unstressed and
stressed response of shielded, or debonded, strands, and code guidelines do not address
fundamental parameters of influence in beam-end cracking and they focus mainly on the possible
reduced shear strength rather than on the potential damaging effects from production. The next
sections provide an overview of current code provisions and research related to debonded strand
in prestressed concrete.

2.5.1 Code Guidelines: Provisions and Limitations
The AASHTO LRFD Bridge Design Specifications [2] on the use of debonded are
mainly based on concerns related to shear performance. ACI-318 [3] does not present any type of
limitations or recommendations related to the use of debonded strands in pretensioned girders
except for the expression to calculate development length. The recommendations provided in
both codes are summarized next.

a) AASHTO LRFD Bridge Design Specifications 2010

The AASHTO LRFD Bridge Design Specifications [2] address debonded strands mainly
by placing limits on the number of partially debonded strands that can be used, which shall not
exceed 25% of the total number of strands. In addition, the number of unbonded strands in any
horizontal row shall not exceed 40% of the number of strands in that row. Moreover, the
debonded length of any strand shall satisfy all limit states considering the total development of
the member’s resistance at any section under consideration. In addition, unbonded strands shall
be in a symmetrical pattern with respect to the beam-center line. The debonded length of strand
30

pairs with respect to the center line shall be the same. Not more than four strands or 40 percent of
the unbonded strands, whichever is greater, shall have the same unbonded length. The AASHTO
Specifications also establish that were bonding does not extend to the end of the concrete
member, the development length of partially debonded strands shall be twice the calculated one
for fully bonded strands. Finally, debonding of strands is limited to interior strands.
According to Shahawy and Hassan [28] if debonded effects are not accounted for in the
design of girders containing shielded strands, cracking in the member and bond failure could
occur during the service life of the girder. This finding has been taken into account in the
AASHTO Specifications [2] by limiting the number of debonded strands to 25% of the total
number in a girder. Thus, the limitations on the percentage on the use of debonded strands in the
AAHSTO Specifications are related to concerns regarding shear performance.

a) ACI-318

The ACI-318 2008 recommendations [3] note that the contribution of debonded strands
to the overall behavior of the prestressed concrete member begins at the point where the fully
effective stress starts, at the end of the transmission zone, and beyond that point. The
development length of debonded strands is to be twice the calculated for fully bonded strands
using Equation 2-6. However, the guidelines also states that at sections where the pretensioned
strands are not fully developed, it is common to assume that both transmission and development
lengths are doubled.
fse
Ld ≥  db + ( fps − fpe )db [ksi, in.]
 3 
 

31

Equation 2-6

c) Florida DOT Structures Guidelines

The Florida Department of Transportation Structures Guidelines [7] establishes that for
the design of prestressed concrete girders containing debonded strands, the distribution of the
shielded strands needs to be even throughout the strand pattern. Whenever possible, debonded
strands should be separated in all directions by at least one fully bonded strand. This
recommendation on the use of staggered debonding pattern has not been adopted by the
AASHTO Bridge Design Specifications or the ACI-318 Recommendations.

2.5.2 Literature Review on Experimental and Numerical Studies on the use of
Shielded Strands in Pretensioned Concrete Beams
Experimental studies on debonded strand behavior and its effect on prestressed concrete
beams are limited. This may follow from the fact that the strand is assumed to be perfectly
debonded from the concrete. Mirza and Tawfik [19] (see Section 2.4.1) included the use of
debonded strands in their experimental studies after end-cracking was experimentally observed.
The use of shielded strand seemed to be beneficial in reducing the restraint effect since the strand
free length increases with the use of unbonded strands. Kannel et al. [13] (see Section 2.4.1)
conducted experimental and numerical studies that included the use of debonded strands. They
found that the greatest benefit of using debonded strands is when the shielded stands are cut last
and some of the fully bonded strands are pre-cut first, such that the high tensile stresses are not
simply shifted to the onset of the fully bonded zone of the blanketed strands. While the studies of
Mirza and Tawfik and Kannel et al. addressed debonded strands, this was not a main objective in
their study. The following summarizes the research studies known to have specifically focused
on issues related to debonded strand.
32

a) Experimental studies
Russell, Burns, and ZumBrunnen, 1994

Russell et al. [27] conducted experiments on AASHTO Type-I beams to evaluate their
flexural resistance. Each beam contained eight 0.5-in. (12.7 mm) diameter strands and four of
those strands were either concurrently debonded or staggered debonded. Concurrent debonded
strands were shielded strands having the same debonded lengths and staggered debonded strands
had different debonding lengths. The variables in the study were embedment length and the
debonded strand lengths, either staggered or concurrent. Debonding was achieved using plastic
tubing made from a semi-rigid plastic with a satisfactory tight fit.
Based on their experimental work, Russell et al. recommended the use of staggered
debonded strands as they found that the use of staggered shielded strands is likely to increase the
resistance of the prestressed concrete member to cracking and that staggered debonding pattern
required shorter embedment length than concurrent debonding pattern. They also concluded that
the appearance of flexural cracking along the transmission region of partially debonded strands
causes the strand to slip, affecting strand anchorage and leading to bond failure. Moreover, they
recommended that for simply supported beams the debonded strand length shall not be greater
than 15 percent of the total span length measured from the ends.

Moon, Kim, Lee, and Kim, 2010

Moon et al. [20] studied the effects of the use of confinement reinforcing, debonded of
some strands using PVC, and detensioning method in the strain history change in the end region
by conducting experimental studies on small-scale pretensioned concrete beams. The units had a
cross section of 6 in. x 6 in. (150 mm x 150 mm) and 13 ft. (4 m) long. Two of the beams had a
33

concentric 0.5-in. (12.7-mm) diameter strand and the rest contained a concentric 0.6-in. (15.2
mm) diameter strand. One of the beams was partially debonded, and the others were fully
bonded. The debonded length was 8 in. (200 mm) from each end, and the confinement was
placed after the debonded region. The detensioning process was flame-cutting and gradual
release with a hydraulic jack. Moon et al. found that the use of debonded strand efficiently
diminished harmful effects at the cut-end from the flame-cutting procedure. When the strand
was gradually released, the effect on the debonded region was negligible.

Pavelchak, 2009

Pavelchak [23] studied the effectiveness of different debonding materials such as
flexible-slit (tight fit) sheathing, un-slit (rigid [oversized]) debonding, and PVC tubing. The
flexible sheathing material was obtained from three different manufacturers. The study was
conducted after damaged was observed on the U-beam girder shown in Figure 1-4 upon strand
detensioning. To assess the efficacy of several debonding materials, small-scale prismatic
prestressed concrete beams were built using different sheathing products to study the
effectiveness of debonded strands. The specimens had a cross section of 4”x4” (100 mm x 100
mm). The length of the beams was 10 ft. (3050 m) and 14 ft. (4270 m). Five of the 10 ft-long
(3050 m) beams contained a concentric partially debonded strand and the blanketed length was 2
ft (600 mm) from each end. The debonded length for the concentric partially debonded strands in
the 14 ft (4270 mm) long beams was 4 ft (1220 mm) from each end and six beams were built
with these characteristics. The strand was 0.5- in. (12.7 mm) diameter and the initial prestressing
force was 0.75 fpu. The test consisted on the gradual release of the prestressing force.

34

Pavelchak noted that the 2-ft (610 mm) debonded length was not sufficient to develop
adequate friction to transfer the prestressing force. He also noted that as the debonded length was
increased, the effectiveness of the debonding appears to decrease. Pavelchak concluded that unsplit (rigid) sheathing shall be used when the efficacy of the debonding is essential since this type
of debonding provides excellent blanketing performance.

Yi Sun, 2010

Sun [31] evaluated basic bond behavior on small-scale beams containing fully bonded
and partially debonded strands. The strand release rate, type of sheathing material (flexible or
rigid), strand debonded length, and strand adjacent effects were the variables incorporated for the
experimental phase of the study. The goal of the study was to understand how these parameters
affect the bond behavior along the unbonded zone and to produce data to calibrate finite element.
In the experimental program four different cross sections with three different strand
configurations were considered: monostrand, twin-strand, and four strand. The strand used for
the experimental phase was 0.6 in. (15.2 mm) diameter and prestressed to 0.75 fpu. The test
consisted of the sudden or gradual release of the prestressing force into the concrete members.
Concrete surface strain profiles were obtained along the neutral axis of the beam units and
internal measurements were taken using an embedded instrument (strain gages) rod at the
centroid of the beams.
Sun observed different strain values between the readings taken from the surface of the
beam and from internal measurements. The reason was determined to be creep and shrinkage
effects, which has also been observed by other researchers [22]; and which was also seen in the
results obtained in the experimental program of this study (see Section 3.6.2). Sun found that
35

sudden release led to longer transfer lengths, ~ 58% compared to values obtained from a gradual
release process. Strand debonding with rigid material exhibited larger transfer lengths compared
to beams with fully bonded strands and about 49% larger than values obtained from beam units
with debonded strand using flexible sheathing. Thus, Sun found that when flexible sheathing is
used the transfer length is not only affected by stored energy along the unbonded zone, but also
by the residual bond stresses along the debonded zone.

b) Numerical Studies
Sun, 2010

Sun [31] developed numerical models based on the laboratory-scale beam models built
and tested for the experimental program (see Section 2.5.2-a) to evaluate basic bond behavior.
The numerical models were developed using the finite element program ABAQUS [1]. The aim
of the simulation of the small-scale beam models was to assess basic bond behavior to
investigate the causes of end cracking in pre-tensioned concrete beams. Flexible sheathing was
simulated providing a tight fit around the strand along the unbonded zone, which continued
throughout the fully bonded zone, and rigid sheathing was simulated by providing oversize hole
around the strand along the blanketed length. Surface contact interaction models (see Section
4.3) were used to simulate strand-concrete bond behavior. The calibrated parameter was the
coefficient of friction and the values were obtained by matching the experimental concrete
internal strain profile with the numerical results. The coefficient of friction along the unbonded
zone was zero for the debonded strands with tight-fitting sheathing and no interaction was
defined for the debonded strands using rigid (oversized) debonding. Results from the study
showed that the tight-fitting debonding caused high stresses along the unbonded zone and that
36

the high stresses extended further into the fully bonded zone. The high stresses along unbonded
zone when using flexible (tight fit) sheathing were attributed to Poisson’s and Hoyer effects
because the strand upon detensioning tries to dilate and the debonding mechanism does not
provide the strand with space to expand.

2.5.3 Literature Review on Numerical Models on Beam-end Cracking on
Full-size Beams with Shielded Strands
Simulations on end cracking in full-size pre-tensioned girders containing shielded strands
are not many. Kannel et al. (see Section 2.4.1) [13] incorporated the use of debonded strands in
their numerical models to study their effect in the stress state in the anchorage region. The model
was validated with experimental data. However, Kannel et al. simulated the strand using truss
elements and along the unbonded zone truss elements were not provided in order to emulate the
debonding mechanism. With this modeling approach, the effect of the use of debonded strands
on release sequence can be accounted for, yet the dilation of the unbonded strand upon release
using a tight fit (flexible sheathing) along the debonded length cannot be taken into account.
Kannel et al. found that the use of debonded strands is beneficial as this increase the strand free
length and reduces the probability of end cracking.

Sun, 2010

Sun [31] investigated beam-end cracking in a skew box-girder containing unbonded
strands for which evidence of damage existed (see Figure 1-3). The box-type cross section beam
is shown in Figure 1-5. Sun created and analyzed large-scale beam models using the finite
element program (ABAQUS) [1] to study the stress state in the anchorage region upon release.
37

The variables considered for the study were: (a) type of debonding material, flexible (tight fit)
and rigid (oversized) sheathing; and, (b) the skew angle of the beam since the beam had a skew
angle of 40°. Linear-elastic material properties were used for the entire model and cracking
damaged was assessed by using an elastic limit of 7.5 f 'ci (psi).
Sun found good agreement between the damage predicted in the simulation and the actual
damage observed during production. However, he observed that a tight fit (flexible sheathing)
seems to significantly contribute to beam-end cracking since the stresses observed from the
model with oversized hole (rigid sheathing) around the strands were lower than those observed
from the model with a tight fit (flexible sheathing). In addition, it seems that the large skew angle
of 40° had a strong influence in the resulting beam-end cracking. Nonetheless, the type of
debonding material was noted to have the greatest effect on end cracking upon strand
detensioning.

38

CHAPTER 3 EXPERIMENTAL ASSESSMENT ON BOND
BEHAVIOR OF BONDED AND SHIELDED
STRAND

3.1 Introduction
This chapter presents the experimental assessment of bond behavior of bonded and
shielded 0.6-in.-diameter (15.2 mm) prestressing strand. The unbonded strands were blanketed
using single and double layers of slit-flexible sheathing. Bond behavior was evaluated for the
stressed and unstressed conditions for normal consolidated concrete (NCC) and selfconsolidating concrete (SCC). Bond strength on unstressed strands was evaluated through pullout tests on simple cylinders and wall-like blocks using fully bonded and debonded strands.
Pull-out tests on cylinders evaluated simple bond strength while the pull-out tests on wall blocks
were used to study the effects of concrete hydraulic head and concrete fluidity. Bond stress
transfer behavior of stressed strands was studied on laboratory-scale prestressed concrete beams
containing a concentric 0.6-in.-diameter (15.2 mm) strand for fully bonded and debonded
(shielded) strand. Results and main findings of the experimental program are presented in this
chapter.

3.2 Description of the Experimental Program
In order to evaluate basic bond behavior, two types of pull-out tests were carried out
containing unstressed 0.6-in.-diameter (15.2 mm-diameter) Grade 270 (1860 MPa) low
relaxation fully bonded and shielded strands. Sixteen 6”x24” (150 mm x 610 mm) cylinders
39

containing concentric debonded strands were built for conducting simple pull-out tests on the
unstressed strands. In addition, two 72” (1830 mm) high wall like blocks with 24”x24” (610 mm
x 610 mm) in cross section were also built for conducting pull-out tests on 18 strands located at
different heights in the block. To assess the transfer of prestressing forces into the concrete
members, twelve 20 ft-long (6.1 m) beams with a 6”x6” (150 mm x 150 mm) cross section were
built and tested by gradually releasing the prestressed strands. The parameters taken into account
for the experimental phase were:
1) bonded and debonded (shielded) strand condition,
2) single vs. double layered slit sheathing for debonding of the strands,
3) concrete fluidity,
4) reinforcing confinement, and
5) hydraulic head effect (top-strand effect).

3.3 Assessment of Strand Quality
The bond quality of the 0.6 in.-diameter (15.2 mm-diameter) strand was verified by
conducting a large block pull-out test (LBPT). The LBPT proposed by Logan [17] containing
six 0.5-in.-diameter (12.7 mm-diameter) Grade 270 (1860 MPa) strands, with an embedment
length of 18” (455 mm), was slightly modified for the 0.6-in. (15.2 mm) diameter strand. The
difference was the strand embedment (bonded) length, which was changed from 18” (455 mm)
to 20” (510 mm). The mix design was kept similar to the one recommended by Logan. The mix
design used for the LBPT in this study is given in Appendix A.
According to Logan, the LBPT first slip force benchmark for 0.5-in.-diameter (12.7 mmdiameter) strands is 16 kip (71.2 kN). Logan also recommended the benchmark for the average
40

peak load to be 36 kip (160.1 kN). The bond stress values for the recommended average first slip
force and peak were calculated to obtain the benchmark values for 0.6-in.-diameter (15.2 mmdiameter) strand and an embedment length of 20” (510 mm). The corresponding average first
slip force and the average maximum pull-out force benchmark values were thus found to be 21
kip (93.4 kN) and 48 kip (213.5 kN), respectively. The geometry and reinforcement details for
the 24”x24” (610 mm x 610 mm) large block pull-out are shown in Figure 3-2 and Figure 3-2.
Specifications for the strand are provided in Appendix A.

a) Front Elevation

b) Side Elevation

1 inch = 25.4 mm
Figure 3-1 Reinforcing details of large block: Front and side elevation

41

a) Top Elevation

b) Photograph of the Top View

Figure 3-2 Reinforcing details of the pull-out block

The block was cast with the strands in the vertical position as shown in Figure 3-2-b. The
average compressive strength obtained for the block at day of the test (~2 days after casting) was
6,400 psi (44.1 MPa). The pull-out test was done with a hollow hydraulic jack (6” [150 mm]
stroke and 200 kip [890 kN] capacity). Load was measured with a 100 kip (445 kN) load cell and
strand movement was measured on the loading (live) and back (dead) ends of the strand with
displacement potentiometers. A schematic and a picture of the test setup are shown in Figure 3-3
and Figure 3-4, respectively.
To conduct the test, the block was rotated to lay on its side so that the strands were in the
horizontal position (Figure 3-4). The loading rate recommended by Logan was about 20
kip/minute (89 kN/minute). However, the loading rate in this case varied between 9 kip (40 kN)
and 13 kip (57.8 kN) per minute. The test was stopped when the maximum pull-out force
dropped significantly or the strand was pulled out about 6 in. (150 mm), whichever occurred
first. In average, this procedure took about 5 to 6 minutes per strand.

42

Figure 3-3 Large-block pull-out test set up

Figure 3-4 Overview of the large block pull-out test set up

Results from the large-block pullout test are shown in Figure 3-5. The average maximum
pull-out force was 55 kip (244.6 kN) and the average first slip pull-out force was 35 kip (155.7
kN). Recalling, the compressive strength at the day of the test was 6,400 psi (44.1 MPa). It shall
be noted that one of the strands performed poorly based on the found benchmark values.

43

Therefore, one of the six values was discarded and the noted average results were obtained from
five (5) strands. The results thus met the pre-qualification benchmark.
40

Pull-out Force at First Slip

Pull-out Force (kip)

140

30

120
25

Benchmark

100

20
80
15

60

10

Pull-out Force (kN)

160

35

40

5

20

0

0
1

2

3

4

5

Strand
a) First slip pull-out force

Maximum Pull-out Force

60

250

Pull-out Force (kip)

200
40
150
30
100

20

50

10

0

0
1

2

3

4

5

Strand
b) Maximum pull-out force

Figure 3-5 Large block pull-out test results
44

Pull-out Force (kN)

Benchmark

50

3.4 Bond Evaluation of Unstressed Strand through Cylinder Pullout Tests
In order to assess the bond behavior between the strand and the surrounding concrete and
assess the effect of concrete fluidity, simple pull-out tests on concrete cylinders containing
concentric non-stressed strand were built and tested using normal consolidated concrete (NCC)
and self-consolidating concrete (SCC). The built and tested cylinders were 6-in.-diameter (15.2
mm) and 24 in.- (610 mm) long, with an embedment length of 20” (510 mm). Some of the
cylinders contained debonded strand using double layer slit sheathing, others were fully bonded,
and the rest were fully bonded with confinement reinforcement.

3.4.1 Test plan, description, and construction
Confinement reinforcement was provided for three cylinders in each batch, NCC and
SCC, using Grade 60 (414 MPa) #3 bars as shown in Figure 3-6. The pitch of the spiral
confinement was 2” (51 mm) and the outside diameter of the spiral was 3” (75 mm). PVC pipes,
0.75-in.- diameter (20 mm) and 2 in.-long (50 mm) were placed from each end, thus leaving an
embedment length of 20 in. (510 mm). The strands were cleaned with a dry towel prior to
placement in the forms. The cylinders were cast in a group with the strand in the vertical
direction (see Figure 3-6). The naming convention given to the pull-out cylinders is given in
Figure 3-7. The amount of pull-out cylinders cast and tested is given in Table 3-1.
The soft “slit” sheathing tubing was placed along the embedment length in the case of the
debonded strands using two layers. The NCC samples were cast in two layers with vibration after
each layer. No vibration was provided to the SCC cylinders and the concrete was poured in a
single step.

45

a)

Reinforcing details

b) Formwork and strand set up

Figure 3-6 Reinforcement and geometry details of pull-out tests on cylinders

Figure 3-7 Test matrix and naming and matrix convention for cylinder pull-out tests

46

Table 3-1 Amount of pull-out cylinders cast per batch

Condition of Strand
FB
FBC
D2
Total

Type of Concrete
NCC
SCC
3
3
2
3
2
3
7
9

3.4.2 Test Procedure
The concrete compressive strength for concrete used in the pull-out cylinder tests was
determined according to ASTM-C873 using 4”x8” (100 mm x 200 mm) cylinders. Three
cylinders were kept in a curing room and the rest were left next to the test units in the laboratory
test bay. The average compressive strength at the day of test (~6 days after casting) was 7,300
psi (50.3 MPa) for the normal consolidated concrete (NCC), and 9,400 psi (64.8 MPa) for the
self-consolidating concrete (SCC). The NCC and SCC mix designs are given in Appendix A.
The set up for the pull-out test on cylinders was similar to the one shown for the largeblock in Figure 3-3. The equipment used for the pull-out test on cylinders was the same as the
one used for the large block pull-out test (see Section 3.4.2). Figure 3-8 shows an overview
picture of the test set-up. Strand-slip was monitored at both ends, pull and dead (see Figure 3-8),
and readings were recorded using a data acquisition system (DAQ). The pull-out test was
stopped when the strand was pulled out about 6 in. (150 mm) or when the peak load suddenly
dropped, whichever happened first.

