Simulation of the Effect of Deck Cracking Due to Creep and Shrinkage in Single Span Precast/Prestressed Concrete Bridges

SIMULATION OF THE EFFECT OF DECK CRACKING DUE TO CREEP AND SHRINKAGE IN SINGLE SPAN PRECAST/PRESTRESSED CONCRETE BRIDGES

SUDARSHAN C KASERA

B.E., SHRI RAMDEOBABA KAMLA NEHRU ENGINEERING COLLEGE, 2008 NAGPUR, INDIA

A THESIS

submitted in partial fulfillment of the requirements for the degree

MASTER OF SCIENCE IN CIVIL ENGINEERING

Department of Civil and Architectural Engineering and Construction Management College of Engineering and Applied Sciences

UNIVERSITY OF CINCINNATI Cincinnati, OH

2014

Committee Chair: Richard Miller, Ph.D.

Committee Members: Gian Andrea Rassati, Ph.D. Ala Tabiei, Ph.D. Abstract

The use of precast/prestressed concrete as a structural building system became prominent in the 1950s especially for bridges. The design of the bridge structures was done as simply supported then. In 1969, an analysis and design method for ‘continuous for live load’ bridges was developed. Extensive research has been conducted since then to study the behavior of ‘continuous for live load’ precast/prestresed bridges and various modifications have been proposed. The current analysis method fails to predict the behavior of the continuous precast/prestressed bridge accurately because of the complex loads such as creep and shrinkage. The experimental procedure to determine the behavior of structures due to creep and shrinkage needs time and lot of resources. The motivation behind the current research is to provide a concept where finite element method can be employed to study the behavior of such structures. This research focuses on studying the behavior of single-span precast prestressed bridge due to long-term loading such as creep and shrinkage. The general purpose finite element program, ABAQUS 6.11-2 is used for analysis in the current project. The concept developed from the current research can be applied to multi-span bridge continuous structure in the future. The effect of creep and shrinkage on the overall system is studied. The results obtained from the analyses conform to the observed behavior in the field. It can be said that the analyses procedure for ‘continuous for live load’ bridges needs to account for the cracking of the deck slab to predict an accurate behavior of the structure.

iii

Table of Contents

Acknowledgements ...... xii

Dedication ...... xiii

Chapter 1. Introduction ...... 1 1.1 Concept of ‘continuous for live load’ bridges ...... 3 1.2 Motivation for the research ...... 5

Chapter 2. Background and Literature Review...... 7 2.1 PCA Research ...... 8 2.2 NCHRP Report 322 (Oesterle et al. 1989) ...... 12 2.3 NCHRP Report 519 (Miller et al. 2004) ...... 14 2.3.1 Analytical study ...... 16 2.3.2 Experimental study on stub specimen ...... 17 2.3.3 Study of full-size specimens ...... 19 2.3.4 Finite element study using ANSYS ...... 21 2.3.5 Conclusions made from the study ...... 21 2.4 Eldhose Stephen Thesis (2006), University of Cincinnati, OH ...... 22 2.5 NYSDOT Bridge Deck Task Force Evaluation of Bridge Deck Cracking on NYSDOT Bridges (Curtis 2007) ...... 24 2.6 Behavior of pre-stressed concrete bridge girders due to time dependent and temperature effects (Debbarma and Saha 2011) ...... 27 2.7 Field monitoring of positive moment continuity detail in a skewed prestressed concrete bulb-tee girder bridge (Okeil et al. 2013) ...... 31

Chapter 3. Finite Element Modeling and Analysis ...... 34 3.1 Use of consistent units ...... 35 3.2 Modeling of geometry in the parts module ...... 35 3.3 Defining and assigning material properties to different parts of the structure ...... 38 3.3.1 Steel ...... 39 3.3.1.1 Reinforcement and strand steel ...... 39 3.3.2 Concrete ...... 39

3.3.2.1 Concrete material model ...... 39 3.3.2.2 Girder concrete...... 43 3.3.2.3 Deck concrete...... 44 3.4 Assembly ...... 44 3.5 Defining steps and loadings ...... 48 3.5.1 Steps ...... 49 3.5.2 Dead load ...... 50 3.5.3 Creep and shrinkage ...... 50 3.5.3.1 ACI – 209 Report ...... 51 3.5.3.1.1 Correction factor for creep and shrinkage coefficient for girder and deck .... 53 3.5.3.2 User subroutine to calculate creep and shrinkage strain ...... 56 3.5.4 Live load ...... 57 3.6 Field output requests ...... 57 3.7 Submission of job for analysis ...... 57 3.8 Post processing in ABAQUS using the visualization module ...... 58

Chapter 4. Analysis approach ...... 60 4.1 Nonlinearity and Convergence ...... 60 4.1.1 Newton-Raphson Method in ABAQUS (Hibbitt et al. 2011a) ...... 61 4.2 Convergence issues in the model ...... 66 4.3 Solution to the convergence issues ...... 69

Chapter 5. Results and Discussion ...... 71 5.1 Effect of girder age on prestressing force in the strands ...... 87

Chapter 6. Conclusions and Future Research ...... 92 6.1 Future research ...... 93

References or Bibliography ...... 95

Appendix A. Validation model for creep and shrinkage subroutine ...... 97

Appendix B. Volume to surface ratio for the girder and the deck ...... 100

Appendix C. FORTRAN subroutine...... 102

Appendix D. Script file for submission of job to the supercomputer ...... 104

Appendix E. MS Excel Macro to create plots...... 107

List of Figures

Figure 1.1 Percentage of bridges built annually with three major construction materials ...... 1 Figure 1.2 Number of new and replaced bridges (both Federal and Non-Federal aid) using different materials from 2003 to 2010 ...... 2 Figure 1.3 Building material for new and replaced bridges in percentages between 2003 and 2010 ...... 2 Figure 1.4 Prestressed girder in the construction yard ...... 4 Figure 1.5 Prestressed girders erected on the supports (piers) at the construction site ...... 5 Figure 1.6 Complete bridge structure with slab, diaphragm and negative reinforcement ...... 5 Figure 2.1 Positive moment in the connection due to creep in the girder ...... 9 Figure 2.2 Negative moment in the connection due to the differential shrinkage in girder and slab ...... 9 Figure 2.3 RESTRAINT Model (Miller et al. 2004) ...... 16 Figure 2.4 Comparison of RESTRAINT with PCA study (Miller et al. 2004) ...... 16 Figure 2.5 Connection capacity (stub girder) specimen (Miller et al. 2004) ...... 17 Figure 2.6 Variation of strain at bottom of diaphragm with time (Miller et al. 2004) ...... 20 Figure 2.7 Comparison of ABAQUS and FHWA data (Stephen 2006) ...... 24 Figure 2.8 Initial temperature rise in bridge decks (Curtis 2007) ...... 26 Figure 2.9 Cross section of single cell precast pre-stressed box girder, Unit = mm (Debbarma and Saha 2011) ...... 27 Figure 2.10 Prediction of shrinkage strain using model codes (Debbarma and Saha 2011) ...... 28 Figure 2.11 Prediction of creep compliance using model codes (Debbarma and Saha 2011) ...... 28 Figure 2.12 Development of strain vs. time in concrete box girder section (Debbarma and Saha 2011) ...... 29 Figure 2.13 Deflection profile of prestressed concrete box girder (Debbarma and Saha 2011) .. 29 Figure 2.14 Pattern of deflection of box girder G-13 at different temperature (Debbarma and Saha 2011) ...... 30 Figure 2.15 Compressive strain vs temperature in soffit slab of box girder bridge at the mid-span (Debbarma and Saha 2011) ...... 30

vii

Figure 2.16 Tensile strain vs temperature in deck slab of box girder bridge at mid-span (Debbarma and Saha 2011) ...... 31 Figure 2.17 Temperature readings in deck and top-and bottom flanges (Okeil et al. 2013) ...... 32 Figure 2.18 Rotation of girder ends (Okeil et al. 2013) ...... 32 Figure 2.19 Strains in hairpin bars at both sides of continuity diaphragm (Okeil et al. 2013) ..... 33 Figure 3.1 Model database in Abaqus...... 34 Figure 3.2 Type of elements used in the model ...... 35 Figure 3.3 Two dimensional sketch of girder and deck ...... 36 Figure 3.4 Three dimensional extruded view of girder and deck part ...... 37 Figure 3.5 Girder and deck part shown as different sections ...... 37 Figure 3.6 Typical steel part ...... 38 Figure 3.7 Response of concrete to uniaxial loading in tension (Hibbitt et al. 2011) ...... 41 Figure 3.8 Response of concrete to uniaxial loading in compression (Hibbitt et al. 2011) ...... 42 Figure 3.9 Girder and deck mesh in cross section ...... 45 Figure 3.10 Three-dimensional view of girder and deck mesh ...... 45 Figure 3.11 Wireframe showing girder with strands and reinforcement ...... 46 Figure 3.12 Wireframe showing slab with strands and reinforcement ...... 46 Figure 3.13 Wireframe of the complete model ...... 47 Figure 3.14 Deck reinforcement with part zoomed ...... 47 Figure 3.15 Strands ...... 48 Figure 3.16 Creep coefficient vs. time for the girder and the deck slab ...... 55 Figure 3.17 Shrinakge strain vs. time for the girder and the deck slab ...... 55 Figure 4.1 Internal and external loads on a body (Hibbitt et al. 2011) ...... 61 Figure 4.2 Newton-Raphson approach to solve a nonlinear problem (Simulia, Dassault convergence class) ...... 63 Figure 4.3 Flowchart showing solution procedure (Simulia, Dassault convergence class)...... 65 Figure 4.4 Screenshot for changing default solution control ...... 66 Figure 4.5 Calibration of Viscosity regularization parameter ...... 70 Figure 5.1 Stress distribution in the girder after release of prestress (camber exaggerated)...... 71 Figure 5.2 Zoomed view of stress distribution near the midspan ...... 72 Figure 5.3 Stress in the girder and the slab at the end of 200 days for 90 -day old girder ...... 72

viii

Figure 5.4 Stress distribution at the end section up to 90 days of creep and shrinkage in the girder ...... 73 Figure 5.5 Stress distribution at the end section with slab dead load and creep & shrinkage in slab (Girder age is 90 days at the time of casting of the slab) ...... 74 Figure 5.6 Stress distribution at the midspan section up to 90 days of creep and shrinkage in the girder ...... 75 Figure 5.7 Stress distribution at the midspan section with slab dead load and creep & shrinkage in the slab (Girder age is 90 days at the time of the casting of the slab) ...... 76 Figure 5.8 Displacement at girder top flange at midspan due to creep and shrinkage ...... 78 Figure 5.9 Stress at bottom of slab at midspan for different age of girders ...... 79 Figure 5.10 Stress at top flange of girder for different age of girders ...... 79 Figure 5.11 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for 10-day old girder at the time of casting of the slab) ...... 80 Figure 5.12 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for a 30-day old girder at the time of casting of the slab)...... 81 Figure 5.13 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for a 60-day old girder at the time of casting of the slab)...... 81 Figure 5.14 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for a 90-day old girder at the time of casting of the slab)...... 82 Figure 5.15 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 10-day old girder at the time of casting of the slab) ...... 82 Figure 5.16 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 30-day old girder at the time of casting of the slab) ...... 83 Figure 5.17 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 60-day old girder at the time of casting of the slab) ...... 83 Figure 5.18 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 90-day old girder at the time of casting of the slab) ...... 84 Figure 5.19 Comparison of stresses for cracked and uncracked slab condition at bottom of slab at midspan (for a 90-day old girder at the time of casting of the slab) ...... 85 Figure 5.20 Comparison of stresses for cracked and uncracked slab condition at top of girder at midspan (for a 90-day old girder at the time of casting of the slab) ...... 86

Figure 5.21 Comparison of displacement for cracked and uncracked slab condition at top of girder at midspan (for a 90-day old girder at the time of casting of the slab)...... 86 Figure 5.22 Stress in the prestressing strands for different ages of the girder ...... 88 Figure 5.23 Comparison of stress in prestressing strands for a 10-day old girder...... 88 Figure 5.24 Comparison of stress in prestressing strands for a 30-day old girder...... 89 Figure 5.25 Comparison of stress in prestressing strands for a 60-day old girder...... 89 Figure 5.26 Comparison of stress in prestressing strands for a 90-day old girder...... 90 Figure 5.27 Comparison of stress by AASHTO refined loss method and ABAQUS analysis .... 91

List of Tables

Table 2.1 Details of positive moment connections in the stub specimens (American Association of State Highway and Transportation Officials., 2012; Miller et al. 2004) ...... 18 Table 2.2 Steps defined in the model (Stephen 2006) ...... 23 Table 3.1 Part and number of instances (Single span system) ...... 44 Table 3.2 Various steps defined in the analysis ...... 49 Table 3.3 Correction factors for the girder ...... 54 Table 3.4 Correction factors for the deck ...... 54 Table 4.1 Default tolerance values for some parameters ...... 66

Acknowledgements

I sincerely express my profound gratitude to Dr. Richard Miller, my advisor, for his guidance, encouragement, and suggestions throughout the period of my research. The joy and enthusiasm he has for his research was contagious and motivational. It was his continuous support and appreciation that kept me focused on my work. My appreciation is extended to the committee members, Dr. Gian Rassati and Dr. Ala Tabiei for reviewing my research work and providing helpful suggestions and comments. I would like to thank Dr. Rassati for providing the ABAQUS computer program and helping out with the problems during the modeling and analysis. I would also like to thank Dr. Tabiei for guiding me on the issues I encountered with the finite element analysis of the problem. My time at the University of Cincinnati was made enjoyable largely due to the many friends and groups that became a part of my life. I thank all my friends at the University of Cincinnati, especially, Abhik, Avdhesh, Kunal, Shekhar, and Suryanarayana. I greatly appreciate the help provided by the Academic Writing Center, University of Cincinnati, especially, Dani Clark, for proofreading my thesis. A special thanks to Mr. Aaron Schwartz from Dassault Systemes for taking keen interest in the analysis and helping in every possible way to resolve the convergence issues in the model during the hands-on workshop in the ABAQUS Convergence class. I would like to express the deepest appreciation to Mr. Umesh Rajeshirke, my previous employer and director at Spectrum Techno Consultants Pvt. Ltd., Navi Mumbai, India, and my supervisors, Mr. Nirav Mody and Mr. Rakesh Varadkar , who always supported my decision to pursue higher education. I have the heartiest gratitude for my dear friends, Abhimanyu, Aditya, Alisha, Deepti and Sumeet. Thank you all for always being there for me and for being so awesome. Finally, I would like to thank my family members for always offering me love and support throughout my life. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center.

xii

Dedication

To my Uncle and Aunt

xiii

Chapter 1. Introduction

The concept of prestressed concrete goes back to 1888 when P.H. Jackson floated the idea and was granted the patent for a prestressed concrete design for the first time in the United States. Due to the unavailability of proper materials such as low relaxation steel, the design concept did not become popular at that time. It was in the 1950s when prestressed/post-tensioned concrete experienced a beginning of a new era. The idea of using prestressed concrete as a structural building system was defined by Eugene Freyssinet. The first major prestressed concrete bridge in the USA was the Walnut Lane Bridge in Philadelphia designed by Gustave Magnel (Dinges 2009). It was completed in 1951 and had a girder length of 160 feet. Since then, prestressed concrete has been one of the most sought after construction processes in the bridge industry. The following figure demonstrates the increase in the use of prestressed concrete in bridges since 1950.

Figure 1.1 Percentage of bridges built annually with three major construction materials (National Bridge Inventory Study Foundation,. 2003)

The following figures show the use of prestressed concrete bridges in recent times.

Number of new and replaced bridges per year 3000 Prestressed 2500 Concrete Reinforced Concrete Reinforced 2000 Concrete Steel

1500 Steel Prestressed concrete 1000

Other Number bridges Number of 500 Other 0 2003 2004 2005 2006 2007 2008 2009 2010 Year Built

Figure 1.2 Number of new and replaced bridges (both Federal and Non-Federal aid) using different materials from 2003 to 2010 (The National Bridge Inventory Database 2011)

Types of new and replaced bridges between 2003 and 2010 Other 2%

Reinforced Concrete Prestressed 34% Concrete 42%

Steel 22% Reinforced Concrete Steel Prestressed Concrete Other

Figure 1.3 Building material for new and replaced bridges in percentages between 2003 and 2010 (The National Bridge Inventory Database 2011)

It is clear from Figure 1.2 and Figure 1.3 that prestressed concrete is the most used material for the construction of bridges. As shown in Figure 1.3, more than 40% of recently built bridges are made of prestressed concrete. This is due to the advantages of using prestressed concrete elements for the construction of bridges and other structures. The construction process is fast, reduces the use of formwork, and improves quality because of the possibility of better quality control at the casting yard. A large amount of research has focused onto understanding the behavior of prestressed concrete under different kinds of loading, and especially time-dependent loading. These studies carried out on prestressed concrete helped understand its behavior over time. The research herein reported aims to understand the long term behavior of a bridge made from precast pre-stressed concrete girders. The pre-stressed concrete bridge behavior is significantly affected by time dependent effects like creep, shrinkage and ambient temperature during its service life (Debbarma, 2011). This study focuses on the response of prestressed concrete bridges to creep in the girder due to prestress, to the effect of concrete shrinkage in the girder, and to differential shrinkage between the girder and the deck. The behavior of the bridge is studied using a three-dimensional finite element model of a single span bridge built using the ABAQUS Standard 6.11-2 computer program (Hibbitt et al. 2011b). While this study focuses on single span bridges, the motivation was to create a model which could eventually be used to study “continuous for live load” bridges.

