Finite Element Modeling of the Load Transfer Mechanism in Adjacent Prestressed. Concrete Box-Beams. a thesis presented to.

Transcription

1 Finite Element Modeling of the Load Transfer Mechanism in Adjacent Prestressed Concrete Box-Beams a thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements of the degree Master of Science Oliver Giraldo-Londoño May Oliver Giraldo-Londoño. All Rights Reserved.

2 2 This thesis titled Finite Element Modeling of the Load Transfer Mechanism in Adjacent Prestressed Concrete Box-Beams by OLIVER GIRALDO-LONDOÑO has been approved for the Department of Civil Engineering and the Russ College of Engineering and Technology by Eric P. Steinberg Professor of Civil Engineering Dennis Irwin Dean, Russ College of Engineering and Technology

3 3 ABSTRACT GIRALDO-LONDOÑO, OLIVER., M.S., May 2014, Civil Engineering Finite Element Modeling of the Load Transfer Mechanism in Adjacent Prestressed Concrete Box-Beams Director of Thesis: Eric P. Steinberg Adjacent box beam bridges have been widely used in the United States for decades due to the ease and speed of construction. Despite the current design requirements, this type of bridge often experiences longitudinal cracking along joints. This is often the result of a deficient load transfer mechanism between beams. In addition, reflective cracking can occur in asphalt overlays or a cast-in-place concrete deck. This can cause chemical agents to leak between the beams, causing corrosion in the reinforcement, which affects the lifetime of the structure. A parametric study on the load transfer mechanism between adjacent prestressed concrete box-beams was developed using ABAQUS. Several finite element models consisting of two adjacent beams connected through a shear key and transverse ties were created for the analysis. The effects of filled and non-filled transverse post-tensioning ducts were included in the models. This was to show the contribution of the dowel action in the load transfer mechanism of adjacent prestressed concrete box-beams. In addition, effects of temperature gradients were considered with the aim of simulating a behavior resembling the reality of bridges in the field. Additional parameters, including grout compressive strength-to-concrete compressive strength ratio, amount of transverse posttensioning (TPT), span-to-depth ratio (L/D), and number of internal diaphragms (N), were included in the models.

4 4 This study was divided into two main parts. The first part consisted of analyzing the behavior of a pair of beams with fixed length and number of internal diaphragms, while increasing the amount of TPT force. The second part consisted of analyzing the behavior of a pair of beams with zero transverse post-tensioning, while varying beam span length and number of internal diaphragms. The significance of the results from all finite element models was statistically analyzed using Analysis of Variance (ANOVA). Results indicated that differential deflections between adjacent beams and principal stresses in the shear key were significantly reduced due to dowel action. Additional results showed that principal stresses in the shear key were significantly affected by temperature gradients, span-to-depth ratio, and grout compressive strength-to-concrete compressive strength ratio. However, principal stresses in the shear key were not strongly affected by amount of TPT and number of internal diaphragms.

5 5 ACKNOWLEDGEMENTS I would like to thank Dr. Eric Steinberg for all his valuable contribution throughout the development of this thesis and his financial support. All his recommendations and corrections were the key to make big improvements during this research process. I also would like to thank Dr. McAvoy for her valuable contribution in the statistical analysis of the results. In addition, I want to express my gratitude to all my other committee members (Dr. Kenneth Walsh, Dr. Shad Sargand, and Dr. Sergio Lopez- Permouth) for their helpful comments. These comments helped improving the quality of my thesis. I also want to thank my colleagues Grace Sallar for all the time she spend critiquing my manuscript, and Jonathan Huffmann for all his help in the initial stage of the finite element modeling. I would also like to thank Colfuturo for its financial support. Finally, and the most important of everything, I would like to thank God, my family and friends, because without their support anything here would not be possible.

6 6 TABLE OF CONTENTS Page Abstract... 3 Acknowledgements... 5 List of Tables... 8 List of Figures Chapter 1: Introduction Objectives Outline Chapter 2: Literature Review Introduction Shear Force Transfer Mechanism Shear Force Transfer Due to Friction between Concrete Elements Shear Force Transfer Due to Shear Key Action Shear Force Transfer Due to Dowel Action Effect of Transverse Post-Tensioning Force in the Mechanical Behavior of Adjacent Box-Beam Bridges Conclusions Chapter 3: Methodology Overview Finite Element Modeling Parts Materials Assembly Mesh Boundary Conditions and Interactions Loading Outputs analyzed from FE models Chapter 4: Results and Discussions... 72

7 4.1 Effects of Amount of TPT Induced Stresses in Shear Key Differential Deflections between Beams Statistical Analysis Effects of Span Length and Number of Internal Diaphragms Stresses Induced in Shear Key Differential Deflections between Adjacent Beams Statistical Analysis Chapter 5: Summary and Conclussions References Appendix I: Induced Stresses in Shear Key as Function of TPT Appendix II: Differential Deflections between Adjacent Beams as Function of TPT Appendix III: Maximum Stresses in Shear Key as Function of Span-to-depth Ratio Appendix IV: Differential Deflections between Beams as Function of Span-to-depth Ratio

8 8 LIST OF TABLES Page Table 1. FEM material properties for first group of FE models Table 2. FEM material properties for second group of FE models Table 3. Basis for temperature gradients (AASHTO LRFD bridge design specifications, 2010) Table 4. Increase in maximum tensile stress in shear key due to positive temperature gradient (%) Table 5. Decrease in max. tensile stress in shear key due to dowel action (%) Table 6. Increase in maximum compressive stress in shear key due to positive temperature gradient (%) Table 7. Decrease in max. compressive stress in shear key due to dowel action (%) Table 8. Increase in max. Von Mises stress in shear key due to positive temperature gradient (%) Table 9. Decrease in max. Von Mises stress in shear key due to dowel action (%) Table 10. Increment in max. tensile stress in shear key due to positive temperature gradient (%) Table 11. Decrease in max. tensile stress in shear key due to dowel action (%) Table 12. Increment in max. compressive stress in shear key due to positive temperature gradient (%) Table 13. Decrease in max. compressive stress in shear key due to dowel action (%) Table 14. Increment in max. Von Mises stress in shear key due to positive temperature gradient (%) Table 15. Decrease in max. Von Mises stress in shear key due to dowel action (%) Table 16. Increment in max. vertical shear stress in shear key due to positive temperature gradient (%) Table 17. Decrease in max. vertical shear stress in shear key due to dowel action (%). 111 Table 18. Increment in max. shear stress in shear key due to positive temperature gradient (%)

9 9 Table 19. Decrease in max. shear stress in shear key due to dowel action (%) Table 20. Increment in max. contact shear in interface between shear key and box-beams due to positive temperature gradient (%) Table 21. Decrease in max. contact shear in interface between shear key and box-beams due to dowel action (%) Table 22. Maximum differential deflections between adjacent beams Table 23. Maximum stresses in shear key for models with no dowel action, and GS = 0.5` Table 24. Maximum stresses in shear key for models with dowel action, and GS = Table 25. Maximum stresses in shear key for models with no dowel action, and GS = Table 26. Maximum stresses in shear key for models with dowel action, and GS = Table 27. Maximum stresses in shear key for models with no dowel action, and GS = Table 28. Maximum stresses in shear key for models with dowel action, and GS = Table 29. Average increase in maximum stresses in shear key due to positive temperature gradients (%) Table 30. Average decrease in maximum stresses in shear key due to dowel action (%) Table 31. Differential deflections between adjacent beams for models with no dowel action, no temperature gradient, and GS = Table 32. Differential deflections between adjacent beams for models with no dowel action, positive temperature gradient, and GS = Table 33. Differential deflections between adjacent beams for models with no dowel action, negative temperature gradient, and GS = Table 34. Differential deflections between adjacent beams for models with no dowel action, no temperature gradient, and GS = Table 35. Differential deflections between adjacent beams for models with no dowel action, positive temperature gradient, and GS =

10 10 Table 36. Differential deflections between adjacent beams for models with no dowel action, negative temperature gradient, and GS = Table 37. Differential deflections between adjacent beams for models with no dowel action, no temperature gradient, and GS = Table 38. Differential deflections between adjacent beams for models with no dowel action, positive temperature gradient, and GS = Table 39. Differential deflections between adjacent beams for models with no dowel action, negative temperature gradient, and GS = Table 40. Differential deflections between adjacent beams for models with dowel action, no temperature gradient, and GS = Table 41. Differential deflections between adjacent beams for models with dowel action, positive temperature gradient, and GS = Table 42. Differential deflections between adjacent beams for models with dowel action, negative temperature gradient, and GS = Table 43. Differential deflections between adjacent beams for models with dowel action, no temperature gradient, and GS = Table 44. Differential deflections between adjacent beams for models with dowel action, positive temperature gradient, and GS = Table 45. Differential deflections between adjacent beams for models with dowel action, negative temperature gradient, and GS = Table 46. Differential deflections between adjacent beams for models with dowel action, no temperature gradient, and GS = Table 47. Differential deflections between adjacent beams for models with dowel action, positive temperature gradient, and GS = Table 48. Differential deflections between adjacent beams for models with dowel action, negative temperature gradient, and GS = Table 49. Maximum stresses in shear key for models with no dowel action, no temperature gradient, and N = Table 50. Maximum stresses in shear key for models with no dowel action, positive temperature gradient, and N =

11 11 Table 51. Maximum stresses in shear key for models with no dowel action, negative temperature gradient, and N = Table 52. Maximum stresses in shear key for models with no dowel action, no temperature gradient, and N = Table 53. Maximum stresses in shear key for models with no dowel action, positive temperature gradient, and N = Table 54. Maximum stresses in shear key for models with no dowel action, negative temperature gradient, and N = Table 55. Maximum stresses in shear key for models with no dowel action, no temperature gradient, and N = Table 56. Maximum stresses in shear key for models with no dowel action, positive temperature gradient, and N = Table 57. Maximum stresses in shear key for models with no dowel action, negative temperature gradient, and N = Table 58. Maximum stresses in shear key for models with dowel action, no temperature gradient, and N = Table 59. Maximum stresses in shear key for models with dowel action, positive temperature gradient, and N = Table 60. Maximum stresses in shear key for models with dowel action, negative temperature gradient, and N = Table 61. Maximum stresses in shear key for models with dowel action, no temperature gradient, and N = Table 62. Maximum stresses in shear key for models with dowel action, positive temperature gradient, and N = Table 63. Maximum stresses in shear key for models with dowel action, negative temperature gradient, and N = Table 64. Maximum stresses in shear key for models with dowel action, no temperature gradient, and N = Table 65. Maximum stresses in shear key for models with dowel action, positive temperature gradient, and N =

12 12 Table 66. Maximum stresses in shear key for models with dowel action, negative temperature gradient, and N = Table 67. Effect of dowel action in average increase in maximum stresses in shear key due to positive temperature gradients (%) Table 68. Effect of L/D in average increase in maximum stresses in shear key due to positive temperature gradients (%) Table 69. Effect of N in average increase in maximum stresses in shear key due to positive temperature gradients (%) Table 70. Effect of GS in average increase in maximum stresses in shear key due to positive temperature gradients (%) Table 71. Effect of temperature gradients in average decrease in maximum stresses in shear key due to dowel action (%) Table 72. Effect of L/D in average decrease in maximum stresses in shear key due to dowel action (%) Table 73. Effect of N in average decrease in maximum stresses in shear key due to dowel action (%) Table 74. Effect of GS in average decrease in maximum stresses in shear key due to dowel action (%) Table 75. differential deflections between beams for models with no dowel action, N = 1, and GS = Table 76. Differential deflections between beams for models with no dowel action, N = 1, and GS = Table 77. Differential deflections between beams for models with no dowel action, N = 1, and GS = Table 78. Differential deflections between beams for models with no dowel action, N = 2, and GS = Table 79. Differential deflections between beams for models with no dowel action, N = 2, and GS = Table 80. Differential deflections between beams for models with no dowel action, N = 2, and GS =

13 13 Table 81. Differential deflections between beams for models with no dowel action, N = 3, and GS = Table 82. Differential deflections between beams for models with no dowel action, N = 3, and GS = Table 83. Differential deflections between beams for models with no dowel action, N = 3, and GS = Table 84. Differential deflections between beams for models with dowel action, N = 1, and GS = Table 85. Differential deflections between beams for models with dowel action, N = 1, and GS = Table 86. Differential deflections between beams for models with dowel action, N = 1, and GS = Table 87. Differential deflections between beams for models with dowel action, N = 2, and GS = Table 88. Differential deflections between beams for models with dowel action, N = 2, and GS = Table 89. Differential deflections between beams for models with dowel action, N = 2, and GS = Table 90. Differential deflections between beams for models with dowel action, N = 3, and GS = Table 91. Differential deflections between beams for models with dowel action, N = 3, and GS = Table 92. Differential deflections between beams for models with dowel action, N = 3, and GS =

14 14 LIST OF FIGURES Page Figure 1. Mechanisms of shear force transfer: (a) shear transfer due to friction force in the interface between elements; (b) shear transfer due to shear key effect; and (c) shear transfer due to dowel action Figure 2. Self-generated compressive force due to tension in steel bars Figure 3. Typical failure modes in connections with shear keys Figure 4. Test specimens: (a) internally reinforced; and (b) externally reinforced (Pruijssers, 1988) Figure 5. Loading frame (Pruijssers, 1988) Figure 6. Failure mechanism according to Ramussen (Pruijssers, 1988) Figure 7. Mechanical model for computing ultimate dowel capacity: a) failure mechanism of dowel; and b) free body diagram at hinge location (Pruijssers, 1988) Figure 8. Distribution of bond stresses (Pruijssers, 1988) Figure 9. Comparison of theoretical and experimental results (Pruijssers, 1988) Figure 10. Schematic cross section view of a precast-prestressed adjacent box-beam bridge: (a) American cross section; and (b) Japanese cross section Figure 11. Required post-tensioning force per unit of length (PCI, 2011) Figure 12. PCI Bridge Design Manual design chart compared to updated design chart (Hanna, et al., 2009) Figure 13. Effect of span-to-depth ratio on post-tensioning force for 0 deg skew angle, and span-to-depth ratio of 30 (Hanna, et al., 2009) Figure 14. Effect of skew angle on post-tensioning force for a bridge with (Hanna, et al., 2009) Figure 15. Comparison between the posttensioning force obtained from the proposed equation and the one obtained from the grid analysis (Hanna, et al., 2009) Figure 16. Experimental setup: a) load distribution test; and b) ultimate load test (Grace, et al., 2010)

15 15 Figure 17. Adequate number of diaphragms vs. span: (a) 36 in width box-beam; and (b) 48 in width box-beam (Grace, et al., 2012) Figure 18. Approximate transverse posttensioning force per diaphragm vs. bridge width. f c for deteriorated slab = 3000 psi, for recently constructed slab = 4000 psi, for special quality slab = 5000 psi (Grace, et al., 2012) Figure 19. Cross section view of beams using in the Finite Element Models Figure 20. Solid extrusion of box-beam Figure 21. Solid extrusion of shear key Figure 22. Solid extrusion of longitudinal reinforcement Figure 23. Sold extrusion of diaphragm Figure 24. Solid extrusion of transverse tie, Figure 25. 3D view of the assembled FE model Figure 26. Partitions made in the parts prior meshing Figure 27. Finite element model after meshing: (a) 3D view of mesh resolution; (b) cross section view; and (c) detail at location of transverse ties Figure 28. Boundary conditions: (a) left end of the bridge; and (b) right end of the bridge Figure 29. Contact pressure-overclosure model: (a) analogy with Hooke s law; and (b) linear pressure-overclosure relationship Figure 30. Friction model with a limit in the shear stress Figure 31. Applied live loads: (a) cross sectional view; and (b) elevation view Figure 32. Transverse post-tensioning application points for the first set of models: (a) cross section; and (b) elevation view Figure 33. Solar Radiation Zones for the United States (AASHTO LRFD Bridge Design Specifications, 2010) Figure 34. Positive Vertical Temperature gradient in Concrete and Steel Superstructures (AASHTO LRFD Bridge Design Specifications, 2010) Figure 35. Proposed failure modes for the joint in adjacent box-beam bridges: (a) due to tensile stress; (b) due to shear stress; (c) due to local crushing; and (d) due to high shear stress at interface between shear key and box-beam

16 16 Figure 36. Maximum tensile stress in shear key as a function of TPT for GS = 1: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 37. Maximum compressive stress in shear key as a function of TPT for GS = 1: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 38. Maximum Von Mises stress in shear key as a function of TPT for GS = 1: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 39. Measured differential deflections between adjacent beams: (a) transverse direction; (b) vertical direction; and (c) longitudinal direction Figure 40. Transverse differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 41. Vertical differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 42. Longitudinal differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 43. Maximum tensile stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout Figure 44. Maximum compressive stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout Figure 45. Maximum Von Mises stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout Figure 46. Maximum vertical shear stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout Figure 47. Maximum shear stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout Figure 48. Maximum contact shear stress between shear key and box beams as a function of span-to-depth ratio (GS = 1, and N = 1): (a) ducts not filled with grout; and (b) ducts filled with grout Figure 49. Transverse differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout

17 17 Figure 50. Vertical differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout Figure 51. Longitudinal differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout

18 18 CHAPTER 1: INTRODUCTION The bridge engineering industry is constantly searching for economical solutions for the construction of short and medium span bridges; that is, bridges spanning up to 100 feet (El-R y et al., 1996). In this case, adjacent precast-prestressed concrete boxbeam bridges have shown to be one of the most attractive and economical solutions (PCI, 2009). This type of bridge has been successful because of its simple design, and also because they are easily and quickly constructed. The superstructure is built by placing precast-prestressed concrete box beams adjacent to each other until the full width of the bridge is completed. Shear keys are then cast between adjacent beams to link them structurally and aid in the load transfer between beams. In some cases, transverse posttensioning (TPT) and/or a cast in place concrete deck are also used to improve this load transfer mechanism. Despite its practicality, there is a frequently reported issue associated with this type of bridge. The main problem has been development of longitudinal cracks along the grouted joints (El-R y et al., 1996; Fu et al., 2011; Grace et al., 2010; Hanna, 2008; Hanna et al., 2009; PCI, 2009; Russell, 2009; Russell, 2011; Sang, 2010). These cracks allow chemical agents to penetrate into the beams, causing corrosion in the reinforcement, which adversely affects the lifetime of the structure. Therefore, as the bridge ages, its ability to transfer loads between adjacent beams is reduced, and, consequently, the bridge s load-carrying capacity is diminished. This reduction in the bridge s ability to transfer loads is due to the degradation of the materials over time and possible shear key cracking due to repetitive loading and thermal cycles.