47

Figure 3-8 Photograph of the cylinder pull-out test setup

3.4.3 Cylinder Pull-out Test Results
Longitudinal cracks along the concrete cylinder (see Figure 3-9) were observed after
hearing a strong noise and observing on the DAQ-computer that the pull-out force had dropped
considerably. The cracks started at the pull-end close to the strand and extended to the surface of
the cylinder. The crack then propagated along the cylinders towards the dead end. In no case did
the crack reach the dead end. Cracking did not occur in the cylinders containing a single
concentric debonded strand.

Figure 3-9 Pull-out cylinder with longitudinal cracking
48

A typical pull-out curve response from the pull-out tests is shown in Figure 3-10. Three
main points are highlighted: a) the first slip force, b) the strand-slip at first slip force, and c) the
peak pull-out load; as described below:
a) First slip force: is the force that causes the initial movement of the strand. The first
slip force coincides with the end of the elastic region along the pull-end curve.
b) Strand slip at first slip force: is the distance that the strand moves at the pull end that
coincides with the initiation of movement at the dead end.
c) Peak Load: is the maximum load reached when pulling the strand and it remains
approximately constant exhibiting a plateau.

Pull-out Force

c) Peak Load

a) First Slip Force
Dead End
Pull End

b) Strand Slip at First Slip Force

.

Strand Slip

Figure 3-10 Typical force-slip response from pull-out test

The obtained average maximum pull-out forces are presented in Table 3-2. It can be seen
that the fully bonded strand with confinement (FBC) exhibited the maximum pull-out force for
each set, NCC and SCC. Figure 3-11 and Figure 3-12 show the pull-out response curve for the
three different strand conditions: fully bonded (FB), fully bonded confined (FBC), and debonded

49

with two layers (D2) (see Figure 3-7) for both sets. Supplementary pull-out curves are given in
Appendix B.
Table 3-2 Summary of maximum pull-out force on cylinders
NCC (f'ci= 7330 psi)
(50.3 MPa)
Maximum Pull-out force -kip
(kN)
Mean
St.dev
51.1
8.16
(227.3)
(36.3)
46.59
7.53
(207.2)
(33.5)
0.26
0.07
(1.16)
(0.31)

Strand
Denomination
FBC
FBU
D2

SCC (f'ci= 9450 psi)
(64.8 MPa)
Maximum Pull-out force -kip
(kN)
Mean
St.dev
49.4
2.02
(319.73)
(8.98)
44.73
1.4
(199)
(6.23)
0.18
0.12
(0.80)
(0.53)
1 kip= 4.448 kN

Strand Slip (mm)
60

0

20

40

60

80

100

120

50
200
40
150
NCFB
NCFBC
NCD2

30
20

100

Pull-out Force (kN)

Pull-out Force (kip)

250

50

10

0

0
0

1

2

3

4

5

Strand Slip (in.)

Figure 3-11 Pull-out response for strand in NCC cylinders

50

Strand Slip (mm)
0

50

100

150

60

50
200
40
150
30
100

20

SCFB
SCFBC
SCD2

10

Pull-out Force (kN)

Pull-out Force (kip)

250

50

0

0
0

1

2

3

4

5

6

Strand Slip (in.)

Figure 3-12 Pull-out response for strand in SCC cylinders

The average maximum bond strength, u, was calculated from the average maximum pullout forces for the corresponding embedment length of 20” (510 mm), and the surface area of the
0.6-in.-diameter (15.2 mm) strand. Equation 2-5 was used to obtain the average bond strength
values. Since the simple pull-out tests on cylinders were conducted using two different types of
concrete, NCC and SCC, the bond stress values were normalized in respect to

f 'ci for each

set. The normalized bond strength values, u’, were obtained using Equation 3-1. The maximum
pull-out force, the calculated bond strength values, and the normalized bond strength are shown
in Table 3-3. Figure 3-13 shows a graph with the normalized bond strength values with respect to

f 'ci .
u' =

u
f 'ci

51

Equation 3-1

Table 3-3 Maximum pull-out forces and bond strength values obtained from pull-out tests
on cylinders
Strand
Denomination
NCFBC
NCFB
SCFBC
SCFB

Maximum Pullout Force - kip
(kN)
51.1
( 227.3)
46.6
(207.3 )
49.4
(219.7 )
44.7
(198.8)

Bond Strength
(u)-psi
(MPa)
1017
(7.0 )
927
(6.4)
983
(6.8)
890
(6.1)

u'
u' NCFB

f'ci
11.9

1.1
10.9
1.0
10.1
0.9
9.2
0.9
1 kip= 4.448 kN
1 MPa= 145 psi

NCC
SCC

12

Normalized Bond Strength, u'

u

u' =

10

8

6

4

2

0
FB

FBU

Strand Condition

Figure 3-13 Normalized peak bond strength values with respect to

52

f 'ci

3.4.4 Bond Evaluation of Unstressed Strand through Cylinder Pullout Tests
From the pull-out test results on simple cylinders it was observed that the strand provided
with spiral confinement reached a greater average pull-out peak load than those without spiral
(Table 3-2). From the NCC tests set, the NCFBC group attained greater pull-out force than the
NCFBU group by 8.8%. For SCC set of cylinders, the SCFBC pull-out tests reached an average
maximum pull-out greater than the SCFBU tests by about 9.5%. In addition, it can be seen in
Table 3-2 that the pull-out tests on SCC had more uniform results as the standard deviation
values were lower compared to the NCC tests. Even though the compressive strength for the
SCC tests was about 2000 psi (13.8 MPa) greater than the NCC set, the average maximum pullout force reached from every subgroup of SCC cylinders (FBC, FBU, and D2) was lower than
for the NCC units.
As shown in Figure 3-11 and Figure 3-12, the fully bonded cylinders with confinement
and fully bonded pull-out cylinders without confinement curves response were quite similar
along the elastic region, indicating similar level of stiffness. From Table 3-2 it can be seen that
the debonding mechanism, double layer flexible slit sheathing, effectively broke the bond
between the unstressed strand and the surrounding concrete, thus paste infiltration did not occur
even when using high fluidity concrete (SCC).

3.5 Evaluation of Hydraulic Head Effect on Bond through Wall-like Blocks
Pull-out Tests
The effect of concrete hydraulic head was evaluated on debonded strands to investigate if
high concrete fluidity can cause bonding of the unbonded strands due to paste infiltration and if
deformations on the sheathing material can take place due to concrete pressure, thus preventing
53

full debonding of the strand and the surrounding concrete when using flexible slit sheathing.
These possible effects were evaluated with pull-out tests on wall-like blocks containing
unstressed debonded strands, where debonding was accomplished using single and double layer
flexible slit sheathing.
Recent studies to evaluate the top-strand effects in prestressed concrete members and its
effect on bond behavior have been conducted by Peterman [24]. It was observed that the main
factor causing the top-strand effect is the amount of fresh concrete cast above the strands. In
addition, Peterman observed that an increase in concrete fluidity will cause a decreased in bond
capacity. This effect was more evident for those strands closer to the top surface of the prestresed
concrete member.

3.5.1 Test plan description and construction
In order to study the hydraulic head effect and its influence on bond behavior for shielded
strand, pull-out tests on unstressed strand were done on two 72” (1830 mm) high wall-like blocks
with a cross section of 24”x 24” (610 mm x 610 mm) using normal consolidated concrete (NCC)
and self-consolidating concrete (SCC). Each block had eighteen (18) strands as shown in Figure
3-15. The strands designated as NCFB and SCFB were fully bonded. Six strands were debonded
using single flexible slit sheathing (NCD1 and SCD1), and the others were debonded by means
of double soft slit sheathing (NCD2 and SCD2).
The naming convention for the wall-like blocks is shown in Figure 3-14. NC and SC refer
to the type of concrete used. NC refers to normal consolidated concrete (NCC) and SC refers to
self-consolidating concrete (SCC). The other two letters correspond to the type of strand, either

54

bonded or debonded. For instance, for NCD2 the NC stands for NCC and D2 for debonded
strand using double soft slit sheathing.

Normal Consolidated Concrete (NCC)
Self-consolidating Concrete (SC)

Fully Bonded
(FB)

Debonded using single
soft split sheathing
(D2)

Debonded using double
soft split sheathing
(D2)

NCFB
SCFB

NCD1
SCD1

NCD2
SCD2

Figure 3-14 Test matrix and naming convention for wall-like block tests

The wall-like blocks were reinforced using bars #4 at each corner as shown in Figure 315a. Ties (#3 bar sizes) were placed every 12” (305 mm) center-to-center (see Figure 3-15). In
total, four #4 bars and five #3 ties were used as reinforcement for each wall-like block. The 0.6in.-diameter (15.2 mm) strands were spaced at every 12” (305 mm). The upper strands were
located at 6” (150 mm) from the top of the block, and the lower strands were placed at 6” (150
mm) from the bottom of the block as shown in Figure 3-15. The blocks were filled with concrete
from the top, with the strands in a horizontal position as shown in Figure 3-16.
The wall-like blocks were demolded about 2 days after the concrete was poured. The
compressive strength of the concrete was verified by testing three 4”x8” (100 mm x 200 mm)
concrete cylinders the day of demolding the wall-block formwork. The wall block cast with NCC
was filled out in three layers with vibration. No vibration was done for the wall block cast with
SCC.
55

Dead End
Top

Row #

Pull End
1

2
0.6-in.φ- Strand

Bars # 3 @ 12# c-c

3

Bars #4
4

0.6-in. - φ Strand
5

2" Sleeve
6
Bottom
NCFB
(SCFB)

a) Side view

NCD1
(SCD1)

NCD2
(SCD2)

b) Front view

Figure 3-15 Wall-like block geometry details: side and front view

3.5.2 Wall-like Block Test Procedure
The concrete compressive strength for the concrete in the wall-like blocks was assessed
with 4”x8” (100 mm x 200 mm) cylinders according to ASTM-C873. Three cylinders were kept
in a curing room and the rest were kept in the laboratory floor next to the wall-blocks. The
average compressive strength at the day of testing (~6 days after casting) was 7,300 psi (50.3
MPa) and 9,400 psi (64.8 MPa) for the NCC and SCC blocks, respectively.
The pull-out test set-up on the wall-like blocks was similar to the ones used for the large
block pull-out test and the simple pull-out test on cylinders. Figure 3-17 shows an overview of
the wall-like block pull-out test set up. Readings from the load cell and linear potentiometers
were recorded with a data acquisition system (DAQ).
56

a) Photograph of construction details

b) Photograph

Figure 3-16 Wall-like block reinforcing details

Figure 3-17 Overview of the wall-like block pull-out test

57

3.5.3 Wall-like Blocks Pull-out Test Results
Typical pull-out force vs. strand slip curves obtained from the NCC wall-like block at a
height of 18 in. (455 mm) (see Figure 3-16) are shown in Figure 3-18. Supplementary results
obtained for each strand are given in Appendix B. Results for the debonded strands, using single
and double slit sheathing layer, in the NCC wall-like block at 18 inches (455 mm) from the block
bottom are seen in Figure 3-18. The observed top-strand effect on both wall-like blocks is clearly
shown in Figure 3-19 for the three different strand conditions, i.e., fully bonded and fully
debonded. It is seen in Figure 3-19 that the fully bonded strands located toward the bottom of the
block exhibited larger peak pull-out forces compared to the strands located toward the top of the
block (first three upper strands). On the other hand, the maximum pull-out force obtained from
the debonded strands was considerably small compared to the fully bonded strand group. Thus,
hydraulic head effect had little or no effect on the behavior of the blanketed strands. The data
indicates that the top strand effect was higher for the NCC block than for the SCC block, which
can be seen in Figure 3-19. The compressive strength at the day of testing and the average peak
pull-out forces for each strand condition are shown in Table 3-4. The bond strength values for
each case were calculated using Equation 2-5 and Figure 3-20a shows the peak bond stresses
from both wall blocks normalized with respect to

58

f 'ci .

Table 3-4 Summary of average maximum pull-out forces: wall-like blocks
Strand
Denomination

FB
D1
D2

NCC (f'ci= 7200 psi)
(49.6 MPa)
Maximum Pull-out Force – kip
(kN)
Average
Standard
Deviation
46.3
7.3
(206.0)
(32.4)
3.4
2.2
(15.3)
(9.8)
0.5
0.2
(2.2)
(1.0)

SCC (f'ci= 9600 psi)
(66.2 MPa)
Maximum Pull-out Force – kip
(kN)
Average
Standard
Deviation
52.5
4.3
(233.4)
(19.3)
1.3
0.5
(5.7)
(2.1)
0.15
0.2
(0.4)
(0.7)
1 kip= 4.448 kN

S tra nd Slip (m m )
0

50

100

150

60

Pull-out Force (kip)

50
200
40
150
30

NC FB
NC D1
NC D2

100

Pull-out Force (kN)

250

20

50

10

0

0
0

1

2

3

4

5

6

S tra nd Slip (in.)

a) Wall-like block cast with NCC
Figure 3-18 Pull-out curve response of the 5th row strand on wall-like blocks

59

Figure 3-18 cont’d

Strand Slip (m m)
0

50

100

150

60
250

200
40
150
30

SCFB
SCD1
SCD2

100

20

Pull-out Force (kN)

Pull-out Force (kip)

50

50

10

0

0
0

1

2

3

4

5

6

Strand Slip (in.)

b) Wall-like block cast with SCC
Height: from bottom to top (m m)
0

200

400

600

800 1000 1200 1400 1600

60

50
200
40
150
30

NCFB
NCD1
NCD2

20

100

50

10

0

Maximum Pull-out Force (kN)

Maximum Pull-out Force (kip)

250

0
0

10

20

30

40

50

60

70

Height: from bottom to top (in.)

a) Wall-like block cast with NCC
Figure 3-19 Top-strand effect on wall-like blocks
60

Figure 3-19 cont’d.
Height: from bottom to top (mm)
0

200

400 600

800 1000 1200 1400 1600

60

50
200
40
150
30

SCFB
SCD1
SCD2

20

100

50

10

Maximum Pull-out Force (kN)

Maximum Pull-out Force (kip)

250

0

0
0

10

20

30

40

50

60

70

Height: from bottom to top (in.)

b) Wall-like block cast with SCC

1400

NCFB
SCFB
8

1000
6
800

600

4

400

Bond Strength (MPa)

Bond Strength (psi)

1200

2
200

0

0
1

2

3

4

5

6

Row #: From Top to Bottom

a) Peak bond stress
Figure 3-20 Bond stresses: Fully bonded strands in wall-like blocks

61

Figure 3-20 cont’d.

NCFB
SCFB

16

Normalized Bond Strength

14
12
10
8
6
4
2
0
1

2

3

4

5

6

Row #: From Top to Bottom

b) Normalized peak bond strength

3.5.3 Hydraulic Head Effect: Wall-like Blocks
The maximum pull-out force reached for the group of debonded strands was very small
compared to the fully bonded strands, which indicates that in this type of setting (unstressed
strand) the debonding mechanism was effective. The top-strand effect was clearly observed as
seen in Figure 3-19, where the upper strands reached lower peak pull-out load values compared
to the strands located toward the bottom of the blocks. In Figure 3-19a, the NCD1 upper strand
exhibited the maximum peak load in that group and this could be attributed to the fast hardening
of the concrete when pouring the last layer of concrete when casting the NCC block. As shown
in Figure 3-19b, the SCFB group of strand performed more uniformly than the NCFB strands.
The three lower SCFBC strands reached about the same maximum pull-out force. This was not
62

observed in the NCC wall-like block since the strand located at 18” (455 mm) from the bottom
had the largest pull-out force among the NCFBC strands. However, in both NCC and SCC wall
blocks, the upper fully bonded strand exhibited a lower maximum pull-out force compared to the
other five fully bonded strands in the blocks. The peak load for the SCFB group of strands
increased gradually from top to bottom, and the three lower strands had a similar maximum pullout load.
The peak pull-out force from the SCFB strands was, on average, about 65% less than the
NCFB strand group as shown in Table 3-4. In Figure 3-20a the peak bond stress values for the
SCFB strands are shown to be higher in most cases since greater peak pull-out loads were
obtained from the SCC wall block for fully bonded strands. However, it is shown in Figure 3-20b
that if the peak bond stresses are normalized respect to

f 'ci the bond strength values for NCFB

strands were higher in most cases.

3.6 Prestress Bond Transfer in Beams
The bond stress transfer behavior of fully bonded and shielded (debonded) 0.6-in.diameter (15.2 mm) prestressing strand on normally consolidated and self-consolidating concrete
(NCC and SCC) was studied by conducting tests on laboratory scale beams. Six beams with a
single concentric strand were tested for each type of concrete. Stress transfer behavior was
assessed by measuring the strains transmitted to the concrete upon gradual release of the
pretensioned strand. The strand was debonded using flexible slit (tight fit) sheathing in either a
single or double layer.

63

3.6.1 Geometry, Reinforcing, and Prestressing Details
The beams units were 6”x6” (150 mm x 150 mm) in cross section and 20 ft long (6.1 m),
with a single concentric 0.6-in.-diameter (15.2 mm) strand (see Figure 3-21). Twelve beams
were tested; cast in two groups of six, one group with NCC and the other with SCC. From the
six beams in a given batch two had fully bonded strands, two had partially debonded strands with
single slit sheathing, and the rest had partially debonded strands with double slit sheathing. When
applicable, the debonded strand length was 24” (610 mm) from each end of the beam. Three of
the six beams had confinement (see Figure 3-21), i.e., one of the beams containing a single fully
bonded strand, one of the beams containing a single debonded strand using one slit sheathing
layer, and one of the beams containing a single debonded strand using double slit flexible
sheathing. The provided confinement reinforcement was the same one as the one used in the
pull-out on cylinders (see Section 3.5.1), and it was placed in the anchorage region as shown in
Figure 3-21. For beams with partially debonded strands, the confinement was placed at the end
of the debonded length, starting at 24” (610 mm) from the end of the beam. For the beam with
fully bonded strand the confinement was placed as close as possible to the releasing end. Figure
3-22 shows a photograph of one of the fully bonded beams containing spiral confinement in the
anchorage region. This picture corresponds to the stage prior to casting the beams.
The test matrix and naming convention for the beams is shown in Figure 3-23. The
naming convention was based on the type of concrete, condition of the strand (bonded or
partially debonded), and the provision or not of confinement. The first two letters, NC or SC,
correspond to the type of concrete. The following two letters refer to the condition of the strand,
i.e., partially debonded using single or double slit sheathing or fully bonded. The last letter, C or
U, refers to the addition of confinement in the beam, where C stands for a beam with
64

confinement and a U refers to a beam without confinement. For instance, NCFBU beam refers to
a beam cast with normally consolidated concrete (NCC), the strand was fully bonded, and no
spiral confinement was provided.

Strand, 0.6-in.-φ
Bar #3

Side View

Front View

Figure 3-21 Geometry of beam with fully-bond strand and with confinement

The test setup for this study was essentially the casting bed for the beams, as shown in the
plan view drawing in Figure 3-24 and the photograph in Figure 3-25. As seen in these Figures,
two beams were placed for each line of strand. This was done for both sets, NCC and SCC.

Figure 3-22 Photograph of end of fully bonded beam with confinement

65

Figure 3-23 Test matrix and naming convention for beam tests

The prestressing action took place approximately two days before casting. The strands
were pretensioned so as to reach a stress level of 0.7fpu after all significant losses. During the
strand tensioning operation the strain in each stand was monitored with a strain gauge and the
pulling load was measured with a load cell. In addition, the stretching of each strand was
monitored by manually measuring the strand elongation and its relaxation after release of the
jacking force. Strain gages were also placed on the strands at dead end, east to the anchor block
(see Figure 3-24) to monitor losses due to relaxation prior to casting, which was typically done 2
days after strand detensioning. The maximum strain due to relaxation prior to release into the
hardened concrete beam was about 80 µ ε. Casting took place two days after strand tensioning for
the NCC beams and fourteen days after tensioning for the SCC beams.