1.1 Concept of ‘continuous for live load’ bridges As the phrase ‘continuous for live load’ suggests, this kind of bridge is continuous for live load and dead loads applied after the slab is cast only. The construction sequence plays an important role on the behavior of ‘continuous for live load’ bridges. The bridge acts as a simple span for the dead loads of the girder and the deck, and as a continuous structure for live loads, superimposed dead loads and other loads during service conditions. ‘Continuous for live load’ bridges have some advantages over simply supported bridges. The joints between adjacent spans are eliminated, which results in a better riding quality, improves the durability of the bridge by eliminating the deck drainage onto the substructure and eliminates the maintenance cost for deck joints. The construction sequence of a ‘continuous for live load’ bridge is explained below:

1. The prestressed girder is cast in the construction yard as any other prestressed concrete girder. The only exception is, some of the strands are left extended at the end of the girder to be used them as positive reinforcement at the connection. The length of extension depends on the design parameters. The extended strands are bent to make a positive moment connection. The same connection can be achieved by embedding mild steel bars instead of strands or by use of mechanical connectors. 2. The girder then is transported to the construction site. The age at which the girder is erected on piers can vary for different projects. The girder is designed to act as simply supported for self-weight and deck dead load. 3. Continuity is achieved by joining the ends of the girder at the pier by pouring the diaphragm and the deck slab concrete. Positive and negative reinforcement are provided to counteract the moment developing at the connection. Negative reinforcement is provided in the slab as a longitudinal continuity bars while positive reinforcement is provided at the bottom part of the girder at the support. The configuration of positive reinforcement varies, most commonly consisting of bent bar or bent strand. The positive moment connection is discussed in detail in the literature review section of this document. Figure 1.4 throughFigure 1.6 depict the construction sequence of a ‘continuous for live load’ bridge.

Prestressing strands Positive reinf.

Figure 1.4 Prestressed girder in the construction yard

Prestressing strands Positive reinf.

End pier/support Pier/support End pier/support Figure 1.5 Prestressed girders erected on the supports (piers) at the construction site Slab Negative reinf.

Prestressing strands Positive reinf.

Diaphragm

End pier/support Pier/support End pier/support Figure 1.6 Complete bridge structure with slab, diaphragm and negative reinforcement (Typical longitudinal sections at pier. Details not shown for clarity)

1.2 Motivation for the research One long term question with continuous for live load bridges is the connection behavior. The girder will camber upward due to creep and shrinkage. This camber will be restrained by both the slab itself and the differential shrinkage between the girder and slab. If the camber is not fully restrained by the slab, a positive moment will develop at the connection. If the connection cracks, there could be a loss of continuity. Experimental evidence of continuity is limited due to the high cost of large specimens. Ideally, an analytical study could answer this question. The prediction of the time-dependent behavior of a ‘continuous for live load’ bridge is a complex procedure. The time-dependent behavior of a concrete structure includes creep and

5 shrinkage in concrete over time, the effects of daily/seasonal temperature changes, relaxation of the steel and time dependent change to the modulus of elasticity of concrete. As previously noted, the overall effect of the time dependent effects may give rise to positive moments at the bottom of the diaphragm at the support. The age of the girder at the time of the casting of the deck also plays an important role for the development of the positive moment at the bottom of the support. Current specifications require an age of 90 days for the girder in order to minimize the development of a positive moment at the bottom of the girder and reduce cracking. The construction sequence of the bridge plays an important role in determining its long-term behavior. The experimental methods to study the long-term effects are time-consuming and require many resources. The available analytical methods fail to accurately predict the behavior in most cases. Therefore, finite element analysis can be a quick and reliable technique to study the behavior of the ‘continuous for live load’ bridges subjected to dead load, creep and shrinkage, temperature loads, and live loads. Finite element programs may or may not have a built-in method to simulate these conditions as needed, but modifications can be made to develop an approach to analyze such problems. An approach was developed previously by Eldhose Stephen (Stephen 2006) in his thesis to analyze such problems using available resources in the finite element program ABAQUS Standard (Hibbitt et al. 2011b). A similar approach is used in the current project to simulate creep and shrinkage in a single span prestressed concrete bridge girder. The current project focuses on simulating the effects of creep and shrinkage on a simply supported single span bridge girder with a deck slab over it. The idea is to provide a basis for the analysis of such bridges using the finite element method. The effect of age variation of the girders at the time of casting the slab on the overall system is also studied in the current thesis. The role of differential shrinkage between the girder and the deck will be investigated. ABAQUS Standard 6.11-2, a general purpose finite element software, is used to create a three-dimensional finite element model of a single span bridge. The procedure followed is documented in detail along with the results and can be used in the future to develop a multiple span model for ‘continuous for live load’ bridges.

Chapter 2. Background and Literature Review

Continuous for live load bridges have been used in the construction industry since the 1960s. This was done to allow the use of shallower sections. During the early 1960s, the performance of these bridges was studied by researchers at the Construction Technology Laboratory (CTL), at the Portland Cement Association (PCA). Various aspects of this kind of bridge were studied: feasibility of the continuity connection, shear connections, flexural strength, creep and shrinkage studies, and performance under live load (Kaar et al. 1960). The study on creep and shrinkage was conducted on a two-span continuous girder by Mattock et al. in 1961. The report was published as “Precast-Prestressed Concrete Bridges 5: Creep and shrinkage studies.” Earlier prestressed concrete bridges were made continuous by providing negative reinforcement in the slab across both the girders. There was no provision for positive steel at the bottom of the girder at the support. The design detail for the positive moment connection was published by PCA in a design guide, “Design of Continuous Highway Bridges with Precast, Prestressed Concrete Girders” (Freyermuth 1969). The design details were based on the outcomes of various long and short term tests carried out during the research. National Cooperative Highway Research Program (NCHRP) report 322, “Design of Simple-Span Precast Prestressed Bridge Girders Made Continuous” was published in 1989 by Oesterle et al. to address uncertainties in the PCA research. This report concluded that the positive moment developed at the support negated any benefit from continuity. Because bridge engineers thought this type of bridge was useful and did not entirely agree with the conclusion that the positive moment negated the benefit, further research was done by NCHRP and the reports were published as NCHRP Report 519, “Connection of Simple Span Precast Girders for Continuity,” (Miller et al. 2004). The scope of the NCHRP Report 519 included a survey of the then existing construction practice followed by an Excel-based analytical model, RESTRAINT, which was used to analyze connection behavior. The experimental models, which included six stub specimens and two full-size specimens, were tested for different connection configurations and construction methods as part of the project. The results of the experimental study were used to create the current language in the American Association of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Design (LRFD) Bridge Design Specifications (AASHTO 2014).

The Bridge Deck Task Force set up by the New York State Department of Transportation conducted an evaluation of the cracking in the bridge decks of continuous bridges. The reasons were mainly attributed to thermal stress due to the restraint of the deck while it cools, live load stresses on continuous bridge, and stress due to concrete shrinkage while restrained by the superstructure. Various other studies were carried out to study the effects of creep, shrinkage and temperature variation on bridges and some of them are presented here.

2.1 PCA Research The performance of precast/prestressed bridges was investigated by the Portland Cement Association (PCA) in the 1960s. The various aspects of bridge construction, which were considered as the part of study , are: girder continuity, horizontal shear connections, bridge design studies, shear strength, flexural strength, creep and shrinkage studies, repeated loading, reverse bending, and half-scale bridge testing (Kaar et al. 1960). Long-term tests were carried out to study the effect of time-dependent loading on twospan continuous girders. The creep and shrinkage study on such bridges was documented in the report “Precast-Prestressed Concrete Bridges 5: Creep and Shrinkage Studies” (Mattock 1961). The construction sequence of continuous girder bridges results in a difference of age between the slab and the girder, as stated before. The girder is older than the deck and has undergone some creep and shrinkage while lying in the casting yard. This results in differential shrinkage between the slab and the girder when the slab is cast over the girder. The freshly cast slab over an older girder tends to shrink more than the girder. This differential shrinkage is restrained by the girder and the reinforcement in the slab. This causes stresses to develop at the girder deck interface. The bottom of the slab is in tension and the top of the girder experiences compression. The creep due to prestress in the girder causes the girder to camber up while the differential shrinkage has a tendency to cause a downward camber in the girder. The deflection of the girder due to creep and differential shrinkage is restrained in a continuous bridge by the continuity connection. The continuity connection experiences a positive restraint moment at the bottom due to creep and negative restraint moment at the top due to the differential shrinkage.

Figure 2.1 Positive moment in the connection due to creep in the girder (Mattock 1961)

Figure 2.2 Negative moment in the connection due to the differential shrinkage in girder and slab (Mattock 1961)

In the study by Mattock, the reaction and strain were measured in the negative continuity reinforcement and the deflections at the midpsan were also measured. The variation of support reaction with time showed an increase in the reaction for the first 35 days after the removal of the deck formwork. Henceforth, the reaction kept on decreasing and became negative at around 220 days and continued to decrease before levelling out. Increase in reaction is caused by the differential shrinkage between the slab and the precast. A similar response was observed with respect to strain in the continuity reinforcement. Strain in the negative continuity reinforcement was initially tensile and reached its peak at around 35 days and started decreasing and eventually became compressive. This is an indication of the slowing down of a negative restraint moment, which gradually becomes positive. This is caused by the dominance of creep due to prestressing in the girder, which eventually negates the effect of differential shrinkage. The report concluded that creep and shrinkage significantly affect the structure but the ultimate load carrying capacity of the structure remains unaffected. The accuracy of the study is questionable because half scale models were used by PCA. In a half scale model, the volume decreases with the cube while the surface decreases as the square of the size. Since the volume-surface ratio is an important parameter to predict creep and shrinkage, scaled models can produce erroneous results. The PCA published a design guide, “Design of Continuous Highway Bridges with Precast, Prestressed Concrete Girders,” (Freyermuth 1969), based on the outcomes of various tests conducted during the study. The design method to determine the prestressing details and positive moment connection details were provided in the guide. The prestressing steel was

9 designed for simple spans, considering the dead load, live load, and impact loads. The design of the positive moment connection was based on the restraint moments set up at the supports due to creep, differential shrinkage, and live/impact loads, after the continuity is established. These restraint moments increase the positive moments at midpsan as well, but the design guide stated that the additional moment requirement was satisfied by the prestressing detail designed for gravity loads (Freyermuth 1969). The restraint moment set up at the support is an important feature of ‘continuous for live load’ bridges, and the detailed guidelines were outlined by the PCA to calculate them. The calculation process can be divided into the following stages: 1. The girder and the deck are assumed to be cast simultaneously and the prestressing and dead loads are applied to a monolithic structure. 2. An elastic analysis of the indeterminate continuous structure provides the moments at the mid-spans and supports. 3. The restraint moments due to creep are calculated by multiplying the elastic moments by a creep factor accounting for the actual difference in age between the girder and concrete. 4. The restraint moments due to differential shrinkage are calculated using a simplified equation for the moment setup along the length of the structure, 푡 푀 = 휀 퐸 퐴 (푒 + ) Eqn. (2.1) 푠 푠 푏 푏 2 2

휺풔 = Differential shrinkage 퐸푏 = Elastic modulus of deck concrete

퐴푏 = Cross-sectional area of deck slab 푡 (푒 + ) = Distance between mid-depth of the slab and centroid of the composite section 2 2 An indeterminate analysis using this moment provides the support moments in the continuous structure. 5. The final restraint moment at the support is calculated as, 1 − 푒−휙 푀 = (푌 − 푌 )(1 − 푒−휙) − 푌 ( ) + 푌 Eqn. (2.2) 푟 푐 퐷퐿 푠 휙 퐿퐿

(푌푐 − 푌퐷퐿) = Elastic moment in continuous, monolithic girder and deck, cast and prestressed simultaneously

(1 − 푒휙) = Factor converting an elastic moment to a restraint moment due to creep in precast girder after continuity is established

푌푠 = Restraint moment due to differential shrinkage

푌퐿퐿 = Positive live load plus impact moment

1−푒−휙 ( ) = Adjusts the moment due to differential shrinkage to account for creep 휙

The positive moment connection at the support was to be designed for the final restraint moment obtained by the above equation. Similarly, a negative reinforcement at the connection was designed for the moment occurring due to live load, impact load, and dead load arising after continuity is established. This design procedure led to the addition of Article 9.7.2 ‘Bridges Composed of Simple-Span Precast Prestressed Girders Made Continuous,’ in the AASHTO Standard Specifications (American Association of State Highway and Transportation Officials., 2002) to design the highway bridges. This was the only design methodology available for designing precast/prestressed bridge girder connections at that time. The economy of the ‘continuous for live load’ bridges along with the convenience of the PCA design procedure led to widespread acceptance of this analysis/design process. Most of the research involving analytical models carried out since then has been based on or validated using the PCA model and data. The PCA study failed to take into consideration certain behavioral aspects of ‘continuous for live load’ bridges: 1. The free shrinkage of the deck is restrained by the top of the girder and the reinforcement, which results in development of tensile stresses in the deck. These stresses can be large enough to cause cracking in the deck. Once the deck cracks, the assumption of the structure being monolithic becomes invalid and the restraint moment developed due to differential shrinkage is released. The negative camber induced by differential shrinkage also gets negated due to deck cracking, and camber in the girder reverses to cause positive camber. 2. The PCA model did not take into account the possibility of tensile creep in the deck caused by the tensile stresses setup due to restraint of differential shrinkage. 3. The effect due to daily temperature variation was not considered in the PCA study.

4. The assumption that the deck and the girder have similar creep and shrinkage properties was incorrect. The above issues needed to be addressed in order to create an effective and efficient design methodology for ‘continuous for live load’ bridges. They were addressed in the further research work carried out by NCHRP and other transportation organizations. Some of them are discussed in the following sections.

2.2 NCHRP Report 322 (Oesterle et al. 1989) The National Cooperative Highway Research Program (NCHRP) Project 12-29 (Oesterle et al. 1989) started working on ‘continuous for live load’ bridges to address issues related to their analysis, design and construction. The scope of this study included surveying the then existing methods of providing continuity in ‘continuous for live load’ bridges and analyzing the effects of the variation in the time-dependent material behavior and in the bridge design parameters on the service moments at mid-span and continuity connections. The effects of variation in bridge design parameters on inelastic distribution of moments were analyzed and computer programs were developed for simplified analysis. Objectives also included providing recommendations for new design procedures. The findings of the project were published in NCHRP report 322 titled, “Design of Precast Prestressed Bridge Girders Made Continuous” in 1989. Based on the survey response, the PCA method was the most common design procedure for designing the positive moment connection at the support while some engineers used standard detailing based on experience. Based on the parametric study for time-dependent effects on continuity, it was found out that live load continuity can range from 0 to 100 percent and construction timing of different elements influences the effective continuity. A combination of young girders, along with late application of live plus impact load, results in maximum loss of continuity. It was also concluded that the midspan service moments were not reduced due to the positive moment connection in the diaphragms. The results from the parametric study on the effect of deck reinforcement and the cross-sectional shape of girders determined a limit of a negative moment reinforcement ratio, . A ratio equal to 0.5 b was determined to ensure ductile behavior and the attainment of the maximum strength of the girders. A computer program, BRIDGERM, was developed to predict restraint moments assuming full structural continuity.

The moments due to live load plus impact were calculated using another computer program, BRIDGELL. Various conclusions were drawn through this study. It was concluded that the advantages of this kind of bridges have made them a common choice of construction regardless of the difficulty, time consumption and costly positive moment connection. The AASHTO specifications are vague for positive moment connection design for continuous for live load bridges and most States used the PCA method to design the positive moment connection. It was also concluded that the creep and shrinkage results based on the calculation of ACI 209 report (ACI Committee 209-Creep and Shrinkage. 2008) gives reasonably accurate predictions. The service moment parametric study concludes that there is no structural benefit of a positive moment connection. The positive moment connection gives rise to a restraint moment at the support which in turn increases the midspan moment making it almost similar to the value as expected in simply supported beam. The age of the girders plays an important role in the continuity of the structure. A delay in the casting of the deck and the diaphragm leads to a high level of positive moment continuity due to higher differential shrinkage. However, too much delay can increase a negative restraint moment at the support demanding more negative reinforcement, which can cause transverse cracking in the deck. The construction sequence affects the development of a restraint moment. A deck cast prior to the diaphragm increases the resultant positive moment at the midspan while the opposite sequence of casting slightly decreases the resultant positive moment at the midspan. Hence there is no economic advantage of casting in sequence and the simultaneous casting of the deck and the diaphragm is the simplest procedure. The conclusions drawn in NCHRP Report 322 identified areas which needed further research to understand and improve the behavior of continuous for live load bridges. It suggested studying the effectiveness of an approach to provide special types of preformed joints at the support and unbonding the deck reinforcement or unbonding the deck to girder interface for a certain length on each side of the joint. Another potential area of research can be a hybrid girder with partial post-tensioning designed to take more dead load moments after continuity being established. The effect of temperature and moisture gradient in the deck and girder section was not considered in this research. Both of these, along with other time-dependent loading, can give rise to a positive moment at the connection and needs further study. There is also a need to

13 develop more appropriate and simplified design procedures for shear in continuous prestressed bridge girders. Along with the above suggestion, it is also important to note that most of the work presented in the report was analytical in nature. While creep and shrinkage tests were performed on the concrete specimen leading to an improved database for the future, the measurement of creep and shrinkage from real time monitoring data of an existing bridge or bridge specimen was not included. The effect of the time-dependent loading needs to be studied experimentally and using finite element methods to help in improving the design of the positive moment connection.