19 19 Understanding the actual behavior of this type of bridge is necessary to accurately predict its load-carrying capacity. However, the existence of longitudinal cracking issues in current design and construction practices indicates lack of complete understanding about the actual load transfer mechanism in these bridges. Due to the limited knowledge, obtaining an accurate value for the load-carrying capacity can be challenging. This indicates that a more detailed analysis on the load transfer mechanism in adjacent precast-prestressed box-beam bridges must be performed in order to better understand their behavior. Truck testing has been one of the techniques used to understand the actual behavior of bridges as well as load rate them (Camino Trujillo, 2010; Chajes, 1997; Huffman, 2012; Setty, 2012). This approach provides an estimate of the load carrying capacity of a single bridge. However, performing these tests is expensive, and results are limited to a particular case (i.e., the bridge analyzed). Thus, results from truck testing performed on a single bridge cannot be used to generalize the behavior bridges with different span, cross section, material properties, or loading conditions. For this reason, the use of Finite Element (FE) models can be a viable solution in this case. Using the results of detailed 3D FE models, a parametric study on the load transfer mechanism of adjacent box-beam bridges can be performed. Different parameters affecting the load transfer mechanism (e.g., bridge span, span-to-depth ratio, material properties, interaction properties, and environmental conditions) can be controlled through this technique. Thereby, the contribution of each variable in the overall response of any bridge configuration can be quantified. Then, results from this study can be generalized to

20 20 evaluate any bridge, and thus, be used as an aid to load rate in a more accurate and realistic way. 1.1 Objectives The main objective of this research is to develop a set of 3D finite element models, using Abaqus/CAE 6.11, to perform a parametric study of the load transfer mechanism in adjacent precast-prestressed box-beams. Results from the finite element models can be used as an aid for better understanding the interaction between beams, and thus have a better understanding of the behavior of this type of bridge. The models must be capable to accurately predict the stresses induced in the shear key considering the following: 1) amount of TPT; 2) beam span-to depth ratio; 3) grout compressive strengthto-concrete compressive strength ratio; 4) number of diaphragms along the bridge span; 4) filled or non-filled transverse post-tensioning ducts (i.e., effects of dowel action); and 5) environmental conditions (i.e., effects of positive and negative temperature gradients through the bridge s depth). 1.2 Outline The remainder of this thesis is organized as follows: Chapter 2 consists of a literature review, in which different parameters affecting the load-transfer mechanism of adjacent box-beam bridges are presented. This chapter covers basic concepts of shear force transfer in concrete structures, as well as advanced concepts of finite element modeling of adjacent box beam bridges. In addition, results from different experimental investigations are shown in this chapter.

21 21 Chapter 3 describes the methodology used to develop the finite element models used to perform the parametric study on the load-transfer mechanism of adjacent boxbeam bridges. This chapter presents a detailed explanation of the creation of each part, material properties, interaction between parts (i.e., defining contact properties between parts), loading and boundary conditions. Chapter 4 shows the results obtained from the finite element models. Different charts showing all the parameters considered affecting the load transfer mechanism in adjacent box-beam bridges are presented. Findings from the parametric study are presented and analyzed in detail. Chapter 5 presents the conclusions and recommendations for construction and load rating obtained from the parametric study.

22 22 CHAPTER 2: LITERATURE REVIEW 2.1 Introduction This chapter presents a comprehensive literature review on the load transfer mechanism in concrete structures, and also about the load transfer mechanism in adjacent precast-prestressed concrete box-beam bridges. This chapter is divided into four sections covering all the important aspects of the variables affecting the load transfer mechanism in adjacent box beam bridges. Contents of Sections 2.2 to 2.4 are explained below. Section 2.2 explains in detail the three main mechanisms controlling the shear force transfer in concrete structures: 1) due to friction forces; 2) due to shear key action; and 3) due to dowel action. In the first subsection, the shear force transfer due to friction forces between concrete elements is explained in detail (fib, 2007; Júlio et al., 2004; and Wan, 2011). Through this subsection, factors affecting the effectiveness of the shear force transfer due to friction are explained. At the end of this subsection, different techniques to improve the performance of shear force transfer due to friction are shown. The second subsection explains the shear force transfer mechanism due to shear key action according to fib (2007). Several failure modes for the shear key proposed by fib (2007) are presented. These failure modes limit the shear force transfer capacity due to shear key action. The last subsection is about the shear force transfer mechanism due to dowel action (fib, 2007; Pruijssers 1988; Ramussen, 1962). Literature review showed that the effect of dowel action in the load transfer mechanism in adjacent precast-prestressed box-beam bridges has not been studied. For this reason, this mechanism is explained in more detail.

23 23 Section 2.3 involves the effect of transverse post-tensioning force in the mechanical behavior of adjacent box-beam bridges. Current transverse design methodologies and design codes are mentioned in the beginning. In addition, different research on the effect of the amount of transverse post-tensioning (TPT) force and TPT force arrangement in the behavior of adjacent box-beam bridges are covered. Section 2.4 is the conclusions section. In this part of the chapter, all the findings from the previous sections are summarized and discussed in detail. The new insights emerging from the chapter are discussed based on the findings. 2.2 Shear Force Transfer Mechanism For certain type of structures made of precast concrete elements (e.g., adjacent precast-prestressed box-beam bridges, precast flooring systems, or precast wall elements) it is important to transfer loads from one element to the other. These loads are in general shear forces in the interface between elements. According to fib (2007) this load transfer between precast concrete elements is governed by three main mechanisms: 1) friction in the interface between elements; 2) shear key effect; and 3) dowel action. These three mechanisms of shear force transfer are shown in Figure 1.

24 24 V Shear key V N F = µn f N Dowels (a) V (b) (c) V Figure 1. Mechanisms of shear force transfer: (a) shear transfer due to friction force in the interface between elements; (b) shear transfer due to shear key effect; and (c) shear transfer due to dowel action Shear Force Transfer Due to Friction between Concrete Elements The first shear force transfer mechanism (i.e., due to friction force in the interface between elements) can only be developed if compressive forces are acting in the interface between the elements in contact (fib, 2007). These compressive forces can be obtained by post-tensioning across the joint, or can be self-generated by tension in steel as the two elements are subjected to shear sliding, as shown in Figure 2.

25 25 σ s σ s σ s shear sliding in crack plane crack width σ s σ s Figure 2. Self-generated compressive force due to tension in steel bars. In many practical cases (e.g., precast flooring systems, or bridges made of precast beams), frictional forces produce a significant contribution to the shear force transfer mechanism between two precast elements. Research has shown that the shear force transfer capacity due to friction is affected by the roughness of the contact surface between elements (fib, 2007; Júlio et al., 2004). This capacity is also influenced by the amount of compressive stress in the interface between elements. Results from different research (fib, 2007; Júlio et al., 2004; and Wan, 2011) showed that not only the roughness of the surfaces and the amount of compressive stress in the interface between elements define the actual strength due to friction. This strength is also affected by the moisture conditions of the surfaces in contact as well as the cleanness of the surfaces. In this manner, assuming that good quality in the surface cleaning is ensured, and moisture conditions in the surfaces are controlled, shear force transfer mechanism due to friction is mainly controlled by interfacial roughness between elements.

26 26 Several methods are typically used to increase the interfacial roughness between precast elements, such as: wire-brushing, sand-blasting, and power washing. There is an additional technique to increase the interfacial roughness, which was discussed in fib (2007). This technique consists of applying curing retardant to the surfaces of precast concrete elements when the concrete is still wet. By using this technique, once the concrete is cured, it is possible to obtain an exposed aggregate surface, with an increased roughness Shear Force Transfer Due to Shear Key Action The second shear force transfer mechanism is present when a shear key is exists in the joint. According to fib (2007), a shear key is formed when grout is cast between two elements with indented joint faces, as shown in Figure 1(b). The shear keys act as mechanical locks to prevent any significant shear sliding between the two elements. For this reason, shear keys have to meet minimum geometric requirements as well as stiffness requirements (fib, 2007). The shear force transfer capacity due to shear key effect is controlled by the shear key failure (fib, 2007). Figure 3 shows different failure modes proposed by fib (2007).

27 27 Figure 3. Typical failure modes in connections with shear keys. The first failure mode is observed when the tooth in the shear key is subject to direct shear. Failure is reached when the shear stress in the direction of the applied shear force exceeds the shear strength of the grout as shown in the left drawing of Figure 3. The second failure mode, depicted in the middle drawing of Figure 3, is obtained by concentration of compressive stresses which exceed the compressive strength of the grout. The third failure mode occurs when interfacial shear stresses in the contact surface between the shear key and the precast element is higher than the interfacial shear strength (fib, 2007) as shown in the right drawing of Figure Shear Force Transfer Due to Dowel Action The third shear force transfer mechanism is due to dowel action. This mechanism only exists if steel bars are crossing the joint interface. The dowel action is produced by the reaction between concrete and steel bars as a result of lateral bar displacement. When transverse steel is present in the joint, the shear force capacity can be increased by enlarging the amount of transverse steel (fib, 2007). However, a large amount of transverse steel can produce local crushing of the concrete surrounding the bars.

28 28 Consequently, fib (2007) asserted that this failure mode (i.e., failure occurring when there is local crushing of concrete surrounding the bars) constitutes an upper limit in the shear force capacity. Because the literature review showed that the effects of dowel action have not been considered in the load transfer analysis of adjacent precast-prestressed boxbeam bridges, special attention to this mechanism is taken into consideration for this thesis. Some theoretical formulations on the effect of dowel action in the shear transfer mechanism of reinforced concrete structures are discussed next. Pruijssers (1988) performed an extensive research on the shear force transfer across a crack in reinforced concrete structures, with special application to offshore structures. This research was divided into two parts. The first part consisted of repeated cyclic shear tests on reinforced concrete specimens. Two groups were studied: 1) a set of internally reinforced concrete specimens; and 2) a set of specimens externally reinforced with external restraint bars. Geometry of both set of specimens is shown in Figure 4(a-b). The dimensions of both sets are the same, and measured in mm. (a) (b) Figure 4. Test specimens: (a) internally reinforced; and (b) externally reinforced (Pruijssers, 1988).

29 29 The amount of reinforcement, applied shear force, initial crack width, and concrete strength were changed to analyze their contribution to the crack behavior. These specimens were loaded using a loading frame as shown in Figure 5. Using this loading frame allowed controlling the amount of applied shear force as well as the number of cycles until failure was reached. For the case of internally reinforced specimens, results from the experimental research allowed obtaining an empirical relationship between the applied stress level and the number of cycles until failure. This expression is shown in Eq. 1 = log Eq. 1 where: = applied shear stress (MPa) = = static shear strength (MPa). = =0.159 = concrete compressive stress (MPa) = steel yielding stress (MPa) = reinforcement ratio = number of cycles until failure

30 30 Figure 5. Loading frame (Pruijssers, 1988). In addition, experimental results from the internally reinforced specimens allowed obtaining empirical expressions to predict the crack width ( ) and the crack displacement ( ), as a function of applied stress level and the number of cycles. Expressions for and are shown in Eq. 2(a), and their corresponding values are in mm. = log 0.20 log Eq. 2(a)

31 31 = log 1.17 log Eq. 2(b) Experimental results for the second set of specimens (i.e., externally reinforced specimens) showed that failure was not influenced the applied shear stress level, but it was affected by the amount of normal stress, σ n. An expression capable to approximate the static shear strength,, was derived using regression analysis. This expression is shown in Eq. 3. = Eq. 3 The behavior of this set of specimens was more difficult to predict, but it was concluded that failure was reached for a less number of cycles than the set of reinforced specimens. Based upon these results, it is concluded that the dowel action produced by the reinforcing bars have an important contribution to the strength of reinforced concrete structures subject to cyclic loading. The second part of this research was a theoretical investigation about the response of cracked concrete to monotonic and cyclic shear loading. For an easy understanding of concepts, the analysis of the dowel bar is shown for the case of monotonically applied load only. Pruijssers s goal was to understand in detail the mechanism of shear force transfer in cracks in plain and reinforced concrete. This mechanism was studied considering two different aspects: 1) the mechanism of aggregate interlock; and 2) the dowel action. For the case of dowel action, Pruijssers (1988) found the failure mechanism to be a combination of concrete crushing and steel bar yielding, when the concrete cover is enough to prevent splitting failure. This concrete crushing is caused by the concentration

32 32 of stress induced by the lateral displacement of the dowel bar. Pruijssers (1988) mentioned that the amount of stress causing concrete crushing can be several times the uniaxial concrete compressive strength. This is because as the bar starts to deform, surrounding concrete produces a confining stress, producing a triaxial state of stresses that allows the concrete to resist stresses higher than the uniaxial compressive strength. One of the main objectives of this research was to predict the dowel force capacity. Some findings in Pruijssers work showed that the dowel force capacity was controlled by the bar diameter, concrete strength, axial stress in steel (which is controlled by bonding between the concrete and steel bars), and steel yield strength. The dowel bar embedded in concrete was analyzed as an infinite flexible beam on elastic foundation, where the concrete is acting as the foundation for the dowel. However, past research showed that linearity was not accurate when predicting ultimate load capacity of the dowel. This is because the concrete behavior is nonlinear. To correct for this nonlinearity, the concrete stress distribution is modified according to the failure mechanism proposed by Ramussen (1962), as shown in Figure 6.

33 33 Figure 6. Failure mechanism according to Ramussen (Pruijssers, 1988). This failure mechanism considered that failure occurs when plastic hinges are produced in the dowel. The mechanical model used by Pruijssers (1988) for the derivation of the dowel force capacity is shown in Figure 7(a-b). (a) (b) Figure 7. Mechanical model for computing ultimate dowel capacity: a) failure mechanism of dowel; and b) free body diagram at hinge location (Pruijssers, 1988).

34 34 The bond force was computed based upon the distribution of bond stresses shown in Figure 8. Pruijssers (1988) showed that the bond stress distribution depends upon steel strains, and normal contact stress between bar and concrete. Pruijssers (1988) mentioned that normal contact stress is small in the vicinity of the shifted neutral axis and increases in locations below this neutral axis. Linear elastic analysis led to conclude that bond stress distribution was function of cos 1.5 (α), as shown in Figure 8. Figure 8. Distribution of bond stresses (Pruijssers, 1988). Based on axial and rotational equilibrium for the dowel, Pruijssers (1988) derived the following expression for the ultimate dowel capacity: = Eq. 4 where: F = ultimate dowel capcity (N) ε= = eccentricity parameter = load eccentricity (in)

35 35 ϕ= bar diameter in [in] f = concrete compressive stress (MPa) f = steel yielding stress (MPa) Pruijssers (1988) compared this expression with experimental results obtained by several authors. Results are shown in Figure 9. Figure 9. Comparison of theoretical and experimental results (Pruijssers, 1988). 2.3 Effect of Transverse Post-Tensioning Force in the Mechanical Behavior of Adjacent Box-Beam Bridges El-R y et al. (1996) compared American and Japanese transverse design methodologies for precast-prestressed concrete adjacent box-beam bridges. Results from this survey revealed that, in general, adjacent box-beam bridges built in the United States

36 36 use relatively small shear keys, and small amount of TPT force. Also, they found that when using high TPT force, most DOTs do not use any theoretical justification. In contrast, when studying Japanese practices, they noticed that a detailed analysis of each bridge and a very large grout key filled with cast-in-place grout, and large amount of TPT force were required. Figure 10(a-b) shows a schematic view of the existing American and the typical Japanese cross section for adjacent box-beam bridges used in the 90 s. It can be noticed how small the shear keys are in the American cross section when compared to the typical Japanese cross section. In addition, it is noticed that the distribution of the TPT force in the Japanese bridges are more uniformly distributed than in the American bridges. Shear keys C L Wearing surface Transverse tie (at location of diaphragms) Depth Width (a) Shear keys C L Wearing surface Transverse ties (at location of diaphragms) Depth (b) Figure 10. Schematic cross section view of a precast-prestressed adjacent box-beam bridge: (a) American cross section; and (b) Japanese cross section. Width Based on all these findings, El-R y et al. proposed a new design methodology for the transverse design of adjacent box-beam bridges. This new approach combined two important aspects: 1) the high performance of Japanese bridge design; and 2) the

37 37 practicality and simplicity of American construction practices. The proposed design required placing diaphragms at quarter points along the bridge span. In addition, a certain amount of TPT force was required to be applied at the location of each diaphragm. As mentioned in Section 2.3, the TPT force is used to improve the load transfer mechanism. Thereby, longitudinal cracking issues along the grouted joints can be prevented. In order to determine the required amount of TPT force, two criteria were considered: 1) the differential deflection between adjacent beams under live load conditions should be less than 0.02 in; and 2) the tensile stress in the shear key is must be zero. To obtain the required amount of TPT, a parametric study based on a grid analysis was performed. Several analytical models were created in which frame elements were used to model both the beams and the diaphragms. Variables as bridge width and beam cross section were considered in the analysis. Then, a design chart with the required amount of TPT force as a function of the bridge width was obtained. This design chart was later adopted by the PCI Bridge Design Manual (2003) and continued to be used in the PCI Bridge Design Manual (2011). The results from this analysis are shown in Figure 11.

