The Pennsylvania State University. The Graduate School. College of Engineering EVALUATION OF BRACING SYSTEMS IN HORIZONTALLY CURVED STEEL I-

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1 The Pennsylvania State University The Graduate School College of Engineering EVALUATION OF BRACING SYSTEMS IN HORIZONTALLY CURVED STEEL I- GIRDER BRIDGES A Dissertation in Civil Engineering by Mohammadreza Sharafbayani 2012 Mohammadreza Sharafbayani Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2012

2 The dissertation of Mohammadreza Sharafbayani was reviewed and approved* by the following: Daniel G. Linzell Shaw Professor of Civil Engineering Dissertation Advisor Chair of Committee Jeffrey Laman Professor of Civil Engineering Andrew Scanlon Professor of Civil Engineering Thomas E. Boothby Professor of Architectural Engineering Peggy A. Johnson Professor of Civil and Environmental Engineering Head of the Department of Civil and Environmental Engineering *Signatures are on file in the Graduate School ii

3 ABSTRACT This work examined the effects of an innovative torsional bracing system, skewed bracing, on the behavior of horizontally curved I-girder bridges with non-skewed abutments. More specifically, the influence of this bracing system was studied on the performance of curved I- girder bridges during construction and while in-service. For this, the performance of the skewed bracing system was compared against that for a common type of bracing for horizontally curved bridges; bracing oriented normal to girder web. Studies were completed using three-dimensional, nonlinear finite element analysis. The first part of the study examined the effect of the proposed skewed braces on the behavior of non-composite horizontally curved I-girder bridges during construction and compared that behavior to existing bracing systems. Studies, first, completed on a single-span bridge having small radii of curvature indicated better load sharing between the girders during construction when skewed bracing was used. The significance of these benefits was extended through a number of parametric studies that investigated varying bridge plan geometries by examining different cross frame connection details, bracing type, brace spacings, girder number, and number of spans. For all cases examined, the skewed bracing, compared to normal-to-web bracing, was found to improve the structural behavior during construction, while generally required fewer intermediate braces. The second part of this research examined the performance of the skewed bracing arrangement on behavior of in-service, composite bridges. First, elastic behavior of the bridge members under unfactored live loading cases was investigated. Similar to the behavior of the noncomposite girders under construction dead load, the skewed bracing, compared with the normal-to-web bracing arrangement, resulted in a slightly more uniform load sharing among the structural members in the composite bridge under the unfactored live loads. Next, the effect of changing the bracing arrangement on inelastic behavior of the girders and the concrete deck under an elevated ultimate loading case was studied. The skewed bracing, although used smaller number of intermediate braces, improved the inelastic behavior of girders and concrete deck. iii

4 TABLE OF CONTENTS LIST OF FIGURES... viii LIST OF TABLES... xvi ACKNOWLEDGEMENTS... xvii Chapter 1. Introduction Background Problem statement Objectives Scope Task list... 5 Chapter 2. Literature Review and Background Bridge stability Lateral Torsional Buckling (LTB) Effect of beam intermediate bracing Computational stability analysis methods Horizontally curved steel I-girder bridges during construction Studies on effects of live loads on behavior of in-service horizontally curved bridges Studies on behavior of bracing systems in horizontally curved and/or skewed I-girder bridges Design and construction guidelines iv

5 2.5.1 AASHTO LRFD Bridge Design Specification (AASHTO 2012) AASHTO/NSBA Guidelines for Design for Constructability (AASHTO/NSBA 2003) Summary Chapter 3. Analytical Studies for Bracing Systems Stiffness Calculation of stiffness components for bracing systems in horizontally curved I- girder bridges Diaphragm stiffness matrix X-type cross frame stiffness matrix K-type cross frame stiffness matrix Comparing bracing systems stiffness Summary and conclusions Chapter 4. Finite Element Modeling FEA modeling for construction studies CSBRP phase one structure CSBRP structure modeling Global modeling validation Brace modeling FEA modeling of composite bridges Concrete deck modeling Material properties Summary and conclusions v

6 Chapter 5. Construction Performance of Skewed Bracing in Horizontally Curved Steel Bridges Skewed bracing in horizontally curved bridges Initial study Finite element models Analysis results and discussion Removal of intermediate cross frames Skewed bracing in horizontally curved bridges Parametric study Case 1 - Cross frame to web connections Case 2- Bracing type Case 3- Brace spacing Case 4- Four-girder bridge model Case 5 Two span bridge Summary and conclusions Chapter 6. Skewed Bracing Performance in In-Service, Horizontally Curved, I-Girder Bridges Studied bridge Analysis results and discussions Bridge behavior under unfactored live loads Bridge behavior under ultimate loading Summary and conclusions Chapter 7. Summary and Conclusions vi

7 7.1 Summary of the findings Individual bracing systems Construction performance of skewed and normal-to-web bracing systems Service and ultimate load performance of skewed and normal-to-web bracing arrangements in composite, horizontally curved bridges Recommendations for future work REFERENCES Appendix A. MATLAB Files for Calculation of Stiffness Matrices A.1 Diaphragm stiffness matrix calculation A.2 X-type cross frame stiffness matrix calculation A.3 K-type cross frame stiffness matrix calculation Appendix B. Comparing FEA and Experimental Results for Girder Strains vii

8 LIST OF FIGURES Figure 2-1 (a) lateral torsional buckling; (b) girder deformed shape at mid-span... 8 Figure 2-2 Bracing stiffness for twin girder cross frames (Yura 2001) Figure 2-3 Assumed deformed shape for bracing systems (Yura 2001) Figure 2-4 CSBRP test structure for phase one, at FHWA Turn-Fairbanks Structures Laboratory (Hartmann 2005) Figure 2-5 CSBRP test structure for phase three, at FHWA Turn-Fairbanks Structures Laboratory (Jung 2006) Figure 2-6 Bracing orientation for bridges with skewed supports Figure 2-7 Connection detail for intermediate skewed cross frames (AASHTO/NSBA 2003). 32 Figure 2-8 X-type cross frame, minimum diagonal angle Figure 3-1 Degrees of freedom in a general bracing system Figure 3-2 Diaphragm analytical model Figure 3-3 Analytical model for X-type cross frame Figure 3-4 Analytical model for K-type cross frame Figure 3-5 Bracing system stiffness components (AASHTO/NSBA 2011) Figure 3-6 General bracing geometry (for diaphragm and K- and X-type cross frames) Figure 3-7 Comparison of X-type and K-type cross frame shear and flexural stiffness elements, varying girder depth, constant girder spacing: S=2.67 m (8.75 ft) Figure 3-8 Comparison of diaphragm and K-type cross frame shear and flexural stiffness elements, varying girder depth, constant girder spacing: S=2.67 m (8.75 ft) Figure 3-9 Comparison of diaphragm and X-type cross frame shear and flexural stiffness elements, varying girder depth, constant girder spacing: S=2.67 m (8.75 ft) viii

9 Figure 4-1 Framing plan for CSBRP structure tested for the initial erection studies (Linzell 1999) Figure 4-2 Mid-span deformation instrumentation (Linzell et al. 2004) Figure 4-3 Girder strain gage location (Linzell 1999) Figure 4-4 CSBRP ABAQUS finite element model Figure 4-5 G3 top flange strains at Station 5L Figure 4-6 G3 bottom flange strains at Station 5L Figure 4-7 G3 web strains at Station 5L Figure 4-8 G3 top flange strains at Station Figure 4-9 G3 bottom flange strains at Station Figure 4-10 G3 web strains at Station Figure 4-11 Brace models Figure 4-12 Boundary conditions Figure 4-13 Shear deformation of brace Figure 4-14 Flexural deformation of brace Figure 4-15 Comparing FEA and analytical diaphragm shear stiffness, varying girder depth, constant girder spacing =2.67 m (8.75 ft) Figure 4-16 Comparing FEA and analytical diaphragm flexural stiffness, varying girder depth, constant girder spacing of S=2.67 m (8.75 ft) Figure 4-17 Comparing FEA and analytical flexural stiffness for K-type cross frames, varying girder depth, constant girder spacing of S=2.67 m (8.75 ft) Figure 4-18 Comparing FEA and analytical flexural stiffness for X-type cross frames, varying girder depth, constant girder spacing of S=2.67 m (8.75 ft) Figure 4-19 Interface modeling between concrete deck and steel girders ix

10 Figure 4-20 Nominal stress-strain response for ASTM A572 Grade 50 steel (Jung 2006) Figure 4-21 Nominal stress-strain response for ASTM A709 Grade HPS70W steel (Jung 2006) Figure 4-22 Nominal compressive stress-strain response for concrete (Jung 2006) Figure 4-23 Nominal tensile stress-strain response for concrete (Jung 2006) Figure 5-1 Normal-to-web bracing in a horizontally curved I-girder bridge Figure 5-2 Representation of skewed bracing in a horizontally curved I-girder bridge Figure 5-3 Single span bridge framing plan, section, and boundary conditions initial study. 83 Figure 5-4 Single-span bridge bracing arrangements initial study Figure 5-5 Finite element model for single-span bridge initial study Figure 5-6 G3 web rotations initial study Figure 5-7 G2 web rotations initial study Figure 5-8 G1 web rotations initial study Figure 5-9 Girder maximum vertical deflections initial study Figure 5-10 G3 normal stress across bottom flange at mid span initial study Figure 5-11 G1 normal stress across bottom flange at mid span initial study Figure 5-12 Cross frame maximum bottom chord normal stresses initial study Figure 5-13 Adding skew effects to the curved girder mid-span sections by using skewed bracing Figure 5-14 Bracing influence on the girders Figure 5-15 In-plane component of the cross frame tension force at the top of the girders Figure 5-16 Skewed bracing arrangement cross frame removal Figure 5-17 G3 web rotations cross frame removal x

11 Figure 5-18 G2 web rotations cross frame removal Figure 5-19 G1 web rotations cross frame removal Figure 5-20 Girder maximum vertical deflections cross frame removal Figure 5-21 G3 normal stress across bottom flange at mid-span cross frame removal Figure 5-22 G1 normal stress across bottom flange at mid-span cross frame removal Figure 5-23 Cross frame maximum bottom chord normal stresses cross frame removal Figure 5-24 Skewed cross frame connection details Case Figure 5-25 Mesh refinement for connection members with tight radius Case Figure 5-26 Material yielding in the bent gusset plate for brace connection to G3 at Station 3L Case Figure 5-27 G3 web rotations Case Figure 5-28 G2 web rotations Case Figure 5-29 G1 web rotations Case Figure 5-30 Cross frame maximum bottom chord normal stresses Case Figure 5-31 Bracing systems in bridge models with varying girder depths Case Figure 5-32 Replacing K-type cross frames with X-type cross frames Case Figure 5-33 G3 web rotations, girder depth=152 cm (60 in), normal bracing Case Figure 5-34 Finite element model, girder depth=152 cm (60 in), skewed bracing, X-type cross frames Case Figure 5-35 G3 web rotations, girder depth=152 cm (60 in), K-type cross frames Case Figure 5-36 Replacing K-type cross frames with diaphragms, normal-to-web bracing Case Figure 5-37 G3 web rotations, girder depth=122 cm (48 in), normal-to-web bracing Case xi

12 Figure 5-38 G1 web rotations, girder depth=122 cm (48 in), normal-to-web bracing Case Figure 5-39 G3 web rotations, girder depth=152 cm (60 in), normal-to-web bracing Case Figure 5-40 G1 web rotations, girder depth=152 cm (60 in), normal-to-web bracing Case Figure 5-41 Finite element model, girder depth=122 cm (48 in), skewed Bracing Case Figure 5-42 G3 web rotations, girder depth=122 cm (48 in), skewed bracing Case Figure 5-43 G1 web rotations, girder depth=122 cm (48 in), skewed bracing Case Figure 5-44 Bridge framing plan, and section Case Figure 5-45 Skewed bracing arrangement Case Figure 5-46 G3 web rotations Case Figure 5-47 G2 web rotations Case Figure 5-48 G1 web rotations Case Figure 5-49 Girder maximum vertical deflections Case Figure 5-50 G1 normal stress across the bottom flange at mid span Case Figure 5-51 G2 bottom flange, exterior tip, normal stress variation along span Case Figure 5-52 Cross frame maximum bottom chord normal stresses Case Figure 5-53 Bridge framing plan, section, and boundary conditions Case Figure 5-54 Skewed bracing arrangement Case Figure 5-55 G4 web rotations Case Figure 5-56 G1 web rotations Case Figure 5-57 Girder maximum vertical deflections Case xii

13 Figure 5-58 G1 normal stress across the bottom flange at mid span Case Figure 5-59 Cross frame maximum bottom chord normal stresses Case Figure 5-60 Two span bridge framing plan, section, and boundary conditions Case Figure 5-61 Skewed bracing arrangement Case Figure 5-62 Girder flange width transition Case Figure 5-63 Two-span bridge concrete deck segments Case Figure 5-64 Two-span bridge model at the end of analysis step 2 Case Figure 5-65 G3 web rotations, Step 2 Case Figure 5-66 G1 web rotations, Step 2 Case Figure 5-67 Two-span bridge model at the end of at analysis step 5 Case Figure 5-68 G3 web rotations, Step 5 Case Figure 5-69 G1 web rotations, Step 5 Case Figure 5-70 G3 normal stress across the bottom flange at mid span of the first span, Step 5 Case Figure 5-71 G1 first span, mid span, normal stress across the bottom flange, Step 5 Case Figure 5-72 G3 normal stress across the top flange at the pier location, Step 7 Case Figure 5-73 G1 normal stress across the top flange at the pier location, Step 7 Case Figure 5-74 Cross frame maximum bottom chord normal stresses, Step 7 Case Figure 6-1 Single-span bridge bracing arrangements Figure 6-2 Single-span bridge concrete deck reinforcement Figure 6-3 Single-span bridge live load cases Figure 6-4 Girder maximum vertical deflections, load Case-2 - unfactored live loads xiii

14 Figure 6-5 G3 mid span, normal stress across the bottom flange, load Case-2 - unfactored live loads Figure 6-6 G1 mid span, normal stress across the bottom flange, load Case-1 - unfactored live loads Figure 6-7 Cross frame maximum bottom chord normal stresses - unfactored live loads Figure 6-8 G3 mid span vertical deflection as a function of total applied load - ultimate loading Figure 6-9 G2 mid span vertical deflection as a function of total applied load - ultimate loading Figure 6-10 G1 mid span vertical deflection as a function of total applied load - ultimate loading Figure 6-11 Mid-span deck vertical deflections, total applied load=5783 kn (1300 kips) - ultimate loading Figure 6-12 G3 mid span normal strains across the bottom flange, total applied load=5783 kn (1300 kips) - ultimate loading Figure 6-13 G2 mid span normal strains across the bottom flange, total applied load=5783 kn (1300 kips) - ultimate loading Figure 6-14 G1 mid span normal strains across the bottom flange, total applied load=5783 kn (1300 kips) - ultimate loading Figure 6-15 Mid-span deck top surface compressive principal strains, total applied load=5783 kn (1300 kips) - ultimate loading Figure 6-16 Concrete deck compressive principal strain contours, top surface of the deck, total applied load=5783 kn (1300 kips) - ultimate loading Figure 6-17 Concrete deck tensile principal strain contours, top surface of the deck, total applied load=5783 kn (1300 kips) - ultimate loading xiv

15 Figure B-1 G1 top flange normal strains at Station Figure B-2 G1 bottom flange normal strains at Station Figure B-3 G1 top flange normal strains at Station 5R Figure B-4 G1 bottom flange normal strains at Station 5R Figure B-5 G2 top flange normal strains at Station Figure B-6 G2 bottom flange normal strains at Station Figure B-7 G2 top flange normal strains at Station 5R Figure B-8 G2 bottom flange normal strains at Station 5R Figure B-9 G3 top flange normal strains at Station 5R Figure B-10 G3 bottom flange normal strains at Station 5R Figure B-11 G3 web normal strains at Station 5R xv

16 LIST OF TABLES Table 2-1 Torsional stiffness of bracing systems (Yura 2001) Table 4-1. CSBRP structure girder plate sizes (Linzell 1999) Table 4-2 Nominal plate sizes for bearing, transverse and connection plate stiffeners Table 4-3. Girder mid span displacements after shoring removal, ES Table 4-4. Compressive damage variables versus nominal compressive stresses and strains (Jung 2006) Table 4-5. Tensile damage variables versus nominal compressive stresses and strains (Jung 2006) Table 5-1. Single span bridge girder nominal plate sizes initial study Table 5-2 Single span bridge nominal plate sizes for bearing, transverse and connection plate stiffeners initial study Table 5-3. Single span bridge sequential analysis erection procedure Table 5-4 Constructability Limit State checks for girder bottom flanges Table 5-5 Bridge girder nominal plate sizes, girder depth=152 cm (60 in) Table 5-6 Bridge girder nominal plate sizes Case Table 5-7 Bridge girder nominal plate sizes Case Table 5-8 Case 5- Bridge girder nominal plate sizes Table 5-9. Two-span bridge sequential analysis erection procedure- Case Table 6-1 Service Limit Sate checks for girder bottom flanges Table 6-2 Strength Limit State checks for girder bottom flanges xvi

17 ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisors Dr. Daniel Linzell for his advice and support in both my research and academics. I would like to thank the Department of Civil and Environmental Engineering for giving me the education needed to fulfill my PhD. Lastly, I would like to thank my friends and family for their support, understanding, and encouragement throughout my academic career. xvii

18 To my wife Maryam Neshastehriz my parents Hamidreza Sharafbayani and Marzieh Asefi, and my sister Maryam Sharafbayani. My great sources of dream, happiness and strength. xviii

19 Chapter 1. Introduction 1.1 Background Horizontally curved, steel, I-girder bridges can offer an economical solution for highway system crossings where roadway alignment and geometry require a smooth, curved transition across the bridge and limited space is available for interior piers. Due to the curved geometry, the centerline of the girder webs at sections away from the end supports in each span are not collinear with a cord between the supports. Resulting eccentricities induce torsional moments which, in turn, cause out of plane deformations and rotations in the girder cross sections. Diaphragms and cross frames are used to brace and interconnect girders in curved and straight steel bridges. According to the AASHTO LRFD Bridge Design Specifications (AASHTO 2012), both are defined as members that connect adjacent longitudinal flexural components, with cross frames being more specifically defined as transverse truss frameworks, and diaphragms presented as vertically aligned solid transverse members. As defined in the AASHTO LRFD, the three main functions of cross frames and diaphragms in steel bridges are: (1) to act as brace points to stabilize the compression flanges of the girders; (2) to resist and transfer lateral forces to the bearings; and (3) to distribute vertical live load and dead load to the girders. In addition to these, in horizontally curved bridges cross frames and diaphragms also have to assist with resisting torsion caused by curvature effects, both during construction and during the bridge s service life. During construction, in the absence of a hardened concrete deck, cross frames and diaphragms play a major role in providing stability to the girders and help to maintain geometric and stress control while concurrently resisting lateral loads. Unpredicted 1

20 construction deformations or stresses caused by the curved girder geometry can result in premature yielding and fit up issues at girder splices or between the girder and bracing members, or in some extreme cases instability issues. Therefore, to mitigate the possibility of these problems in horizontally curved bridges during construction, a large number of intermediate braces are usually used between the girders. Once placed into service, these braces perform a primary role in maintaining bridge integrity and transferring live loads between the curved girders. In fact, as trucks pass over a curved bridge, due to the geometry, large live load forces can be induced in the brace members, which, in turn, can result in large lateral bending moments in the girder flanges. To avoid excessive stresses in these curved flanges, small unbraced lengths are used, again, by increasing the number of intermediate braces. Although the structural weight of bracing members in curved bridge systems in comparison with the total structural weight of the bridges may not be very significant, appreciable cost is generally associated with fabrication and erection of the bracing members (NCHRP/TRB 2011). By using a more effective bracing system between the curved girders it may be possible to reduce the number of intermediate braces in the curved bridges, facilitate fabrication and erection of these bridges, and subsequently produce a more cost effective design. 1.2 Problem statement Bracing members in horizontally curved bridges are commonly placed in a radial direction and normal to the girder webs. This pattern results in smaller unbraced lengths for interior girders, which generally experience smaller deformations and rotations, and larger unbraced lengths for exterior girders that have larger deformations and rotations. Selecting an adequate cross frame spacing for a horizontally curved bridge to limit the effect of curvature on girder deformations and stresses (e.g. out of plane rotations and flange lateral bending stresses) is 2

21 historically governed by the behavior of the exterior girders. Therefore, using a radial arrangement for the cross frames could result in unnecessarily large number of braces between the interior girders, and a less than optimal design. 1.3 Objectives The primary objective of the present study was to study bracing arrangements between the girders in horizontally curved bridges to produce more effective and efficient bracing systems. These bracing systems can then, theoretically, reduce girder stresses, deformations and rotations and concurrently improve the behavior of horizontally curved, steel, I-girder bridges during construction and while in-service, with possibly smaller number of intermediate braces being required within the bridge spans. The possible reduction in the total number of intermediate braces not only can reduce the structural weight, to some extent, but it also can appreciably improve the economy of the design by reducing the amount of material that needs to be fabricated and constructed. 1.4 Scope The present study computationally investigated the behavior of bracing systems in horizontally curved I-girder bridges. In particular, an optimized bracing arrangement, using skewed braces, was introduced and its effect on the behavior of horizontally curved bridges was examined. Both non-composite and composite conditions were considered to examine the behavior of the bridges during construction and in service condition, respectively. The scope for the present research was as follows: 3

22 The angles that were considered for the skewed braces include 10 and 20. These values represented the angle between the braces and a normal to the girder webs. Single and two-span, I-girder bridges were considered. The span length for the single-span bridge and for each span of the two-span bridge was 27.4 m (90 ft). The single span bridges contained three or four girders. The two-span bridge contained three girders. A radius of curvature of 61 m (200 ft) to the centerline of the bridge deck was considered for all bridge models in this study. Also, the spacing between the girders in all bridge models was 2.67 m (8.75 ft). Rollers, guided rollers and non-guided rollers (pin supports) were considered as the different girder support conditions at the end abutments for all bridges and over the middle pier for the two-span bridges. A concrete slab with a thickness of 20.3 cm (8 in) and with an overhang of 0.91 (3 ft) was considered. This concrete slab was included only in the models that were used to study composite, in-service bridge behavior. For the construction studies appropriate dead loads due to the wet concrete self-weight were calculated and applied to the girder flanges. Two different types of cross frames, K-type and X-type, were examined for the bridge models. Angle sections were used for all cross frames members. A built up I-section was considered for the diaphragms that were examined in this study. 4