2.3 NCHRP Report 519 (Miller et al. 2004) There were uncertainties in the design methodology of the positive moment connection at the pier for ‘continuous for live load’ bridges. The conclusion made by NCHRP report 322 is that there are no structural benefits of providing positive moment connection for these type of bridges. This was not accepted universally and engineers were supportive of the fact that positive moment connection has structural benefits such as delaying or eliminating the cracks in the diaphragm. In addition, there was ambiguity about the type of positive moment connection to be provided and the choice of methods to calculate the length and number of bent bar or strands needed for the connection to be effective. There were also questions about the effectiveness of continuity once the girder-diaphragm interface cracks. These questions needed to be addressed. Hence, NCHRP started project 12-53, which submitted its final report titled, “Connection of Simple-Span Precast Concrete Girders for Continuity” in 2004. The objective of this study was to recommend details and specifications for consideration by the AASHTO Highway Subcommittee on Bridges and Structures (HSCOBS) for the design of durable and constructable connections to achieve structural continuity between simple-span precast/prestressed concrete girders. One part of this study was to survey the existing methodology for providing continuous for live load connections in different states. Experimental tests were conducted to determine the capacity and behavior of some typical positive moment connection details. The results from the tests were used to develop design methods for the connection and changes were suggested to the AASHTO Load Resistance Factor Design (LRFD) 2008 specifications for making simple-span bridges continuous for live load.

The survey indicated that the use of positive moment connection reinforcement at the support was prevalent in most states and the most used connection detail was the bent bar and bent-strand type connection. Three-fourths of the respondents overlapped the bar or strand in the diaphragm and half of the respondents provided transverse reinforcement through the beam into the diaphragm. I-girders and bulb-T girders were the ones where these types of connections were mostly used with some of the respondents using it for box girders and other state-specific shapes. Reponses regarding embedment of girder into the diaphragm were also a part of the survey and it was found out that the majority of the responses included embedment of girder into the diaphragm with embedment depth varying anywhere from 2 to 12 inches. The 28-day concrete strength varied from 4000 to 9000 psi for girders while deck/diaphragm concrete had a strength of 3000 to 5000 psi at 28 days. High-performance concrete was also used in some cases. The support conditions in 80% of the responses included bearings placed under the girder ends. The cost to provide a positive moment connection was found to be $200 per girder, which is quite low as compared to the overall cost the girder. The sequence of construction is an important aspect of the continuity connection. The majority of the survey respondents cast the diaphragm and the slab together while few cast the diaphragm or part of it before casting the slab. The minimum age for the girder varied between 28 and 90 days. The common problem was congestion of reinforcement in the diaphragm area leading to improper consolidation of concrete and bending of the strands/bars after the girder is cast. However, the problems were rated as being of minor significance (Miller et al. 2004). The complete study was conducted in three major parts. First, analytical studies were performed by creating a standard spreadsheet program called RESTRAINT, which was a modified version of BRIDGERM. The analytical study was important to get an idea about the configuration of specimens for the experimental studies. Six different types of stub specimen were constructed with different connections. Along with the six stub specimens, two full size specimens were constructed and their responses to different kinds of loading such as prestress load, live loads, time-dependent loads such as creep and shrinkage and temperature loads were documented thoroughly. Three different finite element models (no connection, bent-strand and bent-bar) of the stub specimen were also modeled using finite element program, ANSYS, as a part of the study.

2.3.1 Analytical study The RESTRAINT program models a two-span continuous structure assuming that there is a support at each end of the girder as shown below.

Figure 2.3 RESTRAINT Model (Miller et al. 2004)

The program calculates the internal moments due to creep in the prestressed girder and shrinkage in the girder and the deck on the basis of the creep and shrinkage model documented in the American Concrete Institute 209 report (ACI Committee 209-Creep and Shrinkage. 2008). The loss of prestress force is also calculated using the method outlined in the Precast/Prestressed Concrete Institute Handbook. The result of RESTRAINT was compared to the results of the original PCA tests showing a reasonable agreement.

Figure 2.4 Comparison of RESTRAINT with PCA study (Miller et al. 2004)

It was concluded from the parametric studies that the age of the girders at the time when continuity is established is the most important factor in the behavior of the connection. The amount of positive reinforcement at the connection plays an important role on the performance of the continuity. There has to be a balance in the amount of positive reinforcement to achieve continuity and limit the restraint moment. An increase in the restraint moment can increase the positive moment at the mid-span to such an extent that the girder has to be designed for simply supported condition. It was concluded that the positive reinforcement should be limited, so that the capacity of the connection is at least 0.6 Mcr and does not exceed 1.2 Mcr; where Mcr is the positive cracking moment based on the gross cross section of the girder and slab but the strength of the diaphragm concrete

2.3.2 Experimental study on stub specimen The stub specimens were constructed using two 16-ft long Type II AASHTO I girders joined by a diaphragm as shown in Figure 2.5.

Figure 2.5 Connection capacity (stub girder) specimen (Miller et al. 2004)

Six stub specimens with different configurations of the positive moment connections were tested. The positive moment connections with bent-strand or bent-bar were designed for a capacity of 1.2 Mcr obtained from the analytical study where Mcr is the positive cracking

17 moment. The details of the positive moment connections in the stub specimens are shown in Table 2.1. Table 2.1 Details of positive moment connections in the stub specimens (Miller et al. 2004) Specimen Type of Specimen Diaphragm Girder End Special Cycles to Number Width (in.) Embedment (in.) Feature failure 1 Bent strand 10 0 None 16,000 2 Bent bar 10 0 None 25,000 3 Bent strand 22 6 None 55,000 4 Bent bar 22 6 None 11,600 5 Bent bar 22 6 Extra stirrups 56,000 in diaphragm 6 Bent bar 26 8 Web bars 13,3000

All six specimens were subjected to cyclic loading until failure. The first specimen survived for 16,000 cycles before the concrete on the bottom of the diaphragm split and popped off indicating a slipping and pull-out of the strand/s. Specimen 2 had a bent-bar configuration and it was more difficult to construct. This specimen lasted for 25,000 cycles when diagonal cracks started forming on the faces of the diaphragm and part of the diaphragm spalled off. The bars fractured, showing signs of failure due to fatigue. The third specimen had a bent-strand configuration with the ends of the girder embedded into the diaphragm. It lasted 55,000 cycles before failing. It exhibited cracking and spalling on the face of the diaphragm. Specimen 4 was similar to specimen 2 except the embedment of the girder ended into the diaphragm. It only lasted 11,600 cycles and the reason for early failure was uncertain. There might have been uneven stresses in the bent-bar due to bad construction leading to early failure. Specimen 5 was similar to specimen 4 with additional stirrups in the diaphragm, close to the outer edge of the bottom flange. The specimen failed after 56,400 cycles. The additional stirrups were spanning the crack and prevented failure providing additional ductility at the connection. The sixth specimen consisted of bent-bars and horizontal bars passed through the web of the beam. This certainly added stiffness and capacity to the connection making it last 133,000 cycles. Upon close inspection it was found that the web of the beam had cracked, which is not desirable. From the results, it was evident that bent-strand with embedment can be beneficial. When non-embedded bent-strand was used, the girders separated from the diaphragm without damaging the face of the diaphragm, while an embedded specimen showed a pull-out type of

18 failure. Embedded specimens also have lower strain than their counterpart. It was also clear that an asymmetrical configuration of the bars is undesirable at the connections.

2.3.3 Study of full-size specimens Two full-size specimens were constructed, one with bent-bar configuration and the other with bent-strand configuration for the positive moment connection. The full-size specimens consisted of two 50-feet Type III AASHTO I girders joined with a diaphragm of 10 inch width.

The capacity of the connection was maintained at 1.2 Mcr. As stated earlier, there can be different construction sequences for these types of bridges. For specimen 1, a partial diaphragm was poured at a girder age of 28 days. The idea of partial diaphragm is that, when the slab is cast, the weight of the deck slab concrete will rotate the end of the girder into the partial diaphragm and compress the concrete. Then, any tension caused by the positive moments will simply reduce the compression rather than cause tensile cracking (Miller et al. 2004). The specimens were studied in three stages: monitoring during construction, monitoring after construction and testing after the monitoring period for continuity. The behavior of the structure was recorded for creep, shrinkage, and thermal effects. During the period between the casting of the diaphragm and the deck, an increase in end reaction was observed, indicating development of positive moment at the connection due to creep, shrinkage, and thermal effects. A tensile strain ranging from 60 to 90 x10-6 m developed in the diaphragm during this period. After the casting of the deck, instantaneous compressive stresses developed at the bottom of the diaphragm due to the dead load of the slab. The diaphragm exhibited an interesting response few hours after the pour. The strain at the bottom of the diaphragm increased and became more tensile. Over the next couple of days, the strain decreased resulting in compression at the bottom of the diaphragm. The variation of the strain at the bottom of the diaphragm during the time when the slab was cast is shown in Figure 2.6. It can be seen that the strain values follow the pattern of the slab temperature, which was not expected but made sense. When the deck concrete is poured, the concrete heats up due to heat of hydration and the top flange of the girder also heats up. This causes an expansion in the top flange of the girder and it cambers up causing tension at the bottom of the diaphragm. Once the deck concrete is set, it starts to influence the whole structure. At this point of time, the heat of hydration has reached its peak and the slab begins to cool down. The deck slab contracts upon cooling and causes compression at the

19 diaphragm as seen from Figure 2.6. A similar response was observed at other locations in the system.

Figure 2.6 Variation of strain at bottom of diaphragm with time (Miller et al. 2004)

After the construction process was completed, thermal effects played an important role in the response of the structure. The end reactions due to temperature variation during the day were significant. The changes in reactions caused a positive moment in the diaphragm in the range of +- 250 kip-ft per day. This was more than 50% of the cracking moment value. The differential shrinkage between the girder and the deck was not observed as expected from analytical studies. The observed behavior was consistent with field studies of bridges. The girders were tested for continuity after the monitoring period. It was expected that the cracks formed at the girder-diaphragm interface would reduce the continuity or eliminate it. The reactions and the strains obtained from the continuity test were consistent with the results expected from a continuous system. So cracking at the girder-diaphragm interface did not significantly affect the continuity of the structure. The second full-size specimen was tested for positive moment/continuity and negative moment capacity. The result of the test concluded that continuity was maintained in the system even after positive moment cracking at the joint. Loss of continuity was observed when the connection was about to fail due to cracks in the slab and in the diaphragm.

The negative moment capacity was tested on the same specimen after testing for the positive moment. The negative moment reinforcement was provided in the deck as per AASHTO LRFD specifications. Cracking due to negative moment occurred at a value lower than the calculated value. This was attributed to the fact that the deck was already cracked due to positive moment testing. The bottom of the girder crushed at an applied moment which exceeded the failure moment capacity calculated using actual material properties. So it was concluded that positive moment testing reduced the negative cracking moment but the negative moment capacity of the structure was not affected.

2.3.4 Finite element study using ANSYS A three dimensional finite element model was developed using the finite element program ANSYS (ANSYS® Academic Research, Release 10.0 ) to study the behavior of positive moment connection in a continuous for live load bridge. The finite element simulation included the nonlinear effects of concrete cracking and crushing as well as yielding of steel bars and strands. The modeling of the construction sequence was a challenge and various construction sequences needed to be defined in the model to predict a correct behavior. However, for simplicity in the analysis, the bridge was assumed to be a continuous superstructure system. The FEM did not account for the cold or construction joint at the girder-diaphragm interface. This resulted in a load-deflection curve which did not match the experimental study. The FEM predicted the behavior of the connection with reasonable accuracy, but it was posited that more modifications in the model can provide better predictions. The cold or construction joint at girder-diaphragm interface needs to be modeled. The modeling of interface between the strands and diaphragm concrete should include the slip observed in the experiments. The pullout and fatigue failures need to be accounted for in the model.

2.3.5 Conclusions made from the study Various conclusions were drawn from the study conducted on connections for precast/prestressed concrete girders made continuous for live load. It was concluded that the positive moment connections with a capacity of more than 1.2 times the positive cracking moment of the composite section is not efficient. If the analysis result shows a required capacity in excess of 1.2Mcr then some method of reducing the restraint moment must be employed. One method would be to specify a minimum age for the girder as this will allow the girder to undergo

21 some creep and shrinkage before the establishment of continuity and reduce the formation of positive moment at the connection. The study also concluded that bent strand positive moment connections designed using the equations developed by Salmons et al. (1974) performed adequately and were adopted in the AASHTO LRFD Bridge Design Specifications. The bent-bar connections also performed adequately if they are embedded into the girder with extended hooks embedded into the diaphragm as per the AASHTO LRFD Specifications. The embedment of girder into the diaphragm reduces the positive moment stress but it is difficult to quantify its effects. The placement of additional stirrups in the diaphragm just outside the girder improves the ductility of the connection. The horizontal bars through the web of the girder increase the strength and the ductility of the connection but cause significant cracking in the girders at failure. The idea of limiting the tensile stress at the bottom by partial pouring of the diaphragm was found out to be only partially effective. The report also concluded that temperature effects are significant on the structure. The end reaction can vary +-20% due to the daily temperature variation causing a positive moment at the connection, almost equal to 2.5 times the positive live load moment. The study also concluded that the positive moment cracking in the diaphragm does not necessarily reduce the continuity as opposed to the predictions made from analytical studies. The negative moment capacity of the connection is also not affected by the positive moment cracking until the crack extends into the slab. Based on the findings of the research, various revisions were proposed to the AASHTO LRFD Bridge Design Specifications.

2.4 Eldhose Stephen Thesis (2006), University of Cincinnati, OH The detailed research presented by Eldhose Stephen pertains to studying the effects of creep and shrinkage on a prestressed/precast bridge made from AASHTO Type III – I girder by method of simulation using a finite element program. Since developing a finite element model of a ‘continuous for live load’ bridges can be a complex task, an attempt was made to simulate the behavior of a simply supported single span bridge using a three-dimensional finite element model. The goal of the thesis was to develop the technique to simulate creep, shrinkage and cracking effects in a bridge. Several preliminary studies were conducted to select a proper analysis tool for the simulation. Finally, the ABAQUS software was used to develop the model because of friendly Graphics User Interface, capability to model construction sequences, and its

22 compatibility with FORTRAN subroutines. The results obtained were compared to the existing experimental results to validate the usage of ABAQUS to model such systems. The non-linear material properties were used for the girder and the deck concrete to simulate the tension softening and the compression hardening in the concrete. Different concrete material models were used for the girder concrete and the slab concrete. The Extended Drucker Prager model and the Concrete Damaged Plasticity Model were used for the girder and the deck concrete, respectively. Elastic properties were used for the steel, since the steel stresses were expected to be within the elastic limit. The loads considered for the analysis were prestress load, dead loads of the girder and the deck, and time dependent effects such as creep in the girder and shrinkage in the deck. Creep and shrinkage are concurrent properties but to simplify the analysis, creep was applied only to the girder elements and shrinkage was applied to the deck elements. The sequence of construction of the bridge was taken into consideration in the analysis by using “Model Change” keyword in the ABAQUS program. The details of different steps defined in the model are shown in Table 2.1. Table 2.2 Steps defined in the model (Stephen 2006) Name Type Time Stabilizatio Maximum Increment size period n no. of Initial Minimum Maximum (days) increments DEACTIVATE Static 1 No 100 1 1 1 RELEASE Static 1 No 100 1 1 1 CREEP Static 164 No 1.00E+06 1 1.00E-08 5 DECKDL Static 1 No 100 1 1 1 POURDECK Static 1 No 100 1 1 1 SHRINKAGE Static 500 Yes 1.00E+06 0.1 1.00E-20 5

FORTRAN subroutines were used along with ABAQUS CAE to simulate different behaviors in the model. Subroutine UEXPAN( ) was used to calculate creep and shrinkage in the girder and the deck, respectively, on the basis of the creep and shrinkage model reported in ACI- 209 (ACI Committee 209-Creep and Shrinkage. 2008). The results obtained from the analysis were compared to the study conducted by the Federal Highway Administration (FHWA 2001) and found to be in agreement with the data provided in the FHWA study on a bridge in New Hampshire. A comparison of total strain vs time is shown in Figure 2.7.

Figure 2.7 Comparison of ABAQUS and FHWA data (Stephen 2006) The results from the ABAQUS analysis validated its usage to simulate the response of such structures. A similar approach to study the behavior of a single span prestressed bridge girder is taken in the current project and details about the analysis are discussed in the next chapter.

2.5 NYSDOT Bridge Deck Task Force Evaluation of Bridge Deck Cracking on NYSDOT Bridges (Curtis 2007) Deck cracking is a common phenomenon. A Bridge Deck Task Force (BDTF) was set up by the New York State Department of Transportation (NYSDOT) to evaluate the cracking of the deck on the NYSDOT bridges. The BDTF studied the historical designs of the bridges and conducted a survey on existing decks which included newly constructed decks as well. The decks of two different kinds of girder configurations were investigated. Decks on adjacent prestress concrete girders and decks on spread steel girders were considered for the study. The various factors influencing the deck cracking were noted. Since concrete cracks easily in tension, causes of tensile stresses in the deck were studied and their effects on bridge deck durability

24 were analyzed. Two case studies were investigated and recommendations were made to reduce bridge deck cracking and its effects on the bridge structure. The survey of the existing decks on prestressed concrete box girders demonstrated two types of cracking: longitudinal cracking, which follows the beam joints, and random cracking, with relatively large spacing of the cracks (Curtis 2007). The longitudinal cracks followed the beam keyway and were caused by the differential movement of the beam at the keyways. The random cracking was attributed to the fact that the wet concrete of the deck was placed on dry precast beams. Observations on concrete core samples showed that cracks were initiating and propagating from the bottom of the concrete decks (Curtis 2007). The survey of the then existing bridge decks, which included newly constructed bridges, showed serious cracking issues. 38% of single span bridges and 67% of multiple span bridges had significant cracking (Curtis 2007). Based on the database created from the research conducted on the construction records, as-built plans, and materials records, it was determined that concrete strength, concrete cover, and pour temperature were some of the most influential factors in concrete bridge deck cracking. The deck concrete should meet minimum strength criteria (3000 psi for NYSDOT) but it should not be too strong, to minimize deck cracking. The concrete cover directly controls cracking in the deck and the crack width. To minimize corrosion of the reinforcement from the salt laden water during the winter, increased concrete cover is provided on the top surface of the deck; however, this results in more cracks and larger crack widths. A warmer temperature on the day of pouring the deck concrete can reduce the cracking, but it is difficult to control this parameter. The cause of cracking in the deck was the development of tensile stresses. There are three causes for the development of tensile stresses in the deck (Curtis 2007): 1) Thermal stress due to restraint of the deck while it cools. 2) Live load stresses on the continuous bridge. 3) Stress due to concrete shrinkage while restrained by the superstructure. Thermal stresses develop in the deck as early as the time of initial set of concrete. There is a significant temperature variation in the deck concrete in the first couple of days from the time of pour. There is an increase in the concrete temperature due to the release of heat from the chemical reaction of hydration. The increase in temperature starts from the point when the concrete gains its initial strength but is still not set. The temperature of the top flange of the steel

25 girder also rises with the increase in concrete temperature but it is still cooler than the concrete, as seen from Figure 2.8.