38 38 Figure 11. Required post-tensioning force per unit of length (PCI, 2011). As a recommendation for construction, the authors suggested to apply the posttensioning force after the shear keys are grouted. This helps to reduce the possibility of obtaining tensile stress in the shear keys, which contributes to cracking. In addition, the authors recommended the use of full depth shear keys at each diaphragm, and the posttensioning force equally distributed between the top and bottom of the diaphragm. Although suitable for performing a preliminary estimation of the required TPT force, the obtained design chart is just applicable for small skew angles. When large skew angles are present, a detailed grid analysis is suggested to be performed. Thus, a more reliable value can be obtained. This was the reason for other researchers to proposed new design charts to update the work provided by this research. In this manner, Hanna, et al. (2009) developed a new parametric study, with the aim to update the design chart proposed by El-R y et al. in The analysis was

39 39 also based on grid models. The updated design charts were obtained for various combinations of bridge length, bridge width, skew angle (including high skew angles) and girder depth, using the loading from AASHTO LRFD Bridge Design Specifications (2007). To verify the reliability of the new results, the updated TPT force was compared to the design charts available in the PCI Bridge Design Manual (2003) (i.e., the design charts proposed by El-R y et al. in 1996). To be able to compare, different cross sections were considered, and the effects of bridge skew angle were neglected. Figure 12 shows a comparison of the updated TPT force and the one used in the PCI Bridge Design Manual (2003). It was noticed that, the updated design chart led to significantly higher TPT forces when compared to the PCI Bridge Design Manual (2003). The authors attribute this difference to the use of the AASHTO LRFD live load conditions. Figure 12. PCI Bridge Design Manual design chart compared to updated design chart (Hanna, et al., 2009).

40 40 Additional results were obtained in which the effects of span-to-depth ratio and skew angle were included. As it is shown in Figure 13, the effects of span-to-depth ratio were not significant in the ability of predict the amount of TPT force required to avoid longitudinal joint cracking issues. Figure 13. Effect of span-to-depth ratio on post-tensioning force for 0 deg skew angle, and span-to-depth ratio of 30 (Hanna, et al., 2009). On the other hand, when analyzing Figure 14, they noticed that the skew angle is also a negligible parameter when obtaining the required TPT force. It was obtained that the amount of TPT force increases when the skew angle increases. Figure 14 also reveals that this effect is more important for shallower beams.

41 41 Figure 14. Effect of skew angle on post-tensioning force for a bridge with (Hanna, et al., 2009). Based on these results the authors concluded that the parameters controlling the transverse design of adjacent box-beam bridges were the depth of the beam itself and the bridge width. For this reason, they recommended to initially estimate the TPT force based on the bridge width and beam depth, and then correct this value for span-to depth ratio and skew angle using Figure 13 and Figure 14, respectively. As an additional contribution from this research, a simplified design equation to estimate the effective post-tensioning force P (kip/ft) per unit of length of the bridge was proposed. To obtain the required transverse post-tensioning at each diaphragm, it is necessary to multiply the effective post-tensioning force P by the distance between diaphragms (i.e., P diaphragm = P distance between diaphragms). This equation was obtained from all the possible combinations of skew angle, width-to-depth ratio, and span-to-depth ratio. The equation obtained was:

42 = Eq where: = box depth (ft) = bridge width (ft) = correction factor for span-to-depth ratio = = correction factor skew angle more than 0 deg = = bridge span (ft) = skew angle (deg) Figure 15 shows the comparison between the post-tensioning force obtained from the proposed equation and the post-tensioning force obtained from the grid analysis. By comparing the two values, a good correlation is observed. It showed that the equation can be easily used to estimate the TPT force including any possible combination of cross section, span, bridge width, and skew angle.

43 43 Figure 15. Comparison between the posttensioning force obtained from the proposed equation and the one obtained from the grid analysis (Hanna, et al., 2009). Later, Grace, et al. (2010) proposed a new system in which Carbon Fiber Reinforced Polymer (CFRP) strands were used to apply the TPT force. Unlike previous investigations, a half-scale model tested in a lab was constructed to carry out this research. The adequate number of diaphragms, and the required post-tensioning force per diaphragm were obtained from this research. Two different experimental setups were used to analyze the bridge. The first setup consisted of a load-distribution test which was conducted by applying a single point load of 15 kip on each box beam at the midspan, and varying the TPT force (loads of 20 kips, 40 kips and 80 kips applied at each diaphragm location) as shown in Figure 16a. The second test was performed to determine the ultimate flexural capacity of the bridge. A transverse posttensioning force of 80 kip was applied at each diaphragm location, and an

44 44 eccentric transverse load was applied (as shown in Figure 16b) until the bridge reached failure. Analysis of transverse strain at shear keys, and transverse strain distribution along the bridge length were performed for the different configurations of TPT force. Analysis of vertical deflections at each beam was also analyzed. In addition, while performing the ultimate-load test, a load-deflection response was obtained. From this research it was found that increasing the number of transverse diaphragms do not have a significant influence on transverse strains in the regions between diaphragms. Another important finding was that increasing the post-tensioning force does not affect the load distribution behavior of the bridge in the uncracked deck phase, but significant improvement was obtained during the cracked deck phase.

45 45 (a) (b) Figure 16. Experimental setup: a) load distribution test; and b) ultimate load test (Grace, et al., 2010). More recently, Grace, et al. (2012) performed new research where the effects of temperature gradient through the beam depth were included in the analysis of adjacent box-beam bridges. This research included a combination of experimental data and finite

46 46 element analyses. Results from this research revealed that traffic loads did not appear to be the major factor in the development of longitudinal deck cracking. However, results showed that positive temperature gradients had the most important impact in the initiation of longitudinal cracks along the grouted joints. Also, a recommendation for the number of diaphragms required to avoid longitudinal deck cracking was proposed as a function of bridge span. Figure 17(a-b) shows the number of diaphragms required to eliminate longitudinal deck cracking by considering two different beam cross sections.

47 47 (a) (b) Figure 17. Adequate number of diaphragms vs. span: (a) 36 in width box-beam; and (b) 48 in width box-beam (Grace, et al., 2012).

48 48 Furthermore, transverse post-tensioning forces as a function of the bridge width were obtained. These transverse post-tensioning forces are applied after both shear keys and deck are cast. Different considerations about the slab condition were taken into account. Each slab condition was modeled by varying its concrete strength. Figure 18 shows the obtained transverse posttensioning force per diaphragm as a function of the bridge width. This required transverse post-tensioning force was compared to MDOT design guide (2006), obtaining significantly higher values than those proposed by this design guide. Figure 18. Approximate transverse posttensioning force per diaphragm vs. bridge width. f c for deteriorated slab = 3000 psi, for recently constructed slab = 4000 psi, for special quality slab = 5000 psi (Grace, et al., 2012).

49 Conclusions This chapter presented a comprehensive literature review on the shear force transfer mechanism in concrete structures. The load transfer mechanism in adjacent precast-prestressed concrete box-beam bridges was also presented. This chapter was divided into two main parts. In the first part of the chapter, the three main mechanisms governing the shear force transfer in concrete structures were described. These mechanisms were: 1) friction forces; 2) shear key action; and 3) dowel action. The second part of the chapter analyzed several studies about the transverse design of adjacent precast-prestressed box-beam bridges. The main conclusions obtained from these chapters are shown below. Frictional forces between two elements are only generated when compressive forces are generated in the contact interface between them. For concrete structures, these compressive forces can be externally applied with post-tensioning force or self-generated when there are steel bars across the interface between the two elements. This selfgenerated force is possible when the two elements are subject to shear sliding, and strains are developed in the steel bars, producing axial forces transmitted to the concrete. The effectiveness of the shear force transfer due to friction is also affected by the roughness, cleanness and moisture conditions of the interfacial surface between elements. However, parameters as cleanness and moisture conditions of the interfacial surface between the two elements are more difficult to control in the field. In this manner, the best way to control effectiveness of the load transfer due to friction is increasing the roughness of the contact surface between elements. This is the reason why different

50 50 techniques such as wire-brushing, sand-blasting, and power washing are recommended to be used in order to improve the load transfer mechanism due to friction. Pruijssers s work on the dowel action mechanism shows the importance of the dowel in shear force transfer in concrete structures. Results from this research showed that the dowel force capacity was influenced by the bar diameter as well as the strength of the concrete and steel. These results can be applied to gain a better understanding of the load transfer mechanism in adjacent precast-prestressed concrete box-beam bridges. This is because in this type of bridge there are transverse ties placed at several locations along the bridge span, and used to link the box-beams. These transverse ties are inserted in transverse ducts that can be filled with grout allowing the dowel action mechanism to be possible in this type of construction. The behavior of adjacent box-beam bridges, and initiation of longitudinal cracks along the shear keys are mainly dependent on: 1) the performance of the load transfer mechanism between adjacent beams; 2) environmental conditions (e.g., temperature changes through the beams cross section); and 3) construction techniques (e.g., beams not being completely vertical to generate crown in the road). Research conducted by Grace, et al. (2012) showed that initiation of longitudinal cracking along the shear keys, and consequently loss of load transfer between adjacent beams is more influenced by temperature gradients through the beams depth than it is by truck loading. This finding led to conclude that additional research must be performed in order to understand the effects of temperature gradients in the behavior and shear key cracking issues in adjacent box-beam bridges.

51 51 CHAPTER 3: METHODOLOGY 3.1 Overview A set of 3D finite element models was used to study the load transfer mechanism between adjacent box-beams. The finite element models were developed using Abaqus/CAE All models consisted of two adjacent box-beams connected through a partial depth shear key, and transverse ties located at different locations along the bridge s span. The beam shapes were modeled as beams previously modeled for the bridge FAY in Fayette County, Ohio (Huffman, 2012). Figure 19 shows a cross section view of the two beams used in all finite element models. 3" 4" 3 4 " 3 8 " 11" 26" 3" 3" 21" 36" Figure 19. Cross section view of beams used in the Finite Element Models. 36" Two main groups of finite element models were created to perform the parametric study. The first group of models consisted of pair of box-beams with a fixed length of and two internal diaphragms (Huffman, 2012). Different amount of TPT was applied at the location of each internal diaphragm to analyze its effect in induced stresses in shear key and differential deflections between adjacent beams. The second group of models consisted of a pair of beams, with variable span length (represented with span-todepth ratio, L/D) and number of internal diaphragms, N. Three different span-to-depth

52 52 ratios were considered (e.g., L/D 14.29, 27.33, and 35.71). Also, three different configurations for number of internal diaphragms were used (e.g., N = 1, 2, and 3). For this second group of models, no transverse post-tensioning was applied. The following aspects were considered in all the models used in the parametric study: 1) effects of filled and non-filled transverse post-tensioning ducts (i.e., models with no dowel action, and dowel action, respectively; 2) effects of positive and negative temperature gradients; and 3) effects of grout compressive strength-to-concrete compressive strength ratio, GS. To analyze effects of dowel action, two sets of models were created. The first set of models considered transverse post-tensioning ducts non-filled with grout. For this set of models there was no contact between transverse post-tensioning ducts and transverse tie rods. This was because the diameter of transverse post-tensioning ducts was larger than the diameter of transverse tie rods. The second set of models considered transverse post-tensioning ducts filled with grout. This contact between the transverse tie rods and the grout allowed dowel action. For these models, a tie constraint between the grout and the transverse ties was used to prevent any sliding (i.e., transverse rods were fully bonded). Results from the models with grout-filled ducts were compared with those obtained from the models with ducts not filled with grout. This allowed quantifying the contribution of dowel action. 3.2 Finite Element Modeling This section describes in detail the construction of all the Abaqus/CAE Finite Element Models used for the parametric study. This section was divided into five

53 53 subsections. Section provided a a detailed description of the construction of each part. Next, in Section 3.2.2, material properties used for each part was described in detail. Once parts and materials are explained, Section 0 presented the assembly of the finite element models. This section showed how all the parts were put together. Section was the mesh section. This section presented the procedure used to mesh every part. Section presented boundary conditions and interactions. This section explained how the supports of the beams were modeled, and how the interactions between all parts (i.e., contact properties, constraints, and embedment of reinforcement) were modeled. Finally, Section presented the loads applied to all the models. These loads included truck loading, temperature loading and transverse post-tensioning Parts The following five parts were used to assemble all finite element models for the current study: Box-beams Shear key Longitudinal reinforcement (prestressed and conventional) Diaphragms (internal and at supports) Transverse tie rods The first step to create the parts was to draw the each cross section. Every cross section can be drawn using either the drawing tools provided by Abaqus/CAE, or using Autocad. For easy of construction, Autocad was used to draw the cross section for each part. Each drawing was saved independently with dxf extension, in order to be

54 54 recognized by Abaqus/CAE. Once all drawings were created, each one was imported, and then extruded to make the parts. For the first group of models, the extrusion length was L = as mentioned in the second paragraph of Section 3.1. For the second group of models, three different span-to-depth ratios were used (e.g., L/D = 14.29, 27.33, and 35.71). Based upon these span-to-depth rations, three different extrusion lengths were used (e.g., L = 25, 48 10, and 66 8 ). The diaphragms were extruded 12, and the transverse tie rod was extruded 72 (i.e., the width of the two beams placed one next to other). All parts used for the finite element simulations are shown in Figure 20Figure 24. A larger diameter at ends of transverse ties is observed in Figure 24. This was made intentionally to be able to attach the ends of the transverse ties to the ducts (using a tie constraint). In this way it was possible to transfer force from the transverse ties to the beams. Figure 20. Solid extrusion of box-beam.

55 55 Figure 21. Solid extrusion of shear key. Figure 22. Solid extrusion of longitudinal reinforcement.

56 56 Figure 23. Sold extrusion of diaphragm. Figure 24. Solid extrusion of transverse tie Materials All models were created with linear elastic materials. This was to optimize computing time due to the large amount of models executed in Abaqus/CAE. Material properties for the first group of finite element (FE) models are shown in Table 1.

57 57 Table 1. FEM material properties for first group of FE models Part Material Young's modulus Poisson's (ksi) ratio Beams Concrete Shear key Grout Prestressing strands Steel Transverse ties Steel Material properties for the second group of FE models are shown in Table 2. Table 2. FEM material properties for second group of FE models Part Material Young's Poisson's modulus (ksi) ratio Beams Concrete Grout (GS = 0.5) Shear key Grout (GS = 1) Grout (GS = 2) Prestressing strands Steel Transverse ties Steel Assembly Once every part and material was created, it was required to link each part to its corresponding material. This procedure was performed in two steps. The first step was to create sections in the Sections module. Each section was set as Solid, Homogeneous, and materials were assigned according to each part. For instance, for part beam, a section named beam was created, and material was chosen to be Concrete. Same procedure was used for all the other sections. The second step consisted of using the Section Assignments tool to assign each section to each part. This tool is found in the module Part included in Abaqus/CAE.

58 58 Once a section was assigned to each part, the module assembly was used to import each part. This was performed using the Instances tool. Two instance types were available; however, the option dependent was used for each part. This option allowed creating a mesh in each part separately. Once all parts were imported, editing tools were used to place them in their corresponding places. Figure 25 shows a 3D view of one of the assembled finite element models used for the parametric study. Figure 25. 3D view of the assembled FE model Mesh One of the most critical parts in the finite element modeling was meshing. For all finite element models the parts were meshed dependently. This means that meshing was generated in the module Part. This allowed meshing each part separately. Before start of meshing the parts, different partitions were created. The models were partitioned at the location of each diaphragm and at strategic locations in the cross section. This helped obtaining a more regular mesh, and also ensuring that the nodes shared between different

59 parts match after the mesh is generated. Figure 26 shows a final view of the partitions for one of the models consisting of a span of and two internal diaphragms. 59 Figure 26. Partitions made in the parts prior meshing. To create the mesh, a determined number of elements was assigned to the edges of each part. The number of elements in each edge was selected in order to avoid high aspect ratios (i.e., long side-to-short side ratio). After seeding the edges, all the elements were meshed. Sweep technique was used to generate the mesh for each part. Also, C3D8R (i.e., 8-node linear brick, reduced integration, hourglass control) elements were used in the analysis. Figure 27(a) shows a 3D view of the finite element model after meshing. A cross section view of the mesh resolution can be seen in Figure 27(b). In this figure it can be noticed how the nodes shared between the parts box-beam and diaphragm and box-beam and shear key are matching. Figure 27(c) shows an elevation view of the beam (i.e., side view of the beam) of the mesh resolution at the location of the

60 60 diaphragms. Notice that a finer mesh was used in this location in order to predict more accurately the dowel action. (a) (b) (c) Figure 27. Finite element model after meshing: (a) 3D view of mesh resolution; (b) cross section view; and (c) detail at location of transverse ties.