66

Figure 3-24 Schematic beams layout

Figure 3-25 Overview photograph of the beams layout

67

3.7.2.1 Concrete Surface Strain Profiles

Concrete surface strain profiles (CSSP) were obtained using a detachable mechanical
(DEMEC) strain measuring system with target disks pre-installed at 2 in. (50 mm) on the beam
side along the section centroid. Only one side in one end of each beam was instrumented. The
first metal disk was placed at 1” (25.4 mm) from the end of the beam (see Figure 3-24 and Figure
3-26). The measurement length for beams with fully-bonded strands was about 115” (2920 mm),
and for beams with debonded strands was approximately 150” (3810 mm). An overview of the
test setup showing the prepared/instrumented ends of the beams is shown in Figure 3-27.
Strains were determined from measurements over a gage length of 8 in. (200 mm). Three
readings were taken before and after release of the strand, respectively. The three readings were
averaged (respectively) for post-processing and calculation of strains. Once the strain profile
curves were obtained the curves were smoothed using the procedure depicted in Figure 3-28 as
proposed by Russell et al. [26]. This technique is used to determine the so-called “transfer
length” and it is well known as the 100% average maximum strain (AMS) method [26]. The
procedure is well accepted and has been commonly used by other researchers.

Figure 3-26 Beam side prepared for strain measurement with DEMEC system
68

Figure 3-27 Overview of beam test setup and instrumentation at beam ends

Figure 3-28 Smoothing method for concrete surface strain profiles. Adapted from [32]

3.6.1 Test Procedure: Transfer of Prestressing Force into the Concrete Member
At the day of testing, i.e., gradual release of the prestressed strands, six 4” x 8” (100 mm
x 200 mm) cylinders were tested according to ASTM-C873. Three of the cylinders were kept in
a curing room and the rest of the cylinders were kept in the laboratory floor next to the beams.
69

The strands for all pretensioned beams were gradually released by means of annealing.
The strand release was typically done six days after casting. The delay was due to the time
needed for demolding the specimens and instrumentation setup. The gradual prestress release
into the concrete member was achieved by heat annealing of the strand along the three free
strand regions in a given beam line (see Figure 3-24). The reduction of strand pretension due to
the heat-induced relaxation of the steel was monitored with strain gages connected near one of
the releasing ends. The annealing process took between 12 to 15 minutes. The release sequence
for the NCC set was NCD2C, NCFBU, and NCFBC. Since two beams were placed per strand
line, the prestressing force was transmitted into two beams at the same time. The releasing
sequence for the self-consolidating concrete (SCC) batch was the same as the one described for
the NCC set of beams.

Figure 3-29 Photograph of the three points where the strand was released

70

The time at which the DEMEC strain measurements were taken differed from beam to
beam. Table 3-5 shows a summary of the time at which readings from each beam were taken
after releasing the prestressing strands. The times at which measurements were taken are given
because it is well-documented that transfer length increases with time, particularly during early
age of the concrete beam.
Table 3-5 Times at which final DEMEC were taken after strand release
Beam ID
NCFBC
NCFBU
NCD1C
NCD1U
NCD2C
NCD2U
SCFBC
SCFBU
SCD1C
SCD1U
SCD2C
SCD2U

Time hours (min)
3.00 (180)
5.00 (300)
6.30 (380)
6.00 (360)
0.50
(30)
7.25
(435)
3.50
(210)
5.00
(300)
4.60
(280)
6.30
(378)
0.50
(30)
9.00
(540)

3.6.2 Results of Bond Transfer in Beams
The results obtained from the prestress bond transfer tests are presented next. The
compressive strength at the day of test (release) for the NCC beams was 7,300 psi (50.3 MPa)
and for the SCC beams it was 9,400 psi (64.8 MPa).

3.6.4.1 Concrete Surface Strain Profiles

The set of three DEMEC readings before and after strand release were averaged and then
used to calculate the raw strain profile. A smoothed strain profile was determined by using a
running average of 3 points as defined by Equation 3-2 below.
71

ε i, smooth =

ε i −1 + ε i + ε i +1

Equation 3-2

3

The 100% average measurement strain (AMS) method was used [32] to obtain the
transfer length using the concrete surface strain profiles. This is a subjective technique and the
average strain is calculated from those points near to the plateau of the strain profile. The onset
of the plateau is the transfer length strain point. Figure 3-30 shows unsmoothed vs. smoothed
concrete surface strain profile curves. It can be observed that the smoothed curve is more
uniform and it takes care of peak values that may affect the average strain level. In some cases,
some outlier peak levels were observed and they were eliminated because those points were far
away from the trend.

D is ta n c e fr o m E n d o f th e B e a m (m m )
0

500

1000

1500

2000

2500

3000

2000

-6

Concrete Strain Level (in./in.*10 )

500

400
1500
300

1000
200

500

U n s m o o t h e d C u r v e (N C F B U b e a m )
S m o o th e d C u r v e ( N C F B U b e a m )

100

0

0
0

20

40

60

80

100

120

D is ta n c e fr o m E n d o f th e B e a m (in .)

Figure 3-30 Concrete surface strain profile: smoothed vs. unsmoothed curves

72

Concrete strain profile curves obtained for the NCC beams are shown in Figure 3-31 and
Figure 3-32. Representative results obtained for the SCC beams are shown in Figure 3-33 and
Figure 3-34. In all these figures fpj is the initial prestressing force after sitting losses. It can be
observed from Figure 3-31, Figure 3-32, and Figure 3-34 that some level of prestressing force
was transferred along the debonded length. For the beams containing partially debonded strands,
the transfer length was defined as that from the end of the debonded length until the strain level
became uniform, i.e., a plateau level was reached. The 100% AMS procedure [32] was also
applied.

Distance from end of beam (mm)
0

500

500

1000

1500

2000

2500

3000

3500

L =19"

-6
Strain Level (in./in.*10 )

t

400

300

L =24"
u

200

L =22"
t

100

NCD1U - fpj=198 ksi
NCD1U - 100% AMS
NCD1C - fpj=203 ksi
NCD1C - 100% AMS

0
0

50

100

150

Distance from end of beam (in.)

Figure 3-31 Concrete surface strains profiles for NCD1C and NCD1U beams

73

Distance from end of Beam (mm)
0

500

1000

-6
Strain Level (in./in.*10 )

500

1500

2000

2500

3000

3500

L =29"
t

400

300

L =24"
u

200

NCD2C - fpj=201 ksi
NCD2C - 100% AMS
NCD2U - fpj=201 ksi

L =19"

100

t

NCD2U - 100% AMS
0
0

50

100

150

Distance from end of Beam (in.)

Figure 3-32 Concrete surface strain profiles for NCD2C and NCD2U beams
Distance from end of Beam (mm)
0

500

1000

1500

2000

2500

3000

350

Concrete Strain Level (in./in.*10

-6
)

L =24"
t

300
250
200
150

L =17"

SCFBC - fpj=203 ksi
SCFBC - 100% AMS
SCFBU - fpj=203 ksi
SCFBU - 100% AMS

t

100
50
0
0

20

40

60

80

100

120

Distance from end of Beam (in.)

Figure 3-33 Concrete surface strain profiles for SCFBC and SCFBU beams

74

Distance from End of Beam (mm)
0

500

1000

1500

2000

2500

3000

3500

Concrete Strain Level (in./in.*10

-6
)

400

L =19"
t

350
300
250
200
150

L =24"
u

100

SCD1U - fpj=203 ksi
SCD1U - 100% AMS
SCD2U - fpj=202 ksi
SCD2U - 100% AMS

L =29"
t

50
0
0

50

100

150

Distance from End of Beam (in.)

Figure 3-34 Concrete surface strain profiles: SCC beams containing partially debonded
strand without confinement reinforcement

It can be seen in Figure 3-32 and Figure 3-34 that even though the beams shared the same
strand line the average concrete surface strain level was different, which indicates that creep and
shrinkage effects affected the final strain measurements. Table 3-6 and Table 3-7 summarize the
determined transfer lengths for all beams. The force designated as fpj in Table 3-6 and Table 3-7
is the applied jacking force minus the sitting losses during the pre-tensioning procedure.

Table 3-6 Transfer lengths for NCC beams
NCFBC NCFBU NCD1C NCD1U NCD2C NCD2U
19
23
22
19
19
29

Transfer Length (in)
fpj (ksi)
203
198
1 inch = 25.4 mm, 1 ksi = 6.89 MPa

203

75

198

201

201

Table 3-7 Transfer lengths for SCC beams

Transfer Length (in)

SCFBC SCFBU SCD1C SCD1U SCD2C SCD2U
24
17
16
15
N/A
27

fpj (ksi)
203
203
1 inch = 25.4 mm, 1 ksi = 6.89 MPa

203

203

202

202

The average transfer length for the NCC beams was found to be 22 in. (560 mm) and 20
in. (510 mm) for the SCC beams. These average values fall within two standard deviations for
both sets. The transfer length values were normalized with respect to time since it is well
documented that the transfer length increases with time. Transfer lengths were normalized with
respect to a chosen time of six (6) hours by calculating the creep strains at this time using
Equation 3-3.

Lt norm = Lt measured

CreepStrain6 hours
CreepStrainbeam

Equation 3-3

In Equation 3-3, creep strain at 6 hours is the calculated strain value due to creep effects
for each batch, and creep strain in beam is the creep value calculated for each beam according to
the time at which the DEMEC measurements were taken. The measured and normalized transfer
length values for each beam in each set are given in Table 3-8. The normalized transfer length
for beam NCD2C is not provided since the obtained value was far from the normalized average
transfer length.
The initial stress values, losses due to shortening, and the calculated theoretical transfer
strain values for each beam are shown in Table 3-9. In addition, the theoretical values of
transferred strains obtained after taking into account creep and shrinkage, and the measured
strains at 100% average strain level are also shown in this table. Creep and shrinkage effects
were taken into account since they were found to be significant when comparing the
76

experimental results to code estimates and numerical results. It was found that creep strains were
particularly high. Therefore, creep strain values were added to the calculated strain values caused
by the release of the prestressing force. The experimental strain level for the SCD2C beam is not
shown in Table 3-9 since the results obtained from that beam were not satisfactory. Therefore,
the results were discarded.
Table 3-8 Time normalized transfer lengths
Measured Transfer
Beam ID
Length (in.)
NCFBC
19
NCFBU
23
NCD1C
22
NCD1U
19
NCD2C
19
NCD2U
29
SCFBC
24
SCFBU
17
SCD1C
16
SCD1U
15
SCD2U
27
1 inch = 25.4 mm

Normalized Transfer
Length (in.)
23
25
22
19
28
28
18
17
15
24

As shown in Table 3-8, the determined and normalized transfer length values were below
the 60 times nominal diameter value recommended by the AASHTO LRFD Bridge Design
Specifications [2] for 0.6-in.-diameter (15.2 mm) strands, which is 36 in. (915 mm). In addition,
Table 3-8 shows that the normalized transfer length values for the beams containing fully bonded
strand were slightly larger than the ones obtained for debonded strand. This is attributed to the
partial bond resistance along the sheathed strand length due to strand dilation, as supported by
the numerical simulations presented in Chapter 4. The force transferred along the unbonded

77

length as a percentage of the effective prestressing force was 15% and 17% for the NCC and
SCC beams, respectively.
Table 3-9 Theoretical and experimental transferred strains

Beam ID
NCD2C
NCD2U
NCFBU
NCD1U
NCFBC
NCD1C
SCD2C
SCD2U
SCFBU
SCD1U
SCFBC
SCD1C

fpj
(ksi)
201
201
198
198
203
203
202
202
203
203
203
203

Theoretical
strain level
w/o
Losses due to including
creep strain
shortening
(ksi)
(µs)
7.0
241
7.0
241
6.9
238
6.9
238
7.1
244
7.1
244
6.2
215
6.2
215
6.3
216
6.3
216
6.3
216
6.3
216

Theoretical
strain level
including
creep strain
(µs)
292
391
370
377
360
387
270
335
317
324
307
322

Experimental
strain level
using 100%
AMS method
(µs)
357
435
432
436
535
368
N/A*
300
266
289
286
291
1 ksi = 6.89 MPa

As shown in Table 3-9, the calculated theoretical precompression strain level before
taking into account creep and shrinkage effects did not match the experimental concrete surface
strain levels. Nonetheless, once creep and shrinkage effects were taken into consideration the
theoretical and experimental strain values were much closer (see Table 3-9). Figure 3-35 and
Figure 3-36 show a comparison between the theoretical and experimental concrete prestress
levels for both sets of beams. It is seen from these figures that the experimental results obtained
from the NCC set of beams was quite higher than the theoretical calculations even though creep
and shrinkage effects were included for the calculated values. However, the average difference
between the experimental and calculated concrete strain values is about 13% excluding the

78

results obtained from the NCFBC beam. On the other hand, the experimental concrete strain
values measured from the SCC beam units were lower than the calculated ones about 12%.
600

T h e o re tic a l
E x p e rim e n ta l

-6

Strain Level (in./in.*10 )

500

400

300

200

NCD1C

NCFBC

NCD1U

NCFBU

NCD2U

0

NCD2C

100

B e a m ID

Figure 3-35 Theoretical and experimental strain level comparison – NCC beams

79

400
T h e o r e tic a l
E x p e r im e n ta l

-6

Strain Level (in./in.*10 )

350
300
250
200
150
100
50

SCD1C

SCFBC

SCD1U

SCFBU

SCD2U

0

B e a m ID

Figure 3-36 Theoretical and experimental strain level comparison – SCC beams

3.7 Findings and Discussion
The most relevant findings in the experimental program using normally-consolidated
(NC) and self-consolidating (SC) concrete were:

•

From pull-out tests on cylinders it was observed that the use of confinement
reinforcement improved the bond strength on unstressed strands by about 11%. This was
the average obtained from pull-out tests on cylinders cast with NCC and SCC.

•

The maximum peak pull-out loads on simple cylinders and wall-like blocks obtained
from debonded strands, whether debonding by using single or double layer flexible slit
sheathing, was significantly small compared to the average peak-pull out load obtained
for fully bonded strands.

80

•

The effect of concrete hydraulic head and concrete fluidity was investigated on unbonded
strands through pull-out tests on wall-like blocks. The average peak pull-out load
obtained from the debonded strands was close to zero for wall blocks cast with NCC and
SCC. Thus, flexible slit (tight fit) sheathing effectively eliminates the bond between the
unstressed debonded strand and the surrounding concrete in a unstressed condition.

•

The bond behavior of debonded strands studied using laboratory-scale beams showed that
with the use of flexible slit (tight fit) sheathing, single or double layer, about 17% of the
prestressing force was transferred along the debonded zone based for both, NCC and
SCC beams.

•

The use of confinement reinforcement along the transfer zone of two fully bonded beams
did not show any improvement in bond stress transfer behavior, as judged from the
concrete surface strain profiles obtained along the beam.

•

Measured concrete surface strains showed that creep and shrinkage effects at an early age
greatly influenced the determined transfer lengths (larger values).

81

CHAPTER 4 COMPUTATIONAL ASSESSMENT

4.1 Introduction
Three-dimensional nonlinear finite element models were created to study prestress bond
transfer in small-scale beams containing concentric fully bonded or partially debonded strand
using soft sheathing (tight fit). The models were developed using the finite element program
ABAQUS [1]. The effect of additional stresses caused by the dilation of the partially debonded
strands due to Poisson’s effect was investigated and the influence of confinement reinforcement
was incorporated. The models were calibrated with experimental data obtained from small-scale
beams results (see Section 3.7). The calibrated parameter was the coefficient of friction, which
was used for a nonlinear-friction model that controlled the contact interaction between the
simulated strand and concrete parts in the finite element models. In addition, inelastic-material
behavior of concrete was considered.

4.2 Concrete Damaged Plasticity Model
Concrete behaves as a brittle material under low confining pressure, and when the
concrete is subjected to high confining pressures crack propagation in the brittle material is not
observed [1]. The concrete damaged plasticity (CDP) model, option available on ABAQUS [1],
gives a general capability for modeling concrete and other quasi-brittle materials such as rocks
and ceramics. The model accounts for concepts of isotropic damaged elasticity combining
isotropic tensile and compressive plasticity to represent the inelastic behavior of concrete. The
main formulation aspects behind for the concrete damaged plasticity model are:

• Strain rate decomposition,
82

• Stress-strain relations,
• Hardening parameters,
• Yield function, and
• A flow rule.
The rate strain decomposition for the rate independent model is shown in Equation 4-1
[1].

& &
&
ε = ε el + ε pl

Equation 4-1

&
&
where, the rate strain ε is decomposed into an elastic strain rate, ε el , and a plastic strain rate,

&
ε pl . The concrete damaged plasticity model is one of the possible constitutive models that can
be used to simulate failure and the recognition of crack patterns in concrete [11]. The
fundamentals of any model based on the classical theory of plasticity are the yield criterion, the
flow rule, and the hardening rule. The CDP model is based on the principles of the Mohr
Coulomb and Drucker-Prager, which in their general formulation follow Equation 4-2.

F (σ ) = c
In Equation 4-2, F (σ ) is a function of the stress components and

Equation 4-2

σ

may be identified with the

cohesion or some constant multiplier [18]. The initial part of the yield surface can be described
with c = f

c0

, and failure is reached when

c

attains its maximum along a given loading path

[18]. The formation of microcracks in the concrete is represented macroscopically as softening
behavior of the material, which results in localization and redistribution of strain in a structure
[15].

83

The response from tensile and compressive damage on quasi-brittle materials are
different; thus, it is not realistic to represent all damage states by a single parameter. If two state
variables ( f t and f c ) representing uniaxial tensile strength and compressive strength of the
material, respectively, describe the yield function, the admissible stress states are constrained by
the condition shown in Equation 4-3.

~
F (σ , f t , f c ) ≤ 0

Equation 4-3

According to experimental observations, in most quasi-brittle materials, including
concrete, the tendency is that the compressive stiffness is recovered upon crack closure as the
load changes from tension to compression. In contrast, as the load changes from compression to
tension, the tensile stiffness is not recovered once micro-cracks have developed.
The concrete damaged plasticity model [1] uses a yield condition based on the yield
function proposed by Lubliner et al. [18] and incorporates the modification proposed by Lee and
Fenves [15] to account for different evolutions of strength under tension and compression. The
evolution of the yield surface is controlled by the hardening parameters. The yield function can
be expressed in terms of effective stress, σ , as shown in Equation 4-4 [1].
1 

 ~ pl  ˆ
 ~ pl 
ˆ
F  σ , ε pl  =
 ~ 
 q − 3αp + β  ε  σ max − γ − σ max  − σ c  ε c  ≤ 0




 1−α 



where,

α and γ = dimensionless material constants,
1
p = − σ : I = hydrostatic pressure,
3

q=

3
S : S = equivalent effective stress,
2

S : pI + σ = deviatoric part of the effective stress tensor σ ,
84

Equation4-4

ˆ
σ max = algebraically maximum eigenvalue of effective stress tensor, and

β  ε pl  =






σ c  ε c pl 
~



 (1 − α ) − (1 + α ) .
~ 
 ε pl 
σt  t



~ pl
~ pl
From the last expression, σ t (ε t ) and σ c (ε c ) are the effective tensile and
compressive cohesion stresses, respectively. The variable α is determined using Equation 4-5
[1].

α=

(σ b0 / σ c0 ) − 1
;0 ≤ α ≤ 0.5
2(σ b0 / σ c0 ) − 1

Equation 4-5

From Equation 4-5, σ c0 is the uniaxial compressive yield stress and σ b0 is the
equibiaxial compressive yield stress. Another parameter that needs to be defined is the ratio of
the second stress invariant on the tensile meridian, q (TM ) , to that on the compressive meridian,
q (CM ) , Kc , at initial yield for any given value of the pressure invariant

p such that the

ˆ
maximum principal stress is negative, σ max < 0 (see Figure 4-1) [1].
The plastic flow is governed by a flow potential function G (σ ) according to a
nonassociative flow rule as shown in Equation 4-4.
∂G (σ )
∂σ

&
&
ε pl = λ

Equation 4-6

&
From Equation 4-4, λ is a nonnegative plastic multiplier [1]. The plastic potential is

defined in the effective stress space as shown in Figure 4-2.

85

Figure 4-1 Failure surface on the deviatoric plane. Adapted from [1]

(

1
ˆ
αI1 + 3J 2 + β σ 2
1−α

ˆ
σ2

)= c 0

f ty

ˆ
σ1

( fby , fby )
(

f cy

)

1
αI1 + 3J 2 =c 0
1−α

(

)

1
ˆ
αI1 + 3J 2 + β σ1 = c 0
1−α

Figure 4-2 Proposed yield surface in the stress space by Lubliner (1989). Adapted from [15]

86

The flow potential G chosen for the concrete plasticity model is the Drucker-Prager
hyperbolic function and this is shown in Equation 4-7.
G=

(∈ σ t 0 tanψ )2 + q 2

− p tanψ

Equation 4-7

In Equation 4-7, ψ is the dilation angle measured in the p − q plane at high confining
pressure; σ t 0 is the uniaxial tensile stress at failure; and

∈ is

a parameter related to the

eccentricity that defines the rate at which the function moves toward the asymptote (see Figure
4-3a) [1]. Figure 4-3a and Figure 4-3b show the response of the concrete under uniaxial loading
in tension and compression, respectively.