Figure 2.8 Initial temperature rise in bridge decks (Curtis 2007) Once the concrete reaches the peak temperature it begins to set and cool down. The relative temperature difference between the deck and the top flange of the girder can be up to 54oF (30oC). Since the concrete is set, the deck partially restrains the contraction of the deck, giving rise to thermal stresses. The thermal stress in the concrete due to the hydration process can range from 75 psi (0.5 MPa) to 500 psi (3.5 MPa) with reasonably expected values of 285 psi (2.0 MPa) (Curtis 2007) depending on the strength of the concrete and relative area of the deck to the girder top flange. The daily temperature variation over the course of time also causes thermal stresses in the deck. Live load and concrete shrinkage add to the tensile stresses in the deck. The combination of all the tensile stresses increases the chance of cracking in the deck. Cracking of the deck does not have any noticeable effect on the riding quality or structural condition of the deck (Curtis 2007). But the cracking of the deck will allow water to seep in and cause corrosion of the reinforcement, which is undesirable.

Several recommendations were made to reduce cracking in the concrete bridge deck along with their advantages and disadvantages. Some of the recommendations to reduce the deck cracking are mentioned below: 1) Minimize size of the top flange 2) Reduce effective strength of the deck concrete 3) Shrinkage reducing agents 4) Two course decks 5) Decrease concrete cover In addition methods to treat the bridge deck cracking were also outlined in the paper.

2.6 Behavior of pre-stressed concrete bridge girders due to time dependent and temperature effects (Debbarma and Saha 2011) The paper titled “Behavior of Pre-Stressed Concrete Bridge Girders Due to Time Dependent and Temperature Effects” was presented by Debbarma and Saha at the First Middle East Conference on Smart Monitoring, Assessment and Rehabilitation of Civil Structures (SMAR 2011). The service behavior of a bridge due to time dependent effects such as creep, shrinkage, and temperature variation was presented in the paper. A study was conducted on a prestressed concrete box girder bridge on Airoli Bridge over Thane Creek in Navi Mumbai, India. Vibrating wire strain gauges were embedded in the concrete and data was collected for approximately 800 days after concreting of the box girder section. The cross section of the box girder used for the construction of bridge is shown in Figure 2.9. The length of each girder was about 50 meters (approximately 165 feet).

Figure 2.9 Cross section of single cell precast pre-stressed box girder, Unit = mm (Debbarma and Saha 2011)

The time variation of shrinkage and creep strain predicted from the ACI 209 model (ACI Committee 209-Creep and Shrinkage. 2008)and the CEB-FIP 90 model (Comité euro- international du béton., 1993) using a computer program and are shown in Figure 2.10 and Figure 2.11 respectively. The strain data due to the combine effect of creep and shrinkage was collected in soffit, web and deck slab of the girder. The comparison of predicted data and observed data showed the occurrence of maximum strain in the structure nearly at the same time that is around 350 days from the concreting of the girder. The maximum deflection in the bridge profile was also observed at around 390 days, as seen from Figure 2.13. The deflection at the mid-span was hogging in nature due to the presence of prestressing in the girder.

Figure 2.10 Prediction of shrinkage strain using model codes (Debbarma and Saha 2011)

Figure 2.11 Prediction of creep compliance using model codes (Debbarma and Saha 2011)

Figure 2.12 Development of strain vs. time in concrete box girder section (Debbarma and Saha 2011)

Figure 2.13 Deflection profile of prestressed concrete box girder (Debbarma and Saha 2011)

Thermal effects have an influence on the behavior of the box girder bridge. The effect of change in temperature on concrete creep and shrinkage is basically two-fold. First, they directly influence the rate of creep and shrinkage with time. Second, they affect the rate of aging of the concrete, i.e. the change of material properties due to progress of cement hydration (Debbarma and Saha 2011). The deflection pattern for one of the girders was obtained after 600 days of concreting under no traffic condition. The maximum deflection was observed at the mid-span and the deflection was highest for the maximum value of temperature, as seen in Figure 2.14.

Figure 2.14 Pattern of deflection of box girder G-13 at different temperature (Debbarma and Saha 2011)

The strain values obtained from the vibrating wire gauges installed in the mid-span in soffit and the deck slab showed development of compression in the soffit and tension in the deck slab due to temperature variation. For a temperature increase of 5oC (41o F), the girder deflected by 9.5mm (0.37 inches) at the mid-span. This daily variation of temperature in the structure may give rise to the formation of cracks and causes deterioration at a latter age (Debbarma and Saha 2011) but quite a bit before the end of the design life.

Figure 2.15 Compressive strain vs temperature in soffit slab of box girder bridge at the mid-span (Debbarma and Saha 2011)

Figure 2.16 Tensile strain vs temperature in deck slab of box girder bridge at mid-span (Debbarma and Saha 2011)

The study concluded that creep and shrinkage cause continuous deformation in the bridge with age. This final deformation depends on the initial deformed configuration of the bridge that is after the application of prestress. The daily temperature variation can cause thermal cracks in the structure and can reduce the service life of the structure. Hence, it is important to consider these effects while designing the bridge.

2.7 Field monitoring of positive moment continuity detail in a skewed prestressed concrete bulb-tee girder bridge (Okeil et al. 2013) Okeil et al. performed a study on an existing bridge with a skewed segment constructed using the AASHTO bulb-tee girders (BT-72). The segment chosen for the study was a three-span continuous superstructure having a positive moment connection at the support. The positive moment continuity between the adjacent girders was established using hair pin bars following the recommendations of the NCHRP Report 519. This segment was particularly chosen because NCHRP Report 519 did not consider the skew and bulb-tee girders during its study. The vibrating wire sensors were embedded and surface mounted to measure temperatures, strains, rotations, crack widths and gaps. The sensors were located at the mid-span and on both sides of the diaphragm to capture the data that influenced the continuity of the structure. The study lasted for more than 24 months. The data collected were analyzed and various conclusions were made. The temperature changes due to seasonal variation were in the range of 20 to 115oF (-7 to 46oC). Figure 2.17 shows the temperature readings in the deck, top flange and bottom flange of the girder. The rotation at the girder ends and the strain in the hair-pin bars at both side of the continuity diaphragm girder are shown in Figure 2.18 and Figure 2.19, respectively.

Figure 2.17 Temperature readings in deck and top-and bottom flanges (Okeil et al. 2013)

Figure 2.18 Rotation of girder ends (Okeil et al. 2013)

Figure 2.19 Strains in hairpin bars at both sides of continuity diaphragm (Okeil et al. 2013)

The girder end rotations on both sides of the connection followed the same trend; hence the diaphragm was able to provide continuity to the structure. Similarly the strain in hairpin bars at both sides of the continuity diaphragms is similar. So the continuity detail was able to transfer force across the diaphragm from one girder to another due to long-term effects and live loads. The study concluded that the girder cambers up at the mid-span due to creep and thermal effects, which caused positive moment at the continuity connection. The thermal effects can cause large restraint moment due to seasonal and daily temperature variation and is an important factor in the design. Thermal effects along with creep should be considered to determine the magnitude of the positive restraint moment. The positive restraint moment may cause cracking at the diaphragm and girder interface, which can affect the performance of the girder. The results from the study validate the performance of the positive moment connections recommended by NCHRP Report 519 and also support the idea that the same connection can be used for different configuration of bridges constructed using different kind of girders.

Chapter 3. Finite Element Modeling and Analysis

In order to eventually study continuous for live load bridges, it is necessary to first determine how to model creep, shrinkage and deck cracking in a finite element model. Because of the complexity of modeling a two span system, a non-linear three dimensional finite element model of a single span bridge is created in ABAQUS Standard 6.11-2 as a first step. There are various ways to create a working model in ABAQUS, such as using ABAQUS CAE, keyword version, and Python script. This study uses the ABAQUS interactive version in the form of ABAQUS CAE to create the model. The feature to edit the keywords directly in the input file created using the CAE is also used when needed. The model database is divided into two parts: Model and Analysis. They are further divided into several modules covering different aspects of modeling in each module. The figure below gives a better understanding of the model database division. The various modules in the model database definition are parts, materials, sections, assembly, steps, field output requests, history output requests, interactions, constraints, fields, loads, boundary conditions, and predefined fields. Some of the steps involved in creating a working model are creating parts, defining material properties, defining sections, assembling the parts to create the required structure, defining various steps of analysis, adding boundary conditions, and submitting the job for analysis, etc. After submitting the job, we can monitor the progress of the job. Upon completion of the job, the visualization module can be used for the post processing of the model. The current model is a single span prestressed concrete bridge girder with a deck slab on it. The girder is a 50 foot long Type III AASHTO I girder and represents half the specimen used in NCHRP 519 (Miller, et al. 2004). The steps to be followed to create a successful model in ABAQUS are discussed in this chapter.

Figure 3.1 Model database in Abaqus 34

3.1 Use of consistent units ABAQUS does not have any inherent system of units. The numbers entered by the user need to have a consistent unit throughout the modeling of geometry, material definitions, and loading of the model. The output results correspond to the same unit system as the input. The units used in the current analysis are pounds, inches and days.

3.2 Modeling of geometry in the parts module The geometry can be modeled as small parts in the parts module having different properties and assembled at a later stage to create the desired structure. The cross-section of the parts are created by drawing a two dimensional sketch using the tools and the interface provided in the ABAQUS CAE. The two dimensional sketch is extruded for the required depth to create a three-dimensional shape. The parts need to be seeded prior to meshing. Seeding can be done independently on different edges of the part or to the part as a whole. The seeding size depends on the size of the intended mesh, which affects the results of the analysis. The different parts created in the model are girder and deck, strands, longitudinal reinforcement for the deck, and transverse reinforcement for the deck. The girder and the slab are assigned C3D8 elements while all the strands and the reinforcement are assigned T3D2 elements. The C3D8 and the T3D2 elements are described below: C – Continuum T - Truss 3D – Three dimensional 3D - Three dimensional 8 – 8 nodes 2 – 2 nodes

2 end 2

1 end 1

8-node linear brick stress/displacement, C3D8 2-node linear 3D truss Figure 3.2 Type of elements used in the model

The C3D8 element is a general purpose, fully integrated linear or first-order brick element with 2x2x2 integration points. The node numbering follows the convention shown in Figure 3.2. These elements are stress/displacement elements with three displacement degrees of freedom per node. The C3D8 elements have a better convergence rate in a regular mesh. The T3D2 elements are three-dimensional truss elements having two degrees of freedom per node. They can be to model slender, line-like structures, such as strands, reinforcement, etc. The geometry of parts are shown here.

Figure 3.3 Two dimensional sketch of girder and deck

Figure 3.4 Three dimensional extruded view of girder and deck part

Figure 3.5 Girder and deck part shown as different sections

Figure 3.6 Typical steel part

3.3 Defining and assigning material properties to different parts of the structure Once the parts are created, material properties need to be assigned to different sections of the parts or to the part as a whole. A structure can be made up of different materials and each material needs to be defined with their suitable characteristics. The material definition can consist of any or all of the basic characteristics listed below, which is not exhaustive. Elastic modulus Poisson’s ratio Density Plasticity model The two materials used in the current model are concrete and steel. Steel is used for all the reinforcement and the strands while the girder and the deck are made up of concrete. The properties of the materials used are discussed in detail in the next few sections.

3.3.1 Steel An elastic material property is assigned to steel as the stresses in the steel during the analysis are expected to remain well under the elastic limit. The same material definition is used for both reinforcement steel and prestressing strands, as their properties are the same. The cross- section area of the reinforcement and the strand differ and hence have different section properties. The steel material is defined by the following characteristics in the model: Elastic modulus Poisson’s ratio Density

3.3.1.1 Reinforcement and strand steel The properties of reinforcements and strands are listed below: Elastic modulus: 29000000 psi Poisson’s ratio: 0.25 Density: 0.28 lb/in3 Diameter of strand: 0.5 in Diameter of reinforcement: 0.625 in

3.3.2 Concrete An inelastic material model is used for simulating the behavior of concrete in the current project. It is expected that concrete will reach its stress limit in tension and hence it is important to choose a material that will successfully capture its behavior after it reaches the stress limit in tension.

3.3.2.1 Concrete material model The simulation of reinforced concrete material is a complicated task to perform in a finite element program. The material model definition should be capable of capturing both the elastic and the plastic behavior of concrete in compression and tension. There are two concrete material models available in ABAQUS: Concrete Smeared Cracking model and Concrete Damaged Plasticity (CDP) model. The CDP model is used for both the girder and the deck slab concrete since it can simulate inelastic behavior of the concrete in both tension and compression. There

39 are many features of the CDP model, which makes it appropriate for the current analysis. Some of the features of the CDP model are listed below. The concrete damaged plasticity model: a) provides a general capability for modeling concrete and other quasi-brittle materials in all types of structures b) uses concepts of isotropic damaged elasticity in combination with the isotropic tensile and the compressive plasticity to represent the inelastic behavior of concrete c) is intended primarily for the analysis of reinforced concrete structures d) can be used in conjunction with a viscoplastic regularization of the constitutive equations in ABAQUS/Standard to improve the convergence rate in the softening regime (Hibbitt et al. 2011a) Concrete behaves differently to tensile and compressive loads. The strength of concrete in compression is much greater than its strength in tension. The CDP model assumes that the concrete material fails mainly in tensile cracking and compression crushing. The failure surface 풑풍 풑풍 is controlled by two independent hardening variables: 흐풕 and 흐풄 , which are referred to as tensile and compressive equivalent plastic strains, respectively. The non-linear concrete response to uniaxial loading in tension and compression is shown in Figure 3.7 and Figure 3.8, respectively. The stress-strain response of concrete is linearly elastic under the uniaxial tensile load until the stress reaches a point of failure. Beyond the failure point that is the plastic region, a softening stress-strain response is seen in the concrete. This induces strain localization in the concrete because of formation of micro-cracks in the concrete. To simulate the tensile behavior the user needs to input post failure stress-strain data in the CDP model. The cracking stress vs cracking strain data corresponds to the post failure data. This data can be obtained from experimental results on the concrete material or from numerical models.

Figure 3.7 Response of concrete to uniaxial loading in tension (Hibbitt et al. 2011a)

Figure 3.8 Response of concrete to uniaxial loading in compression (Hibbitt et al. 2011a)

The behavior of concrete under uniaxial compressive load is linear up to the point of initial yield. After concrete has reached the peak stress value, the plastic behavior of concrete is characterized by stress hardening followed by strain softening beyond the point of ultimate stress. Similar to tensile data, cracking stress vs. inelastic strain needs to be provided by the user, which can be obtained from experimental results or numerical models. Although it is a simplified approach, the CDP model captures the important features of concrete behavior in a structure. The parameters needed to define the concrete damaged plasticity model are:   – dilation angle in degrees measured in the p-q plane at high confining pressure   – Flow potential eccentricity, default is 0.1

 bo/co – ratio of the initial equibiaxial compressive yield stress to the initial uniaxial compressive yield stress, default value is 1.16

Kc – ratio of the second stress invariant on the tensile meridian to that on the compressive meridian at initial yield, 0.5 < Kc <= 1.0, default is 2/3

  – viscosity parameter representing the relaxation time of the viscoplastic system, default is 0.0 The failure surface in the Concrete Damaged Plasticity model is governed by parameter

Kc. Kc is the ratio of the cracking stress in triaxial tension to the yield stress in triaxial compression. ABAQUS manual specifies a default value of 2/3 for the parameter, Kc. A similar value was reported by Kupler (1969) from his experimental data. Since conducting laboratory experiments to determine the material parameters was beyond the scope of this project, the default value of 2/3 is used in the analysis. The plastic potential surface in the meridional plane is in a form of a parabola in the CDP model. The eccentricity parameter can be used to adjust the shape of the plastic potential surface. A default value of 0.1 is defined for the concrete materials in the current analysis. The dilation angle characterizes the performance of concrete under compound stress (Kmiecik and Kamiński 2011). It is the angle of inclination of the failure surface towards the hydrostatic axis, measured in the meridional plane. The unquestionable advantage of the CDP model is the fact that it is based on parameters having an explicit physical interpretation (Kmiecik and Kamiński 2011).