61 Boundary Conditions and Interactions The bridge was modeled as simply supported. One of the ends was fixed against translation in all directions. The other end was fixed against vertical and transverse translation, but was free to move longitudinally. No rotational restraints were imposed to the nodes. These restraints were imposed on the bottom edge at the ends of the beams as shown in Figure 28(a-b). (a) (b) Figure 28. Boundary conditions: (a) left end of the bridge; and (b) right end of the bridge. As part of the finite element modeling in Abaqus/CAE, it was necessary to assign interaction properties between the parts that are supposed to be in contact (i.e., interaction between beams and longitudinal reinforcement, shear key and diaphragms, and between the two beams). The interaction between the longitudinal reinforcement and the boxbeams was modeled as an embedment constraint. This type of interaction was used to prevent any slip of the longitudinal bars relative to the box-beams. On the other hand, the interaction between the beams and the diaphragms was modeled as a tie constraint. This prevents any movement between the diaphragms and the beams (Abaqus user s manual, 2006).

62 62 When defining the interaction between the beams and the shear key, a surface-tosurface contact behavior was used. To define the contact, normal and tangential behaviors must be defined. The normal behavior was modeled with a linear contact pressureoverclosure relationship between the two surfaces in contact. This behavior can be understood by analogy with the Hooke s law. Figure 29(a-b) shows graphically the behavior of the linear contact model. When the two surfaces are in contact, Abaqus/CAE allows a finite penetration between the surfaces. In this way, the pressure between them can be obtained by equilibrium using the Hooke s law. Thereby, a contact stiffness value between the two surfaces was required to be assigned. This was the most important parameter to define, since it affects the accuracy and convergence of the model. The contact stiffness must be large enough to minimize the penetration between surfaces (increasing the accuracy of the model), but small enough to avoid convergence issues, King et al., By trial and error, a value of 10 ksi/in was selected as appropriate contact stiffness. This value allowed balancing the convergence efficiency and the amount of penetration between surfaces (i.e., balancing computational time and accuracy of the results).

63 63 F normal normal (a) Pressure 1 k 0 Overclosure (b) Figure 29. Contact pressure-overclosure model: (a) analogy with Hooke s law; and (b) linear pressure-overclosure relationship. The tangential behavior was defined using the Coulomb friction model with a limit on the critical shear stress. The magnitude for the critical shear was chosen as 0.8 ksi according to results of numerous slant shear tests (Wan, Z., 2011). The friction coefficient used in the model was µ = 0.8 according to typical values of concrete-toconcrete friction coefficients (PCI Design Handbook, 2010). The friction model used in the analyses is shown in Figure 30. According to the Abaqus user s manual (2006), regardless the magnitude of the normal stress, sliding will occur if the shear stress reaches the critical shear stress τ.

64 64 Shear stress τ max 1 µ 0 Normal stress Figure 30. Friction model with a limit in the shear stress Loading Different loads were applied to the beams. These loads were divided into mechanical loads and thermal loads. Mechanical loads consisted of live loads and transverse post-tensioning, and thermal loads were applied based on temperature gradients through the beam s cross section. Live loads consisted of AASHTO LRFD design lane load plus the HL-93 wheel loads. These loads were applied to only one of the beams, as shown in Figure 31(a-b). This allowed analyzing load transfer between beams.

65 ksf Wheel loads of HL-93 design truck 21" 36" 16 kips (a) 16 kips 36" 4 kips 14' 14' ksf 21" (b) Figure 31. Applied live loads: (a) cross sectional view; and (b) elevation view. L As discussed in Section 3.1, transverse post-tensioning (TPT) was only applied to the first group of finite element models. To apply this load, a negative temperature field was induced to the tie rods using a predefined field. This is because steel tend to contract under negative temperatures, but restriction of tie rods at the ends transforms this temperature strains into mechanical strains, and thus mechanical stresses can be developed. The temperature required to reach the desired amount of TPT was obtained using based on based on equilibrium of a cable subject to a change of temperature ΔT, and assuming fixed ends. Temperature required to be applied to the tie rods is shown in Eq. 6.

66 Δ = TPT Eq. 6 where: α = Thermal expansion coefficient for steel = / F; E = Young s Modulus for transverse ties = 28,500 ksi; and A t = Area of transverse tie rod = in 2 Figure 32(a-b) shows the location of transverse ties used to apply transverse posttensioning for the first group of finite element models. 66 Transverse post-tensioning duct Transverse tie rod TPT TPT 21" 36" (a) Transverse tie locations 36" 21" L/3 L/3 L (b) Figure 32. Transverse post-tensioning application points for the first set of models: (a) cross section; and (b) elevation view. To prevent issues of straining perpendicular to the transverse tie due to the applied temperature, orthotropic thermal properties were assigned to the steel used for the transverse ties. The thermal expansion coefficient was set to zero in the directions

67 67 perpendicular to the transverse tie, and set to / F in the longitudinal direction of the bar. Thermal loads, consisting of temperature gradients through the beam s cross section were applied according to AASHTO LRFD Bridge Design Specifications (2010). The temperature gradient used in all the models was chosen to be suitable for bridges built in the state of Ohio. According to Figure 33, the state of Ohio falls in the zone 3 of the Solar Radiation Zones in the United States (AASHTO LRFD Bridge Design Specifications, 2010). Figure 33. Solar Radiation Zones for the United States (AASHTO LRFD Bridge Design Specifications, 2010). Once the zone was defined, the temperatures T 1 and T 2 were obtained to define the temperature gradient as shown in Figure 34. From Table 3, the obtained values for T 1 and T 2 were 41 F and 11 F, respectively.

68 68 Figure 34. Positive Vertical Temperature gradient in Concrete and Steel Superstructures (AASHTO LRFD Bridge Design Specifications, 2010). Table 3. Basis for temperature gradients (AASHTO LRFD bridge design specifications, 2010) Zone T 1 ( F) T 2 ( F) Outputs analyzed from FE models Different outputs from the finite element models were selected to analyze the behavior of adjacent box-beams. These outputs were maximum stresses in shear key, maximum shear stress in interface between shear key and box-beams, and maximum differential deflections between adjacent beams. Selection of outputs used to analyze the behavior of adjacent box-beams was based on failure criteria of shear key according to fib (2007), and recommendations in maximum differential deflections based on research by R y et al. (1996).

69 69 According to the International Federation for Structural Concrete, fib (2007), a shear key works as a mechanical lock preventing slip along the joint. When the shear key is loaded, its strength can be broken by cracking or local crushing of the grout. In addition, the joint strength can be broken when the shear stress in the interface between the shear key and the precast element reaches the shear strength. Based on fib (2007), four modes of failure were proposed to determine the performance of the joint for adjacent box-beam bridges. The first failure mode considered is due to cracking in the shear key due to diagonal tension as shown in Figure 35(a). Here, failure occurs when the maximum principal tensile stress in the shear key exceeds the tensile strength of the grout. The second failure mode is shown in Figure 35(b). In this case, failure in the joint occurs when the maximum shear stress in the shear key is larger than the maximum shear strength of the grout. The third failure mode is shown in Figure 35(c). In this case, failure is reached when the grout crushes due high compressive stresses. Thus, to determine if the third failure mode is present, the maximum principal compressive stress in the shear key is compared to the compressive strength of the grout. The three failure modes described above corresponds to failure in the material, but a failure in the interface between the shear key and the box beam can be also an issue. The fourth failure mode considers this case. As shown in Figure 35(d), the fourth failure mode consists on debonding between the shear key and the box beam. This is considered by comparing the maximum interfacial shear stress between the shear key and the box beam with the bonding strength in the same interface.

70 70 (a) (b) (c) (d) Figure 35. Proposed failure modes for the joint in adjacent box-beam bridges: (a) due to tensile stress; (b) due to shear stress; (c) due to local crushing; and (d) due to high shear stress at interface between shear key and box-beam. Based upon these failure modes, the following stresses were analyzed: Maximum tensile stress in shear key: obtained as the maximum value of all maximum principal stresses in shear key. Maximum compressive stress in shear key: obtained as the minimum value of all minimum principal stresses in shear key. Maximum vertical shear stress in shear key: obtained as the absolute value between the maximum and minimum value of in shear key. Maximum shear stress in shear key: obtained as the minimum value of /2 from all elements in shear key. Values of and correspond to maximum principal and minimum principal stress, respectively. Maximum contact stress in interface between shear key and box-beams: obtained as the maximum of contact stress resultants in interface between shear key and box-beams.

71 71 Differential deflections between beams were also analyzed. According to research by El-R y et al. (1996), a maximum vertical differential deflection between adjacent beams was set to 0.02 in. This limit was imposed to prevent reflective cracking issues in the wearing surface of the bridge. However, this value was based on very simplified finite element models. Since detailed 3D finite element models are used in this thesis, the author proposed measuring differential deflections in transverse, vertical, and longitudinal direction. This was because initial results from the finite element models showed that not only vertical differential deflections could be the cause of shear key cracking failure.

72 72 CHAPTER 4: RESULTS AND DISCUSSIONS This chapter presents and discusses the results from the finite element models used to study the shear force transfer mechanism in adjacent box-beam bridges. In order to present the results, this chapter was divided into two sections. Section 4.1 discusses the effect of the amount of transverse post-tensioning (TPT) per rod in the bridge s behavior. Section 4.2 analyzes the coupled effects of bridge s span and number of internal diaphragms in the bridge s response, when TPT = 0. In both sections, the contribution of dowel action in the structural response of the bridge was analyzed for various grout compressive strength-to-concrete compressive strength ratios (GS), and for different temperature conditions. Finally, the importance of each one of these parameters was statistically studied using Analysis of Variance (ANOVA). 4.1 Effects of Amount of TPT In this section, the contribution of dowel action and GS in the load transfer mechanism of adjacent box-beam bridges was studied as function of the amount of TPT. This study was performed by creating several finite element models of a pair of boxbeams coupled through a shear key and transverse tie rods. These finite element models were created using ABAQUS/CAE The geometry of all finite element models (beam s cross section, span, and number of diaphragms) was chosen from the adjacent box-beam bridge studied by Huffman (2012). This included a cross section of 36 21, a span of 47 10, and two internal diaphragms located at third points along the bridge s span (there are no transverse ties located at ends). In order to study the effect of TPT, models were loaded with values of TPT = 0, 5, 10, 15, 20, 30, 40, 50, or 80 kips applied

73 73 at the location of each transverse tie. The maximum value of 80 kips per rod was chosen according to research conducted by Grace et al. (2010). The contribution of dowel action was quantified by creating two independent sets of finite element models. This resulted in models with transverse post-tensioning ducts not filled with grout, and models with transverse post-tensioning ducts filled with grout. In the first case (i.e., models with non-filled ducts), there is no contact between the transverse ties and the duct, and thus, dowel action was not developed. In the second case (i.e., models with filled ducts), the contact between the transverse tie rods and the grout surrounding them allows dowel action to be present. The effects of grout compressive strength-to-concrete compressive strength ratio on model results were considered by selecting different values for GS (e.g., GS = 0.5, 1, and 2). Finally, all results were studied under the influence of positive and negative temperature gradients through the beam s depth, as explained in Chapter Induced Stresses in Shear Key Maximum values for tensile, compressive, and Von Mises stresses in shear key were obtained from all the models. Maximum tensile and compressive stresses were obtained from the maximum and minimum principal stresses in shear key, respectively. Effects of temperature gradients were quantified by computing the increase in maximum stresses induced in the shear key due to positive temperature gradient. Effects of dowel action, on the other hand, were quantified by computing the decrease in stresses induced in shear key due to dowel action. The percentage increase in maximum tensile,

74 74 compressive and Von Mises stresses due to a positive temperature gradient ( ) were quantified, according to Eq. 7 as follows: where % = Eq. 7. = maximum stress in shear key due to positive temperature gradient. = maximum stress in shear key when no temperature gradient is included Variable in Eq. 7 can be replaced for maximum tensile stress, maximum compressive stress, or maximum Von Mises stress in shear key depending upon which variable is being studied. On the other hand, the percentage decrease in the maximum tensile, compressive and Von Mises stresses in shear key due to dowel action ( ) were computed, according to Eq. 8 as follows: where % = 100 Eq. 8 = maximum stress in shear key when transverse posttensioning ducts are not filled with grout. = maximum stress in shear key when transverse posttensioning ducts are filled with grout.

75 75 Notice that the variable in Eq. 8 can be replaced for maximum tensile stress, maximum compressive stress, or maximum Von Mises stress in shear key depending upon which variable is being studied Maximum tensile stresses in shear key Figure 36(a-b) shows the induced tensile stresses in shear key as a function of the amount of TPT, and temperature gradients, for GS = 1 (results for GS = 0.5, and 2 are shown in Appendix I). Figure 36(a) presents these stresses for the condition where transverse post-tensioning ducts are not filled with grout. It is shown that, for zero and negative temperature gradients, maximum tensile stresses decrease for TPT greater than 5 kips per diaphragm. Similar behavior is observed for the case of positive temperature gradient. In this case, maximum tensile stress in shear key decreases when TPT varies from 5 kips per diaphragm to 30 kips per diaphragm. However, for higher values of TPT, this tendency is not very clear, but maximum tensile stresses in shear key tend to increase. Figure 36(b) shows maximum tensile stress in shear key for the case where the transverse ducts are filled with grout. For zero and negative temperature gradients, it is observed that maximum tensile stress in shear key increases as TPT increases. This is not the case for positive temperature gradient, where maximum tensile stress in shear key behaves approximately uniform. From Figure 36(a-b), it can be seen that dowel action has a significant effect in the induced tensile stresses in shear key. It is observed that maximum tensile stresses in shear key are smaller for the case where transverse ducts are filled with grout.

76 76 (a) (b) Figure 36. Maximum tensile stress in shear key as a function of TPT for GS = 1: (a) ducts not filled with grout; and (b) ducts filled with grout. Table 4 presents the increase in maximum tensile stress in shear key due to positive temperature gradients. The percentage of increase in the maximum tensile stresses was computed as a function of the amount of TPT, and for different values of GS. Values shown in Table 4 are obtained using Eq. 7, replacing the variables

77 77., and. by maximum tensile stress in shear key for positive and zero temperature gradients, respectively.. The following observations in Table 4 are made for the case where transverse post-tensioning ducts are not filled with grout: 1) the percentage of increase in maximum tensile stress in shear key is bigger as the value of GS increases. Some exceptions are, however, observed when TPT smaller or equal than 5 kips per diaphragm; 2) in general, the percentage of increase in tensile stress in shear key due to positive temperature gradient grows as amount of TPT increases; and 3) the maximum tensile stress in the shear key increases by more than 15% due to positive temperature gradients. The increase in maximum tensile stress in shear key goes up to almost 80% for high values of TPT. For the case where the transverse post-tensioning ducts are filled with grout, maximum tensile stress in the shear key behaves different than in the case explained above. However, in accordance with the behavior for non-filled ducts, the increase in maximum tensile stress in the shear key is higher as GS increases. An important fact is that, maximum tensile stresses in shear key decrease with amount of TPT. Thus, lower values of TPT produces the maximum increase in tensile stress in shear key. Maximum increases of 55.7%, 99.0%, and 135.5% were observed for values of TPT equal to zero.

78 Table 4. Increase in maximum tensile stress in shear key due to positive temperature gradient (%) TPT Transverse ducts not filled with grout Transverse ducts filled with grout (kips) GS = 0.5 GS = 1 GS = 2 GS = 0.5 GS = 1 GS = The decrease in maximum tensile stress in the shear key due to dowel action is shown in Table 5. These values were obtained using Eq. 8. Overall behavior in Table 5 shows that, as TPT increases, the percentage of decrease in the maximum tensile stress in the shear key due to dowel action also decrease. However, the dowel action occasionally produces increase in maximum tensile stresses in shear key. Fortunately, this increase is relatively small compared with the decrease in the majority of cases where dowel action helps to decrease tensile stress. It should be noted that the increase in maximum tensile stress in the shear key due to dowel action is often observed for higher values of TPT, or in some cases, for higher values of GS (e.g., GS = 2). It is also noticed for lower values of TPT (i.e., TPT less than 20 kips per diaphragm), dowel action produces approximately a 40% decrease in the maximum tensile stress in the shear key. This means that dowel action reduces the possibility of having cracks in the shear key due to tensile stress.