23 Rigid connections between the skewed braces and girders included skewed connection plate stiffeners, normal connection plate stiffeners having bent gusset plates, and split pipe stiffeners. 1.5 Task list The objective of this study was to improve and optimize the behavior of horizontally curved bridges both during construction and while in-service by introducing a more effective bracing system that could reduce the number of required intermediate braces between the girders. To fulfill this objective the following tasks were completed: In Chapter 2, literature was reviewed pertaining to: stability and buckling analysis of plate girders; behavior of horizontally curved bridges during construction and while in-service; and performance of bracing systems in steel bridges with complicated geometry. In Chapter 3, preliminary analytical studies were conducted using matrix structural analysis to obtain necessary stiffness components for three bracing types: diaphragms; K-type cross frames; and X-type cross frames. Results from this study were compared against finite element analysis results from bracing models in Chapter 4 to examine the reliability of techniques and assumptions used for modeling different bracing systems. Results were also used in Chapter 5 to justify the performance of the different bracing types on the behavior of horizontally curved bridges during construction. In Chapter 4, finite element models were developed to examine the influence that different bracing systems had on non-composite curved girder construction behavior and composite in-service bridge behavior. 5

24 In Chapter 5, the influence of changing bracing orientation with respect to the girder web was examined on the construction behavior of non-composite curved bridges having different geometries. In Chapter 6, the performance of different bracing arrangement was investigated for in-service composite horizontally curved bridges. The bridges were first subjected to unfactored live loads and analysis results were compared against the AASHTO (2012) Serviceability and Strength Limit State requirements for bridge members to examine the effect that changing bracing arrangements had on member design checks. Behavior of the bridge models containing different bracing arrangements was then examined under an ultimate loading condition to examine the effects of change in bracing arrangement on inelastic behavior of the girders and concrete deck. In Chapter 7 all significant findings obtained from the studies in the previous chapters were summarized and recommendations for future research were provided. 6

25 Chapter 2. Literature Review and Background In this chapter, general background related to beam buckling and bracing requirements, as they relate to the behavior of steel I-sections, was addressed. Then, horizontally curved bridge construction behavior, recognized as one of the most critical stages with respect to maintaining curved girder stability, was discussed. Studies that investigated issues and challenges in relation to the construction of horizontally curved, steel, I-girder bridges were reviewed. Also studies relating to behavior in-service composite curve bridges under live loads were discussed. In addition, studies that addressed the effect of bracing systems on the behavior of girders during bridge construction and while in service were addressed. Finally, relevant specifications and provisions in current Unites States design codes that pertain to the detailing and design of bracing systems in horizontally curved bridges were discussed. 2.1 Bridge stability This section summarized literature related to steel plate girder stability. First, stability issues for steel, I-girders girders were addressed by discussing research related to Lateral Torsional Buckling (LTB). The effects of intermediate bracing and bracing stiffness on buckling capacity were discussed by summarizing relevant research. Finally, developed computational methods that determine curved I-girder buckling capacity were summarized Lateral Torsional Buckling (LTB) I-sections composed of two flanges and one web, arranged to maximize the major axis moment of inertia and the corresponding stiffness, are commonly used as flexural members in many types of structural systems. Due to their low lateral stiffness when being bent about their major axis, they are susceptible to LTB. Moments due to strong axis bending produce 7

26 compression in one flange and tension in the other in the I-shaped girders. When the compression flange buckles, it translates laterally and this causes the entire cross section to twist (See Figure 2-1). The buckling capacity can be improved by increasing the size of the compression flange or by providing intermediate bracing along the length of the girder to prevent twist at discrete points. (a) (b) Figure 2-1 (a) lateral torsional buckling; (b) girder deformed shape at mid-span For beams with doubly symmetric sections under constant strong-axis bending, Timoshenko (1961) derived a closed-form solution for the buckling capacity (Equation 2-1). This solution was obtained using the assumption that twist was restrained at the beam ends (no intermediate bracing) while the beam was free to warp at those locations. 2 1 Where: = beam buckling capacity = unbraced length = modulus of elasticity = modulus of elasticity = modulus of elasticity 8

27 = polar moment of inertia = warping constant= = beam depth. The torsional stiffness of a cross section is divided into uniform and non-uniform components. In Equation 2-1, the first term under the square root is referred to as the St. Venant term and is related to the uniform torsional stiffness of the cross section. The second term is the warping term that is related to the non-uniform torsional stiffness of the section. The American Institute of Steel Construction s (AISC) Code of Standard Practice for Steel Buildings and Bridges (AISC 2010) uses Equation 2-1 to estimate lateral torsional buckling resistance for I- girders under uniform bending moment. Also, the American Association of State Highway and Transportation Officials (AASHTO) LRFD Bridge Design Specifications (AASHTO 2012) gives an equation in Section for lateral torsional buckling resistance that was also adopted from Equation 2-1. In most situations moment varies along the beam and this variation may result in higher buckling capacities than that for the uniform bending moment case assumed for Equation 2-1. This effect can be accounted for using a moment gradient modification factor,, that can be applied to the expression for buckling capacity in Equation 2-1. Many specifications provide different expressions to estimate for general load cases. The most commonly used expression is provided by AISC (2010) and is shown in Equation 2-2. AASHTO (2012) also used an equation for, based on girder flange stresses, which is given in AASHTO Section and is similar to Equation

28 Where: = absolute value of maximum moment in the unbraced segment A = absolute value of moment at the quarter point of the unbraced segment B = absolute value of moment at the midpoint of the unbraced segment = absolute value of moment at the three quarter point of the unbraced segment Effect of beam intermediate bracing Beam bracing can be generally divided into two categories: lateral bracing and torsional bracing. Lateral bracing prevents the girders from moving laterally while torsional bracing resists twisting of the cross section. Most bridges make use of cross frames and diaphragms for bracing of plate and box girders. These systems fall into the category of torsional bracing since they are designed to predominantly restrain twist of the girder cross sections (Helwig and Wang 2003). Many researchers have investigated the behavior of torsional bracing in I-beams. Winter (1960) investigated the bracing stiffness and strength requirements for straight beams. He examined both discrete and continuous bracing systems and indicated that adequate stability bracing must satisfy both stiffness and strength requirements. He also characterized the effect of initial imperfections on bracing stiffness requirements. Taylor and Ojalvo (1966) also examined the behavior of discrete and continuous bracing for straight beams under various types of loading. They derived an expression for buckling capacity of doubly symmetric beams, with continuous bracing, under uniform bending moment. This expression is shown in Equation

29 2 3 Where: = buckling capacity for beam with no bracing (from Eq. 2.1) = continuous torsional bracing stiffness. Yura et al. (1992) also conducted further experimental and analytical investigations on beam bracing requirements. They examined the effects of type of bracing, bracing stiffness and location, and number of bracings along the girders on girder buckling capacity under a variety of loading conditions. As a result they modified the expression for beams with continuous bracing in Equation 2-3 and proposed the expression in Equation Where: = factor for unbraced beam = factor for fully braced beam = a factor that accounts for location of loading with respect to the height of the beam cross section (1 for mid height loading and 1.2 for top flange loading) = yield or plastic straight of the section = moment corresponding to buckling between the bracing points. Moreover, Yura et al. (1992) proposed the expression in Equation 2-5 for an effective moment of inertia for singly symmetric I-girder sections with smaller top flange and larger bottom flange, which are quite common for composite bridge systems with steel plate girders and a concrete deck. This effective moment of inertial can be used instead of in Equation

30 2 5 Where: = moment of inertia of compression flange about weak axis of the section = moment of inertia of tension flange about weak axis of the section = distance from centroid of compression flange to cross section centroid = distance from centroid of tension flange to cross section centroid. To determine the flexural buckling capacity for a girder having discrete bracing, a modification to the continuous torsional bracing stiffness ( term from Equation 2-4 that gives an equivalent continuous bracing stiffness can be obtained using Equation 2-6 from Yura et al. (1992): 2 6 Where: = number of intermediate bracings = span length = effective torsional stiffness for individual bracing. Yura (2001) also proposed a relationship for the effective torsional stiffness of an individual bracing,, which was obtained by superimposing the stiffness of different components that contribute to the overall stiffness of the bracing in Equation 2-7. As can be inferred from this expression, the effective bracing stiffness is smaller than the stiffness of each individual component Where: = stiffness of the attached bracing 12

31 = stiffness of the cross section web including the attached stiffeners = stiffness of the attached girder. The torsional bracing stiffness,, for some common types of systems has been reported by Yura (2001). Some of these stiffness values are shown in Table 2-1 (see Figure 2-2). These expressions were obtained by assuming identical end forces and moments are applied to the bracing systems from the girders. This assumption results in a brace deformed shapes as shown in Figure 2-3 where it is assumed that the girders have the same out of plane web rotations at their intersection with a cross frame or a diaphragm, and no differential deflections between them. The assumption may be true for systems containing straight girders; however, it is largely invalid for horizontally curved girders where differential rotations and deflections between adjacent girders can be significant. Stiffness components for bracing systems used for horizontally curved girders were discussed in detail in Chapter 3. The findings from that study were used to preliminarily examine if the bracing stiffness of a diaphragm is larger than that for a cross frame having similar geometry. Furthermore, these bracing stiffness values were used in Chapter 4 to evaluate finite element modeling methods for different bracing systems. 13

32 Table 2-1 Torsional stiffness of bracing systems (Yura 2001) Bracing system X-type cross frame (compression system) K-type cross frame Torsional stiffness of bracing 2 8 Diaphragm *Where: = girder spacing = height of cross frame = length of diagonal members = area of diagonal members = area of horizontal (cord) members = moment of inertia of diaphragm 6 X-type cross frame K-type cross frame Diaphragm Figure 2-2 Bracing stiffness for twin girder cross frames (Yura 2001) Figure 2-3 Assumed deformed shape for bracing systems (Yura 2001) The expression for the girder cross section web stiffness ( ) in Equation 2-7 was obtained by Yura et al. (1992) and is shown in Equation 2-8. This equation was obtained for 14

33 full depth girder web stiffeners and it accounts for the effects of flexibility of the web and the stiffeners on the overall stiffness of the braced system. Quadrato et al. (2010) indicated that these effects are insignificant when connection plate stiffeners are welded to both top and bottom flanges of the girders Where: = web depth = bracing contact area with flange = web thickness = stiffener thickness = stiffener width. Helwig et al. (1993) examined the influence of in-plane flexibility of girders in a twin girder system on torsional bracing effectiveness. Yura et al. (1992) also examined this behavior and indicated that, with increasing number of girders, this effect becomes smaller. They proposed the expression in Equation 2-9 for the effect of girder stiffness,, when obtaining the total effective stiffness from Equation Where: = number of girders = girder spacing = strong axis moment of inertia of girder = girder length. 15

34 2.1.3 Computational stability analysis methods In a typical static equilibrium problem the stiffness of the structure is assumed to be independent of the applied load. This assumption is valid in the structure prior to buckling. However, as the applied load approaches the buckling capacity, the stiffness of the structure will decrease until it reaches the bifurcation point, after which a small increase in the applied load will result in large deformations in the structural members. This situation can take the form of an eigenvalue problem as shown in Equation Where: = elastic stiffness matrix of the structure = geometric stiffness matrix of the structure (depends on the applied load) = eigenvalue = applied load = eigenvector (buckling mode shape) associated with. The expression in Equation 2-10 indicates that, when the buckling load is reached, the total structural stiffness matrix becomes singular (det K λk G 0). This problem can be solved numerically using finite element analysis methods. In general, there are two finite element methods reported in the literature that can be used to estimate the flexural buckling capacity,, for straight or horizontally curved girders. These methods are linear eigenvalue buckling analysis, and geometric nonlinear analysis (AASHTO/NSBA 2011). The most efficient method to estimate the buckling capacity of the girder is linear eigenvalue buckling analysis. In this method the buckling capacity of a beam under a specific loading condition is obtained by multiplying the maximum moment under the 16

35 applied loads (from a static analysis) by the associated eigenvalue (from a buckling analysis). The relationship is shown in Equation Where: = eigenvalue obtained from buckling analysis = maximum moment under applied load. Linear eigenvalue analysis assumes that girder displacements prior to buckling are very small. Therefore this method is not suitable for cases such as highly curved girders in which significant displacements occur prior to the buckling load. A geometrically nonlinear analysis method can be performed for those cases to obtain realistic predictions for the buckling capacity. This method, which is also called large deformation or second-order analysis, accounts for the load-deformation ( effects that result in additional (second-order) stresses and deformations in the structure. Stith et al. (2009) compared results from the two analysis methods for buckling analyses of straight and horizontally curved girders during the lifting process. For straight girders an initial imperfection equal to L/500, with L being the girder length, was considered in the models. It was understood from this study that the two analysis methods produced almost the same results for straight girders. However, for horizontally curved girders it was demonstrated that the linear eigenvalue analysis method over predicted their flexural buckling capacity. The primary reason for this discrepancy was that, for horizontally curved girders, due to curvature the girders started to deform significantly at the beginning of loading and deformations became larger as the loads increased and no clear bifurcation point existed. This behavior resulted in higher order deformations that prevented the girders from reaching their predicted idealized eigenvalue buckling capacity. 17

36 2.2 Horizontally curved steel I-girder bridges during construction The first study on the behavior of horizontally curved I-beams was completed by De Saint Venant (1843); however, most curved, steel bridge research occurred more than a century later. More specifically, there has been a growing interest in investigating the behavior of horizontally curved, I-girder bridges over the last fifty years. This section provides an overview of past research on the behavior of these types of curved girders during construction. One of the earliest research projects was conducted by the United States Federal Highway Administration (FHWA) in The project, known as FHWA Consortium of University Research Teams (CURT) project, was sponsored by 25 states and was the biggest project that focused on the behavior of curved bridges to date. Several universities were involved in the project and it was comprised of small scale laboratory tests, mostly conducted at Carnegie Mellon, and various analytical studies, performed at the University of Pennsylvania, Syracuse University and other universities. The tests results were reported by Mozer and Culver (1970) and Mozer et al. (1971 and 1973). These test results were utilized by Brennan (1970; 1971 and 1974) to develop several analytical models to investigate the interaction between curved girders and cross frames. Nasir (1970), Brogan (1974) and Culver et al. (1972; 1973) also studied elastic and inelastic flange local buckling behavior. The main outcome of the CURT project was the first AASHTO allowable stress design (ASD) Guide Specifications for Designing Curved Bridges (AASHTO 1980). Although this project investigated several issues in curved bridges before and after deck placement (Brennan 1971; 1974), it did not specifically address erection behavior of the girders. The FHWA started the Curved Steel Bridge Research Project (CSBRP) in 1992 mainly to address deficiencies of preceding studies that contributed to developing and refining the initial 18

37 AASHTO Guide Specifications publication. The CSBRP project was conducted in three main phases: study of the behavior curved bridges during erection; study of the strength of curved bridge components; to study of the behavior of a composite curved bridge. The erection study phase centered on a full scale experimental study, conducted at the FHWA Turner-Fairbank Highway Research Center, that focused on a single span horizontally curved I-girder bridge and investigated its behavior during construction and under applied loads. Figure 2-4 shows the tested structure. More detailed information about this structure is provided in Chapter 4 (Finite Element Modeling). The studies completed in this phase to examine several erection methods and different shoring scenarios to study girder construction response. Study results were largely published by Linzell (1999), Zureick et al. (2000) and Linzell et al. (2004). Later construction studies examined the effects of erection sequencing on induced stresses and deformations provided improved design guidelines and examined the capability of analysis tools to effectively predict bridge response (Linzell et al. 2004; White and Grubb 2005). Figure 2-4 CSBRP test structure for phase one, at FHWA Turn-Fairbanks Structures Laboratory (Hartmann 2005) 19

38 The component strength phase focused on the ultimate strength and associated behavior of steel I-girder components subjected to: 1) uniform major axis bending; 2) high major axis bending combined with high shear; and 3) high shear combined with low major axis bending. For this phase the three-girder bridge shown in Figure 2-4 was used as the test frame. The components were bolted at mid span of the outermost girder in the test frame and the structure was loaded until failure of the test component (Zureick et al. 2000). Eight components were tested in uniform major-axis bending and four components were tested in high major axis bending combined with high shear. Zureick et al. (2000) also conducted full scale experimental studies at the Georgia Institute of Technology to examine the behavior and strength of four curved steel I-girders under high shear low moment. Several additional component tests were also conducted by Hartmann (2005) to determine the effect of web and flange slenderness and transverse stiffener spacing on curved girder vertical bending capacity. The results showed that stiffener spacing and web slenderness had small effects on the vertical bending capacity but compression flange slenderness had significant effect. The last phase of the CSBRP also involved full scale experimental analysis of a single-span horizontally curved composite I-girder bridge designed in accordance with the AASHTO (2004) LRFD Bridge Design Specifications. The major objective of this phase of the research was to examine system and component behavior under non-composite dead load (Chang 2006), composite live load and ultimate loading (Jung 2006). The overall geometry of the bridge tested in this phase was similar to that of the test frame for the erection study tests in Phase I. However, to make the structure more realistic, smaller I-girders sections were used for the composite bridge tests and the spacing between the cross-frames was increased relative to that 20

39 used in the structure tested for Phase I. Figure 2-5 shows a perspective view of the steel structure for the composite test bridge prior to deck placement. Figure 2-5 CSBRP test structure for phase three, at FHWA Turn-Fairbanks Structures Laboratory (Jung 2006) Since the initiation and completion of these projects, additional experimental and computational studies aimed at addressing some issues pertaining to curved, I-girder construction behavior have been published. One project by Bell (2004) incorporated field study of a multi-span curved bridge. The bridge had experienced unpredicted deformations in the first erected span during placement of the girders, and that span was instrumented to monitor response during construction of additional spans. Measured response was used to calibrate a finite element model employed to examine the influence of girder placement order on behavior. Findings indicated that, when single girder erection was used, the girders should be erected from the inner radius toward outer radius; however, for paired girder erection it was recommended to erect them from outer to inner radius. These findings were confirmed by Linzell and Shura (2010) and Sharafbayani and Linzell (2012) through other field monitoring 21

40 and numerical studies. These studies also highlighted the important role of temporary shoring supports on construction performance. Another study examined a bridge that had experienced severe alignment and fit-up problems during early stages of erection of some of the curved girders segments as discussed by Chavel and Earls (2006a). Based on behavior of these girders during erection, a finite element model was developed and modeling results were validated using CSBRP project data. However, the model was unable to identify the main reasons that caused the construction problems until bridge construction plans were reviewed. The plans identified that the cross frames and girders were inconsistently detailed (i.e. detailed to fit together under different loading conditions). The girders were detailed based on the no-load condition, assuming no deformations and rotations in the girders due to structure self-weight, whereas the cross frames were detailed for the structure s final deformed shape. This information was discussed by Chavel and Earls (2006b), where it was indicated that problems from inconsistently detailed cross frames were more significant for bridges with smaller radii of curvature and for stiffer structures. Howell and Earls (2007) conducted a numerical study to investigate how web out-ofplumbness at the end of erection can affect the behavior of curved girders during deck placement. A bridge finite element model was developed and its response was studied for different girder web rotation (out of plumb) angles. The results showed that high web rotations can induce large deformations and stresses on the girders and the cross frames. According to these results, the authors recommended that girders should be detailed such that their webs hold a plumb position under the full dead load condition. 22

41 Stith et al. (2009) examined issues related to lifting of curved girder segments during construction. The first part of the project was a survey of several bridge contractors from different states to establish their experience with curved bridge construction and identify common strategies for lifting, shoring and erecting curved girder segments. Commonly accepted contractor serviceability and strength criterion during construction were also identified. The field study examined response of curved girders while being lifted. Numerical models of curved girders having varying geometries and support conditions were generated and benchmarked against the lifting data. Two different stability analysis methods were examined for predicting buckling capacity during lifting: linear eigenvalue buckling analysis and fully nonlinear geometric analysis. Findings indicated that the two methods yielded almost the same results for straight girders but different results for curved girders, as discussed previously, and linear eigenvalue analysis was found to be unable to appropriately predict curved girder behavior. Optimal crane pick points on the girder top flanges were also proposed and a spreadsheet based program was developed that could be used by bridge designers and contractors to analyze the lifting performance of curved girder segments. The final study that is summarized here was the basis for the present work and completed by Linzell et al. (2010). This research was sponsored by the Pennsylvania Department of Transportation and the main objective was to develop guidelines for analyzing and designing horizontally curved and skewed, steel I-girder bridges during construction in Pennsylvania. The project involved a set of parametric studies that investigated the effect of several geometric and environmental variables on the behavior of curved bridges during construction. These studies included examining the effects of the following items on construction behavior: erection sequencing; shoring tower locations during erection; web out of plumbness; ambient 23

42 temperature change; support settlement; cross frame inconsistent detailing; and cross frame and diaphragm performance. Findings on the effect of web out of plumbness on skewed bridges and the effects of temporary shoring location on curved bridges were published by Sharafbayani et al. (2011) and Sharafbayani and Linzell (2012). 2.3 Studies on effects of live loads on behavior of in-service horizontally curved bridges In addition to the studies discussed in the previous section for the behavior of noncomposite curved bridges during construction, many studies have been completed on the behavior of in-service, composite, curved bridges under live loads. McElwain and Laman (2000) completed a series of field studies to investigate behavior of three in-service, curved, I- girder bridges under truck loads. They compared lateral bending distribution factors and dynamic load allowance obtained from filed data with those provided by AASHTO LRFD Bridge Design Specifications (1998), and indicated that the AASHTO specifications were conservative for these parameters. Zhang et al. (2005) conducted a series of computational studies to examine the effects of live load on different girders in horizontally curved bridges. The main objective of the study was to develop simplified formulas to predict positive moment, negative moment, and shear distributions for single-lane and multiple-lane live loading on curved bridges. The effects of a number of key parameters on the live load distribution were examined. Radius of curvature, girder spacing and number, overhang length, slab thickness, and longitudinal bending inertia were shown to have significant effects on the distribution factors. However, the effects of cross frame spacing and girder torsional inertia were found to be relatively small. Later, Nevling et al. (2006) conducted a field study on a three-span curved bridge system with skewed abutments under various live loading schemes. The field data was used to evaluate 24