3.3.2.2 Girder concrete The Concrete Damaged Plasticity model is used for defining both the girder and the deck concrete material in the analysis. The material data for the girder and the deck concrete were derived from the previous work done by Eldhose (Stephen 2006) and by studying previous papers on concrete damaged plasticity model. Tensile behavior Compressive behavior Cracking stress Cracking strain Peak stress (psi) Inelastic strain (psi)700 0 8500 0.0 525 0.00005 9100 0.0001 350 0.0001 Other Parameters 262.5 0.00013  56 175 0.00017  0.1

131.25 0.0002 bo/co 1.16

87.5 0.00025 Kc 0.6667 70 0.0003  0.009 61.25 0.0004 

3.3.2.3 Deck concrete Tensile behavior Compressive behavior Cracking stress Cracking strain Peak stress (psi) Inelastic strain (psi)300 0 8500 0.0 225 0.00005 9100 0.0001 150 0.0001 Other Parameters 112.5 0.00013  56 75 0.00017  0.1

56.25 0.0002 bo/co 1.16

37.5 0.00025 Kc 0.6667 30 0.0003  0.25 26.25 0.0004 

3.4 Assembly The next step in modeling is to assemble the different parts to create the desired geometry. The number of instance/s of the same part/s can be created at the assembly stage, such as the number of strands or reinforcements in a structure. The instance/s of the part/s are positioned relative to each other in a global coordinate system. The constraints available in ABAQUS can be used to position the instances relative to each other. After the desired geometry is created, the elements, geometry or nodes can be grouped together into sets, which can be helpful for the application of material properties, loads, boundary conditions, and constraints, etc. at a later stage. The assembly for the current project comprises different parts with multiple instances of some of the parts. The number of instances of each of the parts is listed inError! Reference source not found.. Table 3.1 Part and number of instances (Single span system) Part Number of instances Girder and deck 1 Strand 20 Transverse deck reinforcement 118 Longitudinal deck reinforcement 24 Longitudinal reinforcement at girder top 6

Figure 3.9 Girder and deck mesh in cross section

Figure 3.10 Three-dimensional view of girder and deck mesh

Top reinforcement in the girder

Strands at the bottom of the girder

Figure 3.11 Wireframe showing girder with strands and reinforcement

Longitudinal reinforcement in the slab

Transverse reinforcement in the slab

Figure 3.12 Wireframe showing slab with strands and reinforcement

Figure 3.13 Wireframe of the complete model

Figure 3.14 Deck reinforcement with part zoomed

Figure 3.15 Strands

3.5 Defining steps and loadings The simulation of a structure may consist of one or more steps. Different steps help to follow the changes in the loading and the boundary conditions in the model. The step module in ABAQUS is used to define the analysis steps. Loadings, boundary conditions, field output requests, and model interactions, etc. can be created and/or modified in the step module. Since the current project involves simulating the construction sequence of the bridge, it is important to define several steps in order to capture the behavior of the structure at different stages of construction. The different construction stages of the prestressed concrete bridge modeled in the analysis are prestressing in the girder, creep and shrinkage in the girder, erection of girder and casting of the deck, and creep and shrinkage in the deck. Each of the above sequences is simulated in a different step. The loadings considered in the analysis are prestressing load in the girder, dead load of the girder and the deck elements, and creep and shrinkage in the girder and the deck.

3.5.1 Steps The various steps defined in the current model are listed in Table 3.2. Table 3.2 Various steps defined in the analysis Step Name Type Time Maximum Increment size No. of no. period no. of Initial Minimum Maximum increments (days) increments 1 Initial 2 Prestress Static 1 400 0.1 1E-10 0.2 129, release 137,129, and 137 3 Girder creep Static 10, 30, 500 0.1 1E-10 1 136,195, and shrinkage 60 and 208,and 90 260 4 Slab dead Static 1 100 0.1 1E-05 1 6 load on girder 5 Deck Static 1 100 0.1 1E-10 1 6 activated 6 Slab creep Static 187, 1000 0.1 1E-10 1 192, 206, and shrinkage 177, 137 185, and and 107 156 The number of increments corresponds to 10-days, 30-days, 60-days, and 90-days old girder at the time of casting of the slab. The prestress load can be applied using various methods in ABAQUS. It can be applied by using the predefined field in the Initial step of the analysis by selecting the strand elements and providing the value of the stress. Additionally, stress value can be converted to temperature and the temperature loading can be applied to the strands in the first step to simulate initial stress in the strands. In the current model, the first method is used to apply a prestressing force to the strands using ABAQUS CAE. The input file code for the same is written below:

** PREDEFINED FIELDS *Initial Conditions, type=STRESS Strands_geometry, 202500., 0., 0., where, Strands_geometry is a set representing all the strand elements in the model and the numerical value 202500 is the magnitude of the prestressing force applied in the axial direction. A value of 202500 psi is calculated by using the following formula.

Prestress load = 0.75fpu = 0.75*270 = 202.5 ksi =202500 psi

푓푝푢= yield strength of prestressing strands = 270 ksi

Adequate boundary conditions need to be provided to simulate the release. The strands are connected to the girder using the embedded region constraint. A perfect bond is assumed between the strands and concrete elements since they share the same nodal coordinates.

3.5.2 Dead load The dead load of the girder and the slab are taken into consideration. The dead load is applied as a body force for all the parts stated above. Body force is defined to prescribe loading per unit volume over a body. The value of density of the material is used as the loading value in the body force load assignment and the direction of loading is the gravity direction.

3.5.3 Creep and shrinkage Creep and shrinkage are time dependent occurrences in concrete, which may lead to damaging behavior in concrete bridges such as excessive deflection and cracking. Creep is the strain caused in hardened concrete due to sustained stress. Shrinkage is the change in volumetric strain without the contribution of any externally applied stress. The variation of creep and shrinkage properties in concrete is affected by various factors commonly classified as internal and external factors (Bazant 1985). The variation in the composition and the quality of concrete, such as the slump and percent of fine aggregate can be considered as internal factors while the environmental conditions such as the humidity and temperature can be considered as external factors. The creep and shrinkage of concrete are considered in the prestressed girder and deck in the current model. Creep and shrinkage strains are interdependent properties of concrete. However, they are assumed as independent properties, and creep and shrinkage strains are added together to get the total strain in the current analysis. There are various creep and shrinkage models available for concrete, such as the ACI-209 (ACI Committee 209-Creep and Shrinkage. 2008), the AASHTO LRFD Bridge Design Specifications (American Association of State Highway and Transportation Officials., 2012), the CEB-FIP-90 (Comité euro-international du béton., 1993), and the NCHRP 496 (Tadros, Maher K., National Cooperative Highway Research Program., American Association of State Highway and Transportation Officials., United States., Federal Highway Administration., National Research Council (U.S.).,Transportation Research Board., 2003). Most of the above models include internal and external factors to calculate the creep and shrinkage strain. The ACI-209 model (ACI Committee 209-Creep and Shrinkage. 2008) is used in the current project because it

50 is easy to implement and it does a satisfactory job in predicting the creep and shrinkage strain in the concrete.

3.5.3.1 ACI – 209 Report In 1992, the American Concrete Institute (ACI) recommended a theory to predict the creep and shrinkage effects at any time in the concrete. The report was approved in 1997 (ACI Committee 209-Creep and Shrinkage. 2008). It considered the loading age, ambient relative humidity, member shape and size, and composition of concrete such as slump, cement content, air content, and percent of fine aggregate to calculate the creep and shrinkage in a specimen. This report includes normal weight and light weight concrete, which are composed of Type I and Type III cement and are cured using the moist or the steam curing technique. The procedure to calculate creep and shrinkage strain using the ACI-209 (ACI Committee 209-Creep and Shrinkage. 2008) provisions is outlined here. The total strain is given as: 휎 휀푡 = (휀푠ℎ)푡 + (1 + 휈푡) Eqn. (3.1) 퐸0 where, 휎= applied stress

퐸0 = modulus of elasticity of the concrete at the time of loading

휐푡 = creep coefficient for a loading age of 7 days, for moist-cured concrete and for 1-3 days steam cured-concrete 푡0.60 휐 = 휐 Eqn. (3.2) 푡 10 + 푡0.60 푢

휐푢 = ultimate creep coefficient defined as ratio of creep strain to initial strain 푡 = time after loading

Average values suggested for 휐푢 are given by:

휐푢 = 2.35 훾푐

훾푐 = product of the applicable correction factors The different correction factors account for loading age, ambient relative humidity, size of the member, and concrete composition. The concrete composition includes slump, fine aggregate percentage, and air content. The shrinkage strain after 7 days for moist cured concrete is given as:

푡 (휀 ) = (휖 ) Eqn. (3.3) 푠ℎ 푡 35 + 푡 푠ℎ 푢

(휀푠ℎ)푡 = shrinkage strain at any time, t 푡 = time from the end of initial curing

(휀푠ℎ)푢 = ultimate shrinkage strain

Average values suggested for (휀푠ℎ)푢 is given by: −6 (휀푠ℎ)푢 = 780훾푠ℎ × 10 in./in.

훾푠ℎ = product of the applicable correction factors The various correction factors applicable to the shrinkage strain are ambient relative humidity, size of the member, and concrete composition, which includes slump, fine aggregate percentage, cement content and air content. The calculations of the correction factors for the ultimate creep coefficient and the shrinkage strain are given below. The creep correction factor for loading ages later than 7 days for moist cured concrete is: −0.118 Creep, 훾푙푎 = 1.25(푡푙푎) Eqn. (3.4)

푡푙푎 = loading age in days The creep correction factor for ambient relative humidity greater than 40 percent is:

Creep 훾휆 = 1.27 − 0.0067휆 Eqn. (3.5) The shrinkage correction factors for ambient relative humidity greater than 40 percent are:

Shrinkage, 훾휆 = 1.40 − 0.0102휆, for 40 ≤ 휆 ≥ 80 Eqn. (3.6)

Shrinkage훾휆 = 3.00 − 0.030휆, , for 80 < 휆 ≤ 100 Eqn. (3.7) 흀 = ambient relative humidity in percent The correction factor for the size of the member can be calculated by the average thickness method and the volume-surface ratio method. In this project it is calculated using the second method and the equations for correction factors for creep and shrinkage are as follows. 2 (−0.54 푣/푠) Creep, 훾푣푠 = ⁄3 [1 + 1.13 푒 ] Eqn. (3.8) (−0.12 푣/푠) Shrinkage, 훾푣푠 = 1.2 푒 Eqn. (3.9) 푣/푠 = volume-surface ratio of the member in inches The correction factor for the composition of concrete includes the effect of slump, percent of fine aggregate, cement and air content. The correction factor equation for slump is given by:

Creep, 훾푠 = 0.82 + 0.067푠 Eqn. (3.10)

Shrinkage, 훾푠 = 0.89 + 0.041푠 Eqn. (3.11) 푠 = observed slump in inches The correction factor for percent of aggregate is given by:

Creep, 훾Ψ = 0.88 + 0.0024휓 Eqn. (3.12)

Shrinkage, 훾휓 = 0.30 + 0.014휓 for, 휓 ≤ 50 percent Eqn. (3.13)

Shrinkage, 훾휓 = 0.90 + 0.0024휓 for, 휓 > 50 percent Eqn. (3.14) 흍 = ratio of fine aggregate to the total aggregate of the concrete weight expressed in percent The correction factor for shrinkage in concrete for cement content is given by:

Shrinkage, 훾푐 = 0.75 + 0.00036푐 Eqn. (3.15) 푐 = cement content in pounds per cubic yards The correction factor for creep and shrinkage for air content in concrete is given by:

Creep, 훾α = 0.46 + 0.09훼 ≥ 1.0 Eqn. (3.16)

Shrinkage, 훾훼 = 0.95 + 0.008훼 Eqn. (3.17) 훼 = air content in concrete in percent

Finally, the product of applicable product factors for creep and shrinkage, 훾푐 and 훾푠ℎ, respectively, can be calculated as:

훾푐 = 훾푙푎. 훾휆. 훾푣푠. 훾푠. 훾휓. 훾훼 Eqn. (3.18)

훾푠ℎ = 훾휆. 훾푣푠. 훾푠. 훾휓. 훾푐. 훾훼 Eqn. (3.19)

3.5.3.1.1 Correction factor for creep and shrinkage coefficient for girder and deck Table 3.3 and Table 3.4 give the values of different correction factors for creep and shrinkage, respectively, for this project.

Table 3.3 Correction factors for the girder Condition Creep factor Shrinkage factor

Loading age 푡푙푎 = 1 day 1.250 - Relative humidity 휆 = 70% 0.801 0.686 Volume-surface ratio 푣/푠 = 4.004in 0.753 0.742 Slump 푠 = 2.5in 0.988 0.993 Percent of fine aggregate 휓 = 60% 1.024 1.02 Cement content 푐 = 752 lbs/cu yd - 1.021 Air content 훼 = 7% 1.090 1.006

Applicable correction factor (휸풄)품풊풓풅풆풓 = 0.831 (휸풔풉)품풊풓풅풆풓 = 0.529

Table 3.4 Correction factors for the deck Condition Creep factor Shrinkage factor

Loading age 푡푙푎 = 28days 0.844 - Relative humidity 휆 = 70 % 0.801 0.686 Volume-surface ratio 푣/푠 = 3.647in 0.772 0.775 Slump 푠 = 2.5 in 0.988 0.993 Percent of fine aggregate 휓 = 60 % 1.024 1.02 Cement content 푐 = 752 lbs/cu yd - 1.021 Air content 훼 = 7 % 1.09 1.006

Applicable correction factor (휸풄)풅풆풄풌 = 0.575 (휸풔풉)풅풆풄풌 = 0.552

Ultimate creep coefficient for girder = 2.5 × (훾푐)𝑔𝑖푟푑푒푟 = 1.9538

Ultimate creep coefficient for deck = 2.5 × (훾푐)푑푒푐푘 = 1.35081

Ultimate shrinkage strain for girder = 0.000780 × (훾푠ℎ)𝑔𝑖푟푑푒푟 = 0.00041

Ultimate shrinkage strain for deck = 0.000780 × (훾푠ℎ)푑푒푐푘 = 0.00043

The plot of creep coefficient vs. time and shrinkage strain vs. time is shown in Figure 3.16 and Figure 3.17, respectively, for the girder and the deck slab.

Variation of creep coefficient with time 1.6

1.4

1.2

Girder 1 creep coefficient 0.8 Deck creep

0.6 coefficient Creep Creep coefficient 0.4

0.2

0 0 50 100 150 200 Time (days) Figure 3.16 Creep coefficient vs. time for the girder and deck slab

Variation of shrinkage strain with time 0.0004

0.00035

0.0003

Girder 0.00025 shrinkage strain

0.0002 Deck shrinkage

0.00015 strain Shrinkage Shrinkage strain

0.0001

0.00005

0 0 50 100 150 200 Time (days) Figure 3.17 Shrinkage strain vs. time for the girder and deck slab

3.5.3.2 User subroutine to calculate creep and shrinkage strain The use of subroutines provides an additional advantage to simulate complicated conditions in ABAQUS. All the subroutines for a particular analysis can be combined into a single text file and saved with an extension .f or .for. ABAQUS does not a have built in ACI-209 (ACI Committee 209-Creep and Shrinkage. 2008) creep and shrinkage model to calculate creep and shrinkage strain for concrete, therefore it is achieved by using a FORTRAN subroutine. Creep and shrinkage strain is calculated in a simplified manner by means of UEXPAN subroutine in ABAQUS. UEXPAN subroutine simulates the thermal expansion/contraction in a material, but it can be manipulated to calculate any kind expansion/contraction in a material. The subroutine is called for all integration points for each iteration of an increment in an element whose material definition includes, user-specified thermal expansion, UEXPAN. The strain is calculated as: 0.60 0.60 푡푛 푡푛−1 Δ휀푐푟 = 휐푢휀𝑖 ( 0.60 − 0.60) Eqn. (3.20) 10 + 푡푛 10 + 푡푛−1

∆휀푐푟 = creep strain increment

휐푢 = ultimate creep coefficient

휀𝑖 = instantaneous elastic strain from end of previous increment

푡푛 = time since loading at current increment

푡푛−1 = time since loading at previous increment The instantaneous elastic strain from the end of the previous increment is saved as a state variable using USDFLD subroutine and passed on to the UEXPAN subroutine.

Increment in shrinkage strain at time 푡푛 is calculated as:

푡푛 푡푛−1 Δ휀푠ℎ = 휀푠ℎ푢 ( − ) Eqn. (3.21) 35 + 푡푛 35 + 푡푛−1

∆휀푠ℎ = shrinkage strain increment

휀푠ℎ푢 = ultimate shrinkage strain

푡푛= time since curing in current increment

푡푛−1 = time since curing at previous increment The total strain increment due to creep and shrinkage is calculated as:

Δ휀 = Δ휀푐푟 − Δ휀푠ℎ Eqn. (3.22)

The shrinkage strain is deducted from creep strain due to the sign convention in ABAQUS. Tensile stresses and strains are positive in ABAQUS, hence shrinkage strain is considered negative while the sign of creep strain increment is taken care of by the instantaneous elastic strain allowing for both compressive and tensile creep.

3.5.4 Live load Live load can be added as point loads and distributed loads as required.

3.6 Field output requests The field output requests help us to control the number of output variables to be written and the rate of output to the output database file. The output is created at each node and element level in an analysis, therefore writing too many output variables increases the size of the output database file and slows down the analysis. The interval of increments, or time period at which the output is requested, can also be controlled to minimize storage space and decrease the analysis time. The output variables may change for different problems but some output variables, like stress, strain, and displacement can be common to all the problems. Similarly, for problems involving contact interactions, the output variable related to contact may be needed. For the current problem, stress, strain and displacement output are significant to analyze the results and these output variables are requested for all the models.

3.7 Submission of job for analysis Once all the steps are defined and the boundary conditions are applied to different parts of the structure, an analysis job is created and submitted for analysis. The job module can be used for writing an input file for the model. The input file created can be used to run the simulation from any other computer with faster processors. The analysis can be monitored in real-time after submission at each increment level using the job monitor. The current analysis was run on a supercomputer, which was remotely accessed. The input file created in the job module and a FORTRAN subroutine written in Notepad was submitted for analysis using a script file identifiable by the supercomputer. A sample script file is attached in the Appendix. The analysis approach and the numerical solver involved in the analysis are discussed in detail in the next chapter.