79 Table 5. Decrease in max. tensile stress in shear key due to dowel action (%) TPT (kips) No Temperature Gradient Positive Temperature Gradient Negative Temperature Gradient GS=0.5 GS=1 GS=2 GS=0.5 GS=1 GS=2 GS=0.5 GS=1 GS= Note: negative values indicate increase in stress Maximum compressive stresses in shear key Figure 37(a-b) presents the maximum compressive stress in the shear key as a function of the applied TPT, and for various temperature gradients. Results in this section are presented for GS = 1. For additional information, results for GS = 0.5 and GS = 2 are shown in Appendix I. As shown in Figure 37a and 36 b, it is generally noticed that the maximum compressive stress increased as the amount of TPT increased. Figure 37(a) shows that the rate of increment in compressive stress is maximum for TPT between 0 kips per rod and 10 kips per rod, and minimum for TPT varied between 10 kips per rod to 20 kips per rod. Figure 37(a) also shows that for the negative temperature gradient, the compressive stress slightly decreased when TPT varied from 10 kips per rod to 15 kips per rod. Figure 37(b) shows a more gradual change in the maximum compressive stress in shear key. Figure 37(b) shows that maximum compressive stress is higher for positive temperature gradients and increases as TPT increases.

80 80 (a) (b) Figure 37. Maximum compressive stress in shear key as a function of TPT for GS = 1: (a) ducts not filled with grout; and (b) ducts filled with grout. The increases in maximum compressive stresses in the shear key due to positive temperature gradient are shown in Table 6. These values were obtained using Eq. 7, replacing the corresponding stresses by the maximum compressive stresses in shear key, as previously explained. These values were obtained as a function of the amount of TPT, and for different values of GS. For the case where dowel action is not developed (i.e.,

81 81 models with transverse ducts not filled with grout), results showed that maximum compressive stress in shear key increased up to 42.9%, 48.7%, and 48.5% for values of GS equal to 0.5, 1, and 2, respectively. For the case where dowel action is present (i.e., models with transverse ducts filled with grout), the increase in maximum compressive stress in the shear key rise to maximum values of 61.3%, 66.1%, and 72.8%, for GS equal to 0.5, 1, and 2, respectively. From both sets of results, it is noticed that the increase in maximum compressive stress increases as GS increases. Another important fact obtained from the results is that, when the transverse ducts are filled with grout, maximum compressive stresses are more susceptible to temperature changes. This is because the increase in maximum compressive stresses due to positive temperature gradient is about 20% higher for the case where the transverse post-tensioning ducts are filled with grout. Table 6. Increase in maximum compressive stress in shear key due to positive temperature gradient (%) TPT Transverse ducts not filled with grout Transverse ducts filled with grout (kips) GS = 0.5 GS = 1 GS = 2 GS = 0.5 GS = 1 GS = Table 7 shows values obtained for the decrease in maximum compressive stress in the shear key due to dowel action. These values were obtained using Eq. 8, replacing the

82 82 corresponding stresses by the maximum compressive stresses, as previously explained. Results show that, when a positive temperature gradient is applied to the models, the contribution of dowel action in reducing the maximum compressive stress in the shear key is diminished. Furthermore, Table 7 shows that the maximum decrease in the compressive stress in the shear key is reached for a negative temperature gradient. In addition, results indicate that the decrease in compressive stress in shear key is bigger when GS = 2. This is due to higher reaction stresses in the concrete surrounding the transverse ties. Based on the results in Table 7, for values of TPT smaller than 15 kips per diaphragm, the following reduction in maximum compressive stress are expected to be obtained: 13.9% in average, when no temperature gradient is considered 5.4% in average, when positive temperature gradient is considered 26.4% in average, when negative temperature gradient is considered

83 Table 7. Decrease in max. compressive stress in shear key due to dowel action (%) No Temperature Positive Temperature Negative Temperature TPT Gradient Gradient Gradient (kips) GS=0.5 GS=1 GS=2 GS=0.5 GS=1 GS=2 GS=0.5 GS=1 GS= Maximum Von Mises stresses in shear key Figure 38(a-b) presents the maximum Von Mises stress the in shear key as a function of the applied TPT, and for various temperature gradient. Comparing Figure 37(a-b) and Figure 38(a-b), gives that the behavior of maximum compressive stresses and maximum Von Mises stresses in the shear key are similar. As explained for compressive stresses, Figure 38(a-b) shows that, in general, maximum Von Mises stress increased as the amount of TPT increased. However, and in agreement with the behavior of compressive stresses, the rate of change in the maximum Von Mises stress in shear key decreases as TPT increases.

84 84 (a). (b) Figure 38. Maximum Von Mises stress in shear key as a function of TPT for GS = 1: (a) ducts not filled with grout; and (b) ducts filled with grout. Table 8 presents the increase in maximum Von Mises stresses in the shear key due to the positive temperature gradient. These values were obtained (using Eq. 7) as a function of the amount of TPT, and for different values of GS. For the case with no dowel action (i.e., models with non-filled transverse ducts), results showed that maximum Von Mises stress in the shear key increased up to 43.6%, 48.5%, and 49.4%

85 85 for values of GS equal to 0.5, 1, and 2, respectively. For the cases with dowel action (i.e., models with filled transverse ducts), increase in maximum Von Mises stress in shear key rise maximum values of 61%, 65.8%, and 72.6%, for GS equal to 0.5, 1, and 2, respectively. According to the behavior of compressive stresses, it is noticed that increase in maximum Von Mises stress increases as GS increases. Higher susceptibility to temperature changes is also obtained for models with dowel action allowed. Table 8. Increase in max. Von Mises stress in shear key due to positive temperature gradient (%) TPT Transverse ducts not filled with grout Transverse ducts filled with grout (kips) GS = 0.5 GS = 1 GS = 2 GS = 0.5 GS = 1 GS = Table 9 shows values obtained for the decrease in maximum Von Mises stress in shear key due to dowel action. Results show that, when a positive temperature gradient is applied to the models, reduction in maximum Von Mises stress in shear key is diminished. Furthermore, Table 9 shows that the maximum contribution of dowel action is reached for a negative temperature gradient. In addition, results indicate that the decrease in Von Mises stress in shear key is bigger when the grout compressive strength is higher than the concrete compressive strength. This is due to higher reaction stresses in

86 86 the concrete surrounding the transverse ties. Based on the results in Table 9, for values of TPT smaller than 15 kips per diaphragm, the following reduction in maximum Von Mises stress are expected to be obtained: 14.0% in average, when no temperature gradient is considered 5.7% in average, when positive temperature gradient is considered 26.7% in average, when negative temperature gradient is considered Table 9. Decrease in max. Von Mises stress in shear key due to dowel action (%) No Temperature Positive Temperature Negative Temperature TPT Gradient Gradient Gradient (kips) GS=0.5 GS=1 GS=2 GS=0.5 GS=1 GS=2 GS=0.5 GS=1 GS= Differential Deflections between Beams Differential deflections in the transverse, vertical, and longitudinal direction were obtained from the models as a function of the amount of TPT. A schematic view of the direction of measurement is shown in Figure 39. Different sections along bridge s span were selected to take the model results (i.e., x = 0, L/6, L/3, and L/2, where L is the bridge s span). Results from the finite element models showed that differential deflections are not strongly affected by the amount of GS or by temperature gradients.

87 For this reason, only differential deflections for GS = 1 and for no temperature gradients are presented in this chapter. Results for all cases are shown in Appendix II. 87 δu 1 (a) δu 3 δu 2 (b) (c) Figure 39. Measured differential deflections between adjacent beams: (a) transverse direction; (b) vertical direction; and (c) longitudinal direction. Figure 40(a-b) shows results for the transverse differential deflections between beams as function of TPT. Results in Figure 40(a) indicate that values of TPT less or equal to 10 kips per diaphragm produce significant changes in differential deflections. For TPT = 0 kips, maximum transverse differential deflection is in, and for TPT = 10 kips, maximum transverse differential deflection is in. This represents a reduction of 95.4% in differential deflections when a small TPT is applied to the beams. Figure 40(b) shows a noticeable reduction in the differential deflections due to dowel action. In this case, transverse differential deflections are not highly influenced by TPT. Negative values for transverse differential deflections are observed in Figure 40(a-b). This is because a penalty model to determine the contact between adjacent beams was used. Since contact stiffness was used to model the normal contact in the interface between beams, it is always possible to obtain penetration between surfaces in contact.

88 88 As observed in Figure 40(a-b) the maximum surface-to-surface penetrations (e.g., obtained when δu 1 < 0) are obtained at x = L/3, which corresponded to the location of the transverse ties. Figure 40(a-b) also shows that the magnitude of this penetration increases linearly with the amount of TPT. Smaller penetrations, on the other hand, are obtained at locations more distant from the location of the transverse ties (i.e., x = 0, L/6, and L/2). (a) (b) Figure 40. Transverse differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout.

89 89 Figure 41(a-b) presents the results for vertical differential deflection between beams. Results show that, when no dowel action is considered, maximum vertical differential deflections are in when TPT = 0 kips per diaphragm, and in when TPT = 10 kips per diaphragm. This means a reduction of 78.2% in vertical differential deflections. Based on these results, when no dowel action is considered, differential deflections can be effectively reduced for values of TPT less than 10 kips per diaphragm. Also, it is also noticed that values of TPT greater than 10 kips per diaphragm do no contribute significantly in reducing vertical differential deflections. Figure 41(b) shows the contribution of dowel action. These results show reductions in vertical differential deflections due to dowel action. These reductions were 90.2% when TPT = 0 kips per diaphragm, and 56.4% when TPT = 10 kips per diaphragm.

90 90 (a) (b) Figure 41. Vertical differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout. Figure 42 presents results for longitudinal differential deflections between adjacent beams (i.e., sliding between beams). In accordance with findings for transverse and vertical differential deflections, for the case where no dowel action is considered (Figure 42a), small values of TTP can be used to reduce differential deflections between adjacent beams. In a similar way, when dowel action is incorporated into the model,

91 91 differential deflections are reduced, and not highly influenced by the amount of TPT (Figure 42b). (a) (b) Figure 42. Longitudinal differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout. Based on the results presented in Figure 40-Figure 42, the following can be summarized:

92 92 When transverse post-tensioning ducts are not filled with grout (i.e., no dowel action), differential deflections between adjacent beams can be significantly reduced by applying a TPT = 10 kips per diaphragm. Higher values of TPT do not contribute significantly in reducing differential deflections. For cases where transverse post-tensioning ducts are filled with grout (i.e., dowel action allowed), differential deflections are significantly reduced. Also, these differential deflections are not highly affected by variations in TPT Statistical Analysis Statistical analysis of all results obtained from the finite element models was performed using Analysis of Variance (ANOVA). A significance level of 0.05 was used for all the analyses. This section focused on two main aspects. First, it was determined if all parameters considered in the models (i.e., dowel action, TPT, temperature gradients, and GS) significantly affected maximum stresses in shear key, and differential deflections between beams. Second, it was studied if, when controlled by different factors, dowel action significantly affects maximum stresses in shear key and differential deflections between adjacent beams. The first parameter studied in this section was dowel action. Results from ANOVA tests showed that, when all models were analyzed simultaneously, maximum stresses in shear key (i.e., tensile, compressive and Von Mises) as well as differential deflections in the three components, were significantly affected by dowel action. Based upon mean values obtained from descriptive statistics tables, it was observed that both

93 93 maximum stresses and differential deflections were smaller when dowel action was acting. The second parameter studied was the amount of TPT. Results from ANOVA tests showed that maximum stresses in shear key (tensile, compressive, and Von Mises) were not significantly affected by the amount of TPT. However, differential deflections between adjacent beams showed to be significantly affected by amount of TPT. In order to find which amount of TPT produced a significant decrease in the differential deflections, a Games-Howell post hoc test was performed. Selection of the type of post hoc test was based on equality of variances test. Results from post hoc test indicated that a minimum value of TPT = 10 kips per diaphragm was necessary to obtain a significant reduction in maximum transverse and vertical differential deflections (δu 1,max and δu 2,max, respectively), when compared with differential deflections for TPT = 0. Minimum required TPT to obtain a significant reduction in maximum longitudinal differential deflections (δu 3,max ), was TPT = 15 kips per diaphragm. The third parameter studied was temperature gradients. Results from ANOVA tests revealed that maximum stresses in shear key (tensile, compressive, and Von Mises) were significantly affected by temperature gradients. In order to understand where is the difference between groups (i.e., difference between models with no temperature gradient, positive temperature gradient, and negative temperature gradient), post hoc test was performed. Because equality of variances tests showed that variances were not equal between groups, Games-Howell was used. This test indicated that results from models with positive temperature gradients were different than results from models with zero and

94 94 negative temperature gradients. Confidence intervals helped identifying that, when positive temperature gradients are included in the finite element models, all tensile, compressive, and Von Mises stresses increased compared with models with zero and negative temperature gradients. Results from ANOVA tests, on the other hand, showed that differential deflections were not significantly affected by temperature gradients. The last parameter analyzed was GS. Results from ANOVA tests revealed that all tensile, compressive, and Von Mises stresses were significantly affected by GS. Post hoc tests showed that maximum tensile stress in shear key significantly increased for GS increasing from 0.5 to 2, or 1 to 2, but a significant increase in tensile stress was not found for GS increasing from 0.5 to 1. However, maximum compressive and Von Mises stresses in shear key were significantly increased as GS increased. Differential deflections, on the other hand, were not significantly affected by GS. 4.2 Effects of Span Length and Number of Internal Diaphragms In this section, the coupled effects of bridge s span-to-depth ratio, and number of internal diaphragms along the bridge s span length were studied in detail. All analyses were performed using the same beam s cross section as in the bridge studied by Huffman (2012). An amount of TPT = 0 was used for each model. This was because many of the adjacent box-beam bridges currently constructed have a very small amount or no TPT. Three different beam spans were selected (i.e., L = 25, 47 10, and 62 6 ), which led to span-to-depth ratios of L/D = 14.29, 27.33, and 35.71), and then, for each beam span, different numbers of internal diaphragms, N, were considered (i.e., N = 1, 2, and 3). Dowel action and temperature effects were studied in the same way as in the models

95 95 explained in Section 4.1. Several outputs were studied for every case of span-to-depth ratio, number of internal diaphragms, existence or absence of dowel action, and temperature conditions. These outputs included the stresses induced in the shear key, interfacial stresses in the contact area between the shear key and box-beams, and the differential deflections between adjacent beams Stresses Induced in Shear Key The maximum values for stress in the shear key and the interfacial stress between shear key and box-beams were studied as a function of span-to-depth ratio, number of internal diaphragms, existence or absence of dowel action, and temperature conditions. Increment in stresses due to positive temperature gradients, and decrease in stresses due to dowel action were obtained using Eq. 7, and Eq. 8, respectively. The following sections present, in detail, the analysis performed on each of the outputs studied from the models Maximum Tensile Stresses in Shear Key The analysis of maximum tensile stresses in the shear key as function of bridge s span are presented. In order to have an idea of the overall behavior of all the models, only results for GS = 1, and N = 2 are presented in this section. Additional results are presented in Appendix III. Figure 43(a) presents the tensile stress obtained for the models with transverse post-tensioning ducts not filled with grout (i.e., models with no dowel action). Figure 43 (b), on the other hand, presents results obtained for models with transverse post-tensioning ducts filled with grout (i.e., models with dowel action).

96 96 The results indicated that for higher values of span-to-depth ratio, the maximum tensile stress in the shear key increased. Also, the maximum tensile stress in the shear key was found to increase with positive temperature gradient. Another important aspect found through the results was that maximum tensile stresses tend to decrease when dowel action was included in the models. However, there was an exception for L/D = 35.71and positive temperature gradients. In this case, results showed increase in maximum tensile stress in shear key due to dowel action. Comparing Figure 43 (a-b) shows that the rate of change in maximum tensile stress in shear key as function of L/D is larger for models with dowel action than for models with no dowel action.

97 97 (a) (b) Figure 43. Maximum tensile stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout. Table 10 presents values obtained for the increase in maximum tensile stress in the shear key due to a positive temperature gradient. Values presented in Table 10 were obtained using Eq. 7. Overall results showed that the increase in maximum tensile stress in the shear key was higher for models with dowel action (i.e., models with transverse

98 98 post-tensioning ducts filled with grout). Results showed that, on average, the percentage of increase in maximum tensile stress in the shear key was 28.3% for models with no dowel action and 86.8% for models with dowel action. This implies that dowel action makes bridges more susceptible to an increase in stress due to a positive temperature gradient. Effects of span-to-depth ratio were analyzed by combining the results for N = 1, 2, and 3, and GS = 0.5, 1, and 2. For models with no dowel action, average increases in maximum tensile stress were 39.4%, 21.6%, and 23.8%, for L/D = 14.29, 27.33, and 35.71, respectively. For models with dowel action, average increases in maximum tensile stress were 93.8%, 93.1%, and 73.4%, for L/D = 14.29, 27.33, and 35.71, respectively. Based in these results, increment in maximum tensile stress in the shear key tends to be smaller for longer beam spans. However, no clear conclusion can be made due to the variability of the results. The next parameter analyzed was the number of internal diaphragms. On average, results showed that, for models with no dowel action, increase in maximum tensile stress in shear key due to positive temperature gradients was 21.1%, 30.8%, and 33.0%, for N = 1, 2, and 3, respectively. For models with dowel action, average increases were 74.9%, 96.0%, and 89.5%, for N = 1, 2, and 3, respectively. Based on these results, increase in maximum tensile stress in shear key due to positive temperature gradients tend to be larger for higher values of N. However, for models with dowel action, percentage of increase in maximum tensile stress in shear key was smaller for N =3 than it was for N = 2.