σt
σ t0

Eo

(1− dt )Eo
~
ε t pl

εt

ε t el
a) Tension Response

Figure 4-3 Response of the concrete under uniaxial loading. Adapted from [1]

87

Figure 4-3 cont’d.

σc
σ cu

σ c0

Eo

(1 − dt )Eo
~
ε c pl

εc

ε c el
b) Compression

Under uniaxial tension, the stress-strain curve exhibits a linear-elastic response until the
value of the failure stress, σ t 0 , is reached (see Figure 4-3a). In this study, σ t 0 corresponds to
the modulus of rupture of the concrete, f r = 7.5λ f 'ci , where λ = 1.0 for normal weight
concrete [3]. The failure stress corresponds to the initiation of micro-cracking in the concrete. If
the concrete is subjected to uniaxial compression, the material behaves linear-elastically until the
value of initial yielding σ c0 reached (see Figure 4-3b). In the plastic state, the compressive
response of the concrete is usually identified by stress hardening followed by strain softening
beyond the ultimate stress, σ cu (see Figure 4-3b) [1]. If the concrete is subjected to high
compressive loads, the propagation of cracks is reduced since cracks run parallel to the loading
direction. Nonetheless, the effective load-carrying area is significantly reduced after substantial

88

crushing of the concrete has taken place. The effective uniaxial cohesion stresses, σ t and σ c ,
can be determined using Equation 4-8 and Equation 4-9, respectively.
~
σ t = E o  ε t − ε t pl 





Equation 4-8

~
σ c = Eo  ε c − ε c pl 





Equation 4-9

4.2.1 Parameters and Assumptions on the CDP Model
In order to incorporate the concrete damaged plasticity (CDP) model in the numerical
simulations for this research, plasticity parameters need to be defined. The values of the variables
can be found from experiments or the default values available in ABAQUS [1] can be
preliminary used. A summary of the parameters that need to be defined is presented in the
following.
Dilation angle, ψ , in the p − q plane. The value needs to be entered in degrees.
Eccentricity,

∈,

which is a small positive number that defines the rate at which the

hyperbolic flow potential approaches its asymptote. The default value in ABAQUS is
0.1.
The ratio of initial equibiaxial compressive yield stress ( σ b0 ) to initial uniaxial
compressive yield stress ( σ c0 ), f b0 / f c0 . Experimental values of

f b0 / f c0 range

from 1.10 to 1.16 [18]. The default value in ABAQUS is 1.16.
K c is the ratio of the second stress invariant on the tensile meridian, q(TM ) , to that on
the compressive meridian, q(CM ) , at initial yield for any given value of the pressure

89

invariant p . The following condition must bet satisfy: 0.5 < K c ≤ 1.0 . The default value
in ABAQUS is 2/3.
Viscosity Parameter, µ , use for the visco-plastic regularization of the concrete
constitutive equations [1]. This value was selected by trial and error procedure and
chosen to be 0.02. The viscosity parameter must be kept as small as possible for accurate
results.
The compressive stress data is to be provided as a tabular function of inelastic (crushing)
~ in
strain, ε c . Positive (absolute) values shall be given for the compressive and tensile stress and
strain curves. The stress-strain curve can be defined beyond the ultimate stress into the strainsoftening state [1].

4.3 Description of the Contact Interaction Friction Model in ABAQUS
Surfaces in contact can transmit shear and normal forces. The relationship between these
two types of forces is known as friction, which is usually expressed in terms of the stress at the
interface of the surfaces in contact as shown in Figure 4-4-a.

µ

a) Classical Coulomb friction model

p (σr)
σ

b) Schematic of the strand dilation

Figure 4-4 Schematic of the basic Coulomb friction model and radial pressure due to
dilation of strand upon release
90

The friction model available in ABAQUS [1] gives the option of defining static and
kinetic coefficients of friction with a smooth transition zone defined by a decay curve. The
critical shear stress, τ crit , is the stress at which sliding of the surfaces in contact begins as a
fraction of the contact pressure ( p ) between surfaces, τ crit = µ * p , µ is the coefficient of
friction. This critical shear stress can be defined using the classical Coulomb friction model (see
Figure 4-4-a). A static coefficient of friction can be defined for the friction model and this is
known as the “stiffness (penalty) method” [1]. Once the maximum shear stress is exceeded,
sliding of the surfaces with respect to each other will occur. In the real case of the strand in
contact with concrete, the normal behavior (N as shown in Figure 4-4a) is developed by the
radial pressure exerted to the concrete as the strand dilates upon release (see Figure 4-4-b).
Longitudinal friction forces are developed by the mechanical interlock mechanism and friction
between the two surfaces creating the tangential interaction between the strand and the
surrounding concrete. This behavior is simulated by the aforementioned normal and tangential
models and is referred herein as a nonlinear friction model.
In ABAQUS [1], the stick/slip calculations determine when a point changes from sticking
to slipping or vice versa. The maximum shear stress τ max can be specified when using the
friction model [1]. An acceptable upper bound for estimating the value of τ max is σ y / 3 ,
where σ y is the Von Mises yield stress of the material adjacent to the surface. Figure 4-5
shows the sticking and slipping frictional states of the friction model. As it can be seen, the slope
of the curve is related to the sticking-friction state (linear-elastic region). The slipping friction
state is related to the plastic regime, and this takes place when τ crit is reached [1]. The response

91

shown in Figure 4-5 is comparable to the behavior of an elasto-plastic material without
hardening where κ would correspond to the elastic modulus and τcrit represents the yield stress.

Shear
Stress

Stricking
Friction

crit

Slipping Friction

Total Slip

Figure 4-5 Sticking and slipping friction. Adapted from [1]

The nonlinear friction model is implemented in ABAQUS by using a contact constraint
and such a constraint is activated in the finite element model when the distance separating two
node surfaces defined with the possibility to be in contact becomes zero. If the contact pressure
between the surfaces in contact becomes zero or negative, the constraint is removed and the
surfaces separate. This behavior is referred to as “hard contact” and it is the default contact
behavior in ABAQUS [1].

4.4 Modeling of Prestress Bond Transfer in 2-D and 3-D Models
According the goal of the investigation, prestress bond transfer can be modeled using
different approaches. Mirza and Tawfik [19] developed a one-dimensional (1D) model to study
92

the restraint effect in beam end-cracking and valuable contributions were obtained from this
model, namely, that the short free length of the uncut strands can lead to high stresses in the end
region. Kannel et al. [13] developed 3D numerical models to study the stress distribution in the
end region of pretensioned concrete girders. The strand was modeled as truss elements embedded
in the concrete and the prestressing force was transferred into the concrete at nodes where the
truss elements were connected to the concrete region, as shown in Figure 4-6. The models by
Kannel et al. were developed to show the effect of strand release sequence on the stress state in
anchorage zones. Another modeling technique using 1D elements is to connect the truss elements
to the concrete elements (2D or 3D) by means of springs. This technique allows for distinct
modeling of tensile and compressive stress transfer behavior.

Figure 4-6 Prestress bond transfer in 2D

93

The use of truss elements in a simulation model does not allow for capturing the dilation
of the strand due to the Poisson’s effect from detensioning. One of the goals of this study was to
investigate the stress state in the beam end regions due to the dilation of the partially debonded
strand upon detensioning and the effects that this may have in beam-end cracking. Thus,
modeling of the strand using three-dimensional continuum elements was needed. Figure 4-7
shows an isometric view of a beam and the strand indicating the surfaces to be in contact. In this
figure, the strand was modeled as a simplified circular rod since this facilitated computational
demand and the simulation of prestress bond transfer using surface contact interaction
definitions, as described in Section 4.3.

Figure 4-7 3D Modeling of prestress bond transfer using contact interaction definitions

94

4.5 Numerical Modeling of Laboratory-Scale Beams
Three-dimensional (3D) nonlinear finite element (FE) models that replicated the beams in
the experimental study presented in Section 3.6 were created and analyzed to emulate the bond
stress transfer behavior using the program ABAQUS [1]. The FE models incorporated inelastic
behavior and damage to the concrete material using the built-in material models in ABAQUS.
The high-fidelity three-dimensional models considered surface interaction definitions (see
Section 4.3) between the strand and concrete to model the stress transfer between the two. The
main parameter for the definition of the surface interaction model was the coefficient of friction.
Use of the “hard-contact” surface interaction model in ABAQUS[1] requires the definition of
tangential and normal behaviors. For the tangential behavior, a nonlinear friction formulation and
a coefficient of friction were defined. For the normal behavior, a contact constraint was applied
when the distance separating two surfaces in contact becomes zero such that radial stresses
develop as the strand tries to expand and the two surfaces cannot penetrate each other.
The aim of this simulation was to model bond stress transfer behavior to obtain basic
numerical parameters that control the surface interaction definitions needed to model realistic
bond behavior of prestressing strand in concrete beams. The FE beam models were calibrated by
comparing the numerically calculated and the experimentally measured concrete surface strain
profiles along the beam length. The calibrated parameter was the coefficient of friction, which
controls the nonlinear-friction model. The value of the coefficient of friction obtained from the
calibration was used the case study presented in Chapter 5.

95

4.5.1 Description of the FE Beam Model
The characteristic finite element model developed for this task is shown in Figure 4-8 and
Figure 4-9. The beam had a cross section of 6”x6” (150 mm x 150 mm), as the tested laboratoryscale beams (see Section 3.6). One half of the beam was modeled due to symmetry along the Zaxis, 120” (3050 mm).

Eleven FE beam models were created and analyzed. The models

followed the experimental test matrix given in Figure 3-23. Four models had a concentric fully
bonded strand and the rest had a concentric partially debonded strand. The unbonded length was
24 in. (610 mm) and a tight-fit was provided along the debonded length to simulate the flexible
slit sheathing. With a tight fit between the strand and the surrounding concrete, radial stresses
develop upon release. The strand was modeled as a cylindrical rod with a diameter of 0.5268”
(13.4 mm), which was determined to match the area of the prestressing 0.6-in. (15.2 mm)
2

2

diameter strand (0.218 in. [140 mm ]). The strand was stressed using the initial stress condition
[1]. The input stress values were those obtained from the experimental program (see Table 3-6
and Table 3-7). The model had just one loading step in which the prestressing force was
transmitted into the concrete member.

For the normal behavior of the friction model the

overclosure pressure was defined as “hard contact” or as impenetrable surfaces.

For the

nonlinear friction model, the tangential behavior was defined as penalty, and the coefficient of
friction was the parameter to be calibrated. The maximum shear stress value was not specified.
The coefficient of friction influences the transfer length and the stress transfer rate; however, the
coefficient of friction in these models was an artificial parameter that was used/calibrated to
account for the mechanical interlock and friction bond mechanisms. The coefficient of friction
was changed by trial and error until the transfer length from the numerical model closely
matched (± 1 in. [25.4 mm]) the values experimentally obtained.
96

For the partially debonded beam models, the coefficient of friction along the debonded
length was defined as 4.5 percent of the coefficient of friction given along the fully bonded
region. This followed the experimental observations that a small level of prestressing force was
transferred along the debonded region. The value of 4.5 percent was obtained from the wall-like
block results (see Section 3.5.3) by calculating the ratio of the average maximum pull-out force
for the debonded strand group to the average peak pull-out load for the group of fully bonded
strand, group which was found to be 0.045
Continuum three-dimensional, C3D8 (eight-node linear brick), elements were used for
the concrete and the strand parts in the FE model. The mesh size for the beam was approximately
1”x1”x1” (25.4 mm x 25.4 mm x 25.4 mm), and the elements of the strand varied in cross
section and their length was 1 in. (25.4 mm [see Figure 4-9]). The concrete damaged plasticity
(CDP) model (see Section 4.2) was used in the numerical models. The linear-elastic material
properties of the concrete and the plasticity parameters for the normally-consolidated concrete
and self-consolidated concrete sets of beams are shown in Table 4-1 and Table 4-2, respectively.
For the strand, linear-elastic material properties were defined. The value of the Young’s modulus
was defined as 29,000 ksi (200 GPa) and the Poisson’s ratio was set equal to 0.3. An overview of
the beam model is shown in Figure 4-8 to Figure 4-11, and the end-region of the model is shown
in Figure 4-9. Some of the plasticity parameters shown in Table 4-1 and Table 4-2 were obtained
from the literature. The value of the dilation angle was obtained from Jankowiak and
Lodygowsky [11]. The remaining values were based on the default values recommendations
from ABAQUS [1]. The viscosity parameter was selected by trial and error.
.

97

Figure 4-8 Geometry of the numerical beam model

Figure 4-9 End region of the FE beam model

When the theoretical and experimental concrete strain levels due to the prestress release
were first compared, it was found that the experimental values were higher than the calculated
98

ones. Further inspection indicated that this was due to creep and shrinkage effects (see Section
3.6). To account for those effects in the numerical models, temperature load was applied to the
beam to simulate the additional creep and shrinkage effects. The thermal expansion coefficient
was fixed to one, and temperature was gradually applied along the transfer length and fully
applied beyond the transfer region (see Figure 4-10 and Figure 4-11). The maximum temperature
value that was applied to each beam was obtained from the difference between the maximum
longitudinal strain value of obtained from the FE beam model before applying temperature and
the 100% AMS experimental value obtained for that specific beam

(see Section 3.6).

Temperature load was applied along the transfer zone in increments of 10 percent.
Representative results from the calibrated FE small-scale beam models are shown in Section
4.5.2.

Figure 4-10 Fully bonded beam model

99

Figure 4-11 Typical debonded beam model

Table 4-1 Concrete damaged plasticity model parameters for NCC FE models
Parameters for CDP model

Concrete
f'ci=7300 psi
parameters
Concrete Elasticity

Ec (ksi)

4870

ν

0.2

Concrete compression hardening
Stress (psi)
Crushing strain
1209.0
0
4710.1
0.000026
6480.7
0.000160
7195.5
0.000512
5733.8
0.001314
3652.1
0.002245
1281.6
0.003735
625.8
0.004621
274.3
0.005693
160.5
0.006467

Dilation Angle
Eccentricity

◦

38
0.1

fb0/fc0
1.16
Kc
0.67
Viscosity parameter
0.02
Concrete tension stiffening
Stress (psi)
Cracking Strain
640
0
375
0.000868
288
0.002868
248
0.004868
198
0.009868

1 MPa= 145 psi
100

Table 4-2 Concrete damaged plasticity model parameters for SCC FE models
Parameters for CDP model

Concrete
f'ci=9400 psi
parameters
Concrete Elasticity
Ec (ksi)
5526

ν

0.2

Concrete compression hardening
Stress (psi)
Crushing strain
1381.6
0
5482.2
0.000008
7885.7
0.000073
9287.0
0.000320
9134.3
0.000597
5557.8
0.001744
1556.4
0.003218
442.0
0.004170
147.9
0.004973
24.8
0.006496

Dilation Angle, ψ
Eccentricity, Є
fb0/fc0

◦

38
0.1

1.16

Kc
0.67
Viscosity parameter
0.02
Concrete tension stiffening
Stress (psi)
Cracking Strain
727
0
364
0.001868
291
0.004368
242
0.007868
225
0.009868

1 MPa= 145 psi

4.5.2 Results of Simulations on Bond Transfer on Prestressed Concrete Beams
Representative results of the calibrated beam models are presented herein. Concrete
surface strain profiles were obtained from each beam model along the longitudinal direction. In
addition, plot contours from each model were obtained to illustrate the longitudinal forces and
axial and transverse stress/strain from the strand’s Poisson effect. A complete set of results
obtained from the FE models is given in Appendix C.
The naming convention for stress and strains output from ABAQUS is as follows. The X,
Y, and Z axes are represented by numbers 1, 2, and 3, respectively. The letter E represents strain,
and the letter S means stress [1]. Figure 4-12 shows the normal and shear stresses in a differential
element.
101

Figure 4-12 Differential element subjected to normal and shear stresses

a) Concrete Surface Strain Profile Curves

The concrete surface strain (CSS) profile curve obtained from each model was closely
matched to their respective experimental CSS curves (see Section 3.6). It is well-known that
transfer length increases with time. One of the main contributors to this time-dependent change
is creep. Therefore, it was needed to normalize the coefficients of friction obtained from each
beam model to a common time at which the longitudinal strain measurements were taken during
the tests. The chosen time was six hours after the release of the first strand line. Equation 4-10
shows the expression used to normalize the coefficients of friction,

µ norm =

µ * (ε cbeam )
εc

NCC

102

Equation 4-10

where, µ represents the static coefficient of friction for a specific beam model; ε cbeam is the
creep strain value calculated according to the time at which longitudinal strain measurements
were taken for that specific beam; and, ε cNCC is the creep strain value calculated at a time of
six hours after releasing the first strand. The latter was a common value for all six numerical
beam models for the normally-consolidated concrete (NCC) set of beams. The creep strain value
was different for the SCC set of FE beam models since the compressive strength at release for
both groups was not the same. For the NCC set of beams, the creep strain value was found to be
135 µ s, and for the SCC group it was equal to 100 µ s. A summary of the coefficients of friction
obtained for each beam model, the time at which measurements were taken during the tests, and
the normalized coefficients of friction are shown in Table 4-3.

Table 4-3 Summary of coefficient of friction used in calibration of models
Beam ID
NCFBC
NCFBU
NCD1C
NCD1U
NCD2C
NCD2U
SCFBC
SCFBU
SCD1C
SCD1U
SCD2U

Coefficient
of friction
0.65
0.55
0.65
0.55
0.7
0.45
0.5
0.75
0.85
0.8
0.5

Time @ measurement Normalized coefficient of friction
(minutes)
based on creep strain at 6 hours
180
0.53
300
0.51
380
0.66
360
0.54
30
435
0.47
210
0.43
300
0.71
280
0.78
378
0.79
540
0.56

The purpose of normalizing the coefficients of friction was to obtain an average value
with a lower standard deviation and account for the variation on the experimentally obtained
103

transfer lengths due to creep effects. The coefficients of friction obtained from each beam
model (see fourth column in Table 4-3) were further normalized based on a release strength of
8,000 psi (55 MPa) and an initial prestressing force, fsi, of 202.5 ksi (1396 MPa) using Equation
4-11. The final values are shown in Table 4-4.

* 8 ( ksi) f pi ( ksi)
µ
*
µ '( f si ) = norm
f 'ci (ksi)

202.5(ksi)

Equation 4-11

In Equation 4-11, fpi is the initial prestressing force after sitting losses, and f’ci is the
compressive strength at release. A compressive strength of 8 ksi (55 MPa) and an initial
prestressing force equal to 202.5 ksi (1396 MPa) were selected to normalize the values of the
coefficient of friction since these values were used for the case study presented in Chapter 5. The
coefficients of friction presented in Table 4-3 (last column – right side) were also normalized
based on the above mentioned release strength and on an effective prestressing force, fse, of
189.5 ksi (1,307 MPa) using Equation 4-12, and this value was obtained assuming 13 ksi (89.6
MPa) in instantaneous prestress losses, which was the average value obtained from the FE beam
models.

µ
* 8 (ksi) f se ( ksi)
µ ' ( f ) = norm
*
se
189.5(ksi)
f 'c i (ksi)

104

Equation 4-12

Table 4-4 Normalized coefficients of friction based on f’ci= 8 ksi and on fsi = 202.5 ksi
Values normalized based
on fsi = 202.5 ksi
Beam
ID
NCD1C
NCD1U
SCD1C
SCD1U
NCFBU
SCFBU

Normalized
Coef. of Friction
0.690
0.604
0.724
0.731
0.545
0.658

Statistics on Normalized Coef. Of
Friction

f'ci (ksi) Mean
7.3
0.69
7.3
9.4
9.4
7.3
0.60
9.4

Standard
Deviation

Coef. of
Variation (COV)

0.0585

0.0851

0.08

0.1330
1 ksi = 6.89 MPa

Table 4-5 Coefficients of friction normalized on fse = 189.5 ksi and f’ci=8 ksi
Values normalized based
on fse = 189.5 ksi
Beam
ID
NCD1C
NCD1U
SCD1C
SCD1U
NCFBU
SCFBU

Normalized
Coef. of Friction
0.602
0.507
0.645
0.657
0.52
0.676

Statistics on Normalized Coef. Of
Friction

f'ci (ksi) Mean
7.3
7.3
0.60
9.4
9.4
7.3
0.59
9.4

Standard
Deviation

Coef. of
Variation (COV)

0.068

0.1129

0.1103

0.1845
1 ksi = 6.89 MPa

It can be seen from Table 4-4 and Table 4-5 that the mean value of the coefficient of
friction based on a compressive strength at release of 8,000 psi (55 MPa) was 0.59. This value
was used for the case study presented in Chapter 5. After applying temperature on the concrete
and releasing the prestressing force, the concrete surface strain (CSS) profile curves for the
models were obtained by creating a longitudinal path at mid-height of the beam-part (same
location as the experimentally measured CSS) as shown in Figure 4-13.