3.8 Post processing in ABAQUS using the visualization module ABAQUS has a very user friendly graphical user interface for post processing the results from the analysis. The results can be viewed in the visualization module by opening the .odb (Output Database) file. The .odb file is generated as the analysis progresses, but it is advisable to open the file once the analysis is completed or terminated. Accessing the output database file while the analysis is in progress will slow down the analysis and is not recommended. The visualization module allows the viewing of the results as well as provides tools for diagnosing the errors and warnings in a job. Some of the most commonly used post-processing features are: 1) Contours – The variables, such as stress, strain, and displacement, can be represented as contour in the model for any increment in a step. This makes it easier to understand the behavior of the model. The contours can be plotted on both the deformed and the undeformed shapes. 2) Deformed shape – The nodal deformation pattern of the model can be examined in detail using the deformed shape tool. The scale can be changed to view an exaggerated deformation pattern of the model, which is helpful when the deformation in the model is small. 3) X-Y data – The X-Y data can be read from the field output in the output database obtained from analysis results. The field outputs requested during the analysis are available and can be recorded for an element, nodes, and integration points. Tabular data from the X-Y data can be created, saved and/or copied to Excel or any other program for further analysis. 4) X-Y plots – The plots created from the X-Y data can be formatted using the X-Y plot tool. The title of the plot, axis title, and legends can be labeled and their fonts can be edited as required. 5) Time history animation –The behavior of the structure over time can be displayed as a series of plots shown rapidly, emulating a movie. This feature can be useful to present the result for the whole or part of the model in a presentation. 6) Job diagnostics – ABAQUS terminates the analyses when it encounters severe instabilities or an error. The job diagnostic is a very helpful tool to investigate the reason for the termination. The job history in the job diagnostic tab lists summaries of all the iterations in each increment in a step. The summary of each

58 iteration can be investigated to locate any instabilities/errors in the model. The diagnostic gives the nodes/elements numbers causing instability/error in an iteration. All the warnings and errors at different iterations can be found here and the model can be fine-tuned as required by investigating these warnings/errors.

Chapter 4. Analysis approach

The construction sequence of the bridge is simulated by using a multi-step analysis approach. The complete simulation is divided into six steps where each step corresponds to different real time construction sequences. In most of the simulation programs the whole model is divided into number of nodes and elements. The computation is conducted for each element and node in the system and the results are combined to give the behavior of the system as a whole. There are many computational techniques available and choosing a particular technique for an analysis depends on the problem at hand.

4.1 Nonlinearity and Convergence The current analysis is non-linear with respect to material properties. In a non-linear analysis it is likely that there will be convergence issues and additional care is needed to arrive at a converged solution. Convergence can imply mesh convergence, time integration accuracy, convergence of nonlinear solution procedure, and solution accuracy. For the current problem, convergence of nonlinear solution procedure is significant to obtain an accurate result although there are other factors involved. Mesh convergence refers to the idea that finer mesh provides better results than a coarser mesh. There is a stage when further refining of the mesh does not significantly affect the results. This is what is referred to as a converged mesh. Non-linearity can occur because of geometry, material properties and boundary conditions. Large deflections or deformation lead to geometric non-linearity. Material nonlinearity can include nonlinear elasticity, plasticity and damage in the material. In the current model, the material properties for concrete, as stated earlier, are non-linear. ABAQUS has various numerical analysis techniques like the Newton-Raphson method, arc-length method solver, explicit solver, etc. The time-dependent nature of the current model restricts the use of the arc-length method (Riks analysis) or the explicit solver. Since the time-dependent behavior of concrete is to be analyzed, the time period of the model becomes an important variable. Therefore, Static, General solver, which employs Newton-Raphson solution technique, is used to solve the problem.

4.1.1 Newton-Raphson Method in ABAQUS (Hibbitt et al. 2011a) In non-linear analysis, the total time period in each step is divided into small increments so that the nonlinear path can be followed. The idea is to break the total load into small values. Since the stiffness of the structure changes with deformation, it is not possible to calculate the solution by applying the total load in one attempt. An initial increment size is provided by the user and the subsequent increment is automatically chosen by ABAQUS. The initial increment size should be small so that a stable solution can be achieved in the first increment. An initial increment size of 0.01 or 0.001is acceptable in most of the cases but can vary as per the demand of the problem. The structure is in approximate equilibrium after the end of each successful increment during the analysis. For each increment, iterations are performed until convergence is obtained. If ABAQUS fails to converge during an iteration, it starts a new iteration with a smaller increment size. The smallest increment size possible in an analysis can be controlled by the user. It is important to use the minimum increment size parameters wisely so that the analysis is completed within an acceptable amount of time. The concept of a body to be in equilibrium requires the net force acting at every node to be zero or very close to zero. The internal forces, I, and the external forces, P, must balance each other (Hibbitt et al. 2011a).

P P

P = External load

Ib and Id = Internal forces

Figure 4.1 Internal and external loads on a body (Hibbitt et al. 2011a)

P – I = 0

There are various numerical methods to solve problems by an iterative procedure. The Static, General solver in ABAQUS uses the Newton-Raphson numerical analysis method, which is a robust iterative method to solve a nonlinear problem. In this technique, each iteration involves the formulation and solution of linearized equilibrium equations. Each iteration consists of defining the terms in the equilibrium equations (forming the stiffness matrix) and solving the resulting system.

Ktangent Cu = P – I Eqn. (4.1)

Where Ktangent = tangent stiffness matrix

Cu = correction to displacement, u If the solution in an iteration falls within a defined level of tolerance, then it is accepted as a sufficiently accurate solution. P - I = R(u) Eqn. (4.2) R(u) is the residual or the out of balance force at u for the iteration and it is nonlinear. If the structure is not in equilibrium at the displaced position u then, 푅(푢) ≠ 0

The displacement correction factor, cu is found so that u +cu is in equilibrium,

푅(푢 + 푐푢) = 0 Eqn. (4.3) On expanding the R(u) in a Taylor series about the current displacement u, we get 휕푅 푅(푢 + 푐 ) = 푅(푢) + | 푐 + ⋯ = 0 Eqn. (4.4) 푢 휕푢 푢 푢 Neglecting the higher order terms, we will have, 휕푅 | 푐 = −푅(푢) 휕푢 푢 푢 Substituting the equilibrium equation into the Newton-Raphson scheme: 휕푅 − | 푐 = 푅(푢) 휕푢 푢 푢

퐾푡푎푛𝑔푒푛푡 푐푢 = 푃 − 퐼 ,

The above equation is a linear equation in cu. Since all the other terms are known at a displaced position, cu can be calculated. Once cu is calculated, displacement u can be updated to

푢𝑖+1 = 푢𝑖 + 푐푢 and equilibrium of the system is checked again. If the system is in equilibrium then the convergence is achieved. The two main criteria to test the convergence are that the sum of all the

62 forces acting on each node should be very small and the displacement correction should be very small as well. The tolerance for the above values is set to a small value by default in ABAQUS and can be edited by the user, if needed.

P1 R(u2) R(u ) R(u1) 0 R(u ) K(u2) 3

K(u1)

K(u0)

u u0 u1 u2 u3 u4 Du1 Du2 Du3 Du4

P = load, u = displacement Figure 4.2 Newton-Raphson approach to solve a nonlinear problem (ABAQUS convergence class, Dassault Systemes)

In the above figure, it can be seen that iterations are performed until a point where the residual R(u3) is small enough to approximate the solution as converged. When the residual is less than the defined force residual tolerance at all the nodes, ABAQUS Standard accepts the system to be in equilibrium. The default tolerance value in ABAQUS is 0.5% of the average force in the

63 structure, averaged over time. This tolerance value depends on the accuracy of the results intended for the problem and can be changed by the user. The solution procedure followed by ABAQUS for a problem without contact is shown in Figure 4.3. When the solution for a given increment cannot be found by ABAQUS, a new attempt is made by reducing the magnitude of the load increment (by default to 25% of the previous increment size). If too many attempts are made to reduce an increment (by default 5 attempts), ABAQUS stops the analysis with the following error: Too many attempts made in an increment. When two consecutive increments converge within 5 iterations, ABAQUS automatically increases the size of the increment by 50% (default). ABAQUS controls the automatic incrementation and helps reduce the wastage of the CPU time. In case, the solution does not converge within five (default) iterations, the default value can be increased to a higher value. Increasing the maximum number of cutbacks allowed for an increment can be useful if the problem is highly nonlinear. However, it is important to monitor the analysis intermittently to make sure that the increment size is not so small that there is no significant progress in the analysis for a long period of time. This results in consumption of more CPU resources.

Begin

Begin New Begin new Begin new Begin new Form K step increment attempt iteration tangent

Solve for Cu

Update u

Compute residuals

Yes

Compute new No Convergence No Reduce Dt Converged? DT likely?

Yes No No

End of Yes End of step Output results analysis

Yes

End

Figure 4.3 Flowchart showing solution procedure for analysis without contact (ABAQUS convergence class, Dassault Systemes)

All the default values for tolerances can be changed by the user in the step module under the Other → General solution controls → Manager/Edit and choose the step/s for which the default values need to be changed. The screenshot of the menu instruction is shown in Figure 4.4.

Figure 4.4 Screenshot for changing default solution control

Some of the default parameters and their values are shown in Table 4.1. Table 4.1 Default tolerance values for some parameters Parameters Function Default value 푎 -3 푅푛 Convergence criterion for the ratio of the largest residual to the 5 x 10 corresponding average flux norm for convergence. 푎 -2 퐶푛 Convergence criterion for the ratio of the largest solution correction to 10 the largest corresponding incremental solution value.

퐼0 Number of equilibrium iterations (without severe discontinuities) after 4 which the check is made whether the residuals are increasing in two consecutive iterations.

퐼푅 Number of consecutive equilibrium iterations (without severe 8 discontinuities) at which logarithmic rate of convergence check begins.

퐼퐴 Maximum number of cutbacks allowed for an increment or number of 5 attempts to be made to reduce the time increment.

퐼퐶 Upper limit on the number of consecutive equilibrium iterations 16 (without severe discontinuities), based on prediction of the logarithmic rate of convergence.

4.2 Convergence issues in the model The current model has nonlinear material properties for concrete. Large displacement analysis is used for the whole system in order to capture the response of the system in tension

66 softening. Since both of the above nonlinear features add to the complexity of the problem, there were convergence issues during the analysis. The most common issue was the convergence of the solution when the stress in the concrete reaches the peak tensile stress. The analysis would run up to a point where the material behaved as linearly elastic while being in tension. The increment where tensile stress in the concrete reaches the maximum value, the solution of the analysis failed to converge and the analysis terminated. This was due to the fact that the stress- strain data has a negative slope in the tension softening area. This causes the formation of negative tangent stiffness matrix, which is undesirable in a finite element analysis. The notes, warnings, and errors in the message (.msg), data (.dat), and status (.sta) files created after an analysis can be useful in interpreting the issues in the model. Further details can be obtained from the job diagnostic tool in the visualization module. Even for a successful analysis, the above files can provide more information about the step by step increments, which can be helpful to verify the accuracy of the results and refining the model. Most of the time, warnings in the message file provide a good insight about the issues in the model. Some typical notes, warnings, and errors encountered in the current project with their interpretation are discussed here. Most of them can be termed as classical convergence errors in a nonlinear analysis.

NOTE: THE SOLUTION APPEARS TO BE DIVERGING. CONVERGENCE IS JUDGED UNLIKELY This is a typical convergence error indicating instability in the model. Instability can be due to material properties, contacts, etc. This error can occur due to the onset of cracking in the model. Steps needs to be taken to make the model stable. Use of the automatic stabilization feature in ABAQUS is a good way to overcome this error. However, that technique did not work in the current analysis. The reason can be attributed to the localization of the strain in the current analysis.

ERROR: TOO MANY ATTEMPTS MADE FOR THIS INCREMENT When the number of unsuccessful iterations reaches the maximum number allowed, it results in the above error. The unsuccessful iterations can be due to the instable solution leading to non-

67 convergence. So before increasing the number of unsuccessful iterations for the analysis, message and data files should be studied for any other warnings and notes.

WARNING: THE SYSTEM MATRIX HAS NEGATIVE EIGENVALUES The above warning is typical for a non-converged solution. Negative Eigenvalues refers to some form of loss of stiffness suggesting that the stiffness matrix is assembled about a non-equilibrium state. This can be due to geometrical instability (buckling, compression) and material instability (inappropriate hyperelastic material models, onset of perfect plasticity, tension softening, etc.). If this warning appears in a converged solution, then the solution should be checked for accuracy.

ERROR: TOO MANY INCREMENTS NEEDED TO COMPLETE THE STEP This error may point to convergence issues in the model. This error is generally accompanied by warnings pertaining to negative Eigenvalues, divergence in the solution and/or numerical singularity. If there are no warnings in the message/dat file, then increasing the limit of the number of increments for that step in the analysis may solve the problem.

ERROR: TIME INCREMENT REQUIRED IS LESS THAN MINIMUM SPECIFIED-ANALYSIS ENDS This error may suggest to decrease the minimum time increment in a problem. Warnings in the message file may point to convergence issues, especially in a nonlinear problem. In that case, decreasing the minimum time increment for the step can delay the error in the analysis and result in consumption of more CPU resources.

WARNING: SOLVER PROBLEM. NUMERICAL SIGNULARITY WHEN PROCESSING NODE 1 D.O.F. 3 RATIO=3.141E+15 This warning suggests an unconstrained rigid body motion. The model should be checked for proper constraint. Rigid body motion can also occur when the net section yielding has occurred resulting in large displacements for small increments of load or when buckling has occurred.

4.3 Solution to the convergence issues Many attempts were made to solve the convergence issues in the analysis. Different material properties were used, other solvers available in ABAQUS were tried, boundary conditions were modified, stabilization techniques in ABAQUS were used, etc. None of the above attempts at solving the convergence issues were successful. The automatic stabilization feature is available in ABAQUS to solve convergence issues in an analysis. The automatic stabilization provides a mechanism for stabilizing unstable quasi-static problems through the automatic addition of volume-proportional damping to the model. The applied damping factors can be constant over the duration of a step, or they can vary to account for changes over the course of a step. The latter, adaptive approach is typically preferred (Hibbitt et al. 2011a). The automatic stabilization technique was used for the current analysis, but it failed to address the convergence issue. Finally, the convergence issue was dealt with by introducing the viscosity regularization in the concrete material definition. The concrete damaged plasticity model is equipped to include a viscosity parameter which helps to overcome the convergence problem due to stiffness degradation and softening behavior in a material. Viscosity regularization helps to make the consistent tangent stiffness of the softening material to become positive for sufficiently small time increments (Hibbitt et al. 2011a). It stabilizes the material model by allowing stresses outside the yield surface. ABAQUS uses a generalization of the Duvaut-Lions regularization, 푝푙 according to which the viscoplastic strain rate tensor, 휀휐̇ , which is defined as 1 휀̇푝푙 = (휀푝푙 − 휀푝푙) Eqn. (4.5) 휐 휇 휐 휇 = viscosity parameter representing the relaxation time of the viscoplastic system 휀푝푙 = plastic strain evaluated in the inviscid backbone model A small value of viscosity parameter compared to the typical time increment helps improve the rate of convergence of the model in the softening regime, without the compromising results (Hibbitt et al. 2011a). Values of 0.009 and 0.25 are used for the girder concrete material and the deck concrete material, respectively, for the current analysis. The above values were reached by creating various analysis models with different values of viscosity parameter. A post tensile yield stress vs. cracking strain plot was made for each of the analysis results. The value used in the analysis which gave the closest post tensile yield stress vs. cracking strain curve to the one

69 initially provided was chosen for the final analysis. The post tensile yield stress vs cracking strain plot for different analysis results is shown in Figure 4.5.

Calibration of viscosity regularization parameter 50 - 0.01 800 40 - 0.1 700 40 - 0.05 40 - 0.01 600 40 - 0.02 500 40 - 0.03 40 - 0.02 400 40 - 0.005

300 56 - 0.001 Viscosity parameter = 0.009 40 - 0.015 Tensile stress200 (psi) 40 - 0.017 56 - 0.005 100 56 - 0.009 0 Input curve 0 0.0001 0.0002 0.0003 0.0004 Cracking strain 50 (Dilation angle) – 0.01 (Viscosity parameter) Figure 4.5 Calibration of Viscosity regularization parameter for the girder concrete

Chapter 5. Results and Discussion

The results obtained from the analysis are presented in this section. Stress, strain, and displacements in the slab and the girder are plotted. Various models were created for the different age of each girder at the time of the casting of the slab. The different girder ages considered are 10, 30, 60, and 90 days. The analysis was run for a total time period of 200 for all the models, simulating the behavior of the bridge for the first 200 days from the day of casting the girder in the casting yard. The effects of the different age of the girder at the time of the casting of the deck slab on the system are compared and presented here. It is observed that the age of the girder at the time of the casting of the slab affects the overall system in a significant manner. The behavior of the girder without creep and shrinkage in the slab was also simulated and compared with the system with creep and shrinkage in the slab. The effect of creep and shrinkage on the stress in the prestressing steel is also presented here. The Figure 5.1 shows the stress in the girder after the transfer of the prestressing force to the girder (End of step 1). It can be seen that the bottom of the girder experiences compressive stress (negative stress value in ABAQUS) while the top of the girder is in tension (positive stress value in ABAQUS) after step 1.

Figure 5.1 Stress distribution in the girder after release of prestress (camber exaggerated).

Figure 5.2 Zoomed view of stress distribution near the midspan

Figure 5.3 Stress in the girder and the slab at the end of 200 days for 90 -day old girder

Figure 5.4 throughFigure 5.7 shows the stress distribution at the end section and the midspan section for the girder age of 90 days at the time of the casting of the slab. The stresses are shown for all the major steps in the analysis.