99 99 The last parameter studied was grout strength. Results show that, on average, when GS increases, increment in tensile stress in shear key due to a positive temperature gradient also increases. Average values of 52.4%, 55.3%, and 64.9% were obtained for values of GS = 0.5, 1, and 2, respectively. Table 10. Increment in max. tensile stress in shear key due to positive temperature gradient (%) Transverse GS = 0.5 GS = 1 GS = 2 posttensioning N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 L/D ducts Non-filled Filled Table 11 presents decrease in maximum tensile stress in shear key due to dowel action. Values shown in Table 11 were obtained using Eq. 8. From the results it was found that decrease in tensile stress in shear key due to dowel action was, on average, 37.3%, 10.8%, and 22.0% for models with zero, positive, and negative temperature gradients, respectively. These numbers showed that when positive temperature gradients are applied to the models, dowel action has less effect in reducing maximum tensile stress in the shear key.

100 100 Table 11. Decrease in max. tensile stress in shear key due to dowel action (%) L/D GS = 0.5 GS = 1 GS = 2 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 No Temperature gradient Positive Temperature gradient Negative Temperature gradient Note: negative values indicate increase in stress Maximum Compressive Stresses in Shear Key Figure 44(a) shows maximum compressive stresses in the shear key for GS = 1 and N = 2, and no dowel action. Figure 44(b) shows maximum compressive stresses in the shear key for GS = 1 and N = 2, and no dowel action. Results show that compressive stress in the shear key increased as span-to-depth ratio increased. Results showed that positive temperature gradient increase maximum compressive stress in the shear key, and negative temperature gradients slightly decrease maximum compressive stress in the shear key. It is noticed that curves in Figure 44(b) are more distant one from each other than curves in Figure 44(a). This indicates that models where dowel action is acting are more susceptible to temperature changes. Also, comparing Figure 44(b) and Figure 44(a), it is noticed that dowel action helps decreasing maximum compressive stress in the shear key. However, for the cases with positive temperature gradient, this effect is negligible.

101 101 (a) (b) Figure 44. Maximum compressive stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout. Table 12 shows computed values for the increment in maximum compressive stress in the shear key due to positive temperature gradients. Results showed average increase in maximum compressive stress in shear key of 276.9%, 47.0%, and 29.4%, for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively, and non-filled transverse post-tensioning ducts. The average increases for filled transverse post-tensioning ducts

102 102 were found to be 349.2%, 68.3%, and 43.8%, for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. This indicates that shorter bridges experience higher increase in maximum compressive stress in shear key due to temperature changes. However, regardless the magnitude the increases obtained through these analyses, maximum compressive stress in shear key are still smaller than the grout compressive strengths. Number of internal diaphragms, on the other hand, did not have a clear effect in increasing maximum compressive stress in the shear key due to positive temperature gradients. Average values of increase in maximum compressive stress were 130.5%, 142.2%, and 134.6%, for values of N = 1, 2, and 3, respectively. However, grout strength was found to slightly affect the increase in maximum compressive stress in the shear key due to positive temperature gradients. It was found that as GS increases, the increment in maximum compressive stress also increases. Average values of 123.8%, 135.2%, and 148.3% were found for GS = 0.5, 1, and 2, respectively. Table 12. Increment in max. compressive stress in shear key due to positive temperature gradient (%) Transverse GS = 0.5 GS = 1 GS = 2 posttensioning N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 L/D ducts Non-filled Filled

103 103 Table 13 shows results for the decrease in maximum compressive stress in the shear key due to dowel action. Results showed that, on average, the decrease in compressive stress due to dowel action was strongly affected by span-to-depth ratio, number of internal diaphragms, and temperature gradients. The effects of grout strength were not very influential. The average reduction in maximum compressive stress in the shear key due to dowel action was 24.6%, 13.8%, and 15.1%, for span-to-depth rations of 14.29, 27.33, and 35.71, respectively. This shows that, in general, the maximum compressive stress in the shear key decreases with span-to-depth ratio. Additional findings revealed that increasing the number of diaphragms helped reduce compressive stresses in the shear key. On average, the percentages of reduction of 8.2%, 20.8%, and 24.6%, were found for values of N = 1, 2, and 3, respectively. On the other hand, an average reduction of 12.7%, 0.1%, and 40.7% in the maximum compressive stress in the shear key was obtained for zero temperature gradient, positive temperature gradient, and negative temperature gradient, respectively. Thus, it can be concluded that contribution of dowel action is almost null when positive temperature gradients are acting in the models.

104 Table 13. Decrease in max. compressive stress in shear key due to dowel action (%) L/D GS = 0.5 GS = 1 GS = 2 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 No Temperature gradient Positive Temperature gradient Negative Temperature gradient Note: negative values indicate increase in stress Maximum Von Mises Stresses in Shear Key Figure 45(a-b) shows the maximum Von Mises stress in the shear key as function of span-to-depth ratio, and for different temperature gradients. Based on results, it was found that Von Mises stress increased when span-to-depth ratio increased. In addition, positive temperature gradients increased maximum Von Mises stresses in the shear key, and negative temperature gradients decreased them. Also, it is noticed that curves in Figure 45(b) are more separated than curves in Figure 45(a), which indicates that models with dowel action experience more stress variation than models without dowel action. On the other hand, when comparing Figure 45(a) and Figure 45(b), dowel action effects can be observed. It is noticed that maximum Von Mises stress in the shear key are lower in Figure 45(b). Thus, it is concluded that dowel action has a beneficial contribution. However, contribution of dowel action is lower for models with positive temperature gradient.

105 105 (a) (b) Figure 45. Maximum Von Mises stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout. Table 14 presents computed values for the increment in the maximum Von Mises stress in the shear key due to positive temperature gradients. In accordance with the results presented for the increase in compressive stresses in shear key, it was observed smaller span-to-depth ratios led to higher increase in the maximum Von Mises stresses. Results indicated the average increases in Von Mises stress of 199.2%, 46.8%, and 29.3%, for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively, and non-filled

106 106 transverse post-tensioning ducts. Furthermore, it was found that average increases in Von Mises stress were 348.0%, 68.0%, and 43.5%, for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively, with filled transverse post-tensioning ducts. From these results, it is concluded that models with dowel action experienced higher increases than models without dowel action. Thus, as explained in section above, when dowel action was considered, adjacent box-beam bridges were more sensitive to temperature changes. Table 14. Increment in max. Von Mises stress in shear key due to positive temperature gradient (%) Transverse GS = 0.5 GS = 1 GS = 2 posttensioning N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 L/D ducts Non-filled Filled Table 15 shows the computed decrease in maximum Von Mises stress in the shear key due to dowel action. The average decrease in maximum Von Mises stress due to dowel action was found to be affected by span length, number of diaphragms, and temperature gradients. However, this decrease was not highly affected by GS. Average reductions in maximum Von Mises stress in shear key were 27.0%, 14%, and 15.3% for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. On the other hand, the average reduction in Von Mises stress was found to be 9.9%, 21.7%, and 25.0% for N = 1, 2, and 3, respectively. Analyzing effects of temperature gradient, it was found that

107 107 average percentage of reduction was 18.9% when no temperature gradients were considered, 0.4% when positive temperature gradients were included, and 37.2% when negative gradients were included. These last results indicate that contribution of dowel action is almost negligible under positive temperature gradients. Analysis of GS revealed an average reduction in maximum Von Mises stress of 19.8%, 19.1%, and 17.7%, for values of GS = 0.5, 1, and 2, respectively. Table 15. Decrease in max. Von Mises stress in shear key due to dowel action (%) L/D GS = 0.5 GS = 1 GS = 2 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 No Temperature gradient Positive Temperature gradient Negative Temperature gradient Note: negative values indicate increase in stress Maximum Vertical Shear Stresses in Shear Key Figure 46(a-b) presents obtained maximum vertical stress in the shear key as a function of span-to-depth ratio. Results are shown for zero, positive and negative temperature gradients and for GS = 1 and N = 2. Figure 46(a) shows results for models with no dowel action. In this case it is observed that maximum vertical shear stress in the shear key are unaffected by temperature gradients. Figure 46(b), shows results for models with dowel action. For this case, vertical stress in the shear key was affected by

108 108 temperature gradients. It is observed in Figure 46(b) that a positive temperature gradient increases vertical shear stress in the shear key and a negative temperature gradient decreases it. In both cases, it is observed that maximum vertical shear stress in shear key increases as span-to-depth ratio increases. Finally, it is noticed that, for models with dowel action allowed, the maximum vertical shear stress in the shear key are higher than in models with no dowel action. (a) (b) Figure 46. Maximum vertical shear stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout.

109 109 Table 16 shows the computed increment in maximum vertical shear stress in the shear key due to positive temperature gradients. The increase in stress was studied as function of span-to-depth ratio, number of internal diaphragms, and grout compressive strength-to-concrete compressive strength ratio (GS). Results indicated that temperature gradients had no effect on maximum vertical stress in shear key when transverse posttensioning ducts were notfilled. However, when transverse post-tensioning ducts were filled with grout, an average increment of % was obtained. For this reason, the following observations are for models where dowel action is considered. Based upon average results, the increase in vertical stress in the shear key decreased as span-to-depth ratio increased. Increments of 240.9%, 61.5%, and 29.6% were obtained for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. These results indicated that when dowel action is considered, shorter bridges were more sensitive to temperature changes than longer bridges. In addition, number of internal diaphragms also affected the increment in vertical shear stress. On average, the increments were 79.3%, 55.1%, and 197.6% for N = 1, 2, and 3, respectively. Since there was not a defined tendency in this behavior, further research is suggested to be performed. The last parameter studied was GS. It was observed that, as GS increased, the average increment in vertical shear stress also increased. Increases of 81.1%, 103.8%, and 147.2% were obtained for values of GS = 0.5, 1, and 2. From these results, it can be concluded that stiffer shear keys are more likely to increase induced vertical stresses under temperature changes.

110 110 Table 16. Increment in max. vertical shear stress in shear key due to positive temperature gradient (%) Transverse GS = 0.5 GS = 1 GS = 2 posttensioning N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 L/D ducts Non-filled Filled The computed values for the decrease in maximum vertical shear stress in the shear key due to dowel action are shown in Table 17. Analysis of the results demonstrated that, on average, a reduction in maximum vertical shear stress in the shear key due to dowel action: 1) decreased with bridge s span; 2) increased slightly as number of internal diaphragms increases; 3) reduced due to positive temperature gradient, and grew due to negative temperature gradient; and 4) slightly decreased as GS increased. Average reduction in maximum vertical shear stress in the shear key due to dowel action was 73.2%, 65.9%, and 63.9%, for span-to-depth ratio of 14.29, 27.33, and 35.71, respectively. In addition, it was found that average reduction was 62.8%, 64.9%, and 75.4% for N = 1, 2, and 3, respectively. Based on GS, average reduction was 69.5%, 68.2%, and 65.3%, for GS = 0.5, 1, and 2, respectively. Temperature gradients, on the other hand, showed to have a bigger impact in reduction of maximum vertical shear stress in the shear key due to dowel action. It was found that, when no temperature gradients were implemented in the models, a reduction of 72.8% was found, but when positive gradient was implemented, a reduction of 55.4% was encountered.

111 111 Table 17. Decrease in max. vertical shear stress in shear key due to dowel action (%) L/D GS = 0.5 GS = 1 GS = 2 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 No Temperature gradient Positive Temperature gradient Negative Temperature gradient Maximum Shear Stresses in Shear Key Figure 47(a-b) shows maximum shear stress in shear key. These stresses were obtained based on principal stresses in the shear key as shown in Eq. 9. = 2 Eq. 9 Figure 47(a) represents values obtained for models with no dowel action, and Figure 47(b) for models with dowel action. Results indicate that maximum shear stress in the shear key increase with span-to-depth ratio. Also, in both cases, results showed that these stresses are higher for positive temperature gradient. Finally, by comparing Figure 47(a) and Figure 47(b), it is observed that maximum shear stress in shear key are smaller for models with dowel action. Thus, it is concluded that dowel action contributes to decreasing the maximum shear stresses in the shear key.

112 112 (a) (b) Figure 47. Maximum shear stress in shear key as a function of span-to-depth ratio (GS = 1, and N = 2): (a) ducts not filled with grout; and (b) ducts filled with grout. Table 18 shows values computed for the increment in maximum shear stress in the shear key due to positive temperature gradients. Results showed that, in general, models with dowel action experienced higher increments in stress than models with no dowel action. Average increases of 82.7%, and 153.1% were obtained for models with no dowel action, and dowel action, respectively. Based on all results for models with no dowel action, average increase in maximum shear stress in the shear key was 172.4%,

113 %, and 29.3%, for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. Now, for models with dowel action, average increase in maximum shear stress in the shear key was 347.0%, 68.3%, and 43.9%, for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. These results are consistent in predicting that longer bridges experience less of an increment in maximum shear stress in the shear key due to positive temperature gradients. The number of diaphragms also affected the increase in maximum shear stress in the shear key due to positive temperature gradients. Average values of 111.9%, 123.3%, and 118.5% were obtained for values of N = 1, 2, and 3, respectively. Finally, increasing the value of GS showed an increase in the percentage of increment in maximum shear stress in the shear key due to positive temperature gradients. Average increase values of 107.3%, 116.5%, and 129.9% were obtained for GS = 0.5, 1, and 2, respectively. Table 18. Increment in max. shear stress in shear key due to positive temperature gradient (%) Transverse GS = 0.5 GS = 1 GS = 2 posttensioning N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 L/D ducts Non-filled Filled Table 19 presents the reduction in maximum shear stress in the shear key due to dowel action. Based on the results, the average reduction of shear stress in the shear key

114 114 due to dowel action: 1) reduced as bridge s span-to-depth ratio increased; 2) increased as number of diaphragms increased; 3) decreased as GS increased; and 4) nearly vanished when positive temperature gradients are implemented, and increased when negative temperature gradients were implemented. An average reduction of 31.2%, 16.8%, and 15.7% was obtained for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. Also, the average reductions of 13.1%, 23.6%, and 26.9% were found for N = 1, 2, and 3, respectively. Furthermore, it was obtained that the least variation in contribution of dowel action was found as a function of GS. In this case, average reductions were 22.1%, 21.4%, and 20.1%, for GS = 0.5, 1, and 2, respectively. An average reduction of 1.1%, on the other hand, was obtained when a positive temperature gradient was considered. Table 19. Decrease in max. shear stress in shear key due to dowel action (%) L/D GS = 0.5 GS = 1 GS = 2 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 No Temperature gradient Positive Temperature gradient Negative Temperature gradient Note: negative values indicate increase in stress Maximum Contact Shear Stresses in Interface between Shear Key and Box-Beams Figure 48(a-b) shows maximum contact stress in the interface between the shear key and box beams for models with no dowel action (a) and models with dowel action

115 115 (b). Maximum contact stresses decreased, in most cases, with positive temperature gradients. Also, contact stresses increased with span-to-depth ratio. Finally, when comparing Figure 48(a) and Figure 48(b), it is noticed that dowel action helps decreasing maximum contact shear stress. (a) (b) Figure 48. Maximum contact shear stress between shear key and box beams as a function of span-to-depth ratio (GS = 1, and N = 1): (a) ducts not filled with grout; and (b) ducts filled with grout.

116 116 Table 20 shows the increment in maximum contact stress in the interface between the shear key and box-beams due to a positive temperature gradient. As shown in Table 20, most of the results were negative, which indicates that positive temperature gradient decreased contact shear between shear key and box-beams. For models where dowel action was not considered, average reductions of 14.6%, 7.5%, and 6.5% were obtained for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. However, when dowel action was considered, an increment of 207.1% was observed for span-to-depth ratio of 14.29, but decreases of 9.4% and 14.2% were obtained for span-to-depth ratios of 27.33, and 35.71, respectively. Table 20. Increment in max. contact shear in interface between shear key and box-beams due to positive temperature gradient (%) Transverse GS = 0.5 GS = 1 GS = 2 posttensioning N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 L/D ducts Non-filled Filled Table 21 shows the decrease in maximum contact shear stress in the interface between the shear key and box-beams due to dowel action. Results revealed that average reductions in maximum contact stress due to dowel action: 1) reduced as span-to-depth ratio increased; 2) increased as number of diaphragms increased; 3) reduced as GS increased; and 4) reduced for models with positive temperature gradient. From the

117 117 results, it was obtained that average reduction in maximum contact shear stress was 94.8%, 72.1%, and 68.6% for span-to-depth ratios of 14.29, 27.33, and 35.71, respectively. Average reductions of 69.5%, 78.2%, and 87.8% were found for N = 1, 2, and 3, respectively. In terms of GS, average reductions of 79.7%, 78.6%, and 77.2% were obtained for GS = 0.5, 1, and 2, respectively. In terms of temperature gradient, the behavior was found to be different from results obtained for other analyzed outputs. In this case, reductions in contact stress were 78.6%, 77.3%, and 79.3%, for zero, positive, and negative temperature gradients, respectively. Table 21. Decrease in max. contact shear in interface between shear key and box-beams due to dowel action (%) L/D GS = 0.5 GS = 1 GS = 2 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 N = 1 N = 2 N = 3 No Temperature gradient Positive Temperature gradient Negative Temperature gradient Differential Deflections between Adjacent Beams Differential deflections in the transverse, vertical, and longitudinal direction were obtained from the models as a function of span-to-depth ratio. Different sections along bridge s span were selected to take the measurements (i.e., x = 0, L/6, L/3, and L/2, where L is the span length of the bridge). The maximum absolute value of these

118 118 measurements was used to estimate the maximum differential deflection in each of the three directions mentioned above. Results from the finite element models showed that differential deflections were not affected by the amount of GS, and slightly influenced by temperature gradients. In order to make results easily understood by the reader, only differential deflections for GS = 1 and for no temperature gradients are presented in this section. Results for all cases are shown in Appendix IV. Figure 49(a-b) shows results for the maximum transverse differential deflections between beams as function of span-to-depth ratio. Figure 49(a) shows results for models with no dowel action, and Figure 49(b) shows results for models with dowel action. Results indicated that differential deflections in the transverse direction increase as spanto-depth ratio increase. Only one exception is observed for models with dowel action and N = 3. Also, it was noticed that increasing the number of diaphragms helped decreasing maximum differential deflections between adjacent beams. A final observation is that dowel action contributed significantly in reducing differential deflections between beams.