105

Figure 4-13 Longitudinal path along the FE beam models

Figure 4-14 and Figure 4-15 show the concrete surface strain profiles for two FE beam
models versus the corresponding experimental CSS traces. It can be seen that the FE and
experimental CSS traces match each other well. Calibration of the self-consolidated concrete
(SCC) beams is shown in Figure 4-16 and Figure 4-17.
Figure 4-16 shows the calibrated FE response and the experimentally measured
concrete surface strain profiles for the SCFBU beam. The CDP model was not used in this
particular model due to numerical problems (lack of convergence) attributed to the high
compressive strength and coefficient of friction needed to match the experimental transfer
length. Because of this issue, the results in Figure 4-16 were obtained from a linear-elastic
analysis. The differences between a linear-elastic and an inelastic analysis in terms of stress and
strain values are discussed in Section 4.7. However, it was verified that material behavior of the
concrete does not affect transfer length in the FE models.
106

D is t a n c e f r o m E n d o f B e a m ( m m )
600

500

1000

1500

2000

2500

3000

L =19"

-6

Concrete Surface Strain Level (in./in.*10 )

0
t

500

400

300

200

100

N C F B C B e a m M o d e l (C o e f. o f fric tio n = 0 . 6 5 )
N C F B C B e a m - E x p e r im e n ta l C u rv e

0
0

20

40

60

80

100

120

D is t a n c e f r o m E n d o f B e a m ( in .)

Figure 4-14 NCFBC beam model vs. experimental data

D is t a n c e fro m E n d o f B e a m (m m )
-6

Concrete Surface Strain Level (in./in.*10 )

0

50 0

1000

1500

20 00

2500

30 00

3500

500

L =19"
t

400

300

L =24"
u

200

100
N C D 1 U B e a m M o d e l (C o e f. o f fric tio n = 0 .5 5 )
N C D 1 U B e a m - E x p e rim e n ta l C u rv e
0
0

50

10 0

D is t a n c e fr o m E n d o f B e a m (in .)

Figure 4-15 NCD1U model vs. experimental data

107

150

D is ta n c e fr o m E n d o f B e a m (m m )
0

500

1000

1 500

2000

2 5 00

3000

350

L =17"
-6

Concrete Strain level (in./in.*10 )

t

300

250
200

150

100
50

S C F B U B e a m M o d e l (C o e f. o f F ric tio n = 0 .7 5 )
S C F B U B e a m - E x p e rim e n ta l C u rv e

0
0

20

40

60

80

100

120

D is ta n c e fro m E n d o f B e a m (in .)

Figure 4-16 SCFBU beam model vs. experimental data

As it can be seen from the results in Figure 4-14 to Figure 4-17, the numerical models
were successfully calibrated. It shall be kept in mind that the calibrated parameter was the
coefficient of friction because this variable controls the transfer length. Further, the calibration
process was done by simple comparison of the numerically predicted vs. experimental transfer
length value.

108

Distance from End of B eam (m m )
0

500

1 000

1500

2 000

2500

30 00

3500

-6

Concrete Strain Level (in./in.*10 )

40 0
35 0

L =27"
u

30 0
25 0 L =24"
u

20 0
15 0
10 0
SC D2U Be am M od el (C oef. of F rictio n= 0.5 0)

50

SC D2U Be am - Experim e nta l C urve
0
0

50

10 0

1 50

Distance from End of B eam (in.)

Figure 4-17 SCD2U beam model vs. experimental data

b) Stress and Strain Paths and Plot Contours

The stress and strain values along the cross section at the releasing end of each beam
model were obtained to assess the level of stress and deformation induced by the prestressing
force in the concrete member. A path along the cross section of the beam was selected to obtain
stress and strain levels at the releasing end. In order to understand how the stresses and strains on
the cross section at the releasing end of the beam model can be affected when the strand is fully
bonded or partially debonded with tight fit sheathing, the maximum principal strain, axial and
transverse stress levels are shown in the following. Figure 4-18 shows the maximum principal
strains on the cross-section of the beam at the releasing end for all the NCC numerical models.

109

Distance from edge to center of beam (mm)
0

10

20

30

40

50

60

70

80

0.001
NCFBC Beam Model
NCFBU Beam Model

Strain Level (in./in.)

0.0008

NCD1C Beam Model
NCD1U Beam Model
NCD2C Beam Model

0.0006

NCD2U Beam Model

0.0004

0.0002

0

-0.0002
0

0.5

1

1.5

2

2.5

3

3.5

Distance from edge to center of beam (in.)
Figure 4-18 Maximum principal strain levels along cross section - NCC beams

Representative results of stress and strain contours are discussed next. In the following
figures, positive values denote tension, and negative values mean compression. In addition, the
red color denotes the highest tension value and darkest blue color refers to the largest
compressive value. In order to carry out a better comparison from the simulation, the maximum
and minimum contour values were fixed to the minimum value in tension and in compression for
each group of beams, NCC and SCC, respectively. Thus, the light gray color denotes the tensile
force above the defined tension-limit value, and the dark gray color represents compressive
values beyond the maximum compression-limit established. See Appendix C for supplementary
plot-contour results obtained from each beam model.
To compare the stress and strain levels along the debonded/transfer zone after the
prestressing force was transferred into the concrete member, contour plots comparing a beam
110

with partially debonded and fully bonded strand are compared in the following. Figure 4-19-a
and Figure 4-19-b show the axial strain contours for the NCD1C and NCFBC beam models,
respectively. During the experimental phase the two beams were placed in the same strand line,
thus having the same initial prestressing force. The transverse strain E22 contour plots for the
NCD1U and NCFBU beam models are shown in Figure 4-20-a and Figure 4-20-b, respectively.
The two beams were subjected to the same initial prestressing force of 198 ksi (1365 MPa). The
maximum principal strain contour plots for the NCD1C and NCFBC beam models are shown in
Figure 4-21-a and in Figure 4-21-b, respectively.

111

a) NCD1C beam model

b) NCFBC beam model
Figure 4-19 Axial strain contour plots

112

a) NCD1U beam model

b) NCFBU Beam Model
Figure 4-20 Transverse strains contour plots: front and side views

113

a) NCD1C beam model

b) NCFBC beam model
Figure 4-21 Maximum principal strain contour plots: side and front views

114

As it can be seen from Figure 4-19-a, the axial strain values along the debonded zone
were high and somewhat constant, which indicates that with a tight fit (flexible slit sheathing),
high radial stresses develop at the interface of the two materials. This phenomenon is also
observed in Figure 4-20-a and Figure 4-21-a. On the other hand, for the beam models with a
fully bonded strand, NCFBC and NCFBU models, and the strain values were high close to the
release end and decrease at about 10 in. (250 mm) measured from the releasing end.
The transverse stresses, S22, for the NCD1U and NCFBU beam models are shown in
Figure 4-22-a and Figure 4-22-b, respectively. The maximum principal stresses are shown in
Figure 4-23-a for the NCD2C beam model, and Figure 4-23-b shows the same type of stresses
obtained from the NCFBC beam model. The entire length of the beam is not shown in order to
present a closer view to the stresses and strains distribution on the anchorage zone and along the
transfer region. The maximum principal strain values at the release end were numerically found
to be higher for the models with a partially debonded strand than the ones containing a fully
bonded strand. Since there was nominal friction between the strand and the surrounding concrete
along the debonded zone of the beam models, the radial stresses caused by the dilation of the
unbonded strand were higher as is explained in Figure 4-24.

115

a) NCD1U beam model

b) NCFBU beam model
Figure 4-22 Transverse stresses: side view and cross section at release end

116

a) NCD2C beam model

b) NCFBC beam model
Figure 4-23 Maximum principal stresses contour plots: front and side views

117

The transverse and maximum principal stress contours shown in Figure 4-22-a and Figure
4-23-a also show high values along the unbonded zone. Figure 4-24 is presented as an aid to
explain the phenomenon observed when a tight fit is provided between the partially debonded
strand and the surrounding concrete. At time zero, prior releasing the prestressing force into the
concrete member, the stress in the strand σs is equal along the entire prestressed steel and the
concrete has not been precompressed yet. As the prestressing force Fs is released into the
concrete radial stresses σr develop due to the dilation of the prestressing strand. If the strand is
fully bonded longitudinal bond stresses σb (µσn) develop. The axial strains εx can be related to
radial strains εr by the Poisson’s ratio υ. The prestressing force in the strand is reduced by a
fixed amount ∆Fs, which is composed of the bonding force Fb (σb/A) and the axial force Fx
(σx/A). Thus, as the friction between the strand and the concrete decreases the radial stresses at
the interface increase.

118

∆Fs = Fx + Fb
σ x =

∆ F s − Fb
A

ε r = υε x
εr =

υ

EA

(∆Fs − µσ n )

Figure 4-24 Schematic illustrating of high stresses along debonded zone

4.6 Influence of Confinement Reinforcement on Transfer Zone Stress
State
The distribution of stresses in the end zone of a prestressed concrete member depends on
strand location, the magnitude of prestressing force, the degree of bond between strands and the
surrounding concrete, the amount of draped strands in the end zone, the geometry of the girder,
the compressive strength of the concrete, etc. [33]. Vertical stirrups in the end zone of
pretensioned concrete girders become effective once horizontal cracks develop. Thus, the use of
confinement in the anchorage region could minimize the high stresses cause by the releasing of
the prestressing force. The provision of transverse and confining reinforcement has thus been
recommended to delay bond failure [29]. Confining and transverse reinforcement is a detailing
119

requirement in the design of pretensioned concrete members since it is the only mechanism that
could prevent bursting and spalling cracks in the end zone upon release [27]. The provision of
confinement reinforcement was studied through numerical simulations along the transfer and
debonded zone to investigate the stress state in the end-region upon release, as is presented next.

4.6.1 Numerical Evaluation of Confinement in Prestressed Concrete Beams
The effect of confinement reinforcement in the beam-end region was evaluated through
numerical simulations on beams with single concentric strands. Circular stirrups were placed
along the transfer zone in the case of the beam model containing a concentric fully bonded
strand, and along the debonded region for the beam model containing a concentric partially
debonded strand. Different bar sizes and spiral diameters were considered for the confinement
reinforcement. Two of the eleven FE beam models were considered for this investigation,
namely, the NCFBC and NCD1C beam models. The description for each model is presented
next.

a) NCFBC Beam Model

Numerical analyses were carried out to study the effect of confinement in the end region
of one the laboratory-scale beams containing a concentric fully bonded strand. The NCFBC
beam model was modeled using different confinement conditions. Circular hoops with bar sizes
of #3, 4 and 5 (No. 10, 13, and 16) were considered for this study, and the diameter of the hoops
were 1.5, 3.0, and 4.0 inches (35, 75, and 100 mm). Thus, the NCFBC beam model was run nine
times with the different confinement conditions. Inelastic behavior of the concrete was
incorporated. The results to be shown in this section are the ones obtained from the numerical
120

model with circular hoops with a 1.5 in. (35 mm) diameter and using #3, 4, and 5 (No. 10, 13,
and 16) reinforcing bars. The minimum realistic diameter that was considered was 1.5 in. (35
mm) since the diameter of the strand is 0.6 in. (15.2 mm), leaving only 0.45 in. (11.4 mm) for the
concrete to flow between the strand and the stirrup. The hoops were concentrically placed around
the strand along the transfer region (see Figure 4-25) and spaced at every 2 in. (50 mm) from
center-to-center. The first stirrup was placed at 1 in. (25 mm) from the beam end.

Figure 4-25 NCFBC beam model: close view of the end-zone

b) NCD1C Beam Model

Modeling for the NCD1C (see Figure 3-23) was similar to the one presented in Section
4.5. The dilation of debonded strands upon release can cause transverse stresses along the
debonded length when using soft sheathing (see Figure A-10). In this case, the circular stirrups
were placed along the debonded region to assess if the use of confinement reinforcement in the
121

end-region helped minimize radial stresses along the debonded zone. The confinement
reinforcement options were the same ones considered for the NCFBC beam model (see Section
4.6.1-a). The results presented here are those obtained for the NCD1C beam model containing
#3, 4 and 5 (No. 10, 13, and 16) hoops and diameter equal to 1.5 in. (35 mm). These cases
seemed to be the most effective in reducing high tensile stresses along the transfer length as seen
from the results obtained from the NCFBC beam model with confinement reinforcement. Figure
4-26 shows the NCD1C beam model.

Figure 4-26 NCD1C beam model with circular stirrups along debonded zone

4.6.2 Results
a) NCFBC FE Beam Model
Representative results obtained from the NCFBC beam model with confinement
reinforcement consisting of 1.5” (35 mm) diameter hoops with #3, 4, and 5 (No. 10, 13, and 16)
bar sizes are presented the following. Results along two paths were studied: one along the cross
122

section of the beam at about 4” (100 mm) from the beam end, and another one at the top-center
of the beam along the first 35” (900 mm) from the beam end. Figure 4-27 shows the maximum
principal stresses along the cross-sectional path. The axial strains (E11) along the same crosssectional path are shown in Figure 4-28, and the transverse stresses (S22) are shown in Figure
4-29. The maximum principal stresses along the top-center path are shown in Figure 4-30.

Distance from Edge to Center of Beam (mm)
10

20

30

40

50

60

70

80

500

Stress Level (psi)

4

No Confinement
Confinement - Bar No.3
Confinement - Bar No.4
Confinement - Bar No.5

3.6
3.2
2.8

400

2.4
300

2
1.6

Stress Level (MPa)

0
600

200
1.2
0.8

100
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)
Figure 4-27 Maximum principal stress: NCFBC beam model - cross-sectional path

As it was to be expected, the models with the highest confinement level – smallest
diameter and largest bar size - performed better. At the same time, the differences between the
FE beam model without confinement and the one having the highest confinement level was not
significant. The highest difference in stress value obtained from the model with confinement (1.5
in. [35 mm] diameter hoops with #5 [No. 16]) to that without confinement was about 100 psi,
obtained from the maximum principal stress plot at the cross section of the beam at the releasing
123

end. However, the minimum spacing between prestressing strands established in codes such as
the AASHTO Bridge Design Specifications [2] and ACI-318 [3] is 2 in. (50 mm). If circular
stirrups are used in girders to reduce the high tensile stresses in the anchorage region, the 1.5-in.
(35 mm) diameter stirrups may not be possible to use for girders containing 0.6-in-diameter
strands.
The axial strain (E11) and transverse strain (E22) levels obtained from the beam model
with high confinement (#5 [No. 16] bar and hoop diameter of 1.5 in.[35 mm]) seemed to reduce
the stress values by about 2.5 times compared to the model without confinement. In terms of
maximum principal stresses along the top-center of the NCFBC beam model (see Figure 4-30),
the stresses decreased about 50 psi (345 kPa) compared to those obtained from the NCFBC
beam model without confinement and the one with confinement using a #5 (No. 16) and a hoop
diameter of 1.5 in. (35 mm)
Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.0006

Strain Level (in./in.)

0.0004

0.0002

0

-0.0002
No Confinement
Confinement - Bar No.3
Confinement - Bar No.4
Confinement - Bar No.5

-0.0004

-0.0006
0

0.5

1

1.5

2

2.5

3

Distance from Edge to Center of Beam (in.)

Figure 4-28 Axial strains: NCFBC model

124

3.5

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.0006
No Confinement
Confinement - Bar No.3
Confinement - Bar No.4
Confinement - Bar No.5

Strain Level (in./in.)

0.0004

0.0002

0

-0.0002

-0.0004

-0.0006
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)
Figure 4-29 Transverse strains: NCFBC beam model – path along cross section

Distance from Edge to Center of Beam (mm)
0

100

200

300

400

500

600

700

800

200
1.2
1
0.8
100
0.6
0.4

50

0.2
No Confinement
Confinement - Bar No.3
Confinement - Bar No.4
Confinement - Bar No.5

0

Stress Level (MPa)

Stress Level (psi)

150

0
-0.2

-50
0

5

10

15

20

25

30

35

Distance from Edge to Center of Beam (in.)

Figure 4-30 Maximum principal stresses: NCFBC beam model – path along top beam
center
125

b) NCD1C Beam Model
From the results obtained from the NCFBC beam model, it was observed that the circular
#5 (No. 16) bar hoops performed slightly better than the other two. Thus, only the results
obtained from the beam model with a 1.5 in. (35 mm) diameter hoop with #5 (No. 16) bars are
presented for the NCD1C model. Maximum principal stress values along top-center of the beam
(see Figure 4-31) and equivalent plastic strain in uniaxial tension (PEEQT [see Figure 4-32]) are
shown to compare how the values can decrease when confinement is provided along the
debonded zone to reduce high stresses/strains produced by the dilation of the strand as it tries to
recover its original diameter. As it is shown in Figure 4-24, if tight fit is provided between the
debonded strand and the surrounding concrete the radial stresses increase along the blanketed
region. It can be seen from these figures that the maximum principal stresses along the topcenter of the beam reduced ~10% with the use of confinement (#5 [No.16] hoops), and the
PEEQT values along the cross section at 4 in. (100 mm) from the releasing end only decreased
13% when using the same type of confinement. Thus, the use of confinement did not
significantly affect the high stress/strain values along the unbonded zone; yet, the values seemed
to decrease about 11% in average.

126

Distance from Edge to Center of Beam (mm)
0

200

400

600

800

1000

1200

160
No Confinement
140

1

Stress Level (psi)

120

0.8

100
0.6
80
60

0.4

Stress Level (MPa)

Confinement - Bar No.5

40
0.2
20
0

0
0

10

20

30

40

50

Distance from Edge to Center of Beam (in.)

Figure 4-31 NCD1C model: Maximum principal stress along top-beam path
Distance from Edge to Center of Beam (mm)
0

10

0.00016

20

30

40

50

60

70

80

No Confinement

Strain Level (in./in.)

Confinement - Bar No.5
0.00012

8 10

-5

4 10

-5

0
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)

Figure 4-32 Equivalent plastic strain in uniaxial tension (PEEQT) along cross section (at 4
in. [100 mm] from the releasing end)

127

4.7 Linear-Elastic vs. Concrete Damaged Plasticity FE Models
A comparison between the linear-elastic and inelastic analyses is presented here since one
of the eleven beam models was not calibrated using the concrete damaged plasticity (CDP)
model due to numerical problems. One of the reasons for the problems was because of the high
coefficient of friction needed to calibrate the model and the high compressive strength at release.
The model did converge when using linear-elastic material properties for the concrete part. The
concrete material properties are shown in Table 4-1.
The NCFBU beam model was selected for the comparison since the numerical model that
did not converge was the SCFBU for the above-mentioned reasons. Hence, the NCFBU beam
model was run using the same value for the static coefficient of friction, applying the same level
of temperature to the beam (to simulate creep and shrinkage effects), but one was modeled with
linear-elastic concrete material properties and the other one using the CDP model.
Representative results are shown in the next section.

4.7.1 Results: NCFBU Linear-elastic vs. NCFBU Inelastic FE Beam Models
The representative results shown next were obtained from a path along the beam crosssection at the releasing end and a path on the top-center of the beam along the longitudinal
direction. Other results were also plotted and they are provided in Appendix C. Figure 4-33
shows the transverse stresses (S22) along the cross section of the beam at the releasing end. The
maximum principal stress values are shown in Figure 4-34 and the maximum principal strain
values on the cross section of the beam at the releasing end are shown in Figure 4-35.

128

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

4000
24

Stress Level (psi)

8
0

0
-8

-2000

-16

Stress Level (MPa)

16

2000

-24
-4000
-32

NCFBU Beam Model (CDP)
NCFBU Beam Model (Linear-elastic)

-40

-6000
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)

Figure 4-33 Transverse stresses – elastic vs. inelastic concrete behavior
Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

5000
32

NCFBU Beam Model (CDP)
NCFBU Beam Model (Linear-Elastic)

Stress Level (psi)

24
3000

20
16

2000
12

Stress Level (MPa)

28

4000

8

1000

4
0
0

0.5

1

1.5

2

2.5

3

0
3.5

Distance from Edge to Center of Beam (in.)

Figure 4-34 Maximum principal stress: cross sectional path

129

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.0014
0.0012

Strain Level (in./in.)