43 the accuracy of three commonly employed levels of analysis to predict behavior of horizontally curved, steel, I-girder bridges. Level 1 analysis included the simplified line girder analysis method and the V-load method, while Level 2 and Level 3 analyses were those that used grillage and three-dimensional (3D) finite element models, respectively. According to this study, Level 2 and Level 3 analyses were generally understood to be able to predict girder vertical bending moment distributions more accurately than Level 1. In addition, the bottom flange lateral bending moment distributions that were obtained from Level 1 and Level 3 analyses in this study were shown not to be in good correlation with field test results. In a similar study, Barr et al. (2007) examined behavior of a curved, steel, I-girder, bridge under live loads. A 3D finite element model was calibrated using measured field data. The finite element analysis results for bending moments and distribution factors were then compared against the results based on the V-load method for different boundary conditions. The V-load method was shown to produce unconservative results in some examined cases for positive and negative bending moments when compared against those resulting from the 3D finite element analysis. Furthermore, Kim et al. (2007) conducted a series of parametric studies to examine the effects of different geometrical parameters on live load distribution between girders in horizontally curved bridges. The parameters that most significantly affected live load distribution in curved, I-girder, bridges were found to be radius of curvature, span length, cross frame spacing and girder spacing. On the other hand, the deck thickness, flange width, web depth and including the parapet in the analysis were shown to have relatively small influences on live load distribution. Based on these results a new equation for live load distribution factor 25

44 was developed and shown to accurately predict live load radial moment distribution in curved bridges. Lastly, a field study was completed by Hajjar et al. (2010) to examine the behavior of a composite five-span curved bridge under live truck loadings. Field measurements were compared against results from a linear elastic, grillage analysis model. The grillage model results were found to accurately correlate with the measured displacements and vertical bending stresses in the girders top and bottom flanges. Using this model, flange warping stresses were, however, found to be difficult to predict. At the end of this study some recommendations were provided to improve the accuracy of grillage modeling of composite horizontally curved bridges. 2.4 Studies on behavior of bracing systems in horizontally curved and/or skewed I-girder bridges A limited number of studies have been completed examining the performance of cross frames and diaphragms in horizontally curved and skewed bridges. Keller (1994) and Davidson et al. (1996) conducted numerical studies to investigate the effects of different global bridge geometric parameters, including cross frame and lateral bracing configuration and placement, on the behavior of curved bridges under both construction and service conditions. From those studies curvature effects on girders and cross frames response were shown to be greatest during construction, after addition of the wet concrete dead load to the non-composite girder system. It was also found that cross frame spacing was a primary parameter that influenced the stress gradient across the girder flanges resulting from curvature effects. Maneetes and Linzell (2003) parametrically studied the effects of cross frames on free vibration response of a curved I-girder bridge during construction that included different cross frame types (X-type or K-type), and 26

45 spacing. The findings indicated that different cross frame types had nearly identical effects on girder displacements, stresses and natural frequencies. It was also shown that increasing the number of cross frame members in the bridge reduced lateral displacements. The AASHTO LRFD Specifications (AASHTO 2012) require intermediate cross frames and diaphragms in bridges with non-skewed abutments to be placed in contiguous lines normal to the girder webs. The present study aims to examine the effect of skewed (non-perpendicular to girder webs) cross frames on the behavior of horizontally curved steel I-girder bridges during construction and for in-service conditions. The use of cross frames that are aligned parallel to bridge skew and not normal to the girder web is recommended in AASHTO (2012) for bridges with skew angles less than 20. For larger skew angles, cross frames are recommended to be placed normal to the girder web (see Figure 2-6). Winterling (2007) showed that the effective cross frame stiffness, perpendicular to the girder, reduces significantly for cross frame skew angles larger than 20. This resulted in issues that included fit up problems, member misalignments and excessive girder deformations during construction for a bridge that was examined which contained intermediate cross frames aligned parallel to supports skewed at approximately 60. The bridge required significant retrofitting before the deck was placed and included adding additional cross-frames perpendicular to the girders. Skew 20 Skew 20 Figure 2-6 Bracing orientation for bridges with skewed supports The stiffness of cross frames parallel to the skew was studied by Quadrato et al. (2010) in conjunction with examining two types of cross frame connection details: bent plate stiffeners 27

46 and split pipe stiffeners. Connection stiffness with split pipe stiffeners was found to be larger than that for bent plates, particularly for large skew angles. The influence of bent plate connection stiffness on overall stiffness of a parallel to skew bracing system was also investigated. It was found that, for cross frames with bent plate connections and with skew angles larger than 30, the bracing connection became the most flexible component and reduced overall bracing stiffness. In one of the most recent studies, Sanchez (2011) examined the effect of removing individual members from different cross frame configurations (i.e. X-type with one diagonal removed, X-type with top cord removed, and K-type with top cord removed) on the behavior of horizontally curved and skewed bridges during construction. The study indicated that, for the cases that were examined, removing the top cord from K-type cross frames was the case that resulted in significant stiffness loss and caused the cross frames to be largely ineffective. A simplified method was also developed to be used in conjunction with in grillage (2D) finite element models to more accurately predict the construction behavior of horizontally curved and skewed bridges. Stiffness modification factors were generated for girder sections in the grillage models that incorporated the effect of warping stresses. A procedure was also introduced for determining equivalent stiffness components for different bracing systems that are modeled using single, line, elements in the grillage models. Results from this study are also reported in National Cooperative Highway Research Program (NCHRP) Report 12-79, Guidelines for Analytical Methods and Erection Engineering of Curved and Skewed Steel Deck-Girder Bridges (NCHRP/TRB 2011). 28

47 2.5 Design and construction guidelines There are several bridge design documents that specify requirements for bracing systems in horizontally curved I-girder bridges during construction or while in-service. Some of these specifications provide specific recommendations while others only provide general guidance. In this section a summary of the design provisions for steel girder bracing systems in two national codes, the AASHTO LRFD Bridge Design Specifications (AASHTO 2012) and the Guidelines for Design for Constructability by the AASHTO/NSBA Steel Bridge Collaboration (AASHTO/NSBA 2003) are provided AASHTO LRFD Bridge Design Specification (AASHTO 2012) Most AASHTO LRFD Bridge Design Specification (AASHTO 2012) steel bridge component design criteria are provided in Section 6. A summary of Articles that address designing bracing system details and components is provided in the following sections Section Distortion Induced Fatigue The AASHTO Specification requires that transverse connection plates be bolted or welded to the girder web and flanges when those connection plates are attached to diaphragms or cross frames. Attaching the connection plate to the flanges reduces fatigue problems caused by out of plane distortion in the girder webs around the cross frame regions Section Diaphragms and Cross Frames - General This section discusses investigating the need for diaphragm and cross frames for all stages of construction and in the final condition. The commentary of this section states, Bracing of horizontally curved members is more critical than for straight members. Diaphragms and cross frame members resist forces that are critical to 29

48 the proper functioning of curved bridges. Since they transmit the forces necessary to provide equilibrium, they are considered primary members. Therefore, forces in the bracing members must be computed and considered in the design of these members Section Diaphragms and Cross Frames - I-Section Members This section provides criteria related to the required depth of diaphragms and cross frames for rolled beams or plate girders. It states, Diaphragm or cross frames for rolled beams and plate girders should be as deep as practical, but as a minimum should be at least 0.5 of the beam depth for rolled beams and 0.75 of the girder depth for plate girders. It further states that cross frames in horizontally curved bridges should contain both top and bottom cords. The commentary to this section states that uniform spacing for the intermediate cross frames or diaphragms is more desirable from the standpoint of more efficient structural analysis and also for constructability. However, it is also discussed that closer brace spacing may be required at the critical regions, such as at mid span girder sections, over an interior pier or near skewed supports. For diaphragms it is specified that if their span to depth ratio is larger than 4.0, they may be designed as basic beams. However, for span to depth ratio smaller than 4.0, shear deformation shall be included in the design of the diaphragms. For bridges with non-skewed abutments, this section specifies that the cross frames should be placed in contiguous lines and normal to the girders. The specification allows for intermediate diaphragms or cross-frames on bridges with supports skewed 20 or less to be 30

49 placed parallel to the skew. However, when the supports are skewed more than 20 the intermediate diaphragms or cross-frames are required to be placed normal to the girder webs. For horizontally curved bridges, it is required that the spacing between the intermediate cross frames and diaphragms ( ) shall not exceed the expression in Equation 2-12 (AASHTO Eq ): Where: = limiting unbraced length, determined from Equation 2-13 (AASHTO Equation ) = radius of curvature Where: = effective radius of gyration for lateral torsional buckling, determined from Equation 2-14 (AASHTO Equation C ) = compression flange stresses at the onset of nominal yielding within the cross section Where: = width of the compression flange = thickness of the compression flange = thickness of the web = depth of the web in compression in elastic range. 31

50 In the commentary to this section it is explained that the brace spacing limit of is imposed to prevent elastic lateral torsional buckling of the compression flange AASHTO/NSBA Guidelines for Design for Constructability (AASHTO/NSBA 2003) This document was published jointly by AASHTO and the National Steel Bridge Alliance (NSBA) to provide additional guidelines and recommendations for bridge designers when designing steel bridge girders and components for the Construction Limit State. Section 2 of this document includes information about bracing systems and their connection plates. A summary of relevant Articles that apply to the research project is discussed below Section Connection of Skewed Intermediate Cross Frames (AASHTO/NSBA 2003) This section discusses two common types of details for intermediate skewed cross frames with skew angles less than 20. These two details are shown in Figure 2-7. In the commentary to this section it is stated that the Bent Plate Gusset Plate detail is preferred by most fabricators. Bent Gusset Plate Skewed Connection Stiffener Figure 2-7 Connection detail for intermediate skewed cross frames (AASHTO/NSBA 2003) 32

51 Section Cross Frames and Diaphragms (AASHTO/NSBA 2003) This section provides recommendations, based on current practice, for the selection of bracing system for I-girder bridges. It generally recommends X-type cross frames to be used for bracing. However, if the angle of the diagonal is less than 30 (see Figure 2-8) K-type cross frames are recommended. Moreover, this section recommends using diaphragms as a viable bracing option for girders less than 122 cm (48 ) deep Summary Figure 2-8 X-type cross frame, minimum diagonal angle This chapter has described the current state of knowledge on topics that are relevant to the present research that focuses on the performance of bracing systems in horizontally curved, I- girder, bridges. First studies that investigated stability of steel I-sections were reviewed. Lateral torsional buckling (LTB) was discussed and the effects of intermediate bracing on girder buckling capacity were summarized. Studies that investigated stability issues for curved girders during construction were reviewed next. Also the studies that evaluated the behavior of horizontally curved bridges under live loads were discussed. In addition, researches that examined the performance of bracing systems in curved and skewed bridge systems during construction and for service conditions were summarized. Finally, the current design and 33

52 construction guide specification, in relation with bracing requirements in horizontally curve bridges were addressed. 34

53 Chapter 3. Analytical Studies for Bracing Systems Stiffness Cross frames and diaphragms are primary load carrying members in horizontally curved bridges (AASHTO 2012) and the behavior of girders in these bridges (e.g. magnitudes of radial deformation and lateral bending stress of the girder flanges) is directly dependent on the stiffness of the bracing systems. The objective of the present study was, in part, to examine the performance of cross frames and diaphragms in horizontally curved bridges considering their behavior during construction and while in-service. The most important factor that has the largest influence on the performance of the bracing systems is the bracing stiffness. There is limited information available in the literature regarding the difference between stiffness of cross frames with diaphragms with various geometries, when used in horizontally curved I- girder bridges. This chapter aimed to conduct simplified analytical studies to determine the bracing stiffness of cross frames and diaphragms. Resulting solutions were used to examine if an individual diaphragm is generally stiffer than a cross frame when they both are designed to be used as a bracing system in a horizontally curved, I-girder bridge. Results from studies in this chapter were used in Chapter 4 for validation of bracing system finite element modeling techniques in horizontally curved bridge models that were, in turn, employed in Chapter 5 and Chapter 6. Furthermore, in Chapter 5 findings from examinations that compared individual bracing system stiffness were integrated into an expanded computational study that examined bracing system stiffness influence on curved bridge construction performance. 35

54 3.1 Calculation of stiffness components for bracing systems in horizontally curved I- girder bridges Section indicated that expressions in the literature for establishing bracing system stiffness are obtained based on an assumed deformed shape (see Figure 2-3) in which the bracing experiences similar end rotations and no differential deflections. Although this assumption is valid for straight bridges, it does not fit the behavior of bracing systems in horizontally curved bridges. For these bridges, due to curvature effects, two adjacent girders at equal distances from the bridge supports are subjected to different deflections and out of plane rotations. Therefore, to accurately assess the stiffness of bracing systems that interconnect two curved girders, be they diaphragms or cross frames, at least four degrees of freedom (DOFs), two translational ( and ) and two rotational( and ), must be considered as shown in Figure 3-1. Figure 3-1 Degrees of freedom in a general bracing system The analytical solutions for the stiffness matrices of K- and X-type cross frames and solid plate diaphragms in a curved bridge were obtained based on matrix analysis techniques (Kassimali 2011). All calculations were performed using MATLAB. A copy of the MATLAB calculations is provided in Appendix A. A summary of the assumptions and the procedures used for stiffness matrix calculations for diaphragm, X-type and K-type cross frames is provided in the following sections. 36

55 3.1.1 Diaphragm stiffness matrix To obtain the stiffness matrix for a typical diaphragm, an analytical model was formed as shown in Figure 3-2. The diaphragm was modeled as a single beam element with two nodes following assumptions made by Yura et al. (1992) when determining the stiffness of a diaphragm in a straight girder system as discussed in Section Each node in element had one translational and one rotational DOF with assumed positive directions as shown in Figure 3-2. These DOF s are consistent with the general bracing DOF s assumed in Figure 3-1 ( ; ; ; ). According to AASHTO (2012) a bracing connection plate stiffener is required to be attached to both top and bottom flanges. Therefore, vertical deformation of the girder web, and girder cross section distortion, were very small (Quadrato et al. 2010) and were neglected in the stiffness calculations. As a result of these assumptions, vertical and rotational DOF s of each node represented rigid body vertical deformation of the girder and rigid body rotation of the girder about its longitudinal axis, respectively. Node and element numbers Degrees of freedom Figure 3-2 Diaphragm analytical model The diaphragm cross section in this study was assumed to be a built up I-section with a solid plate web and two flanges at the top and the bottom (see Figure 3-2). This section has commonly been used for diaphragms in steel bridges and has been addressed in some steel bridge codes and standards (e.g. TxDOT 2006; PennDOT 2007). According to the AASHTO 37

56 (2012), since the span to depth ratio of a diaphragm is relatively small ( 4) for common girder spacings and depths, the diaphragm element was assumed to behave like a Timoshenko beam that incorporated the effects of both shear and bending deformations. Expressions for stiffness matrix elements for a Timoshenko beam can be found in a number of sources, such as Kassimali (2011), and the stiffness matrix for a diaphragm appears as shown in Equation Where: = stiffness matrix for diaphragm E = modulus of elasticity = moment of inertia of the diaphragm section about strong axis = girder spacing = shear deformation constant 12 = shear modulus = diaphragm web area ( ) = height of bracing = diaphragm web thickness = shape factor: 1.2 : for rectangular sections (used for the diaphragm section in this study); 1.0 : for wide flange beams bent about the major axis. 38

57 3.1.2 X-type cross frame stiffness matrix In similar fashion to diaphragms, an analytical model was developed for X-type cross frames, as shown in Figure 3-3, and was used to obtain a stiffness matrix. The model was assumed to be comprised of four elements with each element having two nodes at its ends. The X-type cross frame model was assumed to have both top and bottom chords, as required by AASHTO (2012) for horizontally curved bridges. Single angle sections were used for cross frame members, according to the common practice (AASHTO/NSBA 2003). The members were assumed to behave as truss elements for which only axial strength was accounted for and flexural stiffness was neglected. Accordingly, for each node in the models two translational degrees of freedom in the global X and Y directions were considered (see Figure 3-3). Relative vertical deformation between the nodes at the top and bottom of each girder web was neglected according to Yura et al. (1992). As a result, the top and bottom nodes on each side of the cross frame were assumed to have the same degrees of freedom in the Y direction (see Figure 3-3). The stiffness matrix of each element was calculated based on the expression for stiffness of truss elements in the global coordinate system (Kassimali 2011). The final stiffness matrix ( ) was obtained by assembling all individual element stiffness matrices. Node and element numbers Degrees of freedom Figure 3-3 Analytical model for X-type cross frame 39

58 To compare the stiffness components for this type of bracing system against those derived for a diaphragm, the system of 6 DOF s shown in Figure 3-3 was transformed into a model for a general bracing system with 4 DOF s shown in Figure 3-1. Relationships selected to relate the two DOF systems neglected distortion of the girder cross sections and assumed rigid body rotation of the girders on either end of a cross frame. This is valid, again, because connecting cross frame connection plate stiffeners to girders top and bottom flanges, as required by AASHTO, renders girder distortion largely insignificant (Quadrato et al. 2010). Resulting expressions are shown in Equations 3-2. U θ L h U L U θ R h 2 U R U θ L h 2 U θ R h 2 Where: h = height of cross frame. The 6 by 6 stiffness matrix obtained for the X-type cross frame (see Appendix A) was then transformed into a 4 by 4 stiffness matrix using Equation Where: = 4 4 stiffness matrix for X-type cross frame for DOF system of Figure 3-1. = 6 6 stiffness matrix for X-type cross frame for DOF system of Figure

59 T= transformation matrix from DOF system in Figure 3-3 to that in Figure 3-1, shown in Equation 3-4 (obtained from the relationships in Equation 3-2) T T = transpose of the transformation matrix T The resulting stiffness matrix is shown in Equation 3-5 (see Appendix A for detailed calculations) Where: = modulus of elasticity = girder spacing = height of bracing = cross section area of horizontal members = cross section area of diagonal members = length of diagonal members K-type cross frame stiffness matrix The assumed analytical model for the K-type cross frame is shown in Figure 3-4. The model, which was comprised of five elements and five nodes, assumed the cross frame 41

60 members to behave like truss elements and neglected their flexural stiffness. Therefore, similar to the X-type cross frames and following Yura et al. (1992), only translational degrees of freedom in the X and Y directions were considered for each node in the K-type cross frame model. This resulted in a total of 8 degrees of freedom as shown in Figure 3-4. The resulting stiffness matrix was an eight by eight matrix ( ) that was obtained by assembling stiffness matrices for the individual truss elements (see Appendix A). Node and element numbers Degrees of freedom Figure 3-4 Analytical model for K-type cross frame Again, to make the stiffness matrix of the K-type cross frame comparable to those for the X-type cross frame and diaphragm, the system of 8 DOF s shown in Figure 3-4 was transformed into a 4 DOF system for general bracing shown in Figure 3-1. The procedure for this was similar to that explained for X-type cross frames, except that an additional step was taken prior to application of transformation matrix. The eight by eight stiffness matrix; K, obtained for the system of 8 DOF s in Figure 3-4 was first condensed to a six by six stiffness matrix, following the procedure for stiffness matrix degrees of freedom condensation from Kassimali (2011). Degrees of freedom to were selected as independent (external) DOF s and to were considered as dependent or internal DOF s (see Appendix A). The resulting six by six stiffness matrix; was then transformed to a four by four stiffness matrix for the 4 DOF general bracing system (Figure 3-1), using the relationship in Equation

61 3 6 Where: = 4 4 stiffness matrix for K-type cross frame for DOF system of Figure 3-1. = condensed 6 6 stiffness matrix for K-type cross frame for DOF s to of Figure 3-4. T transformation matrix (Equation 3-4). The resulting stiffness matrix is shown in Equation 3-7 (for detailed calculations see Appendix A) Comparing bracing systems stiffness Diaphragm and X- or K-type cross frames have alternatively been used as intermediate bracing for girders in existing bridges. The performance of these bracing systems in bridges is directly related to their stiffness, especially for curved bridges where they have a primary role of maintaining structural equilibrium. This section examines differences between the stiffness of individual bracing systems discussed in Sections through Findings from these comparisons were then used in Chapter 5 to justify the effects of varying bracing stiffness via different bracing configurations on the construction performance of horizontally curved bridge normal-to-web or skewed bracing arrangements. 43

62 To examine the stiffness differences, two representative elements in the stiffness matrices for each type of bracing system were compared for similar bracing geometries. These selected elements were and in each bracing s stiffness matrix as shown in Equations 3-1, 3-5 and 3-7. These elements correspond to the shear stiffness and the flexural stiffness of the bracing system as defined in the Guidelines for Steel Bridge Girder Analysis by AASHTO/NSBA (2011), and subsequently, influence shear or flexural deformations in the bracing, as illustrated in Figure 3-5. Shear Stiffness. Flexural Stiffness Figure 3-5 Bracing system stiffness components (AASHTO/NSBA 2011) To compare the stiffness of different bracing systems, full depth diaphragms or cross frames were arranged to fit between a pair of girders having varying girder spacing (S) and bracing girder depth (D) as shown in Figure 3-6. The examined girder depths and spacings were selected based on statistically significant geometric parameters in existing horizontally curved, I-girder bridges in Pennsylvania, Maryland and New York identified from a previous study (Linzell et al. 2010). 44

63 Figure 3-6 General bracing geometry (for diaphragm and K- and X-type cross frames) The assumed cross sections for the cross frame and diaphragm members for the calculation of their flexural and shear stiffness quantities were selected following design requirements from AASHTO (2012) and AASHTO/NSBA (2003). More detailed information about the design of these components is provided in Chapter 5. A 127x127x13 mm (5x5x½ in) was considered for the cross frame members (X-type or K-type). This angle section is very commonly used in the existing horizontally curved I-girder bridges (Stith et al. 2009). The diaphragm cross section was assumed to be a built up I-section with two 245x13 mm (10x½ in) plates used for the top and bottom flanges and a solid web plate with a thicknesses of 11 mm ( 7/16 in). Figure 3-7 compares the shear (k s =k 11 ) and flexural (k f =k 22 ) stiffness components for X- type and K-type cross frames between two-girders having a constant girder spacing and varying depth. The quantities in these figures are the ratios for the stiffness components (flexural or shear) of the X-type cross frames to those of the K-type cross frames. The results in this figure show that, for the different girder depths that were examined, the stiffness of an X-type cross frame is, in general, close to those of a K-type cross frame (the stiffness ratios are close to one) for similar geometries. Also, by increasing the girder depth, the X-type cross frames become slightly stiffer than the K-type cross frames, especially with respect to shear. 45