119 119 (a) (b) Figure 49. Transverse differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout. Figure 50(a-b) shows results for the maximum vertical differential deflections between beams as function of span-to-depth ratio. Figure 50(a) shows results for models with no dowel action, and Figure 50(b) shows results for models with dowel action. The results indicated that maximum vertical differential deflections: 1) increased as span-todepth ratio increased; 2) decrease as number of diaphragms increased; and 3)were smaller when dowel action was considered. For the case where dowel action is not acting, it was also observed that maximum vertical differential deflections were higher than the value

120 120 of 0.02 in proposed by El-R y et al. (1996). However, this criterion was met in the majority of cases when dowel action was included. This indicates that, when TPT is not applied to the bridge, differential deflections can be controlled by filling transverse posttensioning ducts with grout or by increasing the number of diaphragms. (a) (b) Figure 50. Vertical differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout.

121 121 Figure 51(a-b) shows results for the maximum longitudinal differential deflections between beams as function of span-to-depth ratio. Figure 51 (a) shows results for models with no dowel action, and Figure 51 (b) shows results for models with dowel action. This differential deflection showed to be reduced as number of internal diaphragms increased, or when dowel action was acting in the models. However, tendency shows that this differential deflection is not always increasing with span-todepth ratio. (a) (b) Figure 51. Longitudinal differential deflection between adjacent beams: (a) ducts not filled with grout; and (b) ducts filled with grout.

122 122 The values used to create Figure 49Figure 51 are shown in Table 22. These values showed that models with no dowel action experienced considerably higher differential deflections than models with dowel action. It was found that, on average, the maximum differential deflections for models with no dowel action were: 1) 35 times higher in transverse direction; 2) 14 times higher in vertical direction; and 3) 9 times higher in longitudinal direction. It was also observed that, on average, dowel action helped reduce differential deflections in 83.3%, 91.4%, and 94.9%, for N = 1, 2, and 3, respectively. These results allowed quantifying the contribution that dowel action has in reducing differential deflections between adjacent beams. Table 22. Maximum differential deflections between adjacent beams NO DOWEL ACTION DOWEL ACTION N L/D δu 1,max δu 2,max δu 3,max δu 1,max δu 2,max δu 3,max Statistical Analysis Results from all finite element models were statistically analyzed using ANOVA. As explained in Section 4.1.3, a significance level of 0.05 was used for all the analyses. This analysis was divided into two parts. The first part consisted on studying the

123 123 significance of all parameters considered in the models (i.e., dowel action, temperature gradients, span-to-depth ratio, number of internal diaphragms, and grout compressive strength-to-concrete compressive strength ratio) on both maximum stresses in the shear key and maximum differential deflections between adjacent beams. The second part studied if, when controlled by different factors, dowel action produced a significant contribution in reducing stresses in the shear key and differential deflection between each beam. This included studying if the effects of temperature gradient, span-to-depth ratio, number of internal diaphragms, or grout compressive strength-to concrete compressive strength ratio affected the significance of the contribution that dowel action had in reducing stresses in the shear key and differential deflections between adjacent beams. First, the effect of dowel action on both maximum stresses in the shear key and differential deflections between beams was studied using ANOVA. For this part, all analyses were performed based on results from all finite element models. Results from ANOVA tests showed that maximum tensile stress and vertical stress in shear key as well as maximum contact shear in interface between shear key and box-beams were significantly affected by dowel action. Mean values obtained from the descriptive statistics table showed that models with no dowel action experienced higher tensile and vertical stresses in shear key as well as higher contact stresses in interface between the shear key and box-beams than models with dowel action. On the other hand, results from ANOVA tests showed that maximum compressive, Von Mises, and shear stress in the shear key were not significantly affected by dowel action. Finally, differential deflections between beams were found to be significantly affected by dowel action. Based upon

124 124 mean values obtained from descriptive statistics tables, it was observed that differential deflections were smaller when dowel action was acting. The second parameter studied from the models was the temperature gradient. According to results from ANOVA tests, maximum tensile, maximum compressive, Von Mises, and shear stress in the shear key were significantly affected by this parameter. Descriptive statistics tables showed that, in general, stresses in shear key increased for positive temperature gradients, and decreased for negative temperature gradients. Maximum vertical shear, maximum contact shear, and maximum differential deflections, on the other hand, were not affected by temperature gradients. A more refined analysis was performed for cases where the temperature gradient was significant. This analysis consisted of post hoc tests, which were selected based upon the homogeneity of variances tests results. For cases where variances were significantly different, a Games-Howell post hoc test was used, and for cases where variances were not significantly different, a Tukey post hoc test was used. According to the 95% confidence intervals from the post hoc tests, models with positive temperature gradients experienced significantly higher tensile, compressive, Von Mises, and shear stress in shear key than models with zero and negative temperature gradients. However, no significant difference was found between models with zero temperature gradients, and models with negative temperature gradients. The third parameter studied was the span-to-depth ratio (L/D). Results from ANOVA tests revealed that all maximum stresses in the shear key, contact stresses in interface between the shear key and box beams, and differential deflections between adjacent beams were significantly affected by L/D. In order to understand where the

125 125 difference between groups was (i.e., difference between models with L/D = 14.29, 27.33, and 35.71), post hoc tests were performed. Based on 95% confidence intervals, it was found that, in general, bridges with longer spans experienced higher stresses and higher differential deflections between beams. This increase in stress and deflections was significant for L/D varying from to 27.33, but not significant in all cases for L/D varying from to The next parameter analyzed was the number of internal diaphragms (N). Results from ANOVA tests revealed that only compressive stress in the shear key, contact shear between the shear key and box-beams, and differential deflections were significantly affected by N. Descriptive statistics tables showed that the average compressive stress in the shear key increased as N increased. The maximum contact shear between the shear key and adjacent beams, and differential deflections between adjacent beams, on the other hand, decreased as N increased. For a more detailed analysis, post hoc tests were performed. Results from post hoc tests showed significant difference between N=1 and N=2, in maximum compressive stresses in the shear key and maximum contact stress between the shear key and box-beams. However, there was no significant difference in using N=2 or N=3 in the stresses mentioned above. Differential deflections, on the other hand, were significantly reduced when increasing number of internal diaphragms from N=1 to N=3. The last parameter analyzed was the grout compressive strength-to concrete compressive strength ratio (GS). Results from ANOVA tests revealed that maximum tensile, compressive, Von Mises, and shear stresses in the shear key were significantly

126 126 affected by GS. Maximum vertical stresses, maximum contact shear stress and differential deflections were not significantly affected by GS. Post hoc tests showed that maximum tensile, compressive, Von Mises, and shear stresses in the shear key were significantly increased when GS increased from 0.5 to 2, or 1 to 2, but a significant increase in tensile stress was not found for GS increasing from 0.5 to 1. The next part of the analysis consisted on studying the effects of several factors (e.g., temperature gradient, L/D, N, and GS) on the effectiveness of dowel action. This analysis was performed by creating different groups. These groups were created by separating results from models with dowel action and models with no dowel action. Then, these groups were divided into subgroups, depending on the factor used to control. For instance, if the factor used to control was temperature gradient, each of the two main groups (i.e., dowel action and no dowel action) was subdivided into three categories: 1) no temperature gradient; 2) positive temperature gradient; and 3) negative temperature gradient. Next, the six obtained groups were analyzed through ANOVA testing. The same procedure was used to study the results when controlled by L/D, N, and GS. The first factor used to control was the temperature gradient. Results from ANOVA tests showed that there was a significant difference between groups, for all maximum stresses in the shear key, contact shear stresses in interface between the shear key and box-beams, and differential deflections between adjacent beams. A more detailed analysis was performed to identify where the difference was between groups. This analysis was performed using post hoc tests. Results from post hoc tests indicated that, when there is no temperature gradient in the models, dowel action had a significant

127 127 contribution in maximum tensile and vertical shear stress in the shear key, and maximum contact shear stress in interface between shear key and box beams. Post hoc tests also indicated that there was no significant contribution of dowel action in maximum compressive, Von Mises, and shear stress in the shear key for zero temperature gradients. Furthermore, results showed that, for positive and negative temperature gradients, dowel action had a significant contribution in maximum vertical shear? stress in the shear key and maximum contact shear stress in the interface between the shear key and box beams. However, its contribution was not significant for maximum tensile, compressive, Von Mises, and shear stress in the shear key. Additional observations were made for differential deflection. Results from post hoc tests indicated that differential deflections in the three components were significantly affected by dowel action independently of temperature gradients. The second factor used to control was the span-to-depth ratio. Results from ANOVA tests showed that there was a significant difference between groups for all maximum stresses in the shear key, contact shear stresses in the interface between the shear key and box-beams, and differential deflections between adjacent beams. Post hoc tests showed that dowel action significantly affected maximum tensile stress in the shear key for L/D = and L/D = 27.33, but dowel action was not significantly different from no dowel action for L/D = Maximum compressive, Von Mises, and shear stresses in the shear key were not significantly affected by dowel action independently of the magnitude of L/D. Maximum vertical shear stress in shear key and maximum contact shear stress in shear key, on the other hand, showed to be affected by dowel action for

128 128 L/D = 14.29, 27.33, and Analyzing 95% confidence intervals allowed knowing that for all cases where dowel action had a significant contribution, stresses were reduced due to dowel action. Finally, post hoc tests led to conclusion that differential deflections between adjacent beams were significantly reduced due to dowel action independently of span-to-depth ratio. The third factor used to control was the number of internal diaphragms. Results from ANOVA tests showed that there was no significant difference between groups, when analyzing maximum Von Mises and shear stress in the shear key. Post hoc tests showed that models with dowel action have significantly smaller maximum? tensile stress in the shear key than models with no dowel action, for N=1. However, there was no significant difference in maximum tensile stresses in the shear key for N=2 or N=3. Maximum compressive stress in the shear key was found to be significantly reduced due dowel action, when N=1, but were found to be not significantly affected for N=2 or N=3. Maximum vertical shear stress in the shear key and maximum contact shear stress in the shear key, on the other hand, were significantly reduced due to dowel action for N=1, 2, and 3. Finally, post hoc tests led to conclude that differential deflections between adjacent beams were significantly reduced due to dowel action independently of the number of internal diaphragms. The last factor used to control was GS. Results from ANOVA tests showed that there was no significant difference between all groups, when analyzing stresses in the shear key, contact shear stress between the shear key and box-beams, and differential deflections between adjacent beams. Post hoc tests showed that maximum compressive,

129 129 Von Mises, and shear stress in the shear key were not significantly affected by dowel action independently of GS. Results also indicated that models with dowel action have significantly smaller tensile stress in the shear key than models with no dowel action for GS=0.5, and GS=1. However, there was no significant difference in maximum tensile stresses in the shear key for GS=2. Maximum vertical shear stress in the shear key and maximum contact shear stress in the shear key, on the other hand, were significantly reduced due to dowel action independently of GS value. Finally, post hoc tests led to conclude that differential deflections between adjacent beams were significantly reduced due to dowel action independently of GS.

130 130 CHAPTER 5: SUMMARY AND CONCLUSSIONS A parametric study on the load transfer mechanism between adjacent prestressed concrete box-beams was developed using ABAQUS. Several finite element models consisting of two adjacent beams connected through a shear key and transverse ties were created for the analysis. The effects of filled and non-filled transverse post-tensioning ducts were included in the models. This was to show the contribution of the dowel action in the load transfer mechanism of adjacent prestressed concrete box-beams. In addition, effects of temperature gradients were considered with the aim of simulating a behavior resembling the reality of bridges in the field. Additional parameters, including grout compressive strength-to-concrete compressive strength ratio, amount of transverse posttensioning (TPT), span-to-depth ratio (L/D), and number of internal diaphragms (N), were included in the models. This study was divided into two main parts. The first part consisted of analyzing the behavior of a pair of beams with fixed length and number of internal diaphragms (i.e., L = and two internal diaphragms), while increasing the amount of TPT force per diaphragm. The second part consisted of analyzing the behavior of a pair of beams with zero transverse post-tensioning, while varying beam span-to-depth ratio and number of internal diaphragms. Thus, the effects of dowel action, amount of TPT, temperature gradients, L/D, N, and GS, in both stresses induced in shear key and differential deflections between adjacent beams, were studied in detail. Finally, all the results were statistically analyzed using Analysis of Variance (ANOVA). The main conclusions obtained from all results are presented below:

131 131 Analysis conducted on the results from all finite element models led to conclude that, on average, dowel action significantly reduced maximum tensile, vertical shear, and contact shear stresses in the shear key. However, dowel action had no significant effect in maximum compressive, Von Mises, and shear stress in the shear key. Effects of dowel action on stresses induced in the shear key were strongly affected by temperature gradients. Results showed that, when positive temperature gradients were acting in the models, the decrease in maximum tensile stress in the shear key due to dowel action was around 30% of the decrease obtained for zero temperature gradients. In addition, the decrease in compressive, Von Mises, and shear stress in the shear key due to dowel action was almost null when positive temperature gradients were applied to the models. When positive temperature gradients were applied to the models, average reduction in maximum vertical shear stress in the shear key due to dowel action was around 24% smaller than reductions obtained for models with no temperature gradient. The least affected value was maximum contact shear stress in interface between the shear key and box-beams. The contribution of dowel action in reducing contact shear stresses in the interface between the shear key and box-beams was almost unaffected by positive temperature gradients. In addition, the effects of dowel action on the stresses in the shear key were influenced by span-to-depth ratio. Results showed that contribution of dowel action was bigger in shorter bridges, and smaller in longer bridges of the same depth. The values that were most affected by span-to-depth ratio were the maximum tensile, compressive, Von Mises, and shear stress in the shear key. However, maximum

132 132 vertical stress in shear key and maximum contact stress between the shear key and box-beams was the least affected. Dowel action effects were also affected by the number of internal diaphragms. Increasing the number of internal diaphragms reduced the percentage of decrease in maximum tensile stress in the shear key due to dowel action. In contrast, the decrease in maximum compressive, Von Mises, vertical shear, and shear stress in the shear key as well as decrease in maximum contact shear stress due to dowel action was higher as number of internal diaphragms increased. Additional findings revealed that the effects of dowel action in most of the stresses induced in shear key were not strongly affected by GS. Results led to conclude that the decrease in maximum compressive, Von Mises, vertical shear, and shear stress in the shear key as well as maximum contact shear stress in the interface between the shear key and box-beams due to dowel action was not affected by GS. However, a reduction in maximum tensile stress in the shear key due to dowel action was smaller as the value of GS increased. Finally, results showed that dowel action significantly reduced differential deflections between beams in the three directions (i.e., δu 1, δu 2, and δu 3 ). Results showed that significance of the contribution of dowel action was independent of temperature gradients, L/D, N, or GS, but it was dependent on the amount of TPT. Based on all results, the ratios between maximum differential deflections from models with no dowel action and models with dowel action were 35 for the transverse differential deflections (δu 1 ), 14 for the vertical differential deflection (δu 2 ), and 9 for the

133 133 longitudinal differential deflection (δu 3 ). In terms of TPT, it was concluded that dowel action significantly affected transverse differential deflections (δu 1 ), when TPT 5 kips per diaphragm, vertical differential deflections (δu 2 ), when TPT 20 kips per diaphragm, and longitudinal differential deflections (δu 3 ), when TPT 30 kips per diaphragm. In general, maximum tensile, compressive, Von Mises, and shear stress in the shear key significantly increased due to positive temperature gradients. In contrast, results showed that temperature gradients had no significant effect in the maximum vertical shear stress in shear key only for models with no dowel action. On the other hand, maximum contact stress in the interface between the shear key and box beams was not significantly affected by temperature gradients. The percentage of increase in maximum stresses in the shear key due to a positive temperature gradient was higher from models with dowel action than it was for models with no dowel action. This led to the conclusion that dowel action made shear key stresses more sensitive to temperature changes. The increase in maximum stresses in the shear key due to positive temperature gradient tended to reduce as span-to-depth ratio increased. In other words, for the same beam s depth shorter bridges showed to be more sensitive to temperature variations than longer bridges. Results from the models led to the conclusion that, for the same beam s depth, longer bridges (e.g., L/D = or 35.71) experienced reductions in maximum contact stress between the shear key and adjacent beams for positive temperature gradients.