0.001
0.0008
0.0006
0.0004
0.0002
NCFBU Beam Model (CDP)

0

NCFBU Beam Model (Linear-elastic)
-0.0002
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)

Figure 4-35 Maximum principal strain along front cross-section

The transverse stresses (S22) were obtained along the cross section at the releasing end
and it can be observed from Figure 4-33 that the values from the linear-elastic NCFBU beam
model are significantly higher than the ones obtained when using the CDP model. It shall be
mentioned that the maximum principal stress values were obtained to have a global comparison
in terms of stresses. The maximum value obtained from the linear-elastic analysis was about
3500 psi (24MPa), about 700% greater than the largest value obtained from the analysis
incorporating material nonlinearity for the concrete part.

4.8 Findings
The following summarizes the findings from the numerical simulations of laboratoryscale beams presented in this Chapter.
130

•

The laboratory-scale beam models were successfully calibrated with experimental data so
as to obtain a coefficient of friction parameter to be used for the simulations of largescale beam models to study the stress state in the end-regions of pre-tensioned concrete
girders upon release.

•

Results from the beam models with partially debonded strand showed high stress values
along the debonded length and this can be attributed to the dilation of the strand as it tries
to recover original diameter upon release causing high radial stresses (i.e., it is Poisson’s
effect). These high stresses can continue along the unbonded length because the
debonding mechanism is effective. Thus, this observation can explain the concrete
surface strain levels experimentally measured along the debonded zone of the laboratoryscale beam units.

•

The use of confinement reinforcement along the debonded/transfer zones does not seem
to reduce the high stresses in the end regions. However, confinement and transverse
reinforcement must be provided in detailing pre-tensioned girders as these are essential
elements that can prevent bursting and splitting cracks upon strand detensioning.

131

CHAPTER 5 NUMERICAL SIMULATIONS: STRESS
ANALYSES OF ANCHORAGE REGIONS

5.1 Introduction
In order to evaluate the variables affecting end-cracking in full-size bridge girders
containing partially debonded strands, nonlinear three-dimensional numerical models were
developed using the general purpose FE program ABAQUS [1]. The variables incorporated in
the numerical simulations were the type of sheathing material, flexible (tight fit) or rigid
(oversized), and strand debonding distribution. A skew U-type bridge girder, which contained
partially debonded strands, was chosen as case study because damage attributed to the debonding
of strands was observed during production. The numerical models incorporated inelastic concrete
material behavior and surface contact interaction definitions to model prestress load transfer.
Four models were created incorporating the use of flexible or rigid debonding and a staggered
debonded strand distribution. A description of the theory behind the concrete-inelastic behavior
and contact interaction models was presented in Chapter 4, and the description of the U-beam
models and results are presented in this chapter. It should be noted, however, that the intent of
this study was to obtain qualitative information on the causes behind the observed damage and
assess the role of strand debonding in this behavior. Thus, the aim was not to quantify the stress
state or exactly match the cracking/damage pattern, which is not possible since the number of
parameters involved are too complex and unknown since the only information available on the
damaged girder is a qualitative (photograph) pattern of damage.

132

5.2 Case Study: Skew U-Beam Unit
A skew U-type bridge girder produced for the Indiana Department of Transportation
(INDOT), for which evidence of end cracking believed to be caused by strand release, exists,
was selected as a case study [8][23]. Figure 5-1 shows the damage seen on the U-beam girder
upon release. The dimensions of the U-beam are 56 in. (1420 mm) wide in the bottom flange,
99.5 in. (2525 mm) wide at the top, 61.5 in. (1560 mm) void-wide in the upper part, and 54 in.
(1370 mm) depth, with a skew angle of 18 degrees at both ends. The cross section of the U-beam
girder is shown in Figure 1-6 and Figure 5-2; the region of interest is shown in the same figure.
The length of the U-beam unit is 116’-5” (35.5 m) with a span length of 114’-11.5” (35 m).

Figure 5-1 Damaged observed on skew U-beam during production at beam end [8][23]

The U-beam girder had ninety-three (93) 0.6-in. (15.2 mm) diameter Grade 270 (1860
MPa) strands: 57 straight strands in the bottom flange, and 36 draped strands in the sides of the
beam (see Figure 1-4.) Twenty-one (21) of the 57 strands in the bottom flange were debonded
using flexible (tight fit) slit sheathing. Thus, 37% of the strands were debonded in total and all
133

the shielded strands were placed in the center-bottom flange of the beam (see Figure 1-4 and
Figure 5-2.) The percentage of debonded strands was: 20% in row 1, 43% in row 2, and 43% in
row 3. Such design exceeds limits established in the AASHTO LRFD Bridge Design
Specifications [2] (see Section 2.5.1-a). Thirty-three (33) percent of the unbonded strands had
the same unbonded length, which was less than the 40% limit in the AASHTO Specifications.
The number (ID) below each symbol as shown in Figure 5-3 corresponds to the unbonded
lengths in the U-girder unit as shown in Table 5-1. The symbol without number corresponds to
the fully bonded strands. The concrete compressive strength at release was 8,000 psi (55.2 MPa)
and the initial strand prestressing level was 0.75 fpu (202.5 ksi [1396 MPa]).

Figure 5-2 U-beam cross section. Adapted from [8] [23]

134

5.3 Simulation Case Studies
Nonlinear program ABAQUS [1] to investigate the effects of debonded strands and
sheathing material in the damage seen in the U-beam girder during production (see Figure 5-1).
As earlier mentioned, duplicating the exact observed damage is nearly impossible. Thus, the
intent of this study is rather to gain insight into the role that strand debonding mechanisms and
patterns may have on the observed damage and as potential source of cracking at beam ends in a
qualitative manner.
The study considered the use of soft (tight fitting) sheathing and rigid (oversized hole)
debonding. In addition, the arrangement, or distribution, of debonded strands in the beam cross
section was also evaluated. Four FE models under different conditions were created and analyzed
to study the above-mentioned parameters. The four models were classified as case studies and
are described below:

• Case 1: As-built U-beam model with tight fitting. This model included all the features
of the U-beam unit featuring flexible slit sheathing.

• Case 2: Model with staggered debonding arrangement for the unbonded strands with
flexible (tight fitting) sheathing.

• Case 3: Model with rigid (oversized) sheathing for the debonded strands with the
original (“as-built”) debonded strand arrangement. Rigid debonding was simulated by
providing oversized holes in the concrete part along the strand unbonded (shielded)
length.

• Case 4: Model with a staggered debonding distribution for the unbonded strands with
rigid (oversized) sheathing.

135

The front view of the region of interest and debonding strand configuration for Case 1
and Case 3 is shown in Figure 5-3. The strand unbonded length for Cases 1 and 3 is given in
Table 5-1 and for Cases 2 and 4 is one quarter (1/4) of those shown in Table 5-1. The debonded
length for Cases 2 and 4 was reduced to improve computational efficiency and achieve
convergence in the solution of the model. The arrangement of the partially debonded strands for
Cases 2 and 4 is shown in Figure 5-4.

Figure 5-3 View of the region of interest and debonding strand numbering: Cases 1 and 3

Figure 5-4 Partially debonding strand numbering: Cases 2 and 4
136

Table 5-1 Debonded strand length in U beam [8]

ID
1
2
3

Unbonded Length (ft)
Length
ID
Length
12.0
4
18.0
9.0
5
21.0
15.0
6
24.0
1 ft = 0.3049 m

5.4 Case 1: As Built Condition
This case corresponds to the as-built condition, i.e., the U-beam unit for which evidence
of damage exists (Figure 5-1) with some modeling assumptions/modifications. A tight constraint
is provided to emulate the flexible slit sheathing debonding scheme. Details and schematics of
the finite element model together with representative results are presented in the following

5.4.1 Model Geometry, Mesh, Material Properties and Contact Definition
The cross section of the region of interest was a rectangle at the corner of the cross section
with dimensions of 28 in. (710 mm) by 8.25 in. (210 mm) as shown in Figure 5-2. As it can be
seen from Figure 5-2, the cross section of the U-beam unit was slightly simplified. Since the
strands are symmetrically distributed with respect to the center line of the beam (see Figure 1-4
and Figure 5-2), the high fidelity FE model was refined in a region of interest on one side of the
beam. The U-beam FE model, which is shown in Figure 5-5, was 56 in. (1420 mm) wide, 54 in.
(1370 mm) depth, and 58.33 ft long (17.8 m). The concrete parts were modeled using continuum
solid elements (C3D8R: eight-node linear brick with reduced integration.) Half of the beam unit
was modeled along the longitudinal direction (Z-axis) due to symmetry.

137

The end block (see Figure 5-6) was 56 in. (1420 mm) wide, 45.75 in. (1160 mm) in
height, and 24 in. (610 mm) long (Z-axis). The side parts, region with embedded strands, of the
beam were 7.5 in. (190 mm) wide, 45.75 in. (1160 mm) deep, and 56.3 ft. (17.2 m) long. These
parts are inclined by 14.4° in respect with the vertical (Y-axis). The mesh for these concrete parts
was 4.0 in. x 4.0 in. x 4.0 in. (100 mm x 100 mm x 100 mm).

Figure 5-5 Overview of the U-beam girder as built condition model

The bottom flange was composed of three parts. The first, which was the region of interest
(see Figure 5-2), was 28 in. (710 mm) wide, 8.25 in. (210 mm) deep, and 48 in. (1220 mm) long
in the fully-bonded-strand region to allow transfer of the prestressing force, and 29.2 ft (8.9 m)
long along the partially-debonded-strand region to accommodate the largest debonded length and
additional length for transmission of the prestressing force. The size of the elements in the
138

bottom flange along the region of interest varied in the cross section (see Figure 5-6) and had a
length of approximately 4 in. (100 mm). Beyond the region of interest, the bottom flange was
modeled with solid elements and the mesh size was 4 in. x 4 in. x 4 in. (100 mm x 100 mm x 100
mm). To the right of the region of interest (see Figure 5-5) the concrete part was modeled with a
coarser mesh compared to the region of interest. The partially debonded strands along the
unbonded zone were not accounted for in the model, and fully bonded strands and partially
debonded strands along the fully bonded zone were modeled using discrete embedded truss
elements.

Figure 5-6 End region of the U-beam as built condition model

139

The strand in the region of interest was modeled as a 3-D cylindrical rod with an
unstressed diameter of 0.5268 in. (13.4 mm). The diameter was calculated using the
representative area of a 0.6-in.- (15.2 mm) diameter strand, which is 0.218 in2 (144 mm2). The
fully bonded strands located in the bottom flange (see Figure 5-6) were modeled using
continuum 3-D solid elements (C3D8R) elements up to 4 ft (1220 mm) from the beam end to
accommodate the transmission length of the prestressing force at release. The partially debonded
strands were modeled using C3D8R up to 29.2 ft (8.9 m) from the end of the beam to
accommodate the largest debonded length and the respective transfer lengths. The mesh size
varied on the cross section and was approximately 4 in. (100 mm) long. Beyond these noticed
distances, the strands were modeled as discrete truss elements (T3D2) and the elements were
embedded in the concrete (bottom flange). The draped strands located at the sides of the beam
(see Figure 5-6) were modeled as discrete truss elements embedded in the concrete region as
well, and the hold down point was located at about 47’-3.5” (14.4 m) from the releasing end (Zaxis).
The strand was modeled using linear elastic steel material properties. The elastic modulus
was 29,000 ksi (200 GPa) and the Poisson’s ratio was 0.3. The concrete was modeled using
nonlinear-material behavior. The concrete damaged plasticity (CDP) model [1] was incorporated
in the numerical simulation of the large-scale beams. A description of the CDP model is
presented in Chapter 4. The compressive strength at release was 8,000 psi (55.2 MPa). The
elastic modulus of the concrete was 5098 ksi (35.15 GPa), which was calculated using the
expression, 57000 f 'ci ( psi ) [3]. The parameters used for the CDP model are shown in Table 52.

140

Regarding boundary conditions, the beam model was restrained from vertical
displacement (U2) along the front-bottom edge (see Figure 5-5). Symmetry along the Z axis was
defined and a center point at the back of the beam was restrained in rotation about the Z-axis
(UR3) as shown in Figure 5-5. The model had just one step in which the prestressing force was
transmitted into the concrete member. The initial prestressing force was 0.75 fpu, which
corresponded to 202.5 ksi (1396.2 MPa). This value was obtained from the manufacturer
construction plans [8].
Table 5-2 Concrete damaged plasticity parameters for U-beam model
Parameters for CDP Model

Concrete
f'ci=8000 psi
parameters
Concrete Elasticity

Ec (ksi)

5098

ν

Dilation Angle, ψ
Eccentricity, Є

◦

38

0.1

0.2

fb0/fc0

1.16

Kc
0.67
Viscosity
parameter
0.02
Concrete compression hardening
Concrete tension stiffening
Cracking
Stress (psi)
Crushing strain
Stress (psi)
Strain
1274
0
670
0
7028
0.00012
335
0.001868
7670
0.00025
260
0.004868
7976
0.00044
224
0.007868
7524
0.00077
207
0.009868
4949
0.00178
1 ksi = 6.89 MPa
2675
0.00273
756
0.0041
257
0.0052
84
0.00648

The bond between the strand and the surrounding concrete was simulated using surfacebased contact definitions as described in Section 4.3. The coefficient of friction used for the U141

beam models was 0.59, which was obtained from the calibrated small-scale beam models (see
Section 4.5.2). Along the unbonded zone, the coefficient of friction was defined as zero to
eliminate the bond interaction between the strand and concrete due to the use of sheathing
material. However, flexible (tight fitting) sheathing allows radial stresses to develop upon release
because longitudinal shear stresses do not take place along the debonded length. Thus, normal
interaction at the interface with the concrete is developed, as was explained in Section 4.5.2. All
the concrete parts were interconnected using tie-constraint (surface-to-surface) interaction
definitions. The solid strand and truss strand parts were connected by means of kinematic
coupling constraints (U1, U2, and U3). The truss elements embedded in the concrete were
connected to the concrete parts using embedded region constraints. Table 5-3 shows a summary
of the interaction constraints used for the U-beam models.
Table 5-3 Interaction constraints in U-beam models
Part 1 (Master)
Concrete (solid)
Strand (solid)
Strand (Truss)
Strand (truss)

Part 2 (Slave)
Concrete (solid)
Concrete (solid)
Strand (solid)
Concrete (solid)

Interaction-Definition
Tie-constraint
Contact Friction Model
Kinematic Coupling
Embedded Region

5.4.2 Results
Representative results obtained from the as-built U-beam model, Case 1, are presented in
this section. Contour plots on the cross section of the beam on the region of interest are used to
show general trends that can be correlated to the observed damage on the U-girder unit during
production. Additional contour plots and response traces along different paths at the interface of
the partially debonded strands are shown in Appendix D. The value for the maximum tensile
strength defined for the concrete for the damaged plasticity (CDP) model (see Table 5-2)
142

corresponds to the value of the modulus of rupture, 670 psi (4.6 MPa), which was calculated as
7.5 f 'ci ( psi ) . Figure 5-7 shows the damaged observed during production of the U-beam unit
and the maximum principal plastic strain contour plot on the cross section of the as-built
condition model (Case 1) at the release end. It can be seen in Figure 5-7 that the as-built model
with tight fit satisfactorily predicts, in a qualitative sense, the damaged observed upon
detensioning.

Figure 5-7 Damage seen during production of U-girder unit vs. maximum principal plastic
strains: Case 1

143

The contour plot that shows the maximum principal stresses on the cross section at the
release end of the as-built condition (Case 1) model is shown in Figure 5-8. Again, the maximum
tensile strength in the CDP model was 670 psi (4.6 MPa). It can be seen from Figure 5-8 that the
concrete elements surrounding the partially debonded strands had principal tensile stress values
above 670 psi (4.6 MPa) indicating cracking of the concrete as it was observed in the picture of
damage of the U beam (Figure 5-1).

Figure 5-8 Maximum principal stresses: U-beam as-built condition (Case 1) model

It shall be mentioned that prediction of the exact cracking pattern is not possible with the
CDP model since the concrete softness after the maximum tensile stress (i.e., damage) is
reached. However, Figure 5-9 shows the regions that are beyond the linear-elastic regime of the
concrete tension-response curve (see Figure 4-3-a). It can be observed in Figure 5-9 that the
areas around the partially debonded strands are clearly beyond the elastic regime. Thus, the
144

cracking and spalling off of the concrete observed at release during production of the U-beam
correlates well to these results. Therefore, even though the exact cracking pattern cannot be
duplicated with the modeling assumptions, the overall damage zone-observed during production
of the U-beam girder was well captured by FE simulation (see Figure 5-7 - Figure 5-9).

Figure 5-9 Active Yielding of the concrete at beam cross section at release end: Case 1

5.5 Case 2: Tight Fitting with Staggered Debonding Distribution
The simulation study in Case 2 had two distinct objectives. First, to investigate if a
staggered unbonded distribution could help reduce the localized beam-end damage observed in
the “as-built” model. The second aim was to assess the effect of the distribution of debonded
strands in the beam cross section in terms of longitudinal shear stresses at the beam end. In the
145

“as-built” configuration the partially debonded strands in the U-beam were placed in the center
of the bottom flange, which could lead to high longitudinal shear stresses due to the large
unbalance of forces along the section width. The use of a staggered debonding pattern is required
in the bridge design guidelines of the Florida DOT [7], who recommended to evenly distribute
the debonded strands such that, whenever possible, the shielded strands be separated in all
directions by at least one fully bonded strand. However, the rationale behind the noted
recommendation is not mentioned and could not be found in the available literature. A staggered
strand debonding configuration was thus studied for the U-beam models using both flexible
(tight fitting) sheathing and rigid (oversized) debonding, namely Case 2 and Case 4. The U-beam
model with staggered debonding distribution and tight fit is referred to in this study as Case 2.
The arrangement of the partially debonded strands is shown in Figure 5-4. The modeling details
and representative results are presented next. Results from Case 2 are compared with those from
Case 1 in Section 5.8.1.

5.5.1 Specific Modeling Details
The finite element model developed for Case 2 is similar to the one presented for Case 1
(see Figure 5-5 and Figure 5-6). However, there were two main differences. First, due to the
staggered distribution of the partially debonded strands, the length of the fully bonded strands
was equal to the length of the unbonded strands in the region of interest. Secondly, in order to
maximize computational efficiency, the length of the region of interest was reduced from 29.2 ftlong (8.9 m) to 9.3 ft-long (2.85 m). The unbonded lengths were reduced to one-quarter of the
original debonded lengths (see Table 5-1). The region of interest is shown in Figure 5-10 and as
146

it can be observed from this figure, the strands in the region of interest (both fully bonded and
partially debonded) were modeled as cylindrical rods (solid elements – C3D8R) up to 112”
(2845 mm) from the releasing end. The other parts of the model remain similar to the geometry
showed for Case 1 (see Figure 5-5 and Figure 5-6). Material properties, boundary conditions, and
interaction constraints were also similar to those presented in Section 5.4.1. Symmetry along the
Z-axis was also defined for this U-beam model (Case 2).

Figure 5-10 U-beam Bottom Flange for Case 2: Half-region along X-axis

5.5.2 Results
Representative results for Case 2 in the form of stress and strain contour plots are
presented herein. The maximum principal plastic strains contour plot are compared to the
cracking and spalling damage observed for the U-beam girder in Figure 5-11. From this figure, it
147

can be seen that the simulation predicts a reduced level of damages if one fully bonded strand is
located between partially debonded strands, except for the first column from left to right (front
view) since those three strands were kept as the original distribution for being the center strands.
The strain contour plot in Figure 5-11 shows high strains in the bottom-left region. This behavior
is due to issues with the boundary conditions. The maximum principal stresses contour plot
obtained from Case 2 is shown in Figure 5-12 and the results are also compared to the damage
observed during production. As it is observed from this figure, the maximum principal stresses at
the cross section of the beam model at the beam end are reduced compared to Case 1 (see Figure
5-8) when the partially debonded strands were staggered.
The results obtained from Case 1 and Case 2 are further compared in Section 5.8.1 for a
better understanding of how the debonding strand pattern can minimize high stresses at the beam
end and reduce longitudinal shear stresses.

Figure 5-11 Maximum principal plastic strain vs. damage observed during production:
Case 2

148

Figure 5-12 Case 2: Maximum principal strain vs. damage observed during production

5.6 Case 3: Oversized Debonding with Original Unbonded Strand
Configuration
For Case 3 some modeling simplifications and assumptions had to be made due to
numerical problems with the full-model (similar to Case 1) with rigid (oversized) debonding. To
minimize computational demand, symmetry along the X-axis was assumed for this modeling
case. This modeling simplification implies that the skew angle (18°) was considered small so
that the beam was considered as straight. It is recognized that the assumption of lateral symmetry
for the skew beam cross-section is not technically correct and that it has some drawbacks on the

149

accuracy of the model results. These effects are addressed in Section 5.6.3. Geometric details of
the model and relevant results are presented next.