64 2.0 Girder depth, D (in) Figure 3-7 Comparison of X-type and K-type cross frame shear and flexural stiffness elements, varying girder depth, constant girder spacing: S=2.67 m (8.75 ft) In a similar fashion, Figure 3-8 and Figure 3-9 show the stiffness ratios of diaphragm to K- type or X-type cross frames, respectively, for shear and flexural stiffness values in a two-girder system with constant girder spacing and varying depths. As shown in these figures, for all geometries examined for a two-girder system both the shear and flexural stiffness components for a diaphragm are larger than those for a K- or X-type cross frame, by at least a factor of 2. These results typify those that demonstrate that higher bracing stiffness is provided by a diaphragm when compared to that for a cross frame of similar geometry. As expected, the higher contribution to the overall diaphragm stiffness generally comes from the shear stiffness component (k s =k 11 ). Moreover, the results seem to indicate that the difference between the stiffness provided by a cross frame compared to that for a diaphragm increases for deeper girder cross sections. It must be noted that, in spite of the stiffness advantages that a diaphragm can provide, it is generally heavier than a similarly sized cross frames (more than 50% heavier for the geometries examined Figure 3-7 to Figure 3-9). This item is discussed further in Chapter 5. K XCF /K KCF Ks=K Girder depth, D (cm) Kf=K K XCF /K KCF 46

65 5 Girder depth, D (in) K DIA /K KCF K DIA /K KCF 1 0 Ks=K11 Kf=K Girder depth, D (cm) Figure 3-8 Comparison of diaphragm and K-type cross frame shear and flexural stiffness elements, varying girder depth, constant girder spacing: S=2.67 m (8.75 ft) 5 Girder depth, D (in) K DIA /K XCF K DIA /K XCF 1 0 Ks=K11 Kf=K Girder depth, D (cm) Figure 3-9 Comparison of diaphragm and X-type cross frame shear and flexural stiffness elements, varying girder depth, constant girder spacing: S=2.67 m (8.75 ft) 3.3 Summary and conclusions Cross frames and diaphragms are common bracing systems used for steel bridges. The performance of these bracing systems, especially in curved bridges, is directly related to their stiffness. In this chapter parametric stiffness matrices for diaphragms, X-type cross frames, and K-type cross frames were obtained using matrix analysis method. To determine the difference 47

66 between the stiffness of a cross frames and a diaphragm, two representative elements of the stiffness matrices were compared for different bracing geometries. These elements represented the shear and flexural stiffness of the bracing systems. For all geometries examined, X-type and K-type cross frames were understood to have similar stiffness, with X-type cross frame being slightly stiffer than K-type cross frame in shear and for girder systems with larger depths. Also, for all girder depths diaphragms were found to be generally stiffer than cross frames, particularly in shear. The higher stiffness for this bracing system was more significant for larger girder depths. However, it was recognized that there is a trade-off between structural weight and stiffness for the use of cross frame or diaphragm in a bridge. The parametric study information obtained from this chapter was used in Chapter 4 to validate finite element modeling methods for the different bracing systems in curved bridge models that were then employed for additional computational studies in Chapter 5 and Chapter 6. Also, findings from the studies completed in this chapter were used in Chapter 5 to study the influence of varying bracing stiffness on the construction performance of different horizontally curved bridges containing either normal-to-web or skewed bracing arrangements. 48

67 Chapter 4. Finite Element Modeling The main purpose of this study was to propose optimized bracing arrangements for horizontally curved bridges. To examine the effects that bracing had on the behavior of horizontally curved bridges during construction and service life of the bridge, finite element analysis (FEA) was used. Bridges are commonly modeled using grillage or three-dimensional (3D) finite element modeling methods. A grillage model represents the girders using beam elements and a single beam element is used to model cross frames or diaphragms between the girders. While this modeling method may be more desirable to bridge designers due to its simplicity, it has some deficiencies when used to model horizontally curved or skewed bridges. In particular, the effect of curvature or skew effects on lateral bending stresses in girder flanges cannot be included in the grillage models. There exist some approximate methods proposed in the literature (Linzell et al. 2004; Chang 2006; Sanchez 2011) to eliminate the deficiencies of grillage modeling for horizontally curved or skewed bridges. These methods, however, have some limitations and may not be reliable for all bridge cases. When using a 3D finite element model, bridge girders are usually represented using shell elements for the girder webs and beam or shell elements for the girder flanges. Bracing systems in these models are included using either line (beam or truss) elements or the brace members and their connection plates are modeled in more detail using shell elements. Quadrato et al. (2010) conducted a validation study for modeling bracing systems in skewed bridges and compared analysis results between bracing models using line or shell elements against experimental data. Results showed that line element modeling can sometimes overestimate the stiffness of the bracing systems and, therefore, full shell modeling was indicated to be a better approach for modeling bridge bracing systems. 49

68 In the present chapter full 3D FEA modeling methods were developed to study the behavior of horizontally curved I-girder bridges for construction and in-service conditions. In these models shell elements were used to model all members and components in the bridges. Utilized modeling techniques were either validated against experimental or analytical results in the sections that follow or were adopted from validated models from other studies. 4.1 FEA modeling for construction studies The modeling method developed in the this section is also used for studies in Chapter 5 that examine the effects of changing brace orientation with respect to girder webs on the behavior of non-composite girders during construction. During construction all steel members were designed according to AASHTO (2012) to remain in elastic region. Therefore, for the construction studies a linearly elastic material was used for the steel members in all examined bridge cases, except for one case discussed in Section in which a nonlinear material model was used for the bent gusset plates. Nonlinear geometric effects were, however, considered for all bridge analysis cases to account for the effects of higher order deformations and stresses on behavior. The bridges, used for the construction studies, were composed of non-composite steel plate girders, cross frames or diaphragms, stiffeners, and cross frame connection gusset plates. Prior to initiation of any computational studies, the modeling scheme was validated via examination of curved girders containing the different bracing systems. To validate the overall performance of the curved bridge models, a finite element model of the structure examined for the erection study in the initial phase of the CSBRP project (Linzell 1999) was developed using ABAQUS (ABAQUS 2010), and was compared to experimental data. This structure was selected primarily because it mimicked a tightly curved, plate girder structure and test results were 50

69 readily available (Linzell 1999; Zureick et al. 2000; Linzell et al. 2004). Note that Linzell (1999) also developed and validated a finite element model for the structure tested for the erection study in the CSBRP project. The model developed for the current study differed from the previous model, however. The model discussed in Linzell (1999) was also used for the study of component strengths in the second phase of the project and as such, used a refined mesh for a section located at mid span of the outer most girder. Also, it did not consider geometric nonlinearity effects that were found in a later phase of the CSBRP project to be relatively significant for some analysis results (Jung 2006). The model employed in the present study used uniform element sizes for all girder sections along their spans and incorporated nonlinear geometric effects. As a result, validation of this model was necessary. The bracing system in the CSBRP structure was a K-type cross frame that used pipe sections for the cross frame members. This cross-section is not commonly utilized for cross frame members in existing curved bridges. As a result, bracing systems examined for the bridge models in the current study were different from cross bracing used for CSBRP. As explained in earlier chapters, the current study used angle sections for cross frame members. Also, the effect of diaphragms on behavior was examined using I-shaped sections. Since no actual experimental data was available for these bracing systems in a curved bridge, in an attempt to ensure the validity of finite element results that included various bracing configurations finite element results were compared against results from the analyses from Chapter 3. This was accomplished by creating isolated cross bracing and diaphragm models in ABAQUS and comparing results against values from the Chapter 3 solutions. 51

70 4.1.1 CSBRP phase one structure As explained in Chapter 2, phase one of the CSBRP project involved testing a large-scale horizontally curved structure under various loading and support conditions (Linzell 1999). The initial part of the project included a series of studies during curved girder erection. The structure tested for the erection studies was a single span system having three concentric girders (G1, G2 and G3) spaced at 2.7 m (8.75 ft) on center and having a span length of 27.5 m (90 ft) along the center girder (G2). The radius of curvature to G2 was 61 m (200 ft). Cross section dimensions for the three girders are shown in Table 4-1. K-type cross frames containing tubular sections [12.7 cm (5 ) diameter and cm (1/4 ) thickness] were used at different locations between the girders. Lower lateral bracing was used in the end bays. A WT 6 29 section was used for the bottom lateral bracing members. Figure 4-1 details the CSBRP structure framing plan and dimensions. Table 4-1 CSBRP structure girder plate sizes (Linzell 1999) Girder G1 G2 G3 Flanges b f t f [cm (in)] (16 1 ) (20 1 ) (24 2 ) Web h w t w [cm (in)] (48 ) (48 ) (48 ) 52

71 Figure 4-1 Framing plan for CSBRP structure tested for the initial erection studies (Linzell 1999) Full depth bearing, transverse and connection plate stiffeners were positioned at multiple locations along the girders. Nominal plate sizes for these components are listed in Table 4-2. Spherical bearings were provided at the G1 and G3 supports and theoretically allowed rotation in all directions and translation only in the radial and tangential directions. G2 had similar constraints at its ends, but radial translation was also prevented due to the use of guided bearings. A support frame at the west end (Station 1L) of G2 was connected to the girder web at its mid-depth to prevent tangential movement (see Figure 4-1). 53

72 Table 4-2 Nominal plate sizes for bearing, transverse and connection plate stiffeners Location Structural element Width/Height [cm (in)] Thickness [cm (in)] G1 Bearing stiffener 17.8 (7) 1.9 ( ) G1 Transverse stiffener 12.7 (5) 1.1 ( ) G1 Cross frame stiffener 17.8 (7) 1.6 ( ) G2 Bearing stiffener 22.9 (9) 2.5 (1) G2 Transverse stiffener 12.7 (5) 1.1 ( ) G2 Cross frame stiffener 22.9 (9) 2.1 ( ) G3 Bearing stiffener 22.9 (9) 2.5 (1) G3 Transverse stiffener 17.8 (7) 1.6 G3 Cross frame stiffener 22.9 (9) 2.1 ( ) Extensive instrumentation was used to monitor the behavior of the curved girders during the erection studies. Mid-span girder deformations and rotations were measured using potentiometers and tilt meters. Locations of these transducers on the girders are shown in Figure 4-2. Strain gages were placed at multiple sections along the girders to measure strain distributions across the flanges and through the girder webs. Different strain gage arrangements were used at various sections as shown in Figure

73 Figure 4-2 Mid-span deformation instrumentation (Linzell et al. 2004) 55

74 Figure 4-3 Girder strain gage location (Linzell 1999) 56

75 4.1.2 CSBRP structure modeling Figure 4-4 is an isometric view of the ABAQUS finite element model of the CSBRP structure created for the current research. ABAQUS S4R shell elements, 4-node reduced integration elements, were used to model the girder webs, flanges, stiffeners and connection plates (ABAQUS 2010). These elements use reduced (lower-order) integration to form their matrix; however, mass and distributed loads are integrated exactly. Reduced integration usually provides accurate results while reducing analysis time. This element has been successfully used in previous studies to examine the behavior of horizontally curved girders (Chang 2006; Jung 2006; Chavel and Earls 2006a; Nevling 2008). Figure 4-4 CSBRP ABAQUS finite element model All shell elements in the model had aspect ratios close to 1:1. Therefore, twelve elements were used through the depth of the girder webs. The girder flanges in G1 and G2 used 4 elements across the flange width while the G3 flanges had 6 elements across their width. These element sizes were based on recommendation from Chang (2006). 57

76 As explained earlier, CSBRP testing results were used to validate global curved girder modeling effectiveness. The cross frame system used in the CSBRP structure was not considered in the present study. Therefore, a simplified technique was used for modeling the cross frames in the CSBRP structure. The pipe section cross frame members were modeled using ABAQUS B31 beam elements and were directly connected to the girder connection plates using rigid ties (the gusset plates were not modeled), a technique selected from previously published computational studies (Chavel and Earls 2006a; Chang 2006; Nevling 2008). Lower lateral bracing was also modeled using ABAQUS B31 beam elements connected to the girder bottom flanges with rigid ties. Boundary conditions for the models mimicked actual support conditions. Bottom flange nodes at the web to flange intersection for all three girders were restrained against vertical displacement at the two abutments to represent the effect of the spherical bearings at those locations. This node for G2 at both abutments was also restrained in the radial direction. A tangential restraint was provided at the neutral axis of G2 at the left abutment (Station 1L in Figure 4-1) to represent the effect of the end frame used in the experiment that prevented the structure from rigid body rotation in the tangential direction. Linear elastic material properties were used for all steel components in this model since all construction tests were completed in the elastic range (Zureick et al. 2000; Linzell et al. 2004). As stated earlier, geometric nonlinearities were included in the analyses to account for higher order displacements and resulting elastic stresses that could occur due to curvature effects Global modeling validation To validate the global modeling strategies, two analysis steps were defined in the model to replicate stages examined during one of the CSBRP erection tests. This was the ES3 test 58

77 (Linzell 1999) in which temporary shoring supports that had been initially used to bring the girders to a no-load position were removed from all three girders and girder displacements and stresses under steel member self-weight were obtained. Evaluation of model accuracy occurred by comparing displacements and strains from this model against those from the test. Table 4-3 compares girder vertical deflections and web rotations at mid span from the FEA model against measurements from the test after shoring supports were removed and girders deflected elastically under their self-weight. These FEA results were obtained from nodes in the girder models that were associated with the location of the displacement transducers in the tested CSBRP structure (see Figure 4-3). Table 4-3 Girder mid span displacements after shoring removal, ES3 Girder Vertical deflection [cm (in)] Web rotation (degrees) FEA Experiment Difference FEA Experiment Difference G (0.21) 0.51 (0.20) 5% % G (0.58) 1.63 (0.64) 9% % G (1.00) 2.69 (1.06) 6% % Model results for girder mid-span deformations and rotations show good agreement with the experimental measurements. The maximum discrepancy between measured and the predicted values is about 9% for the mid-span vertical deflections and about 5% for girder rotations. The main reasons behind these disparities will be discussed later in this section. To further examine the capability of the model to predict actual girder responses, the analysis results for normal strains across the girder flanges and through their depth were 59

78 compared against corresponding experimental measurements at instrumented girder sections from Figure 4-3. Figure 4-5 to Figure 4-10 plot normal strains for the G3 flanges and web at Stations 5L and 7 in Figure 4-3. The plots for normal strains in the other girder sections, in Figure 4-3, are provided in Appendix B. Figure 4-5 G3 top flange strains at Station 5L Figure 4-6 G3 bottom flange strains at Station 5L 60

79 Figure 4-7 G3 web strains at Station 5L Figure 4-8 G3 top flange strains at Station 7 Figure 4-9 G3 bottom flange strains at Station 7 61

80 Figure 4-10 G3 web strains at Station 7 Good agreement generally exists between the experimental measurements and model predictions for girder normal strains and these results typify the level of agreement that existed. It is believed that discrepancies could be attributed to idealizing the cross frame and lateral bracing connections to the girders and the simplified support conditions. Also, Linzell et al. (2004) speculated that some experimental conditions that existed during the tests may have caused locked in stresses in structural components that were not quantified. The modeling technique presented in this section was used in Chapter 5 to study the effects of changing bracing plan orientation in curved I-girder bridges on the construction behavior of the noncomposite sections Brace modeling In this section a modeling technique was developed for the cross frames and diaphragms that were used in conjunction with the elastic global model that was validated in the previous section to examine the performance of different bracing systems in horizontally curved I-girder bridges during construction and while in-service. As explained earlier, to ensure that FEA bracing models have a reasonable level of accuracy, FEA results were compared against results 62

81 obtained from the analytical studies in Chapter 3. To do this, isolated brace models were created in ABAQUS for diaphragms and cross frames having different geometries. The models included the brace elements, gusset plates and girder connection plates to account for the effects of structural element stiffness and connection eccentricities on performance. Full depth braces, relative to the girder web depth, were considered and detailed according to the standards for steel bridge systems provided by the Pennsylvania Department of Transportation (PennDOT 2007) and the Texas Department of Transportation (TxDOT 2006). Braces were proportioned to fit between a pair of girders spaced at 2.67 m (8.75 ft) with depths ranging from 122 cm (48 ) to 183 cm (72 ). These dimensions were selected based on statistically significant geometric parameters from existing horizontally curved, plate girder bridges in Pennsylvania, Maryland and New York obtained from a previous study (Linzell et al. 2010). Rolled or built up I- or channel sections are commonly used for diaphragms (AASHTO/NSBA 2003). As explained in Chapter 3, this study uses built up I-sections, comprised of a solid plate web and two flanges attached to the web at its top and bottom (see Figure 4-11) for the diaphragms. As indicated in Figure 4-11, the diaphragm flanges at both ends were coped to make possible connection to the girder stiffeners. This was done according to the common practice for bridge diaphragms with I-sections. 63

82 Diaphragm K-type cross frame X-type cross frame Figure 4-11 Brace models 64

83 Single angle sections were selected for individual cross frame members since these sections are commonly used in existing curved bridges (Stith et al. 2009). Final cross sections used for the cross frame and diaphragm members shown in Figure 4-11 were explained in Section 3.2. The members and components in the bracing systems were assumed to be attached together with welds or bolts. The plate thicknesses used for the gusset plates and connection plate stiffeners were never less than 1.3 cm ( in) as recommended by AASHTO/NSBA (2003). All members and components in the bracing models in Figure 4-11 were created using ABAQUS S4R shell elements. Mesh sizes were selected such that they produced an aspect ratio close to unity for all elements. Rigid ties were used to represent the effect of welded or bolted connections to assemble the brace elements. Note that the girder sections were not included in the brace models and are shown in Figure 4-11 and the figures that follow for clarity. Boundary conditions for the individual brace models were adopted from a study by Quadrato et al. (2010), who examined the effect of different connection types on performance of bracing systems in skewed I-girder bridges. Figure 4-12 shows these boundary conditions applied to nodes on the boundaries of the brace models. Longitudinal restraints were applied to all nodes at the interface between the connection plate stiffeners and girders. These longitudinal restraints were to prevent the braces from moving parallel to the girder longitudinal axis. In addition, a rigid constraint was defined between all nodes along the interface line between the stiffener and the girder and a master node that represented the connection to the girder at each end of the bracing. The rigid constraint forced the edge of the connection plates to rotate or translate as rigid bodies following the master nodes. Additional translational and rotational 65

84 supports were also considered for the master nodes on each end of the brace models to maintain brace equilibrium. Support restraints Rigid constrains Figure 4-12 Boundary conditions Finite element analyses were completed and two representative stiffness components were obtained for comparison against values from Sections through The representative stiffness components were assumed to be the brace shear stiffness ( element in brace stiffness matrices) and flexural stiffness ( element in brace stiffness matrices) as 66

85 presented in Equations 3-1, 3-5 and 3-7 for diaphragms, X-type cross frames and K-type cross frames, respectively. To calculate the shear stiffness components from the FEA models, shear deformation was introduced into these models by imposing a unit vertical support displacement ( ) at the master node on one end of the brace models while all other support displacements were zero (see Figure 4-13). The resulting reaction force in the direction of the applied support translation ( on the master node was obtained from analysis. The shear stiffness component was then calculated using the relationship shown in Equation Figure 4-13 Shear deformation of brace Similarly, for the flexural stiffness component a unit rotation was generated at the master node, at one end of the brace models. This, in turn, resulted in a rigid body rotation (θ) of all nodes in the connection plate stiffener boundary about the master node as shown in Figure The resulting support reaction moment ( ) of the master node was obtained and the flexural stiffness was calculated using Equation

86 4 2 Figure 4-14 Flexural deformation of brace Figure 4-15 and Figure 4-16 show the ratios of FEA results (k FEA ) to the analytical solutions (k Analytical ) for diaphragm shear and flexural stiffness components. The analyses were first performed using a 13 mm (½ in) plate thickness for the connection plate stiffeners. As seen in both figures the FEA and analytical results were found to be in poor agreement for this plate stiffness. Next, the stiffener plate thickness was doubled (25 mm or 1 in) and the analysis was performed again for the individual diaphragm models. Using a thicker stiffener plate largely reduced the discrepancy between the FEA and analytical results. In fact, by increasing the plate thickness the effect of the connection plate stiffness on the overall diaphragm stiffness was reduced and the stiffness of the diaphragm governed the FEA results, and therefore, these results became closer to those from analytical solution that only accounted for the diaphragm stiffness. 68

87 1.5 Girder depth (in) K FEA /K Analitical /2" Stiffener 1" Stiffener Girder depth (cm) Figure 4-15 Comparing FEA and analytical diaphragm shear stiffness, varying girder depth, constant girder spacing =2.67 m (8.75 ft) K FEA /K Analitical 1.5 Girder depth (in) K FEA /K Analitical Figure 4-16 Comparing FEA and analytical diaphragm flexural stiffness, varying girder depth, constant girder spacing of S=2.67 m (8.75 ft) Figure 4-17 and Figure 4-18 compare the FEA and analytical solutions for the flexural stiffness of K-type and X-type cross frames, respectively. Again two different plate thicknesses were examined for the connection plates stiffeners and the gusset plates in the cross frames. Results for cross frame models were similar to those obtained for diaphragm models. When larger plate thicknesses were used for the stiffeners and the gusset plates, results from the FEA solutions approached the analytical solutions. The same behavior was also observed for the cross frames shear stiffness. 1/2" Stiffener 1" Stiffener Girder depth (cm) K FEA /K Analitical

88 1.5 Girder depth (in) K FEA /K Analitical /2" stiffener & 1/2" gusset plate 1" stiffener & 1/2" gusset plate 1" stiffener & 1" gusset plate Girder depth (cm) Figure 4-17 Comparing FEA and analytical flexural stiffness for K-type cross frames, varying girder depth, constant girder spacing of S=2.67 m (8.75 ft) K FEA /K Analitical 1.5 Girder depth (in) K FEA /K Analitical /2" stiffener & 1/2" gusset plate 1" stiffener & 1/2" gusset plate 1" stiffener & 1" gusset plate Girder depth (cm) K FEA /K Analitical Figure 4-18 Comparing FEA and analytical flexural stiffness for X-type cross frames, varying girder depth, constant girder spacing of S=2.67 m (8.75 ft) 4.2 FEA modeling of composite bridges As explained earlier, in addition to studying the performance of bracing systems on horizontally curved bridges during construction under construction dead loads, it was also important to examine their impact on in-service bridge behavior under traffic loads. This section addresses the modeling methods developed in ABAQUS that were used to examine the effect that the skewed braces could have on the behavior of composite curved bridges in 70