Before release at end After release (End of Step 1) 90 days of creep and shrinkage in the girder (End of Step 2)

Figure 5.4 Stress distribution at the end section up to 90 days of creep and shrinkage in the girder

After addition of slab load (End of Step 3)

After creep and shrinkage in slab (End of Step 3, Total time = 200 days) Figure 5.5 Stress distribution at the end section with slab dead load and creep & shrinkage in slab (Girder age is 90 days at the time of casting of the slab)

Before release After release (End of Step 1) 90 days of creep and shrinkage in the girder (End of Step 2)

Figure 5.6 Stress distribution at the midspan section up to 90 days of creep and shrinkage in the girder

After addition of dead load of slab (End of Step 3)

After creep and shrinkage in slab (End of Step 4, Total time = 200 days) Figure 5.7 Stress distribution at the midspan section with slab dead load and creep & shrinkage in the slab (Girder age is 90 days at the time of the casting of the slab)

The girder cambers up (positive camber) after the first step of analysis because of the compressive stresses at the bottom flange of the girder due to the transfer of the prestressed force from the strands to the concrete. A gradual increase in positive camber is observed in the girder for the first few days after the release of the prestressing force. The positive camber in the girder begins to slow down eventually due to the decrease in the creep effect. After the slab is cast, the girder cambers down due to the dead weight of the slab. Once the slab is set, it begins to influence the girder. Then a gradual cambering down of the girder is observed due to the shrinkage of the deck slab concrete. This happens due to differential shrinkage between the girder and the deck slab which is prominent for older girders such as the 60-day and 90-day girder. Shrinkage in the slab is restrained by the reinforcement in the slab and the interaction between the slab bottom and the girder top flange. Due to the restraint shrinkage, tensile stresses start to develop at the bottom of the slab and compressive stress is experienced by the top flange of the girder. The value of the tensile stress at the bottom of the slab depends upon the differential shrinkage between the girder and the slab. When the value of tensile stress in the slab reaches the tensile elastic limit, the slab cracks. The cracking in the slab is simulated by the tension softening behavior of the concrete. As the portion of the slab cracks, the stresses in those areas are released. The cambering down of the girder ceases after the cracking in the slab and the girder starts to camber up again as seen in Figure 5.8. This behavior is observed in the girder, which are 60 days and 90 days old at the time of casting of the slab. This is because of the significant differential shrinkage between the slab and the girder. The cracking in the slab is observed at approximately 60 days and 42 days after the casting of the slab in 60-day and 90-day old girder, respectively. From the above observation, we can say that a slab cast on a relatively old girder tends to experience cracking sooner as compared to slabs cast on relatively young girders. This can be based on the fact that differential shrinkage between the girder and the deck will be more for relatively old girders and hence causing higher tensile stresses in the deck slab. The above fact is also supported by the observations made from the model with girder age of 10 days and 30 days. From Figure 5.9, we can see that the stresses in the deck slab for 10 days and 30 days old girder are well within the tensile stress limit of 300 psi. This can be attributed to smaller differential shrinkage between the slab and the girder. The smaller differential shrinkage also results in the girder and the slab shrinking together causing minimal displacement to the top of the flange of the girder at the midspan. It is observed that the

77 differential shrinkage causes downward camber at the girder midspan. The magnitude of the deformation of the top flange of the girder at the midspan was found to be relatively higher for older girders few days after the activation of the slab, supporting the fact that higher differential shrinkage causes more deformation. The total stress in the top flange of the girder at the midspan is shown in Figure 5.10. The top flange of the girders at the midspan show compressive stresses after the application of dead load of slab in the model. An increase in compressive stress is observed in all the girders due to the shrinkage the deck. Higher compressive stress is experienced by the older girder after the activation of the deck slab in the model. However, these compressive stresses are eventually relieved in older girders because of the cracking in the deck slab, as seen for the girder age of 60 and 90 days. The compressive stress in the top flange of the girder with an age of 10 and 30 days begins to level out after a few days of creep and shrinkage in the girder and the slab together. It can be seen that the final stress state in all the girders is compressive at the end of 200 days of simulation which is desirable as far as the girder concrete is concerned.

Total displacement at girder top flange at midspan 1.8

1.6

1.4

1.2 90 days 1 60 days 0.8 30 days

0.6 10 days Displacement Displacement (in.)

0.4

0.2

0 0 50 100 150 200 Time (days) Figure 5.8 Displacement at girder top flange at midspan due to creep and shrinkage

Total stress at bottom of slab at midspan 300

250

200 90 days 60 days 150 30 days

Stress Stress (psi) 10 days 100

0 0 50 100 150 200 Time (days) Figure 5.9 Stress at bottom of slab at midspan for different age of girders

Total stress at top flange of girder at midspan 800

600

400 90 days

200 60 days

30 days 0 0 50 100 150 200 10 days -200

Stress Stress (psi) -400

-600

-800

-1000

-1200 Time (days)

Figure 5.10 Stress at top flange of girder for different age of girders

A condition without the effect of creep and shrinkage in the deck slab was also simulated for all ages of the girders and the results are compared to the model with creep and shrinkage effect in the deck slab. The dead load of the slab was added in both conditions. Figure 5.11 through Figure 5.14 shows the comparison of stresses in the top flange of the girder at midspan for both the above conditions while Figure 5.15 throughFigure 5.18 depicts the total displacement in the top flange of the girder at midspan. For all ages of girder, it was observed that the stress in the top flange of the girder at the midspan does not vary much after the instantaneous drop in stress due to the slab dead load for the condition without creep and shrinkage effect in the slab. Since there are no long-term effects occurring in the deck slab, it does not influence the behavior of the girder. The displacement plots for the top flange of the girder at midspan without the creep and shrinkage effects in the deck slab shows an increase in positive camber for all the girders after the instantaneous decrease in displacement due to slab dead load. This increase in displacement can be attributed to the domination of creep due to prestress in the girder concrete.

Total stress comparison for 10-day old girder at top flange at midspan 800

600

Creep and 400 shrinkage in slab

No creep 200 and shrinkage in Stress Stress (psi) slab 0 0 50 100 150 200

-200

-400 Time (days)

Figure 5.11 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for 10-day old girder at the time of casting of the slab)

Total stress comparison for 30-day old girder at top flange at midspan 800

600 Creep and 400 shrinkage in slab 200 No creep 0 and 0 50 100 150 200 shrinkage in

-200 slab Stress Stress (psi) -400

-600

-800

-1000 Time (days)

Figure 5.12 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for a 30-day old girder at the time of casting of the slab)

Total stress comparison for 60-day old girder at top flange at midspan 800

600 Creep and shrinkage in 400 slab

200

0 No creep and 0 50 100 150 200 shrinkage in -200 slab

-400 Stress Stress (psi)

-600

-800

-1000

-1200 Time (days) Figure 5.13 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for a 60-day old girder at the time of casting of the slab)

Total stress comparison for 90-day old girder at top flange at midspan 800

600 Creep and shrinkage in 400 slab

200

No creep and 0 shrinkage in 0 50 100 150 200 -200 slab

-400 Stress Stress (psi)

-600

-800

-1000

-1200 Time (days) Figure 5.14 Stress in the top flange of girder due to variation in creep and shrinkage in the slab (for a 90-day old girder at the time of casting of the slab)

Total displacement comparison for 10-day old girder at top flange at midspan 1.6 Creep and 1.4 shrinkage in slab

1.2

1 No creep and shrinkage 0.8 in slab

0.6 Displacement (in.) 0.4

0.2

0 0 50 100 150 200 Time (days) Figure 5.15 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 10-day old girder at the time of casting of the slab)

Total displacement comparison for 30-day old girder at top flange at midspan 1.6 Creep and 1.4 shrinkage in slab

1.2

No creep 1 and shrinkage in 0.8 slab

0.6 Displacement (in.) 0.4

0.2

0 0 50 100 150 200 Time (days) Figure 5.16 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 30-day old girder at the time of casting of the slab)

Total displacement comparison for 60-day old girder at top flange at midspan 1.8 Creep and 1.6 shrinkage in slab

1.4

1.2 No creep and 1 shrinkage in slab 0.8

0.6 Displacement (in.) 0.4

0.2

0 0 50 100 150 200 Time (days) Figure 5.17 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 60-day old girder at the time of casting of the slab)

Total displacement comparison for 90-day old girder at top flange at midspan 1.8 Creep 1.6 and shrinkage

1.4 in slab

1.2 No creep and 1 shrinkage in slab 0.8

0.6 Displacement (in.)

0.4

0.2

0 0 50 100 150 200 Time (days) Figure 5.18 Total displacement in the top flange of girder at mid-span due to variation in creep and shrinkage in the slab (for a 90-day old girder at the time of casting of the slab)

The ideal condition for a concrete bridge is no cracking in the deck slab throughout its service life. Cracking is not only unsightly, but cracks allow water and other chemicals to seep into the slab leading to undesirable effects such as corrosion, freeze thaw damage, leaching or alkali silica reaction. The uncracked slab will also mitigate the formation of positive moment at the diaphragm in a continuous system. A single span bridge model was created where cracking in the slab was restricted by increasing the tensile stress limit of the slab concrete to a higher value in order to see the effects of a large differential shrinkage that was not mitigated by slab cracking. This simulation was conducted only for girder age of 90 days. The comparison of the results from the above model with the one where the slab does not crack is presented in Figure 5.19 throughFigure 5.21. It can be seen that the tensile stress keeps increasing at the bottom of the slab while the top of the girder experiences continuous compression. Continuous decrease in positive camber is observed after the application of slab dead load in Figure 5.21. In a continuous system, this would be advantageous because it would cause the end of the girder to rotate into the diaphragm and increase compressive stresses at the bottom of the

84 diaphragm, prevent formation of positive moment at the bottom of the support and reduce or eliminate tensile cracking at the girder-diaphragm interface. However, in a real system the slab cracks and the beneficial effect of the differential shrinkage in the continuous system is lost. This shows a major shortcoming of analysis methods for continuous bridges which do not account for slab cracking.

Total stress comparison at bottom of slab at midspan for 90-day old girder 400 Cracking 350 in slab

Uncracked 300 slab

250

200

150 Stress Stress (psi)

100

0 80 100 120 140 160 180 200 Time (days) Figure 5.19 Comparison of stresses for cracked and uncracked slab condition at bottom of slab at midspan (for a 90-day old girder at the time of casting of the slab)

Total stress comparison at top of girder at midspan for 90-day old girder 700 Cracking 400 in slab

100 Uncracked slab 0 50 100 150 200 -200

-500 Stress Stress (psi)

-800

-1100

-1400 Time (days)

Figure 5.20 Comparison of stresses for cracked and uncracked slab condition at top of girder at midspan (for a 90-day old girder at the time of casting of the slab)

Total displacement comparison at top of girder at midspan for 90-day old girder 1.8 Cracking 1.6 in slab

1.4 Uncracked

slab 1.2

0.8

0.6 Displacement (in.)

0.4

0.2

0 0 50 100 150 200 Time (days) Figure 5.21 Comparison of displacement for cracked and uncracked slab condition at top of girder at midspan (for a 90-day old girder at the time of casting of the slab)

5.1 Effect of girder age on prestressing force in the strands As soon as the prestressing force is applied to the girder, the strands begin to lose stress. There is an immediate loss due to elastic shortening of the girder and long term losses due to creep and shrinkage of the concrete and relaxation of the strand. Stress gains are also possible due to superimposed dead loads and the effects of differential creep and shrinkage. The gains are controversial. One method for assessing loss of prestressing force in the AASHTO LRFD Bridge Design Specifications is the “refined” method. One aspect of this method is the calculation of a gain in prestressing force due to differential shrinkage of the deck. However, not all states accept this. Some states do not allow inclusion of this gain while others limit it. The stresses in the prestressing strands are shown in Figure 5.22 for different ages of the girders at the time the slab is added. It can be seen that the prestress loss is higher in the first few days for all girders. A gain in prestress is observed due to the dead load of the slab for each of the girder. In addition to the above gain, a minor gain in prestress can be seen for a girder aged 60 or 90 days when the slab is added in the analysis. This gain is due to differential shrkinage of the slab and was eventually lost because of the cracking in the deck slab. The prestress gain is not significant when compared to the prestress losses in the model. The results from the current analysis also show that the prestressing force in the strands is not significantly affected by the creep and shrinkage of the deck slab. The prestressing force in the strands for the model with creep and shrinkage in the slab and the model without creep and shrinkage in the slab are found to be varying by less than 1% for all ages of girder after 200 days of simulation. This can be seen in Figure 5.23 throughFigure 5.26. Cracking of the slab also does not have any significant effect on the stress in the steel. This was observed for all girder ages of 10, 30, 60, and 90 days.

Stress in strands at the midspan for different ages of the girder

210000 10 days old girder 200000 30 days old girder 60 days old 190000

girder 90 days old girder

180000 Stress Stress (psi) 170000

160000

150000 0 50 100 150 200 Time (days) Figure 5.22 Stress in the prestressing strands for different ages of the girder

Stress in strands at the midspan for 10-day old girder 210000

200000

Creep and 190000 shrinkage in slab

180000 No creep and shrinkage in

Stress Stress (psi) slab 170000

160000

150000 0 50 100 150 200 Time (days) Figure 5.23 Comparison of stress in prestressing strands for a 10-day old girder

Stress in strands at the midpsan for 30-day old girder 210000

200000 Creep and 190000 shrinkage in slab

180000 No creep and shrinkage in

Stress Stress (psi) slab 170000

160000

150000 0 50 100 150 200 Time (days) Figure 5.24 Comparison of stress in prestressing strands for a 30-day old girder

Stress in strands at the midspan for 60-day old girder 210000

200000

Creep and 190000 shrinkage in slab

180000 No creep and shrinkage in

Stress Stress (psi) slab 170000

160000

150000 0 50 100 150 200 Time (days) Figure 5.25 Comparison of stress in prestressing strands for a 60-day old girder

Stress in strands at the midspan for 90-day old girder 210000

200000

Creep and 190000 shrinkage in slab

180000 No creep and shrinkage in

Stress Stress (psi) slab 170000

160000

150000 0 50 100 150 200 Time (days) Figure 5.26 Comparison of stress in prestressing strands for a 90-day old girder

A hand calculation was performed to calculate the prestress losses in the system by the AASHTO refined loss method. The transformed section properties were used to for the calculation. The solution obtained from the hand calculation was compared to the result of the simulation performed using the AASHTO creep and shrinkage model (American Association of State Highway and Transportation Officials., 2012). The above analysis was only done for a girder age of 90 days at the time the slab was cast and the total simulation time was kept at 700 days. It was observed that the AASHTO refined loss method gives a conservative value of the prestress losses in the system. The difference between the stress values in the strands after 700 days from the above methods was 5.43 ksi (3.32%) as shown in Figure 5.27. This can attributed to the fact that the AASHTO refined loss method is a single step procedure in which the continuous variation of stress in the system with time is not taken into account.

Stress in the prestressing strands 210000 Abaqus analysis 200000 Hand

190000 calculation

180000 Stress Stress (psi) 170000

160000

150000 0 100 200 300 400 500 600 700 Time (days)

Figure 5.27 Comparison of stress by AASHTO refined loss method and ABAQUS analysis

Chapter 6. Conclusions and Future Research

The key to creating a successful finite element model is to start with a simple model and introduce complications such as loadings, boundary conditions, and nonlinearities one by one and study the output after every new step is introduced in the model. This simplifies the process of the identification of error and helps to efficiently debug the model. Too many steps at once make the model and analysis complex and sometimes makes it difficult to find out the root cause of the error. Studying the results after the introduction of every new parameter helps in understanding the working of the model in a better way. The current finite element model created in ABAQUS Standard 6.11-2 successfully simulated the behavior of single span precast prestressed concrete I-girder bridge subjected to long-term loading, such as creep and shrinkage. The results indicate that ABAQUS Standard 6.11-2 does a good job in simulating the behavior of the prestressed concrete, both in compression and tension. The additional feature of the linking of the subroutine with the CAE model can be used to write a code for various creep and shrinkage models, such as the ACI 209 model (ACI Committee 209-Creep and Shrinkage. 2008), the AASHTO LRFD model (American Association of State Highway and Transportation Officials., 2012), and the CEB-FIP model (Comité euro-international du béton., 1993). From the analysis, it can be concluded that the cracking of the slab depends on the amount of differential shrinkage between the girder and the deck, which depends on the age of the girder at the time of the casting of the slab. For the parameters chosen in this simulation, a girder older than 60 days causes the slab to crack within the first two months after casting of the slab, whereas, girders age 10 days or 30 days at the time the slab is case do not seem to experience cracking of the slab even after 6 months. The current project shows that the age of the girder plays an important role in the long- term behavior of bridges constructed using precast prestressed concrete girders. The older girder will result in more prestress gain, but eventually the slab will crack due to higher differential shrinkage. This causes release in stresses and hence the gain in prestress is lost. The cracking in the slab may also result in the development of the positive moment at the bottom of the support in a continuous girder. On the other hand, if the slab is cast when the girder is relatively young, cracking in the slab is not observed. The girder does not camber up significantly which might be

92 desirable to avoid the development of the positive moment at the bottom of the support in a continuous system. The absence of cracking in the deck slab minimizes the seepage of water in the slab and helps in reducing the corrosion of reinforcement. Hence, a balance has to be found with regard to the age of the girder at the time of the pouring of the deck slab. An appropriate estimation of the desired age of the girder at the time of the casting of the slab from the finite element model can increase the serviceability and durability of the bridge. From the current results, that age may be in between 30 and 60 days. However, it must be noted that the analysis is heavily dependent on the creep coefficients of the girder and slab, the shrinkage potential of the girder and slab and the tensile strength of the slab. Thus, re-running the analysis with different parameters would like result in very different girder ages when the slab could be cast without cracking. The ages cited in this work are to simply illustrate results from a single case and should not be extrapolated to other cases. It can be concluded that the AASHTO refined loss calculation gives a conservative estimate of the prestress losses in the structure. It can also be concluded that the gain in prestressing force calculated by that method is small and may not occur if the deck cracks. Thus, the need for including this calculation is questionable.