134 134 In most cases, the maximum increase in shear key stresses was found for models with two internal diaphragms. Exceptions were found for maximum vertical stress in the shear key and maximum contact shear stress in the interface between the shear key and box beams. The average increase in shear key stresses due to positive temperature gradient became larger as the value of GS increased. This led to the conclusion that stiffer shear keys are more sensitive to positive thermal gradients. The average percentage of increase in contact shear stress in the interface between the shear key and box beams, on the other hand, was not affected by GS.

135 135 REFERENCES ABAQUS. (2006). User s manual - Version Pawtucket, RI: Abaqus Inc. American Association of State Highway and Transportation Officials (AASHTO). (2010). LRFD Bridge Design Specifications (5th ed). Washington, DC. Barr, P. J., Stanton, J. F., & Eberhard, M. O. (2005). Effects of temperature variations on precast, prestressed concrete bridge girders. Journal of Bridge Engineering, 10(2), Camino Trujillo, S. J. (2010). Analytical Evaluation of Damaged Prestressed Concrete Box Beams Bridge Girders. Masters Thesis, Ohio University, Russ College of Engineering and Technology. Chajes, M. J., & Shenton III, H. W. (2006). Using diagnostic load testing for accurate load rating of typical bridges. Bridge Structures, 2(1), Dong, X. (2002). Traffic Forces and Temperature Effects on Shear Key Connections for Adjacent Box Girder Bridge. Doctoral dissertation. University of Cincinnati. El-R y, A., Tadros, M. K., Yamane, T., & Krause, G. (1996). Transverse design of adjacent precast prestressed concrete box girder bridges. PCI Journal, 41(4), Frenay, J. W., Reinhardt, H. W., & Walraven, J. C. (1991). Time-dependent shear transfer in cracked concrete: Part II. Journal of Structural Engineering, 117(10), Fu, C. C., Pan, Z., & Ahmed, M. S. (2011). Transverse posttensioning design of adjacent precast solid multibeam bridges, Journal of performance of constructed facilities, 25(3),

136 136 Grace, N. F., Jensen E. A., Enomoto T., Matsagar V. A., Soliman E. M., & Hanson J. Q. (2010). Transverse diaphragms and unbonded CFRP posttensioning in box-beam bridges. PCI Journal, 55(2), Grace, N. F., Jensen E. A., & Bebawy M. R. (2012). Transverse post-tensioning arrangement for side-by-side box-beam bridges. PCI Journal, 57(2). Gulyas, R. J., Wirthlin, G. J., & Champa, J. T. (1995). Evaluation of keyway grout test methods for precast concrete bridges. PCI Journal, 40(1), Hanna, K. E. (2008). Behavior of Adjacent Precast Prestressed Concrete Box Girder Bridges. Doctoral Dissertation. University of Nebraska-Lincoln. Hanna, K. E., Morcous, G., & Tadros, M. K. (2009). Transverse post-tensioning design and detailing of precast-prestressed concrete adjacent box-girder bridges. PCI Journal, 54(4), He, X.G., & Kwan, A.K.H. (2001). Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Journal of Computers and Structures, 79(6), Huffman, J. (2012). Destructive Testing of a Full-Scale 43 Year Old Adjacent Prestressed Concrete Box Beam Bridge: Middle and West Spans. Masters Thesis, Ohio University, Russ College of Engineering and Technology. International Federation for Structural Concrete (fib). (2007). Structural connections for precast concrete buildings (1st ed.). Lausanne, Switzerland.

137 137 Júlio, E. N. B. S., Branco, F. A. B., & Silva, V. D. (2005). Concrete-to-concrete bond strength: influence of an epoxy-based bonding agent on a roughened substrate surface. Magazine of Concrete Research, 57(8), Kim, J. H. J., Nam, J. W., Kim, H. J., Kim, J. H., & KEUN, J. B. (2008). Overview and applications of precast, prestressed concrete adjacent box-beam bridges in South Korea. PCI Journal, 53(4), King, S., & Richards, T. (2013, March). Solving Contact Problems with Abaqus. Coventry, UK. Precast/Prestressed Concrete Institute (PCI). (2009). The state of the art of Precast/ Prestressed Adjacent Box Beam Bridges (1 st ed.). Chicago, IL. Precast/Prestressed Concrete Institute (PCI). (2010). PCI design handbook (7th ed.). Chicago, IL. Precast/Prestressed Concrete Institute (PCI). (2011). PCI bridge design manual (3rd ed.). Chicago, IL. Pruijssers, A. F. (1988). Aggregate interlock and dowel action under monotonic and cycling loading. Delft University Press. Netherlands. Ramussen, B. H. (1962). Strength of transversely loaded bolts and dowels cast into concrete. Laboratoriet for Bugningastatik, Denmark Technical University. Russell, H. G. (2009). Adjacent precast concrete box beam bridges: connection details. National Cooperative Highway Research Program (NCHRP). Russell, H. G. (2011) Adjacent precast concrete box-beam bridges: state of the practice. PCI Journal, 56(1),

138 138 Sang, Z. (2010). A Numerical Analysis of the Shear Key Cracking Problem in Adjacent Box Beam Bridges. Doctoral dissertation. The Pennsylvania State University. Setty, C. J. (2012). Truck Testing and Load Rating of a Full-Scale 43-Year-Old Prestressed Concrete Adjacent Box Beam Bridge. Masters Thesis, Ohio University, Russ College of Engineering and Technology. Sharpe, G. P. (2007). Reflective cracking of shear keys in multi-beam bridges. Doctoral dissertation. Texas A&M University. Wall, J. S., & Shrive, N. G. (1988). Factors affecting bond between new and old concrete. ACI Materials Journal, 85(2), Wan, Z. (2011). Interfacial Shear Bond Strength Between Old and New Concrete. Master Thesis. Beijing University of Technology. Wight, J. K., & MacGregor, J. G. (2012). Reinforced concrete: Mechanics and design. Upper Saddle River, New Jersey: Pearson Education, Inc. X.G. He, & A.K.H. Kwan. (2001). Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Journal of Computers and Structures, 79(6).

139 APPENDIX I: INDUCED STRESSES IN SHEAR KEY AS FUNCTION OF TPT 139 Table 23. Maximum stresses in shear key for models with no dowel action, and GS = 0.5` Maximum Maximum Von Maximum compressive TPT Mises stress in tensile stress stress in shear shear key in shear key key (kips) (psi) (psi) (psi) No temperature gradient Positive temperature gradient Negative temperature gradient

140 Table 24. Maximum stresses in shear key for models with dowel action, and GS = 0.5 Maximum Maximum Von Maximum compressive TPT Mises stress in tensile stress in stress in shear shear key shear key key (kips) (psi) (psi) (psi) No temperature gradient Positive temperature gradient Negative temperature gradient

141 141 Table 25. Maximum stresses in shear key for models with no dowel action, and GS = 1 Maximum Maximum Von Maximum compressive TPT Mises stress in tensile stress stress in shear shear key in shear key key (kips) (psi) (psi) (psi) No temperature gradient Positive temperature gradient Negative temperature gradient

142 Table 26. Maximum stresses in shear key for models with dowel action, and GS = 1 No temperature gradient Positive temperature gradient Negative temperature gradient TPT Maximum Von Mises stress in shear key Maximum tensile stress in shear key 142 Maximum compressive stress in shear key (kips) (psi) (psi) (psi)

143 143 Table 27. Maximum stresses in shear key for models with no dowel action, and GS = 2 Maximum Maximum Von Maximum compressive TPT Mises stress in tensile stress stress in shear shear key in shear key key (kips) (psi) (psi) (psi) No temperature gradient Positive temperature gradient Negative temperature gradient

144 Table 28. Maximum stresses in shear key for models with dowel action, and GS = 2 No temperature gradient Positive temperature gradient Negative temperature gradient TPT Maximum Von Mises stress in shear key Maximum tensile stress in shear key 144 Maximum compressive stress in shear key (kips) (psi) (psi) (psi)

145 145 Table 29. Average increase in maximum stresses in shear key due to positive temperature gradients (%) Models with no dowel action Models with dowel action Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Table 30. Average decrease in maximum stresses in shear key due to dowel action (%) Zero temperature gradient Positive temperature gradient Negative temperature gradient Maximum tensile stress Maximum compressive stress Maximum Von Mises stress

146 146 APPENDIX II: DIFFERENTIAL DEFLECTIONS BETWEEN ADJACENT BEAMS AS FUNCTION OF TPT Table 31. Differential deflections between adjacent beams for models with no dowel action, no temperature gradient, and GS = 0.5 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 32. Differential deflections between adjacent beams for models with no dowel action, positive temperature gradient, and GS = 0.5 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

147 147 Table 33. Differential deflections between adjacent beams for models with no dowel action, negative temperature gradient, and GS = 0.5 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 34. Differential deflections between adjacent beams for models with no dowel action, no temperature gradient, and GS = 1 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

148 148 Table 35. Differential deflections between adjacent beams for models with no dowel action, positive temperature gradient, and GS = 1 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 36. Differential deflections between adjacent beams for models with no dowel action, negative temperature gradient, and GS = 1 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

149 149 Table 37. Differential deflections between adjacent beams for models with no dowel action, no temperature gradient, and GS = 2 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 38. Differential deflections between adjacent beams for models with no dowel action, positive temperature gradient, and GS = 2 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

150 150 Table 39. Differential deflections between adjacent beams for models with no dowel action, negative temperature gradient, and GS = 2 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 40. Differential deflections between adjacent beams for models with dowel action, no temperature gradient, and GS = 0.5 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

151 151 Table 41. Differential deflections between adjacent beams for models with dowel action, positive temperature gradient, and GS = 0.5 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 42. Differential deflections between adjacent beams for models with dowel action, negative temperature gradient, and GS = 0.5 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

152 152 Table 43. Differential deflections between adjacent beams for models with dowel action, no temperature gradient, and GS = 1 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 44. Differential deflections between adjacent beams for models with dowel action, positive temperature gradient, and GS = 1 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

153 153 Table 45. Differential deflections between adjacent beams for models with dowel action, negative temperature gradient, and GS = 1 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 46. Differential deflections between adjacent beams for models with dowel action, no temperature gradient, and GS = 2 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

154 154 Table 47. Differential deflections between adjacent beams for models with dowel action, positive temperature gradient, and GS = 2 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 48. Differential deflections between adjacent beams for models with dowel action, negative temperature gradient, and GS = 2 TPT x = 0 x = L/6 x = L/3 x = L/2 (kips) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

155 APPENDIX III: MAXIMUM STRESSES IN SHEAR KEY AS FUNCTION OF SPAN- TO-DEPTH RATIO Table 49. Maximum stresses in shear key for models with no dowel action, no temperature gradient, and N = 1 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) 155 Max Contact Shear (psi) Table 50. Maximum stresses in shear key for models with no dowel action, positive temperature gradient, and N = 1 L/D GS Max σ Mises,max σ Tensile, max σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi)

156 156 Table 51. Maximum stresses in shear key for models with no dowel action, negative temperature gradient, and N = 1 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi) Table 52. Maximum stresses in shear key for models with no dowel action, no temperature gradient, and N = 2 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) Max Contact Shear (psi)

157 Table 53. Maximum stresses in shear key for models with no dowel action, positive temperature gradient, and N = 2 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi) Table 54. Maximum stresses in shear key for models with no dowel action, negative temperature gradient, and N = 2 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi)

158 Table 55. Maximum stresses in shear key for models with no dowel action, no temperature gradient, and N = 3 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) 158 Max Contact Shear (psi) Table 56. Maximum stresses in shear key for models with no dowel action, positive temperature gradient, and N = 3 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi)

159 Table 57. Maximum stresses in shear key for models with no dowel action, negative temperature gradient, and N = 3 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi) Table 58. Maximum stresses in shear key for models with dowel action, no temperature gradient, and N = 1 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi)

160 Table 59. Maximum stresses in shear key for models with dowel action, positive temperature gradient, and N = 1 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) 160 Max Contact Shear (psi) Table 60. Maximum stresses in shear key for models with dowel action, negative temperature gradient, and N = 1 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) Max Contact Shear (psi)

161 161 Table 61. Maximum stresses in shear key for models with dowel action, no temperature gradient, and N = 2 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi) Table 62. Maximum stresses in shear key for models with dowel action, positive temperature gradient, and N = 2 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) Max Contact Shear (psi)

162 Table 63. Maximum stresses in shear key for models with dowel action, negative temperature gradient, and N = 2 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) 162 Max Contact Shear (psi) Table 64. Maximum stresses in shear key for models with dowel action, no temperature gradient, and N = 3 L/D GS Max σ Mises,max σ Tensile, σ Compressive, max σ 12,max τ max = (σ 1 - Contact (psi) max (psi) (psi) (psi) σ 3 )/2 (psi) Shear (psi)

163 Table 65. Maximum stresses in shear key for models with dowel action, positive temperature gradient, and N = 3 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) 163 Max Contact Shear (psi) Table 66. Maximum stresses in shear key for models with dowel action, negative temperature gradient, and N = 3 L/D GS σ Mises,max (psi) σ Tensile, max (psi) σ Compressive, max (psi) σ 12,max (psi) τ max = (σ 1 - σ 3 )/2 (psi) Max Contact Shear (psi)

164 164 Table 67. Effect of dowel action in average increase in maximum stresses in shear key due to positive temperature gradients (%) Models with no dowel action Models with dowel action Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress Table 68. Effect of L/D in average increase in maximum stresses in shear key due to positive temperature gradients (%) L/D Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress Table 69. Effect of N in average increase in maximum stresses in shear key due to positive temperature gradients (%) N Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress

165 165 Table 70. Effect of GS in average increase in maximum stresses in shear key due to positive temperature gradients (%) GS Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress Table 71. Effect of temperature gradients in average decrease in maximum stresses in shear key due to dowel action (%) Zero temperature gradient Positive temperature gradient Negative temperature gradient Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress Table 72. Effect of L/D in average decrease in maximum stresses in shear key due to dowel action (%) L/D Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress

166 Table 73. Effect of N in average decrease in maximum stresses in shear key due to dowel action (%) N Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress Table 74. Effect of GS in average decrease in maximum stresses in shear key due to dowel action (%) GS Maximum tensile stress Maximum compressive stress Maximum Von Mises stress Maximum vertical shear stress Maximum shear stress Maximum contact shear stress

167 167 APPENDIX IV: DIFFERENTIAL DEFLECTIONS BETWEEN BEAMS AS FUNCTION OF SPAN-TO-DEPTH RATIO Table 75. differential deflections between beams for models with no dowel action, N = 1, and GS = 0.5 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 76. Differential deflections between beams for models with no dowel action, N = 1, and GS = 1 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

168 168 Table 77. Differential deflections between beams for models with no dowel action, N = 1, and GS = 2 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 78. Differential deflections between beams for models with no dowel action, N = 2, and GS = 0.5 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

169 169 Table 79. Differential deflections between beams for models with no dowel action, N = 2, and GS = 1 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 80. Differential deflections between beams for models with no dowel action, N = 2, and GS = 2 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

170 170 Table 81. Differential deflections between beams for models with no dowel action, N = 3, and GS = 0.5 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 82. Differential deflections between beams for models with no dowel action, N = 3, and GS = 1 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in)

171 171 Table 83. Differential deflections between beams for models with no dowel action, N = 3, and GS = 2 x = 0 x = L/6 x = L/3 x = L/2 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) δu 1 (in) δu 2 (in) δu 3 (in) Table 84. Differential deflections between beams for models with dowel action, N = 1, and GS = 0.5 x = 0 x = L/6 x = L/3 x = L/2 L/D δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 No temperature gradient Positive temperature gradient Negative temperature gradient

172 172 Table 85. Differential deflections between beams for models with dowel action, N = 1, and GS = 1 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 x = 0 x = L/6 x = L/3 x = L/2 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu Table 86. Differential deflections between beams for models with dowel action, N = 1, and GS = 2 x = 0 x = L/6 x = L/3 x = L/2 L/D No temperature gradient Positive temperature gradient Negative temperature gradient δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu

173 173 Table 87. Differential deflections between beams for models with dowel action, N = 2, and GS = 0.5 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 x = 0 x = L/6 x = L/3 x = L/2 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu Table 88. Differential deflections between beams for models with dowel action, N = 2, and GS = 1 x = 0 x = L/6 x = L/3 x = L/2 L/D No temperature gradient Positive temperature gradient Negative temperature gradient δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu

174 174 Table 89. Differential deflections between beams for models with dowel action, N = 2, and GS = 2 x = 0 x = L/6 x = L/3 x = L/2 L/D No temperature gradient Positive temperature gradient Negative temperature gradient δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu Table 90. Differential deflections between beams for models with dowel action, N = 3, and GS = 0.5 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 x = 0 x = L/6 x = L/3 x = L/2 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 ( 10-3 in) δu 3 δu 1 δu 2 δu

175 175 Table 91. Differential deflections between beams for models with dowel action, N = 3, and GS = 1 No temperature gradient Positive temperature gradient Negative temperature gradient L/D δu 1 x = 0 x = L/6 x = L/3 x = L/2 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu Table 92. Differential deflections between beams for models with dowel action, N = 3, and GS = 2 x = 0 x = L/6 x = L/3 x = L/2 L/D No temperature gradient Positive temperature gradient Negative temperature gradient δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu 3 δu 1 δu 2 δu

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