89 Chapter 6. Unlike the models developed for the construction studies in the previous section, for the in-service bridge models nonlinear material properties were considered in conjunction with nonlinear geometric effects. This was done to account for possible material nonlinearities that could occur under the application of live traffic loads in the composite bridge models. Similar, fully nonlinear, models were used to examine the effects of the different bracing arrangements on the inelastic behavior of girders and the concrete deck in Chapter 6. The models included both the steel superstructure members and a concrete deck. The steel members, which included the plate girders, braces, stiffeners and connection plates, were modeled using the same element type and mesh sizes that were discussed for the construction bridge models in Section 4.1. All the steel members were assembled together using rigid ties. The modeling procedure for concrete deck is adopted form Jung (2006) and addressed in the following section. The nonlinear material properties used for the steel members and the concrete deck in the composite bridge models were also selected based on the material models developed by Jung (2006), which were based on test results from the composite curved bridge examined for the CSBRP project Concrete deck modeling As explained earlier, modeling techniques used for concrete decks in the composite bridge models were adopted from a composite bridge modeling method that was validated against experimental results for a portion of the CSBRP project that studied the inelastic behavior of horizontally curved bridges (Jung 2006). Accordingly, the concrete deck was created using S4R shell elements having aspect ratios closed to one. A uniform mesh density was considered for the entire deck with the mesh size being approximately twice the size of the mesh used for the 71

90 plate girder flanges. The reference surface for these shell elements was located at mid-thickness of the concrete deck. The steel reinforcement in the deck was incorporated in the model using the ABAQUS REBAR LAYER command, following Jung (2006), which made it possible to define multiple rebar layers through the shell element thickness. Using this command, the rebar is modeled as a section property within the concrete deck elements and is represented using a smeared stiffness modeling approach. To model the interface between the concrete deck and the steel girders in the composite bridge, beam type multipoint constraints (MPCs) were used to couple the girder top flange nodes to the nodes of concrete deck, following Jung (2006). The beam type MPCs provide rigid links between the two set of nodes in the model, which constrain all of their translational and the rotational degrees of freedom to each other. The MPCs are shown in Figure Figure 4-19 Interface modeling between concrete deck and steel girders 72

91 4.2.2 Material properties As explained earlier, fully nonlinear modeling techniques were used for the composite bridge models, including both nonlinear geometric effects and material nonlinearities. Subsequent material properties used for the steel superstructure members, concrete deck and steel reinforcement in the composite bridge models, adopted from Jung (2006), are addressed in the following sections Material model for steel member elements The inelastic material model that was assumed for the steel members in the composite bridge was based on Von Mises yield criterion, which allows for isotropic yielding. Isotropic strain hardening was also incorporated into the steel member material models by defining multi-linear stress-strain curves. In general, for composite bridge models in the present study, similar to what used for the composite structure in the CSBRP project, two different steels were used; ASTM A572 Grade 50 and ASTM A709 Grade HPS70W. The multi-linear nominal (engineering) stress-strain curves for these two materials are shown in Figure 4-20 and Figure These curves were developed by Jung (2006) using material testing results that were reported by Beshah (2006). 73

92 Figure 4-20 Nominal stress-strain response for ASTM A572 Grade 50 steel (Jung 2006) Figure 4-21 Nominal stress-strain response for ASTM A709 Grade HPS70W steel (Jung 2006) ABAQUS employs a large strain formulation for the S4R shell elements that were used for all structural members in the composite bridge models. Therefore, when defining the material stress-strain curves true stresses and true strains must be considered. The nominal stress and strain quantities shown in Figure 4-20 and Figure 4-21 were converted to true quantities using the relationships shown in Equations 4-3 and

93 1 4 4 Where σ and ε are the true stress and strain, respectively, and the corresponding nominal (engineering) values are σ and ε Material model for concrete deck elements The concrete constitutive model assumed for the composite bridge models was the Concrete Damaged Plasticity model following Jung (2006). This plasticity model, which was originally formulated by Lee and Fenves (1998), considers the combination of isotropic (scalar) damage elasticity and isotropic tensile and compressive plasticity to represent the inelastic behavior of concrete. It uses a fracture-energy-based damage concept and corresponding degradation to the elastic stiffness. This degradation is characterized by a scalar degradation variable, d. That is, if the initial (undamaged) elastic stiffness of the concrete is, the damaged stiffness in compression is 1. The stiffness degradation variable is, in turn, a function of two scalar damage variables, d t and d c, defined to help establish compressive and tensile damage strain levels. The damaged plasticity constitutive model can be implemented in ABAQUS using the CONCRETE DAMAGED PLASTICITY command. Concrete compressive and tensile behavior was defined using the corresponding multi-linear curves developed by Jung (2006). These curves, which are shown in Figure 4-22 and Figure 4-23, were obtained using results from 298- day concrete cylinder tests of the slab used in the CSBRP phase III structure (Beshah 2006). 75

94 Figure 4-22 Nominal compressive stress-strain response for concrete (Jung 2006) Figure 4-23 Nominal tensile stress-strain response for concrete (Jung 2006) Table 4-4 and Table 4-5 show damage variables as a function of the nominal stresses and strains for concrete in compression and tension. These quantities were again adopted from Jung (2006) and were input in the composite bridge models for this study. Similar to what was explained for the steel stress-strain curves in the previous section, the stress and strain values in Figure 4-22 and Figure 4-23, as well as in Table 4-4 and Table 4-5, are based on nominal 76

95 (engineering) quantities and corresponding true quantities were obtained using Equation 4-3 and 4-4. Table 4-4 Compressive damage variables versus nominal compressive stresses and strains (Jung 2006) Stress [MPA (ksi)] Strain (%) Compressive damage variable, d c (2.192) (4.206) (4.804) (4.87) (4.684) (4.075) (3.522) (2.985) (2.62) (2.347) (2.018)

96 Table 4-5 Tensile damage variables versus nominal compressive stresses and strains (Jung 2006) Stress [MPA (ksi)] Strain (%) Tensile damage variable, d t (0.501) (0.446) (0.405) (0.365) (0.325) (0.284) (0.244) (0.203) (0.162) (0.122) (0.081) Material model for concrete deck reinforcement As explained earlier, steel reinforcement was incorporated as rebar layers into the concrete deck shell elements using the REBAR LAYER command. Using this modeling technique, the steel reinforcement behavior is considered independently from the concrete. For the concrete deck reinforcing steel in the composite bridge models, following Jung (2006) an elasticperfectly plastic stress-strain response was considered with nominal yield strength of 414 MPa (60ksi). 78

97 4.3 Summary and conclusions Two finite element modeling schemes were developed in this chapter for examining the effect of different bracing systems on non-composite horizontally curved bridges during construction in Chapter 5 and on composite bridges examined in Chapter 6. For both bridge models ABAQUS S4R shell elements were used to model the members. Linear elastic material behavior was assumed for all the elements in the non-composite bridge model for the construction studies. For the composite bridge model, nonlinear steel and concrete material properties were used. Nonlinear geometric effects were included in both models. To examine the validity of this modeling technique for curved girders, analysis results were compared against experimental data available from an erection study completed for the CSBRP. Generally good agreement existed between the models and test results. To examine the reliability of FEA modeling method for different brace systems, FEA results for individual braces were compared against simplified analytical solutions from Chapter 3. Good agreement between the results for the two solution methods was obtained when stiffer connection elements were used, thereby making the overall stiffness for each bracing system from the FEA results approach the simplified analytical solutions that did not account for connection plate stiffness. The concrete deck in the composite bridge model was created using shell elements and composite action between the deck and steel girders was modeled using MPC elements. These modeling techniques, as well as the selected nonlinear material models for the steel and concrete members in the composite bridge, were adopted from a bridge model which was validated by Jung (2006) against experimental results from a structure that was tested for the 3 rd phase of CSBRP project. 79

98 Chapter 5. Construction Performance of Skewed Bracing in Horizontally Curved Steel Bridges Bracing is commonly placed between girders in a radial direction in horizontally curved bridges, normal to the girder webs, as shown in Figure 5-1. This pattern results in smaller unbraced lengths for interior girders, that are generally more stable and experience smaller deformations and rotations, and larger unbraced lengths for exterior girders having less stable conditions and larger deformations and rotations. The use of braces that are not perpendicular to the girder webs (i.e. skewed relative to the web) has been proposed by AASHTO (2012) for intermediate braces in bridges having skew angles less than 20 to mitigate skew effects. Since the effects of skew can be equated to the effects of horizontal curvature, it is of interest to study if orienting bracing in a skewed condition relative to the girder web in horizontally curved structures provides any benefits. Figure 5-1 Normal-to-web bracing in a horizontally curved I-girder bridge The first part of this chapter employed nonlinear finite element modeling strategies developed in Section 4.1 to study the effects of using skewed braces on a single span horizontally curved, I-girder bridge construction behavior. The bracing was oriented such that it produced larger cross frame spacings for the bridge interior girders at intermediate sections (see Figure 5-2). This pattern was selected for study because it may be possible to remove some bracing lines from the interior girders (for the girder sections adjacent to the end supports). 80

99 Reduction in the number of intermediate braces could reduce superstructure weight, expedite bridge fabrication and erection, and improve the economy of the design. Bracing inclination angles were limited to 20 relative to a normal to the girder webs in the curved bridge models. This angle is the limit for the use of skewed bracing in bridges with skewed abutments, according to AASHTO (2012). Study results were compared against conventional braces oriented normal to the webs by looking at key bridge construction response quantities, such as: girder deformations and forces and cross frame forces. Findings were then extended by examining the influence of key bridge geometric parameters on the performance of skewed bracing through a set of parametric studies. The performance of skewed bracing arrangements for composite horizontally curved bridges is discussed in Chapter 6. Figure 5-2 Representation of skewed bracing in a horizontally curved I-girder bridge 5.1 Skewed bracing in horizontally curved bridges Initial study The study summarized in this section was completed for a single-span, three-girder, bridge to examine the effects that changing bracing plan orientation, based on the pattern shown in Figure 5-2, had on girder response during construction. The single-span bridge model was selected for this initial study because curvature effects have been shown to be greater for this type of bridge when compared to a multi-span bridge (Nevling 2008). The performance of 81

100 skewed bracing in a two span, curved bridge model at different stages of construction was examined and results are discussed later in this chapter. Figure 5-3 shows a plan view and boundary conditions of the studied single-span bridge model. Selected girder global geometric properties were taken from the structure that was tested for the third phase of the CSBRP project (Jung 2006). These geometry values were adopted primarily because they mimicked a tightly curved plate girder structure. The bridge had three concentric curved girders with a radius of curvature equal to 61 m (200 ft) for the middle girder. Although most actual bridges have more than three girders, it is believed that the findings from this study can also apply to the curved bridges with larger number of girders. Moreover, the performance of a skewed bracing arrangement for a four-girder bridge was examined as one of the parametric studies in the second part of this chapter. Table 5-1 shows cross section dimensions used for the three girders in the one span bridge model. These girders were designed for the AASHTO Strength Limit State by Jung (2006), assuming that all girder components were made of ASTM A572 Grade 50 steel (f y =345 MPa or 50 ksi), except the bottom flange of G3, for which ASTM A709 HPS 70W (f y =482 MPa or 70 ksi) was used. Higher strength steel was used for the bottom flange of G3 mainly to reduce the size of the flange and optimize the weight. The section sizes were modified in the present study to satisfy the requirements of the AASHTO LRFD Bridge Design Specifications (AASHTO 2012). The nominal sizes used for bearing transverse and connection plate stiffeners in the single-span bridge model are also listed in Table

101 Bridge framing plan and boundary conditions Bridge section Figure 5-3 Single span bridge framing plan, section, and boundary conditions initial study 83

102 Table 5-1 Single span bridge girder nominal plate sizes initial study Top flange Web Btm. flange Girder bf tf h w t w b f t f [cm (in)] [cm (in)] [cm (in)] G (12 1) ( ) (12 1) G (14 1) ( ) (16 1) G (18 1) ( ) ( ) Table 5-2 Single span bridge nominal plate sizes for bearing, transverse and connection plate stiffeners initial study G1 G2 G3 Stiffener width cm (in) thick. cm (in) width cm (in) thick. cm (in) width cm (in) thick. cm (in) Bearing stiffeners Outside curvature Inside curvature 18 (7) 13 (5) 1.9 (0.75) 1.9 (0.75) 22.5 (9) 22.5 (9) 1.9 (0.75) 2.5 (1) 22.5 (9) 22.5 (9) 2.5 (1) 2.5 (1) Transverse stiffeners 11.5 (4.5) 1.6 (0.63) 14 (5.5) 16 (0.63) 15 (6) 16 (0.63) Connection plate stiffeners Outside curvature Inside curvature 18 (7) 12.5 (5) 1.6 (0.63) 1.6 (0.63) 22.5 (9) 22.5 (9) 21 (0.82) 21 (0.82) 22.5 (9) 22.5 (9) 21 (0.82) 21 (0.82) 84

103 Constant unbraced lengths equal to 6.9 m (22.5 ft) were used along all girders for the CSBRP Phase 3 bridge. This resulted in a subtended angle between the cross frames of L b /R = , which is slightly larger than the maximum limit of 0.10 in AASHTO (2012). The bridge model studied in this section considered five intermediate cross frames between the girders that produced a cross frame spacing of 4.6 m (15 ft) and a subtended angle of L b /R = for the model having cross frames oriented normal to the girder webs (see Figure 5-4). Also as shown in Figure 5-4, bridge models containing skewed cross frames maintained the same cross frame spacing for G3 as that in the model with normal bracing. Larger spacings existed for G1 and G2 when cross frames were placed in 10 and 20 degree skewed patterns. Normal to girder webs cross frames Skewed cross frames Figure 5-4 Single-span bridge bracing arrangements initial study 85

104 As discussed earlier, members in the K-type cross frames of the CSBRP Phase 3 structure used pipe sections, a section type that is uncommon for cross frames in existing bridges. As a result, for the single-span bridge model studied here, K-type cross frames with individual members consisting of single L mm (5 5 ½ in) angle sections, were used. According to Stith et al. (2009) these angles are the most commonly used sections for cross frame in horizontally curved bridges. The cross frame members were assumed to be made of ASTM A572 Grade 50 steel (f y =345 MPa or 50 ksi), and were proportioned to meet the requirements of AASHTO (2012) for Strength Limit State. The cross frame members were assumed to be connected to the girders connection plate stiffeners using gusset plates, as shown in Figure 5-3. When skewed cross frames were used, the cross frame members were attached to the girders using skewed connection plate stiffeners (see Figure 2-7). Skewed stiffeners have been shown to be a more common connection type for braces with skew angles not larger than 20 (AASHTO/NSBA 2003). The performance of a skewed bracing arrangement using other types of skewed connections (connections utilizing bent gusset plates and split pipe stiffeners) was also examined as one of the parametric studies and results are reported later in this chapter Finite element models Detailed nonlinear finite element models of the single bridge structure with the bracing arrangements explained in the previous section were created in ABAQUS. Figure 5-5 depicts an isometric view of the finite element model for the single span bridge with normal bracing. The model was composed of non-composite steel plate girders, cross frames, stiffeners, and gusset plates, created using the modeling techniques developed in Section 4.1. Linear elastic material properties were used for all steel components in the models employed in this chapter, 86

105 since all structural elements were designed to remain in the elastic range during construction. Geometric nonlinearities were included in the analyses to account for higher order displacements and stresses that could occur due to curvature effects. Figure 5-5 Finite element model for single-span bridge initial study For all bridges studied in this chapter, sequential analyses were performed by creating multiple ABAQUS analysis steps that mimicked each superstructure erection step. During each step, the program analyzed the structure having the appropriate structural components, loads, and boundary conditions. Table 5-3 shows the erection scheme that was employed for the single-span bridge models in this study. This procedure was based on the documented erection method reported by Jung (2006). For the single span bridges it was understood that girders experienced the largest stresses and deformations under the so called full dead load condition; the construction stage after placement of the concrete deck, while concurrently not receiving any stiffness benefit from the hardened concrete via composite action. This condition corresponds to analysis Step 3 in Table 5-3. Therefore, for the single span bridges in this chapter analysis results are discussed only for this analysis step. For this step, distributed dead loads were applied to the girder top flanges to represent the weight of the forms, reinforcement 87

106 and wet concrete. These dead loads were calculated assuming a concrete deck having a thickness of 20 cm (8 in) with 0.91 m (3 ft) overhangs. These dimensions were adopted from those for the concrete deck used for the CSBRP Phase 3 study (Jung 2006). In addition, the effects of deck overhangs were incorporated by considering uniformly distributed couples for the outermost and the innermost girders, G1 and G3. The couples were formed using two horizontal uniform loads along the girders. The top load was applied at the top flange and the bottom load 91.5 cm (36 in) below the top flange to represent the location of overhang bracket supports on the girder webs (Jung 2006). Table 5-3 Single span bridge sequential analysis erection procedure Analysis step Step 1 Step 2 Step 3 Placement of G1, G2, and cross frames G3, and cross frames Concrete deck Analysis results and discussion As stated earlier, skewed bracing was utilized in this study in an attempt to optimize cross frame arrangement and numbers between girders in horizontally curved bridges. This section discusses the effects that the proposed bracing arrangement had on the behavior of the single span horizontally curved bridge discussed earlier during construction. As was also stated earlier, to accomplish this, frame orientation angles of 10 and 20 relative to a normal to the girder web were examined. Results for girder deformations and stresses were then compared against those from a model that had cross frames oriented normal to the web. 88

107 The first studied response quantity was girder out of plane web rotations. This parameter is important for girders in curved bridges because larger rotations and corresponding web out of plumbness have been found to cause higher displacements and stresses in bridge girders and cross frames during construction (Howell and Earls 2007). They also can result in fit-up problems during erection (Chavel and Earls 2006b). Figure 5-6 to Figure 5-8 plot out of plane rotation angles for all girders in the single-span bridge model for the three cross frame orientation angles that were examined. For G3 and G2, the largest radius and middle girder, using skewed cross frames resulted in smaller girder rotations along their entire length. This reduction is more pronounced in G3, which, as the largest radius girder, generally experienced larger rotations. The amount of reduction was also greater for larger cross frame skew angles. For a 20 skew angle relative to the case where cross frames were placed normal to the girder web, maximum G3 rotation decreased by about 20% when compared against the results for the normal to web case (see Figure 5-6). Note that the cross frame spacing and the resulting subtended angle for G3 is the same for the models with normal to web or skewed cross frames. However, G2 has larger cross frame spacing when skewed cross frames were used compared to the normal-to-web case (Figure 5-4). Unlike G2 and G3, web rotation for innermost girder G1 did not significantly change as a result of the change in cross frame plan orientation. In fact, G1 mid span rotation was slightly reduced with skewed cross frames but small rotation increases were observed at other sections. 89

108 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=10 Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-6 G3 web rotations initial study Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=10 Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-7 G2 web rotations initial study 90

109 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=10 Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-8 G1 web rotations initial study In similar fashion to girder web rotations, Figure 5-9 compares maximum girder deflections in the span for the three cross frame orientations. Skewed cross frames resulted in smaller (approximately 5%) vertical deflections in G3 and larger (approximately 8%) deflections for G1. The maximum vertical deflection in G2 was practically unchanged. These results indicated that using skewed cross frames resulted in more uniform load sharing between the girders. As a result the overall rotation of the bridge section at mid span was reduced. In spite of the increase in vertical deflection for G1, the maximum deflection for all girders in the bridge, which occurred in G3, was reduced. Note that the provisions for the girders deformations and rotations during construction in the current bridge design codes and specifications (e.g. AASHTO) are largely qualitative and no unanimously acceptable quantitative limit exists for these items. 91

110 Max deflection (cm) Normal to web Skew=10 Skew=20 G1 G2 G Max deflection (in) Figure 5-9 Girder maximum vertical deflections initial study To study the influence that using skewed cross frames had on superstructure stresses during construction, stress distributions across the girders bottom flanges were examined. Figure 5-10 and Figure 5-11 show normal stress distributions across the bottom flange for G3 and G1 at the mid span location in bridge models with cross frames normal to the girder web and skewed at 20. Stresses were compared at the mid span for the girder bottom flanges, as this location corresponded to the location of maximum normal stresses along the girders Flange Width (in) Stress (MPa) Normal to web Skew= North Stress (ksi) Flange Width (cm) Figure 5-10 G3 normal stress across bottom flange at mid span initial study 92

111 Stress (MPa) Flange Width (in) Normal to web Skew=20 4 North Stress (ksi) Flange Width (cm) Figure 5-11 G1 normal stress across bottom flange at mid span initial study The normal stresses across the girder bottom flanges in Figure 5-10 and Figure 5-11 are composed of uniform vertical bending stress and linearly varying lateral bending stress. As observed in Figure 5-10, a small reduction (approximately MPa or 2 ksi) occurred in the bottom flange maximum normal stress for outermost girder, G3, with 20 skewed cross frames when compared to results for the bridge with normal bracing. The maximum stress reduction occurred at the bottom flange exterior tip of G3, where the normal stresses were generally higher. This reduction resulted in a more uniform stress distribution across the flange when 20 skewed bracing was used. In spite of the reduction in G3 bottom flange stresses, as a result of changing the cross frame arrangement, a relatively significant rise (approximately MPa or 5 ksi) was observed in the bottom flange exterior tip stress, as well as in the stress gradient across the flange, for innermost girder, G1, at mid span (see Figure 5-11). This rise was mainly caused by increasing the unbraced length at the mid span sections for this girder as a result of using 93