6.1 Future research The current thesis includes a single span precast prestressed concrete girder bridge. There is a scope for modeling and analyzing multiple spans or a continuous system. Different parameters like deck thickness, girder types, temperature loading, and moving loads need to be further investigated to assess their effects in the reported results. A complete cross-section of a single span bridge consisting of multiple girders can be analyzed. The continuous girder system with the slab and the diaphragm can be modeled to come up with an analysis procedure to find out the positive moment at the connection in a ‘continuous for live load’ bridge. The continuity of the system for live load after the cracking at the girder-diaphragm interface needs to be studied. The current research work can be used as a basis for such analyses. The effect of the temperature field on the system needs to be studied since it plays a significant role on the behavior of the overall structure. The different creep and shrinkage models can be included and their effects on the prestress losses/gain can be studied. The current study is based on finite element results. In the future, the results from the finite element model can be compared to the

93 bridge monitoring data for a multiple span bridge. A complete bond between concrete and reinforcement was assumed in the current research. More realistic bonding characteristics between the concrete and the strands might be of interest in the future.

References or Bibliography

1. ACI Committee 209-Creep and Shrinkage. (2008). "Guide for modeling and calculating shrinkage and creep in hardened concrete." Rep. No. 209.2R, American Concrete Institution, Farmington Hills, MI. 2. American Association of State Highway and Transportation Officials.,. (2012). "AASHTO LRFD bridge design specifications." American Association of State Highway and Transportation Officials, Washington, DC. 3. American Association of State Highway and Transportation Officials.,. (2002). "AASHTO LRFD bridge design specifications." American Association of State Highway and Transportation Officials, Washington, DC. 4. ANSYS® Academic Research, Release 10.0. "ANSYS® Academic Research, Release 10.0." . 5. Comité euro-international du béton.,. (1993). CEB-FIP model code 1990 : design code. T. Telford, London. 6. Curtis, R. H. (2007). "NYSDOT Bridge Deck Task Force Evaluation of Bridge Deck Cracking on NYSDOT Bridges." . 7. Debbarma, S. R., and Saha, S. (2011). "Behavior of pre-stressed concrete bridge girders due to time dependent and temperature effects." First Middle East Conference on Smart Monitoring, Assessment and Rehabilitation of Civil Strucutres, . 8. Dinges, T. (2009). "The history of prestressed concrete: 1888 to 1963". Master of Science. Kansas State University, Manhattan, Kansas. 9. Freyermuth, C. L. (1969). "Design of continuous highway bridges with precast, prestressed concrete girders." Journal of the Prestressed Concrete Institute, also Printed as PCA Engineering Bulletin EB014.01E, 14(2), 14-29. 10. Hibbitt, H. D., Karlson, B. I., and Sorenson, E. P. (2011a). ABAQUS 6.11 Analysis User's Manual. Hibbitt, Karlson & Sorenson, Providence, RI. 11. Hibbitt, H. D., Karlson, B. I., and Sorenson, E. P. (2011b). "ABAQUS Version 6.11, Finite Element Program." Hibbitt, Karlson & Sorenson, Providence, RI, 6.11.

12. Kaar, P. H., Kriz, L. B., and Hognestad, E. (1960). "Precast-prestressed concrete bridges: 1. Pilot tests of continuous girders." Journal of the PCA Research and Development Laboratories, 2(2), 21-37. 13. Kmiecik, P., and Kamiński, M. (2011). "Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration." Archives of Civil and Mechanical Engineering, 11(3), 623-636. 14. Mattock, A. H. (1961). "Precast-prestressed concrete bridges 5: Creep and shrinkage studies." Journal of the PCA Research and Development Laboratories, 3(2), 32-66. 15. Miller, R. A., Castrodale, R., Mirmiran, A., and Hastak, M. (2004). "NCHRP Report 519 Connection of Simple-Span Precast Concrete Girders for Continuity." Rep. No. 519, Transportation Research Board, Washington, DC. 16. National Bridge Inventory Study Foundation,. (2003). "NBI Report 2003." . 17. Oesterle, R., Glikin, J., and Larson, S. (1989). "NCHRP Report 322: Design of Precast Prestressed Bridge Girders Made Continuous." Transportation Research Board, Washington, DC. 18. Okeil, A. M., Hossain, T., and Cai, C. (2013). "Field monitoring of positive moment continuity detail in a skewed prestressed concrete bulb-tee girder bridge." PCI J., . 19. Stephen, E. (2006). "Simulation of the long-term behavior of precast/prestressed concrete bridges". Master of Science. University of Cincinnati, Cincinnati, Ohio. 20. Tadros, Maher K., National Cooperative Highway Research Program., American Association of State Highway and Transportation Officials., United States., Federal Highway Administration., National Research Council (U.S.).,Transportation Research Board.,. (2003). Prestress losses in pretensioned high-strength concrete bridge girders. Transportation Research Board, Washington, DC.

Appendix A. Validation model for creep and shrinkage subroutine

The validation for the creep and shrinkage subroutine in ABAQUS is provided in this appendix. A solved example from book titled, Reinforced concrete structures, by Omar Chaallal and Mohamed Lachemi, chapter-1, Example 1.4, is taken for reference.

A-1 Problem Statement Consider a 3-m-high reinforced concrete column with a cross-section of 400 mm × 400 mm. It is reinforced with 4 No. 30M steel bars. The column is subjected to an axial compression load of 1600 kN after one week of moist curing. a) Calculate the instantaneous compressive and tensile stresses in concrete and steel and the corresponding instantaneous strain. b) What is the shortening of the column after 180 days of loading? Use: fc′ (at 7 days) = 20 MPa; Type GU cement (300 kg/m3); relative humidity = 60%; air content = 5%; slump of fresh concrete = 125 mm; sand = 670 kg/m3; coarse aggregate = 1000 kg/m3. A-1.1 Solution Calculated instantaneous stress in concrete = 8.65MPa Calculated instantaneous strain in concrete = 430 x 10-6 mm/mm Strain due to creep = 533 x 10-6 mm/mm Strain due to shrinkage = 348 x 10-6 mm/mm

A-2 ABAQUS model and results

Figure A.1 Creep and shrinkage strain in ABAQUS model

Calculated instantaneous stress in concrete = 9.91219 MPa Calculated instantaneous strain in concrete = 473.481 x 10-6 mm/mm Strain due to creep = 567.005 x 10-6 mm/mm Strain due to shrinkage = 348.224 x 10-6 mm/mm

Table A.1 Comparison of the ABAQUS output with the calculated result ABAQUS value Calculated value Percent Error Creep strain 567.005 x 10-6 mm/mm 533 x 10-6 mm/mm 6.98 Shrinkage strain 348.224 x 10-6 mm/mm 348 x 10-6 mm/mm 0.06

Appendix B. Volume to surface ratio for the girder and the deck

B-1 Volume-to-surface ratio of the girder 16.0 0 . 7 5 . 4 4.5 0 . 9 0 .

7.0 1 5 4

7.5 5 . 7 0 . 7

22.0 Figure B.1 Cross-section of the AASHTO Type III, I girder

Cross-sectional area of the girder = 560 sq. inch Length of the girder = 600 inches Volume of the girder = Area x Length = 560 x 600 = 336,000 in.3 Area of the top face of the girder = 16 x 600 = 9,600 in.2 Area of the bottom face of the girder = 22 x 600 = 13,200 in.2 Area of the side faces of the girder = 50 x 600 = 30,000 in.2 Area of the end faces of the girder = 2 x 560 = 1,120 in.2 Total surface area of the girder = 53,920 in.2 Volume to surface ratio of the girder = Total volume/Total surface area = 336000/53920 = 6.2314 in.

100

96.0 0 . 8

Figure B.2 Cross-section of the deck slab

Cross-sectional area of the deck = 8 x 96 = 768 in.2 Length of the deck = 600 inches Volume of the deck = Area x Length = 768 x 600 = 460,800 in.3 Area of the top face of the deck = 96 x 600 = 57,600 in.2 Area of the bottom face of the deck = 96 x 600 = 57,600 in.2 Area of the side face of the deck = 8 x 600 = 4,800 in.2 Area of the end faces of the deck = 8 x 96 = 786 in.2 Total surface area of the deck = 120,786 in.2 Volume to surface ratio of the deck = Total volume/Total surface area = 460800/120786 = 3.8150 in. 96.0 0 . 8

Figure B.3 Cross-section of the bridge

101

Appendix C. FORTRAN subroutine

SUBROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELENT, 1 TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV,NOEL,NPT,LAYER, 2 KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMATYP,MATLAYO, 3 LACCFLA) C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 CMNAME,ORNAME CHARACTER*3 FLGRAY(15) DIMENSION FIELD(NFIELD), STATEV(NSTATV), DIRECT(3,3), T(3,3), TIME(2) DIMENSION ARRAY(700), JARRAY(700), JMAC(*), JMATYP(*), COORD(*) C CALL GETVRM('THE',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,MATLAYO, LACCFLA) THE = ARRAY(3) STATEV(1)=THE C C Reading instantaneous elastic strain in direction 33(axial) CALL GETVRM('EE',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,MATLAYO, LACCFLA) EE = ARRAY(3) STATEV(3)=EE C RETURN END C SUBROUTINE UEXPAN(EXPAN,DEXPANDT,TEMP,TIME,DTIME,PREDEF,DPRED, STATEV,CMNAME,NSTATV,NOEL) C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 CMNAME C DIMENSION EXPAN(*),DEXPANDT(*),TEMP(2),TIME(2),PREDEF(*), DPRED(*),STATEV(NSTATV),ARRAY(15) THE=STATEV(1) EE=STATEV(3) C C Calculation of creep and shrinkage strain in girder concrete C IF ((CMNAME.EQ.'CONCRETE_GIRDER').AND.(TIME(2).GT.1)) THEN T=TIME(2)-1 TL=T-DTIME IF(TL.LT.0) THEN

102

TL=0 END IF CREEPSTRAIN=1.800*EE*(T**0.6/(10+T**0.6)-TL**0.6/(10+TL**0.6)) SHRINKAGESTRAIN=-T/(35+T)*0.00061+TL/(35+TL)*0.00061 EXPAN(3)=CREEPSTRAIN+SHRINKAGESTRAIN END IF C C Calculation of creep and shrinkage strain in slab concrete C IF((CMNAME.EQ.'CONCRETE_SLAB_AND_DIAPHRAGM').AND.(TIME(2).GT.63)) THEN C C 63 is used for 60 day old girder. Will change as per the age of C the girder C T=TIME(2)-63 TL=T-DTIME IF(TL.LT.0) THEN TL=0 END IF CREEPSTRAIN1=1.35*EE*(T**0.6/(10+T**0.6)-TL**0.6/(10+TL**0.6)) SHRINKAGESTRAIN1=-T/(35+T)*0.000608+TL/(35+TL)*0.000608 EXPAN(3)=CREEPSTRAIN1+SHRINKAGESTRAIN1 END IF C C Writing output for a single girder element in data file for C verification C IF ((NOEL.EQ.75750)) THEN WRITE (6,'(I5, F16.11,F16.11,F16.11,F16.11,F16.11,F16.11, F16.11)') NOEL,T,DTIME,TL,EE,CREEPSTRAIN,SHRINKAGESTRAIN, EXPAN(3) END IF C C Writing output for a single deck element in data file for C verification C C IF ((NOEL.EQ.9164)) THEN WRITE (6,'(I5, F16.11,F16.11,F16.11,F16.11,F16.11,F16.11, F16.11)') NOEL,T,DTIME,TL,EE,CREEPSTRAIN1,SHRINKAGESTRAIN1, EXPAN(3) END IF C RETURN END

103

Appendix D. Script file for submission of job to the supercomputer

D-1 Script file #PBS -l walltime=6:00:00 #PBS -l nodes=2:ppn=12 #PBS -N my_ABAQUS_job #PBS -j oe #PBS -l software=ABAQUS+19 # # The following lines set up the ABAQUS environment # module load ABAQUS/6.11-2 # # Move to the directory where the job was submitted # cd $PBS_O_WORKDIR cp *.inp $TMPDIR/ cp *.f $TMPDIR/ cd $TMPDIR # # Run ABAQUS, note that in this case we have provided the names of the input files explicitly # ABAQUS job=M0312Twospan_30days.inp user=M0312Twospan_30days_fort.f cpus=24 interactive # # Now, move data back once the simulation has completed # mv * $PBS_O_WORKDIR

After submitting the script file, a job ID is created for each submission. The job ID can be helpful in checking the status of the job, deleting the job, altering the walltime of the job, etc. The #PBS statement in the above script is a special comment statement used to specify job parameters to PBS. The PBS commands for the above functions are provided in section D-2.

104

D-2 Explanation of the script file Role of each line in the script file Command name Description #PBS -l walltime=HH:MM:SS This directive specifies the maximum walltime (real time, not CPU time) that a job should take. If this limit is exceeded, PBS will stop the job. Keeping this limit close to the actual expected time of a job can allow a job to start more quickly than if the maximum walltime is always requested. #PBS -l nodes=N:ppn=M This specifies the number of nodes (nodes=N) and the number of processors per node (ppn=M) that the job should use. PBS treats a processor core as a processor, so a system with eight cores per compute node can have ppn=8 as its maximum ppn request. Note that unless a job has some inherent parallelism of its own through something like MPI or OpenMP, requesting more than a single processor on a single node is usually wasteful and can impact the job start time. #PBS -N my_ABAQUS_job #PBS -j oe Normally when a command runs it prints its output to the screen. This output is often normal output and error output. This directive tells PBS to put both normal output and error output into the same output file. #PBS -l software=ABAQUS+19 This specifies the number of tokens to be requested for running the job. module load ABAQUS/6.11-2 This command loads the ABAQUS 6.11-2 module. cd $PBS_O_WORKDIR This changes the directory we submitted the script from. cp *.inp $TMPDIR/ This copies the input file to the TMPDIR (temporary directory) location. cp *.f $TMPDIR/ This copies the Fortran file to the TMPDIR (temporary directory) location. cd $TMPDIR This changes the directory. ABAQUS These commands specify the name of the input file for the analysis job= and the Fortran subroutine file to be linked with the analysis. The user= number of CPUs to be used can be found by multiplying the number of nodes requested by the number of processors available cpus= interactive per node. mv * $PBS_O_WORKDIR This command directs the computer to move the output files in the working directory. The directory can be changed if needed.

105

PBS command for submitting a job, checking job status, and deleting a job Command name Description qsub This command submits the PBS script to the PBS to make the script eligible to run. qstat This command shows the status of all the PBS jobs. The time displayed is the CPU time used by the job qstat-u This command shows the status of all the PBS jobs submitted by the user . The time displayed is the walltime time used by the job qstat-f This command shows detailed information about the Job ID. qdel This command deletes the job identified by the Job ID. qdel $(qselect-u ) This command deletes all the jobs belonging to the provided User ID. PBS job states State Meaning Q It indicates that the job is queued and is waiting to start. R It indicates that the job is currently running. E It indicates that the job is currently ending. H It indicates that the job has a user or system hold on it and will not be eligible to run until the hold is removed.

106

Appendix E. MS Excel Macro to create plots

E-1 VBA Code The following Macro is customized for the data and plots used in the current project.

Sub Plots() ' Creatinging plots Macro ' Selects the cells for plotting ' Keyboard Shortcut: Ctrl+Shift+R 'ActiveCell.Offset(0, 2).Range("A1").Select m = 1 l = 40 Do While Cells(4, m).Value <> "" Application.ScreenUpdating = False 'Dim 10day As Worksheet 'Set 10day = Wb.Sheets("10 day_plots") Set objChart = ThisWorkbook.Sheets("90 day_plots").ChartObjects.Add(25, l, 500, 300).Chart objChart.ChartType = xlXYScatterLines 'Set objSeries = objChart.SeriesCollection.NewSeries q = m Do While Cells(4, m).Value <> "" ActiveSheet.Cells(4, m).Select x = 0 y = 0 Do While ActiveCell.Value <> "" x = x + 1 y = y + 1 ActiveCell.Offset(1, 0).Range("A1").Select Loop ActiveCell.Offset(-x, 0).Range("A1").Select a = ActiveCell.Row b = ActiveCell.Column c = a + x - 1 d = b + 1 Set objSeries = objChart.SeriesCollection.NewSeries

107

objSeries.Name = Cells(2, m).Value objSeries.XValues = Range(Cells(a, b), Cells(C, b)) objSeries.Values = Range(Cells(a, d), Cells(C, d)) objSeries.MarkerStyle = 0 m = m + 2 Loop objChart.SetElement (msoElementChartTitleCenteredOverlay) objChart.ChartTitle.Caption = Cells(1, q).Value With objChart.ChartTitle.Format.TextFrame2.TextRange.Font .NameComplexScript = "Arial" .NameFarEast = "Arial" .Name = "Arial" End With objChart.ChartTitle.Format.TextFrame2.TextRange.Font.Size = 12 objChart.Axes(xlCategory).HasMajorGridlines = True objChart.SetElement (msoElementPrimaryValueAxisTitleRotated) objChart.SetElement (msoElementPrimaryCategoryAxisTitleAdjacentToAxis) objChart.Axes(xlValue).AxisTitle.Caption = Cells(3, q + 1).Value With objChart.Axes(xlValue).AxisTitle.Format.TextFrame2.TextRange.Font .NameComplexScript = "Arial" .NameFarEast = "Arial" .Name = "Arial" End With objChart.Axes(xlValue).AxisTitle.Format.TextFrame2.TextRange.Font.Size = 10 objChart.Axes(xlCategory).AxisTitle.Caption = Cells(3, 1).Value With objChart.Axes(xlCategory).AxisTitle.Format.TextFrame2.TextRange.Font .NameComplexScript = "Arial" .NameFarEast = "Arial" .Name = "Arial" End With objChart.Axes(xlCategory).AxisTitle.Format.TextFrame2.TextRange.Font.Size = 10 objChart.Axes(xlCategory).MaximumScale = 200 objChart.PlotArea.Height = 250 objChart.PlotArea.Width = 400 objChart.PlotArea.Top = 30 objChart.PlotArea.Left = 20 m = m + 1

108

l = l + 400 Loop End Sub

109