5.6.1 Model, Geometry, Mesh, and Generalities
An oversized hole was defined in the concrete part of the model to simulate the rigid
debonding material. The hole was 0.95 in. (24.1 mm) in diameter around the 3D cylindrical
strand, which was greater than the diameter of the unstressed strand (0.5268 in. [13.4 mm]).
Thus, the rigid sheathing was successfully simulated by fully decoupling the strand and the
concrete parts. The size of the hole was selected so to not affect the elements in aspect ratio (i.e.,
cross-sectional area to length). The model developed for Case 3 is shown in Figure 5-13. The
region of interest is similar to in Case 1 (see Figure 5-3). The main difference between Cases 1
and 3 is that in Case 3 symmetry was assumed along the X-axis. The total length of the model
was 58.3 ft (17.8 m), half of the length of the U-beam girder, due to symmetry along the Z-axis.
The concrete part was modeled using solid elements (C3D8R) as it was described in Case
1, and the size of the elements was approximately 3.5”x3.5”x3.5” (90 mm x90 mm x 90 mm).
The strand in the region of interest was modeled as a cylindrical rod with a diameter of 0.6-in
(15.2 mm) as was previously described for Cases 1 and 2. However, for Cases 3 and 4, no strand
elements were defined along the unbonded length to minimize computational demand. Rather,
only about 6 in. (150 mm) of strand measured from the fully bonded zone toward the unbonded
length were modeled using C3D8R elements to allow for the dilation and relative slip of the
strand at the onset of the fully bonded zone upon release. The size of the elements for the strand
(cylindrical rod) varied in cross section as it was earlier mentioned and the length of the elements
was about 3.7” (94 mm). The elements in the strand part were given with a slightly higher mesh
150

size because, as described in Table 5-3, the concrete part was defined as the slave surface, and
this surface needs to be smaller or equal to the master surface (cylindrical strand part). The endzone for the Case 3 is shown in Figure 5-14. As seen in this figure, no major changes were made
between Cases 1 and Case 3 except for the symmetry definition along X-axis. The material
properties and constraints are similar as described in Section 5.4.1.

Figure 5-13 Skew U-beam model with oversized hole and original debonded distribution:
Case 3

151

Figure 5-14 End-zone view: U-beam model – Case 3 (Half-model along X-axis)

5.6.2 Results
Contour plots that show results from Case 3 are presented herein. However, it should be
noted that the assumption of symmetry along X-axis affected the accuracy of the results,
especially at the beam end. This is addressed with detail in Section 5.6.3. Nonetheless, the results
obtained from Case 3 gave valuable information on how the use of rigid (oversized) sheathing
can help minimize high stresses in the end region of the U-beam girder upon detensioning. A
detailed comparison of the results obtained from Case 1 and Case 3 are shown in Section 5.8.2.
The maximum principal strain contour plot at the beam end for Case 3 is shown in Figure
5-15. It is seen from this figure that at the bottom-right side (front view) of the U-beam model,
152

shows a high concentration of strains. This is not a real effect and is attributed to issues with
symmetric boundary condition defined along X-axis. This same issue is observed in the contour
plots obtained for the equivalent plastic strain in uniaxial tension (PEEQT) shown in Figure
5-16. However, Figure 5-16 also shows that at a distance of 12 in. (300 mm) from the beam end
the strain values significantly drop. The difference is attributed to the fact that at a distance of 1
ft. (300 mm) from the skew-end the effects of symmetry in X-axis are reduced. A top view of the
end of the beam model is shown in Figure 5-17 for a better understanding of this issue.

Figure 5-15 Case 3: Maximum principal plastic strain at release end

153

Figure 5-16 Case 3: Equivalent plastic strain in uniaxial tension

Figure 5-17 Top view of the end of the bottom flange of the beam model with symmetry
along X-axis

154

5.6.3 Effects on Symmetry Assumptions
In order to evaluate the effects on symmetry along the X-axis made in the model for Case
3 model (see Figure 5-13), the results from a model similar to Case 1 but with symmetry along
X-axis are compared to the results from the full model from Case 1 (see Section 5.4.1). Contour
plots at the cross section from both models at the beam end are used as the basis for the
comparison.
In Figure 5-18, the maximum principal plastic strain contours are shown for both
models. It can be seen that the assumption on symmetry along the X-axis increased the plastic
strain values, particularly around the location of the partially debonded strands.
Contour plots that show the equivalent plastic strains in uniaxial tension (PEEQT) are
presented in Figure 5-20. The activated yielding flag on the concrete elements (tensile stresses
beyond elastic regime) is shown in Figure 5-19. The maximum principal stresses for Case 1 and
the skew beam model with symmetry along X-axis and tight fitting debonded strand is shown
Figure 5-22. In
Figure 5-20, Figure 5-19, and Figure 5-22 similar results are obtained in terms of higher
values due to the assumption of symmetry along X-axis for the skew-beam model. That is, the
assumption of symmetry along the X-axis increases the stresses and strains at the end of the
beam model especially toward the location of the partially debonded strands.

155

Figure 5-18 Maximum principal plastic strain: Case 1 vs. Skew- Symmetry Case 1

However, if contour plots are obtained at a small distance from the release end such as 12
in. (300 mm), where the beam becomes certainly straight (or where the effects due to symmetry
along X-axis for a skew beam somewhat decrease) similar response is obtained from both
models. This is shown in Figure 5-21.

156

Figure 5-19 Effects on skew-symmetry boundary condition: Actively yielding

157

Figure 5-20 Case 1 vs. Skew-symmetry conditions: Equivalent plastic strain in uniaxial
tension at beam end cross section

158

Figure 5-21 Equivalent plastic strain in uniaxial tension at 12 in. from release end

159

Figure 5-22 Maximum Principal Stresses: Case 1 vs. Skew-Symmetry

5.7 Case 4: Oversized Debonding with Staggered Unbonded Strand
Distribution
In the case of the model with staggered debonding distribution and oversized hole,
similar issues with convergence of the full numerical beam model (similar to Case 1) were
encountered as described in Section 5.6 for Case 3. Therefore, the assumption of symmetry along
160

the X-axis was also made for Case 4. A brief description of the model is following presented
since the U-beam model developed for Case 4 has similarities to Cases 2 and 3.

5.7.1 Generalities of the Model
The model developed for Case 4 is similar to Case 2 in terms of the pattern distribution of
the partially debonded strands (see Figure 5-4), and the dimensions of the region of interest are
also similar to those presented in Section 5.5.1. On the other hand, Case 4 is comparable to the
one in Case 3 since symmetry along X-axis was assumed. The impact of this assumption for a
skew-beam was presented in Section 5.6.3. The geometry for Case 4 is shown in Figure 5-23.

Figure 5-23 Overview of the model for Case 4

161

In Case 4 the fully bonded and partially debonded strands along the region of interest
were modeled as 3-D cylindrical rods. However, along the unbonded zone, the strand part was
almost eliminated; only about 6 in. (150 mm) [measured from the fully bonded zone toward the
debonded zone] of cylindrical was rod was modeled as discussed in Section 5.6.1. All the
concrete parts were modeled using continuum solid elements, and discrete embedded truss
elements were used to modeled the strand beyond the region of interest and the draped strands
located in the side part of the U-beam girder. As it can be seen from Figure 5-23, the model for
Case 4 is similar to that of Case 2 (see Section 5.5) and Case 3 (see Section 5.6). Representative
results from Case 4 are given next.

5.7.2 Results
Figure 5-24 shows the maximum principal plastic strain at the beam end and at a cross
section 12 in. (300 mm) into the beam (see Figure 5-17). It is seen from Figure 5-24 that just few
inches from the skew-release end the plastic strains decay quickly. This is also observed from the
contour plots that show the equivalent plastic strain in uniaxial tension (PEEQT) in Figure 5-25.
Thus, these two figures show that while the assumption of symmetry along X-axis leads to an
increase on the response at the beam end, that this effect decreases rapidly into the beam. Thus,
the simulation results are considered reliable in qualitatively shown that the use of a staggered
debonding pattern and oversized strand sheathing decrease the high stresses in the end region
upon release. This is further discussed in Section 5.8.2, where results from the four skew-beam
models are compared to each other for a better understanding on the improvement on the results
upon the use of a staggered debonding distribution and rigid (oversized) strand sheathing.
162

Figure 5-24 Results for Case 4: Maximum principal plastic strain

163

Figure 5-25 Equivalent plastic strain in uniaxial tension: Case 4

5.8 Comparison of Modeling Results
To better understand the results obtained from each of the numerical models just
presented, the cases with common features are compared in the following sections. Case 1 and
Case 2 are compared first because these two models had the same dimensions. Second, the four
skew-beam models are compared using contour plots at the cross section and then response
traces along different paths at the interface of partially debonded strands are compared according
to the unbonded lengths in common, i.e., Case 1 vs. Case 3, and Case 2 vs. Case 4.
164

5.8.1 Case 1 vs. Case 2
It can be seen in Figure 5-26 that the use of a staggered distribution of debonded strand
helps reduce the high stresses in the beam-end region upon release. In this particular case, a skew
U-beam model, it is seen from the results obtained from Case 2 (see Figure 5-26) that the stresses
at the cross section at the release end are still above the concrete modulus of rupture, 670 psi
(4.6MPa), but the values are lower than those obtained from Case 1 (as-built condition model).
Further, the high stress region is localized only at the beam end and decays rapidly.

Figure 5-26 Maximum principal stresses: Case 1 vs. Case 2

165

The equivalent plastic strain in uniaxial tension (PEEQT) contour plots for Cases 1 and 2
are shown in Figure 5-27. In this Figure, Case 2 was divided in Case 2-a and Case 2-b because it
was observed that the model had an spurious highly stressed zone in the bottom left corner due
to boundary conditions. Since this effect does not affect the region of interest around the partially
debonded strands, the highly stressed corner was removed from the results display as shown in
Case 2-b. If Case 1 and Case 2-b are compared (see Figure 5-27), it can be observed that the use
of a staggered debonding distribution noticeably decreases the high stresses in the end region
upon release of the pretensioning force.

166

Figure 5-27 Equivalent plastic strain in uniaxial tension: Case 1 vs. Case2

The decrease in the high strain values is also seen by comparing the yielded concrete
elements (those above the elastic state) as shown in Figure 5-28. Again, the spurious high stress
region in the bottom corner needs to be neglected. While the staggered debonding pattern still
167

indicates damage to the concrete, the number of affected elements is reduced and the yielded
state decays much more rapidly into the beam for the Case 2 compared to Case 1.

Figure 5-28 Yielding in concrete: Case 1 vs. Case 2

The effect of a staggered strand debonding pattern on the resulting longitudinal shear
stresses was studied by examining the shear demands along an XZ plane located between the
fully bonded and partially bonded strands in the “as-built condition” as shown in Figure 5-29. It
is seen from this figure that the shear stresses are noticeably reduced in Case 2 (staggered
168

debonded pattern) compared to the condition in Case 1 (“as-built.”) This reduces the overall
stress state in the anchorage region upon release, which can improve overall performance of the
girder under service an ultimate demand. Therefore, the use of a staggered debonding
distribution helps to control not only the high stresses in the beam-end region due to the dilation
of the debonded strands, but also longitudinal shear stresses induced from strand force unbalance
upon release.

Figure 5-29 Shear Stresses on XZ Plane: Case 1 vs. Case 2

169

5.8.2 Four Skew-symmetry Beam Models
The four skew U-beam models with the assumption of symmetry along the ‘Z’ and ‘X’
axes are compared in this section. Contour plots at the beam-end cross section are compared as
well as trances along three different partially debonded strands, #1, #9, and # 11 (see Figure
5-30).

Figure 5-30 Partially debonded strands selected for response traces

5.8.2.1 Contour Plots at Beam-end: Four skew-symmetry U-beam models

Representative results comparing the stresses and strains at the beam-end cross section
and at a distance of 12 in. (300 mm [see Figure 5-17]) into the beam are shown next. Figure 5-31
presents contour plots for the actively yielding concrete elements. This figure shows that the
original strand configuration with rigid (oversized) strand sheathing (Case 3) presents the lowest
“yielding” (demand beyond the elastic regime) among the four models.

170

Figure 5-31 Actively yielding: Four skew U-beam models

Maximum principal strain contour plots at a distance of 12 in. (300 mm [see Figure
5-17]) from the release end are shown in Figure 5-32. It can be seen from this figure that the
models with rigid (oversize) sheathing debonding had the lower strain demands compared to the
models featuring tight fitting debonded strands. Similar results can be observed from the
maximum principal stress contour plots shown in Figure 5-33. In addition, the longitudinal shear
stresses contour plots for the four skew-symmetric U-beam models are compared in Figure 5-34.

171

Figure 5-32 Skew U-beam models: Maximum principal plastic strain at 12 in. (300 mm) from release end
172

Figure 5-33 Maximum principal stresses at 12 in. (300 mm) from the release end
173

Figure 5-34 Shear stresses on the XZ Plane: Four skew-symmetry U-beam models
174

5.8.2.2 Paths along partially debonded Strands: Skew-symmetry beam with tight fit
and original configuration vs. Case 3

Response traces along the interface of some partially debonded strands (No.1, No.9, and
No.11 [ see Figure 5-30]) comparing the results from the skew U-beam with X-symmetry, tight
fitting debonding option, and original debonding configuration vs. Case 3 are presented herein.
The maximum principal stresses along the interface of the partially debonded strand No.1 (see
Figure 5-30) as shown in Figure 5-35. It can be observed from this Figure that the stresses along
the debonded region for Case 3 are lower than the results obtained from the skew-symmetry
model with tight fitting debonding material and original debonded strand pattern. However, at
the release end the stresses are high for Case 3 and this can be attributed to boundary condition
effects, that is, the X-symmetry definitions for the skew beam.
A similar pattern is observed from the response traces along partially debonded strand
No.9, as shown in Figure 5-36. The maximum principal plastic strains (PE) are relatively
constant and low along the unbonded/fully bonded region for Case 3, except for the response
obtained close to the release end as previously mentioned. The maximum principal PEs from the
model with symmetry along the X-axis with original debonded pattern decrease at the onset of
the fully bonded zone, and the values are about the same level from those obtained for Case 3.
The response traces for the equivalent plastic strains in uniaxial tension (PEET) obtained
at the interface of the partially debonded strand No.11 for the model with symmetry along Xaxis, tight fitting debonding, and original debonding distribution and the results obtained from
Case 3 are shown in Figure 5-37. The strain values obtained from both models at the release end
are quite high; however, the response traces obtained from the Case 3 model are lower. The
remaining of the response traces for these two beams models are presented in Appendix D.

175

0

Skew - original configuration - tight
Skew - original configuration - rigid

Stress Level (psi)

1000

6

800
600

4

400
2
200

Stress Level (MPa)

1200

Distance from end of beam (mm)
4
2000
4000
6000
8000
1 10
8

0

0
-200
0

50

100

150

200

250

300

350

400

Distance from end of beam (in.)
Figure 5-35 Longitudinal path along partially debonded strand 1: Maximum principal
stresses – Skew-symmetry tight model vs. rigid and original configuration

0.01

0

Distance from end of beam (mm)
4
2000
4000
6000
8000
1 10

Strain Level (in./in.)

Skew - original configuration - tight

0.008

Skew - original configuration - rigid

0.006
0.004
0.002
0
0

50

100 150 200 250 300 350 400
Distance from end of beam (in.)

Figure 5-36 Response traces along partially debonded strand No.9: Maximum plastic
strains – Skew-symmetry U-beam models tight vs. rigid with original configuration
176

Distance from end of beam (mm)
0

2000

4000

6000

8000

1 10

4

0.0035

Skew - original configuration - tight

0.003

Strain Level (in./in.)

Skew - original configuration - rigid
0.0025
0.002
0.0015
0.001
0.0005
0
-0.0005
0

50

100

150

200

250

300

350

400

Distance from end of beam (in.)
Figure 5-37 Longitudinal path along partially debonded strand No.11: Equivalent plastic
strain in uniaxial tension – Skew-symmetry tight vs. rigid with original unbonded pattern

5.8.2.3 Longitudinal paths along partially debonded Strands: Skew U-beam model
with tight fit and staggered debonding pattern vs. Case 4

Response traces along the interface of partially debonded strands No.1, No.9, and No. 11
(see Figure 5-30) for the beam model with symmetry along the X-axis, tight fitting strand
debonding option, and staggered debonding distribution vs. Case 4 are presented in this section.
In all the response traces, high values were observed from both models at the release end due
boundary conditions effects. As previously mentioned, these are considered spurious effects
from the symmetry boundary condition and are neglected in the interpretation of the results.

177

The maximum principal stress traces for the above-mentioned models are shown in
Figure 5-38. The values observed in this Figure close to the release end are above 670 psi (4.6
MPa), which is the calculated concrete modulus of rupture for a compressive strength of 8,000
psi (55.2 MPa). However, as it is observed from Figure 5-35, Figure 5-36, and Figure 5-37, the
values obtained from the model with rigid (oversized) debonding quickly decay into the beam,
approximately 12 in. (300 mm) measured from the release end (see Figure 5-17 and Figure
5-38). The maximum principal stress values become somewhat similar for both models at the
onset of the fully bonded region.
The maximum principal plastic strains and equivalent plastic strain in uniaxial tension
response traces for Case 4 and the model with X-symmetry, tight fitting debonding, and
staggered unbonded pattern are presented in Figure 5-39 and Figure 5-40. Both figures show
similar trends, high values at the release end and a rapid decay further into the beam for the
model with rigid (oversized) sheathing (Case 4).

178

Distance from end of beam (mm)
0

500

1000

1500

2000

2500

3000

1400

Skew -staggered pattern - tight
Skew -staggered pattern - rigid

8

1000
6
800
600

4

400

Stress Level (MPa)

Stress Level (psi)

1200

2
200
0
0

20

40

60

80

100

0
120

Distance from end of beam (in.)

Figure 5-38 Maximum principal stresses: response trace along interface of partially
debonded strand No.1 – skew-symmetry model tight vs. rigid with staggered pattern

Distance from end of beam (m m )
0

500

1000

1500

2000

2500

3000

3500

0.004

Skew -staggered pattern - tight
Skew -staggered pattern - rigid

Strain Level (in./in.)

0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0

50

100

150

Distance from end of beam (in.)

Figure 5-39 Maximum principal plastic strains along longitudinal path at the interface of
partially debonded strand No.9

179

Distance from end of beam (mm)
0

500

1000

1500

2000

2500

3000

0.007

Skew -staggered pattern - tight
Skew -staggered pattern - rigid

Strain Level (in./in.)

0.006
0.005
0.004
0.003
0.002
0.001
0

0

-0.001
0

20

40

60

80

100

120

Distance from end of beam (in.)

Figure 5-40 Equivalent plastic strain in uniaxial tension along longitudinal path at the
interface of partially debonded strand No.11

5.8.3 Discussion
From the comparison of the results between Case 1 (“as-built” condition model) and Case
2 (skew U-beam model with a staggered debonding distribution and tight fitting debonding) it is
seen that the use of a staggered unbonded pattern reduces the high stresses in the beam-end
region created by the dilation of the debonded strand as shown in Figure 5-26, Figure 5-27, and
Figure 5-28. In addition, the implementation of a staggered debonding distribution also decreases
the longitudinal shear stresses (see Figure 5-29). To better understand how the longitudinal shear
stresses are minimized with the use of a staggered debonding pattern, Figure 5-41 is presented as
an aid. In this figure, a rectangular section of a beam is presented with four fully bonded strands,
two at each side, and three partially debonded strands located in the center of the beam. The
prestressing force in the fully bonded strands is immediately transmitted into the beam, at the
180

onset of the release end until the end of the transfer length. However, the force in the partially
debonded strands is transferred further into the beam, at the end of the debonded zone and along
this unbonded region there are no (or very small) forces in the center of the beam to counteract
the longitudinal shear stresses generated from the transmission of the prestressing force from the
fully bonded strands. Thus, in the case study presented in this research, skew U-beam, the
concentration of partially debonded strands in the center of the bottom flange of the beam leads
to high (and probably) undesirable longitudinal shear stresses, as it is seen from comparing the
results from Case 1 and Case 2.
On the other hand, it was observed from the results presented in Section 5.8 that the
assumption of X-symmetry for the skew U-beam models had negative effects on the results
obtained from the numerical models, especially at the beam end. However, the general trend that
can be observed from these results seems obvious: that the use of rigid (oversized) strand
debonding is beneficial to reduce the high stresses at the end region upon release. This general
trend was observed from the response traces (see Section 5.8.2.1 and Section 5.8.2.2) at the
interface of the partially debonded stands, and it was consistent with all the response traces
obtained at the interface between the strands and concrete.