112 skewed bracing (see Figure 5-4). Finally, no significant change was observed in G2 maximum stresses as a result of the variation in cross frame plan orientation. The AASHTO (2012) Constructability Limit State requirement for distinctly braced tension flanges is provided in Article , and is shown in Equation 5-1 (AASHTO Equation ). The equation indicates that the flange maximum lateral bending stress,, and vertical bending stress,, are directly summed. According to AASHOT, these flange stresses must be obtained under non-composite dead loads and then multiplied by a load factor from the AASHTO load combination Strength IV (load factor = 1.5) and are compared against maximum permissible stress limit shown on the right hand side of Equation 5-1. This comparison is shown in Table 5-4 for the G3 and G1 bottom flanges in the models having skewed or normal-to-web cross frames. Column 3 in this table contains nondimensionalized ratios between the maximum factored normal stress in the girder bottom flanges from the models and the maximum allowable stress according to the AASHTO (2012) Constructability Limit State. 5 1 Where: = flange lateral bending stress. = flange vertical bending stress calculated without consideration flange lateral bending. = resistance factor for flexure (1.0) = nominal flexural resistance of the tension flange = = hybrid factor, specified in Article of AASHTO (2012) = nominal yield strength in the tension flange. 94

113 Table 5-4 Constructability Limit State checks for girder bottom flanges Bottom flange of girder Bracing type (1) f f [MPa (ksi)] (2) [MPa (ksi)] (3) Nondimentionalized ratio (1)/(2) G3 G1 Normal-to-web (35.1) (69.3) 0.51 Skewed (32.4) (69.3) 0.47 Normal-to-web 55.1 (8.0) (50.0) 0.16 Skewed 98.5 (14.3) (50.0) 0.28 As explained earlier, design of the girders in the single-span bridge model was governed by the AASHTO (2012) Strength Limit State. Therefore, nondimensionalized ratios in Table 5-4 were much less than one for both girder flanges. Also form this Table 5-4 it is understood that, even though G1 flange tip stresses did increase as a result of using skewed bracing, the final stress was still much less than the maximum strength for this flange according to the Constructability Limit State. More discussion about the effect of change in the bracing arrangement on girder flange stresses is provided in Chapter 6 for the Service and Strength Limit States of AASHTO (2012). The last response quantity examined for the single-span bridge models containing both normal and skewed bracing was stress levels in the cross frame members. Figure 5-12 shows normal stresses in bottom chord members for intermediate cross frames at lines 2L, 3L and 4 (see Figure 5-4) for the bridges with both normal to web and skewed bracing arrangements. Maximum cross frame normal stresses for all bridge models occurred in the cross frame bottom chord at mid span (Line 4) between G2 and G3. This maximum stress was slightly smaller for 95

114 the bridge having the skewed bracing arrangement (about 5% smaller for the model with the 20 skewed cross frames) when compared to the bridge model with normal to web braces. No significant change was observed in the normal stresses for the cross frames at lines 2L and 3L between G2 and G3 between the different bracing arrangements. Also, the maximum normal stress for the bottom chord member in Cross Frame 4 between G1 and G2 was reduced with skewed bracing. However, stresses in other intermediate cross frames between G1 and G2 (2L and 3L) were higher for bridge models with the skewed bracing patterns when compared to those for the model with normal bracing. As discussed previously, this behavior showed that, as a result of change the bracing arrangement to a skewed pattern, loads appeared to be more effectively transferred between the girders via the cross frames. 96

115 Stress (MPa) Normal to web Skew=10 Skew= Stress (ksi) L 3L 4 0 Between G2 and G3 Stress (MPa) Normal to web Skew=10 Skew= Stress (ksi) L 3L 4 0 Between G1 and G2 Figure 5-12 Cross frame maximum bottom chord normal stresses initial study As explained earlier, following the common design practice, a uniform cross section was used for all cross frame members in the single-span bridge model studied here. Therefore, design requirements were checked for the cross frame members using the maximum member forces generated in all cross frame members. This was accomplished for the single-span bridge model that used normal-to-web bracing arrangement. The maximum cross frame forces for this model occurred for the mid span cross frame (Station 4) between girders G2 and G3. After 97

116 changing the arrangement to skewed bracing, maximum cross frame forces slightly decreased and their location along the bridge span remained unchanged. This indicates that changing the bracing arrangement will not adversely affect cross frame member selection in the single-span bridge. A similar discussion is provided in Chapter 6 regarding the effect of changing bracing orientation on cross frame member design forces for the AASHTO Strength Limit State (AASHTO 2012). Examination of the single span bridge model results indicated that, by changing the bracing orientation between the girders, the overall stability of the bridge during construction appeared to be enhanced, given the understanding that stability here refers to large girder deformation and rotations which could lead to unpredicted problems during construction. Moreover, with skewed bracing, stresses decreased in the critical structural members (girders and cross frames) when compared to the structure with normal bracing, and increased in those that initially had smaller stresses, so that better load sharing between structural members have been obtained. The observed beneficial effects in the single span bridge are envisioned to have caused largely by two items. The first is adding skew effects to the girder sections at mid span as a result of changing in the brace orientation. That is, the skewed bracing arrangement induced torsional moments to the girder sections at the mid span of the bridge, as shown in Figure These moments counteracted the global torsional moments due to the curved geometry of the girders, and therefore, mitigated the curvature effects in the structural members and made the curved bridge responses appear more like those of a straight bridge. 98

117 Figure 5-13 Adding skew effects to the curved girder mid-span sections by using skewed bracing The second item is the effect of in-plane skewed bracing forces on girder vertical bending stresses and deformations. According to the analyses, the cross frames in the single span horizontally curved bridge model were in tension at the top and compression at the bottom, which resulted in a couple acting on the girders at their intersections (see Figure 5-14). For the skewed braces, the resulting couple had an in-plane (tangential) component relative to the plane of the girder webs. The in-plane couple counteracted the effect of vertical bending moments for the outermost girder, G3, and reduced the deformations and stresses in that girder. More specifically, the in-plane couple acting on girder G3 at the intersection with the skewed braces resulted in a tensile stresses in the top flange and a compressive stresses in the bottom flange and these stresses were in opposite direction to those produced by the vertical bending moment. Unlike the outermost girder, the in-plane component of the couple acting on the innermost girder, G1, at its intersection with the skewed braces was in the same direction as the vertical bending moments and resulted in larger deformations and stresses for this girder. Figure 5-15 depicts these effects in the girders of the single span bridge that was studied in this section. 99

118 This figure shows the effect of skewed bracing tension force at the top of the girder top flanges. A similar effect can be conceived for the bracing compression force at the bottom of the girder. As observed for the skewed bracing arrangements, a portion of the loads for the normal to web arrangement were transferred from the exterior girder to the interior girder. In other words, the exterior girder was pulled upward by the interior girder and this resulted in smaller stresses and deformations in the exterior girder and larger stresses and deformations in the interior girder. Also, for middle girder G2 the in-plane components of the couples from the braces on both sides of this girder canceled each other out and, therefore, no significant change was observed in its behavior as a result of changes to the bracing orientation. Figure 5-14 Bracing influence on the girders Figure 5-15 In-plane component of the cross frame tension force at the top of the girders 100

119 5.1.3 Removal of intermediate cross frames Using the skewed cross frame arrangement instead of normal-to-web braces for the singlespan bridge model produced larger unbraced lengths for the interior girders at mid span. However, the skewed cross frames resulted in smaller unbraced lengths for interior girder sections adjacent to the abutments (see Figure 5-4). This attribute was obviously more significant for the 20 skew case. Given the understanding that the curvature effects and resulting deformations and stresses were generally small near the end supports in this single span structure, having a small unbraced length at those locations seemed unnecessary. Therefore, it was of interest to further examine effects of the skewed cross frame arrangement by removing the two frames adjacent to the abutments (2R and 2L) as shown in Figure Figure 5-16 Skewed bracing arrangement cross frame removal In similar fashion to the findings reported in the previous section, Figure 5-17 to Figure 5-19 plot girder out of plane rotations for the single-span bridge model with 20 skewed cross frames after removing the cross frames at 2R and 2L between G1 and G2. These results were compared against results from the bridge models with normal to web bracing and with 20 skewed bracing before removing the two cross frames. It can be observed from these comparisons that, after removing the two cross frames from the bridge model with the

120 skewed frames, rotations for all girders slightly increased, especially for G3. However, the maximum web rotation in all girders still occurred at mid span of G3, and it was still smaller (approximately 18%) than that for the model with normal to web cross frames. Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=20 Skew=20 Removed CF Arc length along centerline of bridge (m) Figure 5-17 G3 web rotations cross frame removal Rotation angle ( ) Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=20 Skew=20 Removed CF Arc length along centerline of bridge (m) Figure 5-18 G2 web rotations cross frame removal Rotation angle ( ) 102

121 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=20 Skew=20 Removed CF Arc length along centerline of bridge (m) Figure 5-19 G1 web rotations cross frame removal Rotation angle ( ) Similar behavior was observed for maximum girder deflections when the two cross frames were removed between the interior girders. Figure 5-20 shows these values for normal to web bracing and skewed bracing before and after cross frame removal. As seen in this figure, although G3 s maximum vertical deflection slightly increased after removing the cross frames from the model with the 20 skewed bracing; it was still lower (approximately 4%) than that in the model having normal bracing. Max deflection (cm) Normal to web Skew=20 Skew=20 Removed CF G1 G2 G Max deflection (in) Figure 5-20 Girder maximum vertical deflections cross frame removal 103

122 No appreciable change was identified in girder flange normal stresses for outermost girder G3 when the cross frames were removed relative to the original single-span bridge that contained the 20 skewed bracing (see Figure 5-21). Not surprisingly, flange stresses increased slightly for middle girder G2 after removing the two cross frames. However, as shown in Figure 5-22, stresses for innermost girder G1 slightly decreased as a result of the cross frame removal. In fact, after removing the two cross frames between G1 and G2, the effectiveness of skewed braces for transferring loads imposed on the exterior girder to the interior girder was reduced, and this resulted in slightly smaller stresses in G1 and slightly larger stresses in G2 when compared against results for the original, skewed bracing, case Flange Width (in) Stress (MPa) Normal to web Skew=20 Skew=20 Removed CF North Stress (ksi) Flange Width (cm) Figure 5-21 G3 normal stress across bottom flange at mid-span cross frame removal 104

123 Stress (MPa) Flange Width (in) Normal to web Skew=20 Skew=20 Removed CF 4 North Stress (ksi) Flange Width (cm) Figure 5-22 G1 normal stress across bottom flange at mid-span cross frame removal Finally, normal stresses in intermediate cross frame bottom chord members for the skewed bracing case after cross frame removal are compared in Figure 5-23 against skewed bracing stresses before cross frame removal and against stresses for the normal bracing case. The maximum cross frame normal stress, again, occurred at the bottom cord of mid-span frame between G2 and G3 (Line 4) and it largely remained unchanged after removing selected skewed cross frames. As expected, a significant rise in normal stresses occurred for cross frames adjacent to the removed frames, those at 3L between G1 and G2 and 2L between G2 and G3 (see Figure 5-23). In spite of the increase in the normal stresses and corresponding cross frame forces after cross frame removal, maximum member normal stresses and forces still occurred for the cross frame members at Station 4 between G2 and G3 and they were slightly smaller than those with normal-to-web bracing. Therefore, new cross frame member sections did not need to be selected after cross frame removal. 105

124 Stress (MPa) Normal to web Skew=20 S20 Removed CF 2L 3L 4 Between G2 and G Stress (ksi) Stress (MPa) Normal to web Skew=20 S20 Removed CF 2L 3L 4 Between G1 and G Stress (ksi) Figure 5-23 Cross frame maximum bottom chord normal stresses cross frame removal Findings from studies discussed in this section indicated that, although two intermediate cross frames were removed from interior girders in the single span bridge model, using a skewed bracing arrangement produced girder rotations and vertical deflections that were still smaller (approximately 5%) than those in the model with normal to web cross frames. In addition, relatively large increases were observed in normal stresses for cross frames adjacent to the removed frames; however, these cross frames originally experienced smaller axial forces 106

125 when compared to forces carried by other intermediate frames in the bridge model. Therefore, by removing the two cross frames from the bridge model with skewed bracing, it appears that more efficient use of the brace members resulted and, as a result, the intermediate braces more equally transferred loads between the girders and more effectively maintained the bridge geometry. In addition to these beneficial effects, the reduction in number of intermediate braces can reduce the weight and could speed up the construction process. 5.2 Skewed bracing in horizontally curved bridges Parametric study The study in the previous section demonstrated that changing bracing orientation with respect to normal to girder webs in a single span bridge model resulted in smaller deformations and, therefore, could theoretically enhance stability during construction by mitigating the possibility of unexpected large deformations and fit-up problems. Although the single span bridge studied incorporated a number of key attributes pertaining to horizontally curved, composite, I-girder, bridge structural systems, it does not include all important physical attributes that may have a significant influence on the performance of different bracing systems on these types of structures. Therefore, the base single-span bridge FEA model was used as a starting point for single and multi-variant parametric FEA investigations into the effects of different bracing arrangements on the behavior of other curved, I-girder bridge systems. This was accomplished by modifying selected parameters within the initial, base model that were understood from the literature to have important effects on the behavior of horizontally curved bridges and bracing systems Case 1 - Cross frame to web connections The effect of different cross frame connection details on the performance of end diaphragms in straight and skewed bridges with different skew angles has been recently 107

126 examined by Quadrato et al. (2010). They showed that different connection types can significantly change the effectiveness of these diaphragms for maintaining the stability of girders during construction, especially for large skew angles ( 30 ). For the base single-span bridge, skewed connection plate stiffeners were used to connect the cross frames to the girders. According to AASHTO/NSBA (2003), this connection is more common for skewed braces having skew angles less than 20. Another skewed connection detail discussed in AASHTO/NSBA (2003) is a bent gusset plate and the differences between these two connection types were discussed in Chapter 2. Also, Quadrato et al. (2010) proposed using split pipe stiffeners for the end diaphragms in straight and skewed I-girder bridges, with skew angles larger than 30. They indicated that for these large skew angles, the split pipe detail was much stiffer than that for a bent gusset plate. Also, since a pipe was much stiffer than a plate with respect to torsion, this stiffener type enhanced the girder warping resistance. For this parametric study case (Case 1), influence of the type of connection detail on the performance of skewed bracing in horizontally curved bridges was studied via replacing skewed stiffener details with the detail having bent gusset plates and with one having split pipe stiffeners (see Figure 5-24). This was accomplished for the largest angle that was considered in this study, 20. The single span bridge model from Section 5.1 was used for analysis. The framing plan and girder sizes for this bridge are shown in Figure 5-3 and Table 5-1. The optimized skewed bracing arrangement, having removed cross frames between the two interior girders near the end supports, was considered (see Figure 5-16). 108

127 a) Connection with skewed stiffener b) Connection bent gusset plate c) Connection with split pipe stiffener Figure 5-24 Skewed cross frame connection details Case 1 For the three connection types that were examined, as shown in Figure 5-24, a 12 mm (1/2 ) gusset plate was used for the gusset plates and connection plate stiffeners for all details. Split pipe stiffeners having a wall thickness of 12 mm (1/2 ) were considered. Based on Quadrato et al. (2010) the diameter of the pipe stiffeners was also selected to be two inches less than the width of the smaller flange for each girder to provide enough room for welding. 109

128 For bent plates, the AASHTO LRFD Bridge Construction Specification (AASHTO 2011) specifies a minimum bend radius of 1.5 (where is the plate thickness), measured to the concave face of the plate. For the bent gusset plate in this study, two bend radii were examined; 19 mm (3/4 ) and 38 mm (1.5 ). The first radius was set equal to the minimum radius allowed by AASHTO (2011) and the second radius was chosen to be arbitrarily double the first. This was performed to investigate the influence of bend radius on connection stiffness and on the subsequent performance of the skewed bracing. A preliminary investigation of the analysis results, however, showed no pronounced difference between behaviors for the two bend radii. Therefore, analysis results for the model with the smaller bend radius were discussed herein. The modeling procedure developed in Section 4.1 was again employed here. For modeling the bent plate gusset plates and the pipe stiffeners, ABAQUS S4R shell elements were used with aspect ratios close to one. Acceptable mesh sizes were utilized for these parts to adequately model the curved bend radii surfaces, as seen in Figure 5-25 (girder flanges were removed in this figure for clarity). Similar to the studies in previous sections, analysis results were obtained for the full dead load condition (analysis Step 3 in Table 5-3). 110

129 Connection with bent gusset plate Connection with split pipe stiffener Figure 5-25 Mesh refinement for connection members with tight radius Case 1 A preliminary check of the analysis results indicated that, for the model that used bent gusset plates, stresses exceeded yield in small portions of some of the bent gusset plates that connected the cross frame bottom cord members to the girder webs. Figure 5-26 details the yielded area in the bent gusset plate that connected the cross fame bottom cord at Station 3L to G3. This location experienced the largest area of yielding under the full dead load condition. The main reason for these large stresses was the load eccentricity that existed in the bent plate connections, which resulted in appreciable bending moments and twist in the connecting elements. To better address the effects of yielding on the performance of the bent plate connection detail, the nonlinear material model discussed in Section for ASTM A527 Grade 50 steel was used for the plates. Although material yielding can be considered as a significant disadvantage regarding the use of bent gusset plates for skewed brace connections in the curved bridges, as seen in Figure 5-26 even when material nonlinearities were considered, the yielded area was quite localized relative to the overall dimensions of the bent plate. 111

130 Yielded area Figure 5-26 Material yielding in the bent gusset plate for brace connection to G3 at Station 3L Case 1 To examine the effects of the different brace connection details on bridge global behavior, Figure 5-27 to Figure 5-29 compare girder web rotation analysis results for 20 skewed braces, with the three connection types. Very small differences are evident for the different connection types for the curved bridge model that was studied here. This is mainly due to the fact that, for a relatively small skew angle of 20, difference between the stiffness of the different connection types was fairly insignificant. This also confirms findings from Quadrato et al. (2010) for the behavior of different brace connection types in bridges having small skew angles. Not surprisingly, using split pipe stiffeners made the girder rotations somewhat smaller (approximately 2%) than those for the skewed stiffener models. Moreover, in comparison to the base model that had the skewed connection plate stiffeners, girder web rotations are slightly larger (approximately 4%), especially for G3, when the bent gusset plate detail was used. The main reason for this is believed to be, again, the load eccentricity that exists for this connection type, which makes it more flexible than a concentric connection detail that used skewed stiffener plates. Also, material yielding that occurred in the bent gusset plates reduced, to some 112

131 extent, the effectiveness of the skewed bracing by decreasing the stiffness of the brace connection to the girders. Despite the fact that the detail with the bent gusset plates resulted in slightly larger girder rotations than those for the other connection details, all results are still smaller than those for the model having bracing oriented normal to the girder web. Rotation angle ( ) Arc length along centerline of bridge (ft) North S20 Skewed Stiffener S20 Bent Gusset PL S20 Split Pipe Stiffener Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-27 G3 web rotations Case 1 Rotation angle ( ) Arc length along centerline of bridge (ft) North S20 Skewed Stiffener S20 Bent Gusset PL S20 Split Pipe Stiffener Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-28 G2 web rotations Case 1 113

132 Rotation angle ( ) Arc length along centerline of bridge (ft) North S20 Skewed Stiffener S20 Bent Gusset PL S20 Split Pipe Stiffener Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-29 G1 web rotations Case 1 Similar effects to those discussed for girder web rotations were also observed for vertical deflections and flange normal stresses for the models having the different connection details. Generally, no pronounced differences were observed between results for the different connection types. Figure 5-30 compares maximum normal stresses in the cross frame bottom cord members for the different connection types. As observed, no significant difference existed between these values for the skewed stiffeners and split pipe stiffener details. However, maximum stress in the bottom cord for the skewed cross frames (at Stations 2L and 3L) in the model that had bent gusset plate connections was slightly higher than the other two cases. The reason for this is that the connection eccentricity with bent gusset plates induced more bending and torsion into the lower chord members, which thereby generated larger normal stresses when compared to the skewed and split pipe stiffeners concentric connection details. Although the maximum cross frame normal stresses for the model with bent gusset plates, which occurred at Stations 3L and 3R, exceeded those in the models with normal-to-web cross frames or with skewed cross frames that used skewed or split pipe stiffeners (maximum stresses for these cases occurred at Station 4), they were still less than the AASHTO (2012) permissible 114

133 limits for the bridge model studied here. However, in other bridge cases if bent gusset plates are used for the cross frame connections, a significant rise in stresses may require using a larger cross section for the cross frames than what is required for the other connection types. This can be considered a significant disadvantage for bent gusset plate connection detail over the other details in the skewed bracing arrangement. Stress (MPa) S20 Skewed Stiffener S20 Bent Gusset PL S20 Split Pipe Stiffener Stress (ksi) L 3L Between G2 and G3 Stress (MPa) S20 Skewed Stiffener S20 Bent Gusset PL S20 Split Pipe Stiffener 2L 3L Stress (ksi) Between G1 and G2 Figure 5-30 Cross frame maximum bottom chord normal stresses Case 1 115

134 5.2.2 Case 2- Bracing type According to Yura et al. (1992), the total bracing stiffness in a bridge section is a combination of the stiffness of the individual brace systems, the stiffness of the attached girders and the stiffness of brace to girder connections. These stiffness components are combined in series (see Equation 2-7) and the cumulative bracing stiffness is smaller than that of the three constituents. Although it is recognized that increasing the stiffness of each of the three individual brace components will increase the overall brace stiffness, when using this approach stiffening the component with the smallest relative stiffness magnitude has the largest effect on overall stiffness. The study in Section compared the influence of different bracing connection types having varying stiffness on the overall performance of skewed bracing arrangements proposed in this study. In an attempt to take the relative stiffness investigation one step further, the study in the current section aims to examine the performance of different bracing types on behavior of horizontally curved bridges during construction. The single-span bridge model in the initial study used K-typed cross frames as bracing. As shown in Chapter 3, for individual braces a diaphragm is stiffer than an X or K-type cross frame for cases having similar geometry and placed between two girders with a fixed spacing and depth. For parametric study Case 2, the K-type cross frames in the base single span bridge model were replaced with X-type cross frames and with diaphragms and the performance of the different bracing types in the different bracing arrangement schemes (normal or skewed bracing) were examined. As discussed in Chapter 2, AASHTO/NSBA (2003) recommends using K-type cross frames and diaphragms for girder sections with small depths and X-type cross frames for girders with large depths. However, information available regarding the performance of these different 116

135 braces in curved girders with different girder depths is quite limited. Therefore to address this research gap, the Case 2 studies are completed for the base single-span bridge model, introduced in Section 5.1, with girder depth of 122 cm (48 in) and for a second single-span bridge model with girders having a depth of 152 cm (60 in). As explained earlier, the girder depth of 122 cm (48 in) was chosen for the first single-span bridge to make it close to the limit for this parameter according to AASHTO (2012). The 152 cm (60 in) girder depth was selected because existing research indicates that it is one of the more common girder depths in excising bridges (Quadrato et al. 2010). Global geometric and boundary conditions for the bridge with the larger girder depth were the same as those in the base, single-span bridge model (see Figure 5-3, Table 5-1). The girders in this bridge were designed following AASHTO (2012) Strength Limit State and the final cross sections are shown in Table 5-5. Again, for this bridge, A572 Grade 50 steel (f y =345 MPa or 50 ksi) was assumed for all girder components except the bottom flange of the outermost girder, G3, for which ASTM A709 HPS 70W (f y =482 MPa or 70 ksi) was used. Table 5-5 Bridge girder nominal plate sizes, girder depth=152 cm (60 in) Top flange Web Btm. flange Girder b f t f h w t w b f t f [cm (in)] [cm (in)] [cm (in)] G ( ) ( ) ( ) G (14 1) ( ) (14 1) G (16 1) ( ) (16 1) 117

136 As explained in Chapter 3, single angle sections were used for the K-type and X-type cross frames. These sections are the most commonly used sections for cross frames in existing horizontally curved bridges (Stith et al. 2009). Also, a built-up I-section was used for the diaphragms between the girders with varying depths. The bracing systems were designed as primary members based on AASHTO LRFD provisions (AASHTO 2012) for the Strength Limit State, which was found to be the controlling case for the cross frames and diaphragms. The final designs are shown in Figure As observed in this figure, mm (5 5 in) angles were selected for all cross frame members in the K-type and X-type cross frames, for the two girder depths. The built up diaphragms consisted of two mm (10 in) plates for the top and bottom flanges and a solid web plate having a thicknesses of 11 mm ( ), and a depth corresponding to the overall girder depth. All bracing members and components were assumed to be made out of A572 Grade 50 steel (f y =345 MPa or 50 ksi). 118

137 a) K-type cross frame b) X-type cross frame c) Diaphragm Figure 5-31 Bracing systems in bridge models with varying girder depths Case 2 Similar to the studies completed in the previous sections, two bracing arrangements were considered: normal-to-web and skewed bracing. The skewed bracing arrangement again used a 20 skew angle and had two cross frames removed between the interior girders near their end supports (see Figure 5-16). Modeling procedures for the curved girders and bracing systems developed in Section 4.1 were again employed here and the full dead load condition (analysis Step 3 in Table 5-3) was considered. Results are discussed in two sections, with the first comparing the behavior for bridges containing K-type cross frames against those with X-type cross frames and the second comparing bridges with K-type cross frames against those containing diaphragms. 119

138 K-type cross frame or X-type cross frame - effects on a curved bridge during construction This section compares single-span bridge performance for structures containing K-type and X-type cross frames during construction. As stated earlier, two different girder depths were considered. To complete the comparison, all K-type cross frames in the bridges that were studied were replaced with X-type cross frames (see Figure 5-32). Figure 5-32 Replacing K-type cross frames with X-type cross frames Case 2 From the simplified analytical studies that compared individual brace stiffness in Chapter 3, it was understood that an X-type cross frame was slightly stiffer than a K-type cross frame having a similar geometry. However, no significant difference was observed for the effects of these two types of cross frames on girder rotations and vertical deformations in the studied bridges. As an example, Figure 5-33 compares G3 web rotation for the bridge models having the larger girder depth with K-type or X-type cross frames in normal bracing arrangement. As shown in this figure, the two cross frame types practically resulted in the same girder web rotations. Rotations were slightly smaller for the X-type cross frames due to their larger 120

139 stiffness as discussed in Section 3.2. The difference is less pronounced for the other girders in this bridge or for all girders in the bridge model having the smaller girder depth. Rotation angle ( ) Figure 5-33 G3 web rotations, girder depth=152 cm (60 in), normal bracing Case 2 The performance of skewed bracing arrangements was also examined for varying girder depths and with K-type or X-type cross frames. Figure 5-34 depicts the finite element model with X-type cross frames and with skewed arrangement. As explained earlier, a 20 skew angle was used for the skewed bracing systems and the two cross frames between the two girders near the end supports were removed. Arc length along centerline of bridge (ft) North K type cross frames X type cross frames Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-34 Finite element model, girder depth=152 cm (60 in), skewed bracing, X-type cross frames Case 2 121

140 The skewed bracing arrangement, when compared against the normal-to-web bracing, either with X or K-type cross frames, resulted in smaller girder rotations and deformations for all bridge models that were examined. An example is shown Figure 5-35, where G3 web rotations are compared for models with K-type bracing, for normal-to-web or skewed bracing arrangements. Again no marked benefit was observed by selecting one cross frame type over the other for the girder depths that were examined. Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-35 G3 web rotations, girder depth=152 cm (60 in), K-type cross frames Case K-type cross frame or diaphragm - effects on a curved bridge during construction Studies in the previous section demonstrated that, in spite of differences in bracing stiffness reported in Section 3.2, K-type and X-type cross frames performed almost identically in bridges that were studied. Section 3.2 also indicated, however, that diaphragms were substantially stiffer than cross frames of similar geometry. This larger stiffness can theoretically reduce girder deformations and result in a more stable condition for the girders during construction. Again it must be noted that improvement in stability in this study refers to mitigating the possibility that the curved girders experience large deformations, which can, in turn, result in unexpected and costly construction issues. It was also recognized in Section

141 that, for similar geometries, a diaphragm having a solid plate web is heavier (50-100% heavier for different the girder depths) than a cross frame. This larger weight could result in larger girder deformations, especially when replacing cross frames with diaphragms at mid-span locations could possibly counteract increased stiffness benefits. Therefore, this section examines the effects of replacing cross frames with diaphragms at various locations along the girders on behavior during construction. This was accomplished for the bridge models with two different girder depths using the conventional normal-to-web bracing and the optimized skewed bracing system proposed in this study. The main objective of this study was to determine whether apparent beneficial effects from using a diaphragm instead of cross frame at different locations along a bridge span outweighed adverse effects caused by their increased weight. Figure 5-36 shows the single-span bridge models, with normal-to-web bracing, that were developed for examination in this section. Similar to the studies in the previous sections, the base model included only K-type cross frames with no diaphragms between the girders. Cross frames were then replaced with diaphragms at two sections: the abutments and mid-span. To denote locations where diaphragms replaced cross frames in the models three letter acronyms were used, with each letter denoting the location that was examined for cross frame replacement. Locations where cross frames were replaced with diaphragms are denoted with D, and, for cross frames, C. For example, a bridge for which the cross frames at the mid span section are replaced with diaphragms was named as CDC. 123

142 No diaphragm(ccc) Diaphragm at a mid-span (CDC) Diaphragm at a abutments (DCD) Figure 5-36 Replacing K-type cross frames with diaphragms, normal-to-web bracing Case 2 Figure 5-37 to Figure 5-40 compare web rotations for G3 and G1 in bridge models having different girder depths with and without diaphragms at the abutments and mid-span for the normal-to-web bracing arrangement. Girder rotations at the abutment were small and, as a result, the contribution of the cross frames and diaphragms to the torsional stiffness at these locations was insignificant. Therefore, no appreciable change was observed for girder rotations for models that replaced cross frames with diaphragms at the abutments (DCD) when compared 124

143 to the models having no diaphragms (CCC). On the other hand, replacement of cross frames with diaphragms at mid-span (CDC), where torsion and the subsequent cross frame forces were large, caused an appreciable reduction in girder rotations at intermediate sections for both girder depths examined here. This reduction was more pronounced for G3, the highest radius girder with, subsequently, the highest rotations. For instance, web rotation magnitude at midspan of this girder in the CDC model with girder depth of 122 cm (48 in) was about 14% smaller than that in the CCC model (see Figure 5-37). Almost the same amount of reduction can be seen for the G3 maximum rotation in Figure 5-39 for the model with girder depth of 152 cm (60 in) having normal-to-web bracing. As stated earlier, it was recognized that the diaphragms are % heavier than the cross frames they replaced in the models with different girder depths. However, this translates into a very small total increase in bridge weight (approximately 0.2 % of the total for both girder depths) when cross frames at mid span were replaced with diaphragms. Given the deformation control that is offered, the benefits that result from reducing construction time may outweigh any increased material costs. Rotation angle ( ) Arc length along centerline of bridge (ft) North CCC DCD CDC Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-37 G3 web rotations, girder depth=122 cm (48 in), normal-to-web bracing Case 2 125

144 1.2 Arc length along centerline of bridge (ft) Rotation angle ( ) North CCC DCD CDC Rotation angle ( ) Arc length along centerline of bridge (m) 0 Figure 5-38 G1 web rotations, girder depth=122 cm (48 in), normal-to-web bracing Case Arc length along centerline of bridge (ft) Rotation angle ( ) North CCC DCD CDC Rotation angle ( ) Arc length along centerline of bridge (m) 0 Figure 5-39 G3 web rotations, girder depth=152 cm (60 in), normal-to-web bracing Case 2 126

145 1.2 Arc length along centerline of bridge (ft) Rotation angle ( ) North CCC DCD CDC Rotation angle ( ) Arc length along centerline of bridge (m) 0 Figure 5-40 G1 web rotations, girder depth=152 cm (60 in), normal-to-web bracing Case 2 In similar fashion to girder web rotations, as expected no significant change was observed for girder displacements when cross frames at the bridge abutments were replaced with diaphragms. Also, when comparing girder deflections for the bridge model with no diaphragms (CCC) to that with mid span diaphragms (CDC) for the normal-to-web bracing arrangement, it was interestingly observed that, in spite of the use of heavier diaphragms at mid-span, vertical deflections for all girders were slightly smaller (about 2%) in the models with diaphragms. These findings help to reiterate that benefits from the stiffness increase provided by strategic diaphragm placement appear to outweigh any weight increases that result. The final case studied in this section examined the use of the optimized, skewed bracing system for the bridge models where K-type cross frames at mid span where replaced with diaphragms. It was of interest to see the combined effect that the optimized skewed bracing arrangement coupled with strategic placement of diaphragms had on girders deformations and rotations. For the optimized skewed bracing arrangement a 20 skew angle was used. The two 127

146 cross frame between the interior girders near the end supports were removed and the cross frames at mid span between all three girders were replaced with diaphragms (see Figure 5-41). Figure 5-41 Finite element model, girder depth=122 cm (48 in), skewed Bracing Case 2 Figure 5-42 and Figure 5-43 show web rotations for G3 and G1 in the bridge model with a girder depth of 122 cm (48 in) that used the skewed bracing arrangement and a diaphragm at mid span (CDC-Skew=20). These results were compared against those from the model that had normal bracing and a cross frames at every location (CCC-Normal to web). As expected, the combined effects of using skewed bracing and replacing cross frames with diaphragms at mid span resulted in significantly smaller girder rotations for the outermost girder G3. The maximum G3 rotation is about 26% smaller in the CDC-Skew=20 model than that in the CCC- Normal to web model. Also, for innermost girder, G1, although the brace spacing increased, replacement of the cross frames with diaphragms at mid span resulted in smaller mid span girder rotation when compared against the model having no diaphragms and normal bracing. Similar behavior was observed for the bridge with the larger depth girders. 128

147 Rotation angle ( ) Arc length along centerline of bridge (ft) North CCC Normal to web CDC Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-42 G3 web rotations, girder depth=122 cm (48 in), skewed bracing Case 2 Rotation angle ( ) Arc length along centerline of bridge (ft) North CCC Normal to web CDC Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-43 G1 web rotations, girder depth=122 cm (48 in), skewed bracing Case Case 3- Brace spacing Variation of brace spacing has been shown by many researchers (e.g. Maneetes and Linzell, 2003; Stith et al. 2009) to have significant effects on behavior of curved girders during construction. The subtended angle between the braces (L b /R) within the single-span bridge model studied in the first part of this chapter was radians, which was less than the limit of 129

148 0.1 allowed by AASHTO (2012). For parametric study Case 3, the effect of using larger spacings on behavior of bridges with skewed bracing was investigated by comparing analysis results against those from corresponding cases having normal-to-web bracing. For this, a brace spacing of 6.9 m (22.5 ft), and the associated subtended angle of radians was considered for the single-span bridge model. This brace spacing was taken from the structure that was tested for Phase 3 of the CSBRP project (Jung 2006). Figure 5-44 shows the framing plan and section for the single bridge model studied here. Boundary conditions considered for the girders in this bridge are the same as those in Figure 5-3. The girder sections were modified for the increased brace spacing according to the AASHTO (2012) Strength Limit State, and the final girder designs are shown in Table 5-6. To make the bridge designs consistent with the design of the initial (base) single-span bridge model studied earlier in this chapter, ASTM A572 Grade 50 steel (f y =345 MPa or 50 ksi) was assumed for all steel members except the bottom flange of G3, for which ASTM A709 HPS 70W (f y =482 MPa or 70 ksi) was used. 130

149 Bridge framing plan Bridge section Figure 5-44 Bridge framing plan, and section Case 3 Table 5-6 Bridge girder nominal plate sizes Case 3 Top flange Web Btm. flange Girder b f t f h w t w b f t f [cm (in)] [cm (in)] [cm (in)] G ( ) ( ) ( ) G (16 1) ( ) (22 1) G (24 1) ( ) ( ) 131

150 To create the skewed bracing arrangement, similar to what explained for previously examined cases, the unbraced length was kept constant for G3 and it increased for the inner girders by changing the orientation of the cross frames at the quarter points (Stations 2L and 2R) with respect to a normal to G3 s web. A 20 skew angle was used for the skewed braces. For this arrangement, which is shown in Figure 5-45a, the skewed braces are collinear between all three girders. As observed in this figure, because of the collinear bracing pattern the unbraced length for the innermost girder, G1, at mid span became significantly large (subtended angle was 0.14 radians) when compared against the maximum allowable subtended angle (0.1 radians) according to AASHTO (AASHTO 2012). This was expected to provide ineffective bracing and large deformations for the innermost girder mid span sections and, as a result, the braces were staggered between the interior girders to create more reasonable spacings. This modified pattern was referred to as the staggered skewed bracing pattern here, and is shown in Figure 5-45b. 132

151 a) Collinear bracing b) Staggered skewed bracing Figure 5-45 Skewed bracing arrangement Case 3 The finite element modeling technique that was developed for construction studies in Section 4.1 was employed for analyses in this section. The full dead load condition was again considered as the critical loading condition for the girders in the models. Figure 5-46 to Figure 5-48 show the girder web rotation angles for the normal-to-web and skewed bracing arrangements with collinear and staggered skewed bracing patterns. Similar to what observed for the base single-span bridge model with smaller unbraced lengths, the skewed bracing arrangements, when compared against normal-to-web bracing, resulted in smaller rotations for outermost girder G3. This reduction was, more pronounced with the collinear pattern (with approximately 20% smaller rotations) than the staggered pattern (with 133

152 approximately 16% smaller rotations). For G2, the collinear skewed bracing resulted in slightly larger web rotations when compared against those from the normal-to-web bracing. However, when the staggered skewed bracing was used, although the unbraced length for the G2 mid span sections remained the same (see Figure 5-45), due to the effects of skewed bracing arrangement web rotations were smaller when compared against normal-to-web bracing. Benefits from the staggered skewed bracing arrangement over collinear bracing are more significant for G1 web rotations. For this girder, as seen in Figure 5-48, the collinear bracing pattern, as expected, resulted in excessively large girder rotations (approximately 30% larger) at the mid span sections when compared against the normal-to-web bracing. In fact, the unbraced length for the innermost girder sections at mid-span with collinear skewed bracing became significantly larger than the allowable limit. Therefore, while this bracing arrangement produced the smallest web rotations for G3, it may result in lateral torsional bucking (LTB) issues for G1 and, if that girder remains stable, excessive deformations and rotations could still occur. The use of the staggered skewed bracing pattern mitigated issues for G1 and did reduce G3 web rotations, but not as significantly as the collinear skewed bracing. 134

153 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=20 Collinear Skew=20 Staggered Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-46 G3 web rotations Case 3 Arc length along centerline of bridge (ft) Rotation angle ( ) North Normal to web Skew=20 Collinear Skew=20 Staggered Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-47 G2 web rotations Case 3 135

154 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew=20 Collinear Skew=20 Staggered Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-48 G1 web rotations Case 3 Girder maximum vertical deflections are compared in Figure 5-49 for the different bracing arrangements. Similar behavior to that observed for the single span bridge with smaller brace spacings is also observed here for the bridge studied in this section. Skewed bracing arrangements resulted in a more balance deformed shape and smaller differential deflections between the girders in the bridge by reducing the vertical deflection in the outermost girder and increasing it for the innermost girder. These effects were again more pronounced for the collinear skew pattern. However, possible LTB issues and unpredicted deformations for the interior girders during construction makes the use of this skew bracing arrangement unreasonable for the bridge studied here. 136

155 Max deflection (cm) Normal to web Skew=20 Collinear Skew=20 Staggered G1 G2 G Max deflection (in) Figure 5-49 Girder maximum vertical deflections Case 3 As explained earlier, the collinear skewed bracing arrangement resulted in unbraced lengths for the interior girders that were significantly larger than the allowable limit by AASHTO (2012). This shortcoming was slightly modified with the use of the staggered skewed bracing arrangement. Therefore, it was of interest to see how this modification affects the stress levels in the interior girder flanges. Figure 5-50 depicts the mid span normal stresses across the bottom flange of G1 for the normal-to web and the collinear and staggered skewed bracing cases. As expected, the stress gradient and the maximum normal stress across the bottom flange of this girder are larger for the cases with skewed bracing than normal-to-web bracing. This effect was also reduced with the staggered skewed bracing pattern that created smaller brace spacings for the girder s mid span sections. Also note that, increased stresses for both skewed bracing cases were still much less than the girder s maximum permissible stress based on the AASHTO (2012) Constructability Limit State (Equation 5-1), even though the AASHTO unbraced length allowable limit was largely violated. 137

156 Flange Width (in) Stress (MPa) Normal to web Skew=20 Collinear Skew=20 Staggered North 15 Stress (ksi) 23 Flange Width (cm) Figure 5-50 G1 normal stress across the bottom flange at mid span Case 3 One issue of concern with respect to the use of staggered skewed bracing in the bridge model studied here was the effects that G2 bracing offsets at Stations 2L and 2R could have on flange stresses in that girder. Figure 5-51 compares flange normal stresses along the exterior tip of the G2 bottom flange for the different bracing arrangements. As shown in this figure, no significant difference existed between stress levels at Stations 2L and 2R. Also, the staggered pattern produced slightly lower maximum stresses in the G2 flange at mid span (Station 3), which was because of the reduced unbraced length due to staggering the cross frames. 138

157 276 Arc length along centerline of bridge (ft) Stress (MPa) L 3 2R North Stress (ksi) Arc length along centerline of bridge (m) 0 Normal to web Skew=20 Collinear Skew=20 Staggered Figure 5-51 G2 bottom flange, exterior tip, normal stress variation along span Case 3 Finally, Figure 5-52 compares maximum normal stresses in the cross frame bottom cord members for the bridge model studied in this section. Similar to what was discussed for the cross frame stresses in the base bridge, the skewed bracing arrangements resulted in better load sharing between the cross frame members in this bridge model by reducing the stresses in the members that, initially, had larger normal stresses and by increasing it in the members that initially had smaller stresses. Comparing the performance of the two skewed bracing patterns, the most significant difference in the cross frame normal stresses occurred at Station 2L between G1 and G2, where the staggered bracing produced smaller normal stresses than the collinear bracing. This finding is similar to that obtained by Quadrato (2010), who examined staggered and collinear bracing arrangements in straight and skewed bridges and showed that cross frame forces are generally smaller for the staggered arrangement. 139

158 Stress (MPa) Normal to web Skew=20 Collinear Skew=20 Staggered Stress (ksi) L Between G2 and G3 Stress (MPa) Normal to web Skew=20 Collinear Skew=20 Staggered Stress (ksi) L Between G1 and G2 Figure 5-52 Cross frame maximum bottom chord normal stresses Case Case 4- Four-girder bridge model All the single-span bridge models studied in the previous sections were composed of three girders. However, to accommodate at least two lanes of traffic, bridges commonly have more than three girders. For parametric study Case 4, a four girder horizontally curved bridge was designed as shown in Figure Four girders were selected based on statistically significant parameters from existing horizontally curved, I-girder bridges in Pennsylvania, Maryland and New York discussed in a previous study (Linzell et al. 2010). For this bridge, the girders were 140

159 again concentric with a radius of curvature of 69 m (200 ft) to the centerline of the bridge. Five intermediate braces were considered and similar boundary conditions that were assumed for the girders in the three-girder, base, bridge model were also considered here for the four girders (see Figure 5-53). Girder sections for this bridge were sized according to the Strength Limit State for AASHTO (2012), which was the governing load case. The final girder designs are shown in Table 5-7. Again, in similar fashion to the previous cases ASTM A572 Grade 50 steel (f y =345 MPa or 50 ksi) was assumed for all steel members except the bottom flange of the two outermost girders, G3 and G4, for which ASTM A709 HPS 70W (f y =482 MPa or 70 ksi) was used, to optimize structural weight for these two girders that experienced larger stresses. Bridge framing plan and boundary conditions Bridge section Figure 5-53 Bridge framing plan, section, and boundary conditions Case 4 141

160 Table 5-7 Bridge girder nominal plate sizes Case 4 Top flange Web Btm. flange Girder b f t f h w t w b f t f [cm (in)] [cm (in)] [cm (in)] G (12 1) ( ) (12 1) G (14 1) ( ) (14 1) G (16 1) ( ) (16 1) G (45.7 1) ( ) ( ) To create the skewed bracing arrangement, similar to the what explained for previously examined cases, the unbraced length was kept constant for the outermost girder (G4) and it increased for the inner girders by changing the orientation of the cross frames with respect to a normal to G4 s web (see Figure 5-54). A 20 skew angle was again used for the skewed braces. Also, similar to the bridge case studied in Section 5.2.3, to avoid creating large unbraced lengths for G1, as seen in Figure 5-54, the braces were staggered between the interior girders to create more reasonable spacings. To maximize the effectiveness of the skewed bracing for the exterior girders, the braces were maintained in a collinear pattern between girders G2 and G4. The skewed braces at Stations 3L and 3R between G1 and G2 were shifted towards bracing Station 4 so that nearly uniform unbraced lengths were created along G1. Note that, to avoid creating unnecessarily stiffened girder sections near the end supports, two cross frames at Stations 2L and 2R between G1 and G2 were removed. 142