181

Figure 5-41 Discussion of un-balanced forces due to strand debonding distribution

5.9 Findings
It is recognized that the finite element models presented herein are far from reproducing
the exact damage pattern observed during production of the U-beam girder in the presented case
study. This follows from the following complex parameters involved in the problem, which
include:

182

•

Mechanical interlock bond resistance of the seven-wire twisted strand: in the models the
strand was modeled as a smooth cylindrical rod with nonlinear friction interaction
surface-to-surface with the surrounding concrete.

•

Heterogeneity of concrete: in the FE models the concrete was simulated as a
homogeneous material. This has an effect in the localization, initiation and propagation of
cracking and the development of spalling regions.

•

Construction procedure and release pattern: the prestressing force was released
simultaneously and no interaction (i.e., friction) with the casting bed was simulated. Sole
plates and other local reinforcement in the beam section were not considered.

•

Concrete cracking and damage simulation: the concrete damage plasticity (CDP) model
permits modeling the inelastic response of the concrete, but it does so in a homogeneous
sense and damage is introduced by softening of the material and not discrete opening of
cracks. This has an effect on the simulated damage propagation.

•

Strand-concrete bond simulation and damage: strand bond was simulated through a
friction model between the interacting surfaces of a smooth rod (strand) and concrete.
Bond resistance thus depends on the normal pressure generated between the interacting
surfaces. Upon damage the CDP model will soften the concrete elements, an action
which in turn will minimize the constraint onto the expanding strand part and thus
decrease the normal surfaces between the interacting surfaces and thus bond resistance.
Such a decrease in bond resistance is likely not present in the mechanical interlock
component of bond even if microcracks become present along the transfer region.
It follows from the points noted above that while the presented finite element models

have many features (positive features) and allow a high degree of fidelity, they are not able to
183

replicate the exact cracking and damage pattern seen in the case study. Thus, the stress and strain
values presented in this chapter are to be used as qualitative measures for the choices on strand
debonding options and distribution and not as exact predicted responses. Nonetheless, it is felt
that the numerical models provide “enhance” qualitative data. Thus the following findings are
presented:

•

It was found from the results of Case 1 that the high stresses developed around the
partially debonded strands upon release are comparable in general sense to the damage
observed during production of the U-beam girder in the case study. Thus, the dilation
resulting from the flexible slit sheathing debonding material and the concentration of
strands in a group at the center of the section bottom flange are attributed as the cause of
the damage.

•

It is believed that the use of a staggered debonding pattern can decrease high stresses in
the end region and longitudinal shear stresses into the beam depth as shown from the
results from Case 2.

•

The use of rigid (oversized) strand sheathing can minimize the high stresses (those
intruded from the dilation of debonded strands with flexible slit sheathing) at the beam
end and along the entire unbonded zone (across the section and along the debonded
length) upon detensioning.

•

The implementation of strand debonding with rigid oversized sheathing in a staggered
distribution pattern provided the best option for reducing high dilation stresses at the
beam end as well as longitudinal shear stresses due to unbalanced forces generated when
the partially debonded strands are concentrated in the center of the beam.

184

CHAPTER 6 RECOMMENDATIONS AND CONCLUSION

6.1 Relevant Findings
Based on the experimental and numerical studies presented in this thesis, the following
relevant findings are offered.

•

Experimental results on pull-out tests on simple cylinders with a concentric partially
debonded or fully bonded strand showed that confinement reinforcement had only a
minor effect (~10 %) on bond strength.

•

Flexible-slit sheathing, single or double layer, is an effective debonding mechanism
based on experimental data from pull-out tests on cylinders and wall-like block with
unstressed strands.

•

The hydraulic head effect does not affect the debonding mechanism provided by the use
of a single or double layer of flexible slit (tight fit) sheathing based on experimental
results on pull-out tests on unstressed strands in wall-like blocks.

•

Experimental results on laboratory-scale beam models with a concentric partially
debonded strand using single or double layer slit sheathing indicated that flexible
sheathing did not fully break the “bond” between strand and concrete since an average of
16% of prestressing force was transferred along the “unbonded” length. Numerical
simulations on the same beams showed that the axial stress transfer follows from the
wedge effect that develops as the blanketed strand dilates due to the lack of bond
resistance, and it creates a wedging anchorage mechanism with the concrete due to the
tight fit created with the soft (and collapsible) flexible sheathing.

185

6.2 CONCLUSION
•

The use of flexible sheathing in either one or two layers seems to be an effective
debonding mechanism on unstressed strands based on experimental data. Paste
infiltration did not take place in debonded strands as the maximum pull-out load was
close to zero.

•

Concrete fluidity did not affect the effectiveness of slit sheathing on unstressed strands as
the maximum pull-out load reached from wall-like blocks was significantly small
compared to fully bonded strands.

•

Experimental results on laboratory-scale beam models showed that a nominal amount of
prestressing force was transferred along the debonded length when using a double slit
sheathing. This is attributed to the strand’s dilation from a Poisson’s effect as the strand
tries to expand and creates a wedging anchorage mechanism against the tight fitting
concrete.

•

Numerical studies on small-scale and large-scale beam models (U-beam model) showed
that the use of flexible (tight fit) sheathing allows the development of high radial
pressure at the interface of the strand and concrete and that these stresses continue along
the debonded length decreasing only after the transfer zone. With the use of rigid
(oversized) debonding the strand is fully debonded as the strand and concrete are not on
contact upon release. Thus the use of rigid sheathing decreases the high stresses in the
end-region of pretensioned concrete members upon detensioning.

•

A staggered strand debonding configuration seems to reduce the shear stresses in the
longitudinal direction of the beam upon strand release. It was observed that the use of a
staggered debonded strand distribution and rigid (oversized) sheathing can decrease the
186

high tensile and shear stresses in the anchorage region of pretensioned concrete girders
upon strand detensioning.

187

APPENDICES

188

APPENDIX A – Materials Testing and Concrete Mix Designs
The strand specifications and the mix designs used for both sets, normal-consolidated
concrete and self-consolidated concrete, are presented in this section. In addition, the mix design
used to cast the large block to verify the quality of the strand is also presented.

Strand Specifications
Table A-1 Seven-wire 0.6-in. (15.2 mm) diameter strand specifications

Ultimate breaking strength
60.13 kip (267 kN)
Load @ 1% Extension
54.14 kip (241 kN)
Ultimate elongation, %
7.4
Representative area
0.218 in2 (140.6 mm2)
Actual area
0.2175 in2 (140 mm2)
Average modulus of elasticity 29000 ksi (200 GPa)

Mix Design for the Large Block Pull-out
Table A-2Concrete Mix Design – Large Block
Design

Type III cement

660 lbs/c.y.

Fine Aggregate (2NS sand)

1269 lbs/c.y.

Coarse Aggregate (#4 limestone)

1900 lbs/c.y.

Water

292 gal/c.y.

ASTM C-494 (Type A)

106 oz/c.y.

Water/Cement Ratio = 0.44
Concrete Unit Weight = 146.3 PCF

189

Normal Consolidated Concrete (NCC)
Table A-3 Concrete mix design - NCC
Design

Type III cement

658 lbs/c.y.

Fine aggregate (2NS sand)

1356 lbs/c.y.

Coarse
Aggregate
limestone)

(6AA
1779.6 lbs/c.y.

Water

16 gal/c.y.

HRWR

52.8 oz/c.y.

Air-entraining Admixture

2 oz/c.y.

Self-Consolidated Concrete (SCC)
Table A-4 Self-consolidating concrete mix design
Design

Type III cement

658 lbs/c.y.

Fine Aggregate (2NS sand)

1511 lbs/c.y.

Coarse Aggregate (6AA limestone)

1463 lbs/c.y.

Water

31.6 gal/c.y.

HRWR

106 oz/c.y.

Delvo Type D

39.5 oz/c.y.

Rheomac VMA

9.9 oz/c.y.

Air-entraining Admixture

2.6 oz/c.y.

Water/Cement Ratio = 0.4
Concrete Unit Weight = 143.8 PCF
Total Air = 6.0% +/- 1.5

190

APPENDIX B – Supplementary Results from Experimental Program:

NCC Normalized Average Peak Bond Strength

Pull-out on Cylinders

1.2

1

0.8

0.6

0.4

0.2

0
NCFBC

NCFBU

SCFBC

SCFBU

Concrete type

Figure A-1 Normalized average peak bond strength on pull-out cylinders

191

Pull-out on Wall-like Blocks
Strand Slip (mm)
0

50

100

150

2.5

2
8
1.5
6
1

NCD1

4

NCD2
0.5

Pull-out Force (kN)

Pull-out Force (kip)

10

2

0

0
0

1

2

3

4

5

6

Strand Slip (in.)

Figure A-2Response of debonded strands in wall-like block cast with NCC – 5th row

Height: from bottom to top (mm)
50

100

150

200

250

300

60

1500

Maximum Pull-out Force (kip)

2

y = 56.863 - 0.29295x R = 0.81345

1400

55

NCFB

1300

50
1200
45
1100
40

1000

Maximum Pull-out Force (kN)

0

900

35
0

10

20

30

40

50

60

70

Height: from bottom to top (in.)
Figure A-3 Least square fit fully bonded strand in wall-like block cast with NC
192

Height: from bottom to top (mm)
0

200

400

600

800

1000 1200 1400 1600

SCFB
56

250

54

240

52

230

50
220
2

48

y = 58.753 - 0.17443x R = 0.81429
210

Maximum Pull-out Force (kN)

Maximum Pull-out Force (kip)

58

46
0

10

20

30

40

50

60

70

Height: from bottom to top (in.)

Figure A-4 Least square fit fully bonded strand: top-strand effect on wall-like block cast
with SCC

193

APPENDIX C – Results from numerical small-scale beam models
D is ta nc e fro m E n d o f B e a m (m m )
0

500

1000

1500

2000

2500

3000

-6

Concrete Strain Level (in./in.*10 )

350

L = 2 4"
t

300
250
200
150
100
50

S C F B C B e a m M o d e l (C o e f. o f F rictio n = 0 .5 0 )
S C F B C B e a m - E xp e rim e n tal C u rve

0
0

20

40

60

80

100

120

D is ta n c e fro m E n d o f B e a m (in .)

Figure A-5 SCFBC numerical model vs. experimental data
D is ta n c e fro m E n d o f B e a m (m m )
0

500

1000

1500

2000

2500

3000

3500

S C D 1 C B e a m M o d e l (C o e f. o f F ric tio n =0 .85 )
S C D 1 C B e a m - E xp e rim e n ta l C u rv e

-6

Concrete Strain Level (in./in.*10 )

500

400

L =16"
t

300

200

L =24"
u

100

0
0

50

100

D is ta nc e fro m E n d o f B e a m (in .)

Figure A-6 SCD1C beam model vs. experimental results
194

150

D is ta n c e fro m E n d o f B e a m (m m )
0

500

1 000

1 500

2 000

2500

30 00

35 00

L =19"
Concrete Strain Level (in./in.*10 )

t

-6

400

300

L =24"

200

u

100
N C D 2 C B e a m M o d e l (C o e f. o f fric tio n = 0 .7 0 )
N C D 2 C B e a m - E xp e rim e n ta l C u rve
0
0

50

100

150

D is ta n c e fro m E n d o f B e a m (in .)

Figure A-7 NCD2C Beam Model vs. Experimental Data

D istan c e from E nd of B e a m (m m )
0

500

1 0 00

1 5 00

2 0 00

2 500

30 00

35 00

L = 29 "
t

-6

Concrete Strain Level (in./in.*10 )

500

400

L = 24 "
u

300

200

100

N C D 2 U B ea m M o de l (co ef. o f friction = 0 .4 5 )
N C D 2 U B ea m - E xp e rim e nta l C u rve

0
0

50

10 0

150

D ista n ce fro m E n d o f B ea m (in .)

Figure A-8 NCD2U Beam model vs. experimental curve
195

Distance from edge to center of beam (mm)
0

10

20

30

40

50

60

70

80

1000
5

Stress Level (psi)

0

0

-500
-5
-1000
-10

-1500
NCFBC Beam Model
NCFBU Beam Model
NCD1C Beam Model
NCD1U Beam Model
NCD2C Beam Model
NCD2U Beam Model

-2000
-2500

-15

-20

-3000
0

0.5

1

1.5

Stress Level (MPa)

500

2

2.5

3

3.5

Distance from edge to center of beam (in.)

Figure A-9 Axial stresses along cross-section of beam at releasing end - NCC set
Distance from edge to center of beam (mm)
0

10

20

30

40

50

60

70

80

1000
5

0

Stress Level (psi)

0
-500

-5
NCFBC Beam Model
NCFBU Beam Model
NCD1C Beam Model
NCD1U Beam Model
NCD2C Beam Model
NCD2U Beam Model

-1000
-1500
-2000

-10

Stress Level (MPa)

500

-15

-2500
-20

-3000
0

0.5

1

1.5

2

2.5

3

3.5

Distance from edge to center of beam (in.)

Figure A-10 Transverse stresses along cross-sectional Path: NCC set of beams

196

Figure A-11 NCFBC Beam Model: Transverse Strains

Figure A-12 NCFBC Beam Model: Maximum principal plastic strains

197

Figure A-13 NCFBC Beam Model: Equivalent plastic strain in uniaxial tension

Figure A-14 NCFBC Beam Model: Axial stresses

198

Figure A-15 NCFBU Beam Model: Transverse strain

Figure A-16 NCFBU Beam Model: Maximum principal plastic strains

199

Figure A-17 NCD1C Beam Model: Transverse strain

Figure A-18 NCD1C Beam Model: Maximum principal stresses
200

Figure A-19 NCD1U Beam Model: Axial stresses

Figure A-20 NCD1U Beam Model: Equivalent plastic strain in uniaxial tension (PEEQT)
201

Figure A-21 NCD2C Beam Model: Axial stresses

Figure A-22 NCD2C Beam Model: Maximum principal strain
202

Figure A-23 NCD2U Beam Model: Maximum principal plastic strain

Figure A-24 NCD2U Beam Model: Transverse stresses
203

Figure A-25 NCD2U Beam Model: PEEQT

Figure A-26 SCFBC Beam Model: Maximum principal strains
204

Figure A-27 SCFBC Beam Model: Transverse stresses

Figure A-28 SCFBC Beam Model: Maximum principal stresses

205

Figure A-29 SCD1C Beam Model: Equivalent plastic strain in uniaxial tension (PEEQT)

Figure A-30 SCD1C Beam Model: Axial stresses

206

Figure A-31 SCD1U Beam Model: Axial strains

Figure A-32 SCD1U Beam Model: Transverse strains
207

Figure A-33 SCD2U Beam Model: Maximum principal stresses

Figure A-34 SCD2U Beam Model: Maximum principal plastic strains

208

Figure A-35 SCD2U Beam Model: PEEQT

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.00012
No Confinement

Strain Level (in./in.)

0.0001
8 10

-5

4 10

-5

2 10

Confinement - Bar No.4

-5

6 10

Confinement - Bar No.3

-5

Confinement - Bar No.5

0
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)

Figure A-36 PEEQT: NCFBC beam model – path along the cross section of the
beam at 4 in. (200 mm) from releasing end

209

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.0006
No Confinement
Confinement - Bar No.3
Confinement - Bar No.4
Confinement - Bar No.5

Strain Level (in./in.)

0.0005
0.0004
0.0003
0.0002
0.0001
0
-0.0001
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)
Figure A-37 Maximum Principal Strain – NCFBC Beam Model along cross section (4 in.
[100 mm] from the releasing end)
Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

1000
6

Stress Level (psi)

2
0

0
-2

-500

-4
No Confinement

Stress Level (MPa)

4

500

-6

Confinement - Bar No.3

-1000

Confinement - Bar No.4

-8

Confinement - Bar No.5
-10

-1500
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)

Figure A-38 NCFBC beam model – Transverse stresses along cross section (at 4 in. [100
mm] from releasing end)

210

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.001
No Confinement

Strain Level (in./in.)

0.0008

Confinement - Bar No.5

0.0006

0.0004

0.0002

0

-0.0002
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)
Figure A-39 NCD1C beam model – Maximum Principal Strain along cross section (at 4 in.
[100 mm] from releasing end)

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

700

4.8

4

500
3.2
400
2.4
300
1.6
200
No Confinement

100

Stress Level (MPa)

Stress Level (psi)

600

0.8

Confinement - Bar No.5
0
0

0.5

1

1.5

2

2.5

3

0
3.5

Distance from Edge to Center of Beam (in.)
Figure A-40 Maximum principal stress along cross section – NCD1C beam model

211

Distance from Edge to Center of Beam (mm)
0

10

20

30

40

50

60

70

80

0.001

Strain Level (in./in.)

No Confinement
Confinement - Bar No.5

0.0005

0

-0.0005

-0.001
0

0.5

1

1.5

2

2.5

3

3.5

Distance from Edge to Center of Beam (in.)
Figure A-41 NCD1C beam model – Transverse Strain along cross section

212

APPENDIX D –Results from Numerical Case Study: U-beam Models

Figure A-42 Case 3: Maximum principal stresses at beam end

Figure A-43 Results for Case 4: Yielding of concrete elements at beam cross section
213

Figure A-44 PE Max Principal: Case 1 vs. Case 2 at beam end

214

0.0002

Distance from end of beam (mm)
2000
4000
6000
8000

0

1 10

4

Strain Level (in./in.)

0
-0.0002
-0.0004
-0.0006
Skew - original configuration - tight

-0.0008

Skew - original configuration - rigid

-0.001
0

50 100 150 200 250 300 350 400
Distance from end of beam (in.)
Figure A-45 Response traces along partially debonded strand No.9: Axial strains – Skewsymmetry U-beam models tight vs. rigid with original configuration

0.01

0

Distance from end of beam(mm)
2000
4000
6000
8000

1 10

4

Strain Level (in./in.)

Skew - original configuration - tight

0.008

Skew - original configuration - rigid

0.006
0.004
0.002
0
0

50 100 150 200 250 300 350 400
Distance from end of beam(in.)

Figure A-46 Response traces along partially debonded strand No.9: Maximum principal
strains – Skew-symmetry U-beam models tight vs. rigid with original configuration
215

0.01

Distance from end of beam (mm)
0
2000
4000
6000
8000

4

Skew - original configuration - tight
Skew - original configuration - rigid

0.008
Strain Level (in./in.)

1 10

0.006
0.004
0.002
0
-0.002
0

50

100 150 200 250 300 350 400
Distance from end of beam (in.)

Figure A-47 Response traces along partially debonded strand No.1: Maximum principal
plastic strains – Skew-symmetry U-beam models tight vs. rigid with original configuration

0.0035

0

Skew - original configuration - tight

0.003
Strain Level (in./in.)

Distance from end of beam (mm)
4
2000
4000
6000
8000
1 10

Skew - original configuration - rigid

0.0025
0.002
0.0015
0.001
0.0005
0
-0.0005
0

50 100 150 200 250 300 350 400
Distance from end of beam (in.)

Figure A-48 Response traces along partially debonded strand No.9: PEEQT – Skewsymmetry U-beam models tight vs. rigid with original configuration

216

0.007

0

Distance from end of beam (mm)
2000
4000
6000
8000

4

Skew - original configuration - tight
Skew - original configuration - rigid

0.006
Strain Level (in./in.)

1 10

0.005
0.004
0.003
0.002
0.001
0
-0.001
0

50

100 150 200 250 300 350 400
Distance from end of beam (in.)

Figure A-49 Response traces along partially debonded strand No.11: Maximum principal
plastic strains – Skew-symmetry U-beam models tight vs. rigid with original configuration

Figure A-50 Response traces along partially debonded strand No.9: PEEQT– Skewsymmetry U-beam models tight vs. rigid with staggered configuration

217

Figure A-51 Response traces along partially debonded strand No.9: Maximum principal
strains – Skew-symmetry U-beam models tight vs. rigid with staggered configuration

Figure A-52 Response traces along partially debonded strand No.11: Equivalent Axial
stresses – Skew-symmetry U-beam models tight vs. rigid with original configuration
218

Figure A-53 Response traces along partially debonded strand No.11: Maximum principal
plastic strain – Skew-symmetry U-beam models tight vs. rigid with staggered configuration

219

Figure A-54 Maximum principal stresses: Four skew U-beam models
220

Figure A-55 Equivalent plastic strain in uniaxial tension at release end: Four skew U-beam models
221

Figure A-56 PEEQT: Four skew-symmetry U-beam models at 12 in. from release end
222

Figure A-57 Axial stresses: Four skew-beam models

223

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224

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