161 Figure 5-54 Skewed bracing arrangement Case 4 Construction performance of the skewed bracing arrangement shown in Figure 5-54 was examined using the finite element modeling approach from Section 4.1. The sequential analysis was employed by defining multiple steps in the bridge model. The erection method assumed for this bridge was similar to that explained for the base single-span bridge model shown in Table 5-3, except that for analysis Step 2 G4 was also added in the model in conjunction with G3 and the cross frames. Again, the full dead load condition (analysis Step 3 in Table 5-3) was found to result in the largest deformations and rotations for this bridge and the analysis results are presented for this step. Figure 5-55 and Figure 5-56 compare web rotations for the outermost and the innermost girders in the bridge with the different bracing arrangements. As seen in these figures the skewed bracing, when compared against normal-to-web bracing, produced smaller (approximately 14%) web rotations for G4. In addition, although a smaller number of intermediate braces were used between the interior girders, the staggered skewed bracing arrangement shown in Figure 5-54 resulted in only small increases in web rotation for G1. 143

162 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-55 G4 web rotations Case 4 Rotation angle ( ) Arc length along centerline of bridge (ft) North Normal to web Skew= Arc length along centerline of bridge (m) Rotation angle ( ) Figure 5-56 G1 web rotations Case 4 Figure 5-57 shows girder maximum vertical deflections with the different bracing arrangements. Similar to the previous cases, the optimized skewed bracing pattern reduced maximum vertical deflections in the four-girder bridge model at mid span of G4. 144

163 Max deflection (cm) Normal to web Skew=20 G1 G2 G3 G Max deflection (in) Figure 5-57 Girder maximum vertical deflections Case 4 As expected, the skewed bracing arrangement produced a more significant gradient for normal stresses across the G1 bottom flange at mid span, and, as a result, the maximum normal stress was larger for the skewed bracing cases (see Figure 5-58). Similar to the previous cases, increased stresses in the G1 flanges were smaller than the permissible limit for this girder according to the AASHTO (2012) Constructability Limit State (see Equation 5-1) Flange Width (in) Stress (MPa) North 10 Normal to web 0 Skew= Flange Width (cm) Figure 5-58 G1 normal stress across the bottom flange at mid span Case 4 Finally, normal stresses in the cross frame bottom cord members for the four-girder bridge model were studied for each bracing arrangement. Figure 5-59 shows the maximum normal stresses for cross frame bottom cords between the two exterior girders (G3 and G4) and the two Stress (ksi) 145

164 interior girders (G1 and G2). As shown in this figure, in similar fashion to the previous cases maximum normal stresses occurred for the cross frame between the two exterior girders at mid span (Station 4). This maximum value is slightly smaller with skewed bracing. Stress (MPa) Stress (MPa) Normal to web Skew=20 2L 3L 4 Between G3 and G4 Normal to web Skew= Stress (ksi) Stress (ksi) 0 2L 3L Between G1 and G2 Figure 5-59 Cross frame maximum bottom chord normal stresses Case Case 5 Two span bridge All of the above cases examined single-span, simply-supported bridges. To study the performance of skewed bracing on multi-span curved bridges, a two-span bridge model was considered for Case 5 and the effect of changing brace orientation angle on bridge behavior 146

165 during construction was examined. Given the understanding that critical stages of construction in multi span bridges occur during early stages of girder erection and concrete deck placement within the initial erected spans, it was assumed that any findings from the two-span bridge model, studied here, could also be applied to multi-span bridges having larger span numbers. The two-span bridge model was created by reflecting the plan geometry of the girders from the single span bridge for each of the two spans as shown in Figure As also indicated in this figure, the same boundary conditions that were considered for the girders in the base single span bridge model in this study (see Figure 5-3) were also used for the girder supports at the abutments of the two-span bridge model. In addition, at the pier all three girders were translationally restrained in vertical direction with G2 also being restrained against translation in the radial direction. 147

166 Bridge framing plan and boundary conditions Bridge section Figure 5-60 Two span bridge framing plan, section, and boundary conditions Case 5 Similar to the single-span bridge models studied in the previous sections, the girders and the cross frames in the two-span bridge model were designed according to the AASHTO (2012) Strength Limit State. Also, ASTM A572 Grade 50 steel (f y =345 MPa or 50 ksi) was assumed for all steel members except the bottom flange of G3, for which ASTM A709 HPS 70W (f y =482 MPa or 70 ksi) was considered in the design to produce a more optimized section for this 148

167 girder. Table 5-8 shows the final girder sections. As shown in this table, to optimize the two span girder designs all girders were composed of two sections: one for positive flexure and one for negative flexure. Section transitions occurred near the dead load contraflexure locations in each span (see Figure 5-60). Table 5-8 Case 5- Bridge girder nominal plate sizes Girder Girder section Top flange b f t f [cm (in)] Web h w t w [cm (in)] Btm. flange b f t f [cm (in)] G1 Positive Negative (12 0.5) (12 1) ( ) ( ) (12 0.5) (12 1) G2 Positive Negative (12 0.5) (14 1) ( ) ( ) (12 0.5) (14 1) G3 Positive Negative ( ) (16 1) ( ) ( ) ( ) (18 1) The skewed bracing arrangement examined for each of the spans in the two span bridge model is shown in Figure As shown in this figure, similar to the skewed bracing for the single span bridge cases in the previous sections, the cross frame spacing for the outermost girder, G3, in the two span bridge model was kept constant, and it was increased for the interior girders by changing the cross frame orientation with respect to the normal to G3 s web. A

168 angle was used for the skewed braces in the two-span bridge. Also, to avoid unnecessary stiffened girder sections for the interior girders, two cross frames were removed at each span, between G1 and G2, near their support locations at bridge abutments and the pier (Stations 2L, 6L, 6R, and 2R). This cross frame removal resulted in an almost uniform brace spacing for the entire length of the innermost girder, G1 (Figure 5-61). Figure 5-61 Skewed bracing arrangement Case 5 Finite element modeling techniques from Section 4.1 were again used to model the girders and cross frames in the two span bridge models. For girders G2 and G3, the transition in the girder flange widths at the splice locations were model by tapering the girder flanges at these locations in the finite element model as shown in Figure The slope of this flange transition is 1:2.5, which was adopted from published standard details by PennDOT (2007). 150

169 Figure 5-62 Girder flange width transition Case 5 Similar to the single-span bridge models studied in the previous sections, a sequential analysis was performed for the two-span bridge by creating multiple ABAQUS analysis steps that mimicked each superstructure erection step. Because of crane capacity limitations and geometric limits, it is usually not possible to erect an entire girder length in a two-span bridge at once. Therefore, girders in this two-span bridge model were divided into two segments: from Abutment 1 to the Field Splice; and from the Field Splice to Abutment 2 (see Figure 5-60). Table 5-9 details the considered erection scheme. As shown in this table, the first segments of the first two girders were erected as a pair, followed by single erection of the first segments of G3. A similar approach was also assumed for the erection of the second segments of the girders. This constitutes the first four analysis steps in the model. Girders segments in Table 5-9 were identified using two numbers: the first being the girder number and the second the segment number for that girder. For instance, the first segment for G3 was G

170 Table 5-9 Two-span bridge sequential analysis erection procedure- Case 5 Analysis step Step 1 Step 2 Step 3 Step 4 Placement of G1-1, G2-1, and cross frames G3-1, and cross frames G1-2, G2-2, and cross frames G3-2, and cross frames Step 5 Concrete deck segment 1 Step 6 Concrete deck segment 2 Step 7 Concrete deck segment 3 As also indicated in Table 5-9, three analysis steps were considered for placement of concrete deck: first within the positive flexure regions of each span followed by the negative flexure region over the pier. This placement procedure is usually used for concrete decks in multi-span bridges to minimize the deck cracking in the negative flexure regions. Figure 5-63 shows the resulting concrete deck placement segments. As shown in the figure: the first considered segment was between Abutment 1 and the girder section transition in the first span; the second was from the girder section transition point in the second span to Abutment 2; and the final segment is between the section transition points. The concrete deck in the two-span bridge was again assumed to remain plastic during the entire placement sequence. Although this assumption is not necessarily representative of actual field conditions, incorporation of time dependent changes in the concrete modulus of elasticity has been shown to insignificant effects on resulting deformations and stresses in the superstructure during deck placement (Choo et al. 2005). Therefore, for each deck placement steps in Table 5-9 only the dead loads 152

171 from the concrete deck segment self-weight were placed onto the girder top flanges in the model. Figure 5-63 Two-span bridge concrete deck segments Case 5 An initial issue of concern associated with constructing a multi-span span curved bridge relates to field splice section rotations during girder erection. This issue has been addressed by many researchers (e.g. Stith et al. 2009; Sharafbayani and Linzell 2012). Large girder rotations at the splice locations can cause fit-up problems between girder segments. To examine the effect of the skewed bracing arrangements on these girder rotations, rotations were compared after erection of the first segment of all three girders for bridge models having skewed and normal-to-web bracing arrangements. This condition corresponds to the analysis Step 2 in Table 5-9 and is shown in Figure 5-64 for the normal-to-web and skewed bracing arrangements. 153

172 a) Normal-to-web bracing b) Skewed bracing Figure 5-64 Two-span bridge model at the end of analysis step 2 Case 5 Figure 5-65 and Figure 5-66 show G3 and G1 web rotations at the end of analysis Step 2 for the bridges with the normal-to-web and skewed bracing arrangements. As shown in these figures for G3, skewed bracing resulted in approximately 25% smaller girder web rotations along the entire girder length, at the end of Step 2. The reduction in web rotation at the splice location for this bridge is of high importance because of the previously explained issues that large girder rotations at this location can cause. For G1, however, although changing the bracing orientation and removing cross frames near the girder supports produced larger unbraced lengths (see Figure 5-64b) no significant change is observed for girder rotation at intermediate sections and at the splice location. 154

173 Rotation angle ( ) Arc length along centerline of bridge (ft) North Pier Normal to web Abut. 1 Skew=20 Splice Arc length along centerline of bridge (m) Figure 5-65 G3 web rotations, Step 2 Case Rotation angle ( ) Rotation angle ( ) Arc length along centerline of bridge (ft) North Pier Normal to web Abut. 1 Skew=20 Splice Arc length along centerline of bridge (m) Figure 5-66 G1 web rotations, Step 2 Case Rotation angle ( ) Another issue of concern related to construction of a two-span bridge is girder deformations and rotations during deck placement. Excessive deformations during deck placement can cause misalignment of superstructure components or violation of different design goals, such as desired deck final elevations and prescribed tolerances for final girder web plumbness. As opposed to the single-span bridge cases, for the two-span bridge maximum girder deformations and rotations did not occur after full dead load was applied at the end of deck placement. In fact, for the full dead load condition, because of the continuity effects, loads acting on each 155

174 span counterbalanced each other and, therefore, reduced resulting deformations and stresses. It is noteworthy to mention that this conclusion was obtained using models for which the concrete deck was assumed to conservatively remain plastic during all stages of deck placement. For the two-span bridge model studied here, maximum girder deformations and rotations occurred after adding dead loads caused by placement of the initial concrete deck segment on the first span. This condition is represented by analysis step 5 in Table 5-9 and is also depicted in Figure Figure 5-67 Two-span bridge model at the end of at analysis step 5 Case 5 G3 and G1 web rotations at the end of analysis step 5 are shown in Figure 5-68 and Figure 5-69 for the normal-to-web and skewed bracing arrangements. As observed in these figures, in similar fashion to results for analysis Step 2, skewed bracing resulted in up to approximately 20% smaller G3 rotations. The skewed arrangement also had no significant effect on G1 rotations, even though cross frame spacing on this girder was larger than the normal-to-web arrangement. 156

175 Rotation angle ( ) Arc length along centerline of bridge (ft) Normal to web Skew=20 North Abut. 1 Pier Abut Arc length along centerline of bridge (m) Figure 5-68 G3 web rotations, Step 5 Case Rotation angle ( ) Rotation angle ( ) Arc length along centerline of bridge (ft) Normal to web Skew=20 North Abut. 1 Pier Abut Arc length along centerline of bridge (m) Figure 5-69 G1 web rotations, Step 5 Case Rotation angle ( ) To further examine the effects of changing bracing arrangement on girder behavior at different two-span bridge construction stages, girder flange stresses were also studied. For sections in the first span positive flexure region, maximum stresses also occurred at the end of analysis Step 5. Figure 5-70 and Figure 5-71 compare the normal stresses across G3 and G1 flanges at mid-span of the first span. Similar to what was observed for the single-span bridges, when compared to the normal-to-web bracing arrangement, skewed bracing resulted in a slight 157

176 decrease (approximately 13.8 MPa or 2 ksi) in the maximum normal stress in the G3 top flange at the mid span of the first span. It also resulted in a MPa (8 ksi) increase in G1 top flange maximum normal stresses at mid span. Similar to the single-span bridge cases, the observed increase in normal stress in G1 at mid-span is still much smaller than the permissible stress limit according to the AASHTO (2012) Constructability Limit State (see Equation 5-1) Flange Width (in) Stress (MPa) Normal to web Skew= North Stress (ksi) Flange Width (cm) Figure 5-70 G3 normal stress across the bottom flange at mid span of the first span, Step 5 Case Flange Width (in) Stress (MPa) Normal to web Skew= North Stress (ksi) Flange Width (cm) Figure 5-71 G1 first span, mid span, normal stress across the bottom flange, Step 5 Case 5 158

177 For girder sections in negative flexure, maximum flange stresses during construction occurred at the end of the last analysis step; Step 7. This analysis step represented the full dead load condition in the bridge model. Figure 5-72 and Figure 5-73 compare the normal stresses across the G3 and G1 top flanges over the pier. As shown in the figures, G3, top flange, normal stresses over the pier are 21 MPa (3 ksi) lower for the model having skewed bracing. For G1, the skewed bracing arrangement removed two cross frames between G1 and G2 adjacent to the pier location and increased the unbraced length (see Figure 5-61). However, only a small increase (approximately 34.5 MPa or 5 ksi) in the G1 top flange normal stresses was observed over the pier in Figure This slight increase did not violate the AASHTO (2012) Constructability limit (Equation 5-1) Flange Width (in) Stress (MPa) Normal to web Skew= North Stress (ksi) Flange Width (cm) Figure 5-72 G3 normal stress across the top flange at the pier location, Step 7 Case 5 159

178 Flange Width (in) Stress (MPa) Normal to web Skew= North Stress (ksi) Flange Width (cm) Figure 5-73 G1 normal stress across the top flange at the pier location, Step 7 Case 5 The last examined response quantity for the two-span bridge model was cross frame normal stresses. It was understood during design for this bridge that maximum cross frame forces occurred in cross frame between G2 and G3 over the pier (Station 7) under the full dead load condition (at the end of analysis Step 7). Therefore, Figure 5-74 shows the normal stresses in the bottom cord members of the cross frames between girders in the first span at the end of this analysis step. As shown, changing the bracing arrangement reduced, approximately 5%, the maximum stress in the most critical cross frame member; between G2 and G3 at Station 7. For other cross frame locations, stresses with the skewed bracing arrangement were, more or less, larger than those for normal-to-web bracing. This rise in cross frame stresses is obviously more significant for the frames between G1 and G2 that were adjacent to the removed cross frames. Similar behavior was observed for cross frame member stresses for other analysis steps in the bridge model. So, in similar fashion to the single span bridges that were studied, it could be stated that the skewed bracing arrangement resulted in more balanced load sharing between cross frames in the two-span bridge. 160

179 Normal to web 24.0 Stress (MPa) Skew= Stress (ksi) 0 2L 3L 4L 5L 6L Between G2 and G3 Normal to web 24.0 Stress (MPa) Skew= Stress (ksi) 0 2L 3L 4L 5L 6L Between G1 and G2 Figure 5-74 Cross frame maximum bottom chord normal stresses, Step 7 Case Summary and conclusions Modified plan arrangements for bracing in horizontally curved, single-span, three I-girder bridges were examined in the first portion of this chapter. The bridges that were studied had a relatively tight radius of curvature of 61 m (200 ft) to the middle girder and contained five intermediate braces between the girders. In the modified arrangements, bracing was placed in two skew angles relative to a normal to the girder webs, 10 and 20, such that they either produced equal unbraced length for all girders in the bridge or produced larger lengths for the interior girders. Sequential analysis was performed to simulate the anticipated construction 161

180 stages. Results for the most critical construction stage; under the full dead load, were compared for bridges containing the skewed bracing, against those for cases with the conventional, normal-to-web bracing. For this portion of the study, the following conclusions were obtained: For the outermost (largest radius) girder, G3, using the optimized cross frame plan orientation resulted in smaller girder web rotations (approximately 20% smaller with 20 skewed bracing), vertical deflections (approximately 5% smaller with 20 skewed bracing) and flange stresses (approximately 8% smaller with 20 skewed bracing). These reductions were more pronounced for the larger cross frame skew angle. For the middle girder, G2, web rotations were slightly reduced (approximately 6%) and insignificant changes were observed for vertical deflections and flange stresses as a result of modifying the cross frame orientation. Both girder web rotations and vertical deflections slightly increased for the innermost (smallest radius) girder, G1, as a result of changing bracing plan orientation. A relatively large increase (from 34.5 MPa or 5 ksi to approximately 68.9 or 10ksi) in maximum flange tip stress was observed for the model with bracing skewed at 20 when compared to the case with normal-to-web bracing. However, the increased stress did not exceed the AASHTO (2012) permissible stress. By changing bracing arrangement in the bridge model and using skewed bracing, normal stresses in cross frame members at mid span, which generally have larger stresses in more traditional bracing arrangements, were reduced (more than approximately 5%) whereas those in other intermediate cross frames increased. The increased stresses were, however, less than the maximum calculated resistance for these members. 162

181 To further explore optimizing bracing placement using skewed orientations relative to the girder web, two cross frames adjacent to the abutments were removed from the bridge containing 20 skewed frames. It was shown that, by removing the two cross frames, girder rotations and vertical deflections slightly increased for the outermost girder, however, these results were still approximately 18% smaller than those in the model with normal-to-web cross frames. No significant change was observed in flange stresses for G3. Flange stresses for G2 slightly increased and those for G1 decreased. Moreover, appreciable increases in normal stresses for cross frames adjacent to the removed frames were observed. However, these stresses were still smaller than allowable design values. In summary, changing bracing plan orientation appeared to result in more uniform load sharing between girders for the single span bridge that was studied. Using skewed bracing appeared to more effectively transfer the loads, in part, from the exterior girders to the interior girders and, as a result, deformations and stresses decreased in more critical components that experienced larger horizontal curvature effects. In addition, it appears that effective use of skewed bracing systems in horizontally curved bridges could possibly reduce the number intermediate cross frames while still maintaining necessary stress and geometric control during construction. To extend the finding from these initial studies, a set of parametric studies were conducted. For these studies the single-span, three-girder, bridge model was considered as the initial (base) case, and five additional cases were examined by changing some key parameters in the base case model. For each parametric case section designs were modified, if necessary, following AASHTO design requirements (AASHTO 2012). Findings from these studies are summarizes below. 163

182 Case 1: Cross frame to web connections The effects of different skewed brace connections to the girders on skewed bracing performance were studied for Case 1. To accomplish this, three common connection details for skewed braces were examined in the base, single-span bridge model. The details were those that used: (1) skewed connection plate stiffeners; (2) bent gusset plates; and (3) split pipe stiffeners. The largest angle considered for the skewed braces in this study, 20, was used for this case. The following conclusions were obtained: Generally no significant performance differences were observed for the different connection details in the skewed bracing configuration, with the skewed arrangements producing 4% to 6% smaller girder deformations and 15% to 20% smaller girder rotations when compared to the normal-to-web arrangement. Localized yielding was observed in small portions of some of the bent connection plates in the skewed configuration. This resulted from load eccentricity that existed in the bent plate connections and produced a more flexible connection that, in turn, resulted in approximately 4% larger girder rotations when compared to the other two connection types. Also, due to connection eccentricities that existed with the bent gusset plates, larger normal stresses were created in the skewed cross frame members. The split pipe stiffener connection generated approximately 2% smaller girder rotations than the other connection types, behavior that was mainly attributed to this details larger stiffness. 164

183 Case 2: Bracing type Analytical studies completed in Chapter 3 compared the stiffness of individual cross frames and diaphragms. They indicated that, for different geometries, X-type cross frames are slightly stiffer than K-type frames; however, the diaphragms had significantly larger stiffness than both cross frame types. Difference between the performances of horizontally curved bridges containing these bracing systems was investigated in Case 2 for two single-span bridges having different girder depths; 122 cm (48 in) and 152 cm (60 in). Both skewed bracing and normal-to web bracing arrangements were examined. The following conclusions were obtained for this case: The skewed bracing arrangement produced a better bracing system than the normalto-web arrangement for the studied bracing system types and girder depths. This system resulted in smaller girder deformations and rotations and reduced the required number of intermediate braces. Although AASHTO/NSBA (2003) recommends using K-type cross frames for smaller girder depths and X-type cross frames for larger girder depths, no significant difference was observed between the effects of these two cross frame types on girder response for the girder depths that were examined. Replacing K-type cross frames with diaphragms at the bridge abutments had no significant influence on girder response. Replacing cross frames with diaphragms at mid span resulted in torsionally stiffer bridge sections and the beneficial stiffness increase exceeded any adverse effects from increased weight caused by the diaphragms. As a result, out of plane girder 165

184 rotations reduced for all girder sections (15% reduction for G3) in the vicinity of the diaphragms for the smaller and the larger girder depth bridges. In general, it was understood from the parametric studies in this case that girder response only varied when the stiffness of the bracing system changed significantly. Also the best performing bracing arrangement for the single-span bridges that were studied in this chapter was the one that utilized a skewed bracing arrangement having a diaphragm at mid span and K or X-type cross frames everywhere else. Case 3: Brace spacing The main purpose for the Case 3 studies was to demonstrate the desirable construction performance of skewed bracing, even, for a bridge with very large unbraced lengths. Therefore, for this case cross frame spacing was increased by reducing the number of intermediate cross frames from five in the base case model to three. This resulted in unbraced lengths that were slightly larger than allowable limits according to AASHTO (2012). The plan geometry for the bridge studied in this case is shown in Figure Two skewed bracing arrangements were examined. The first, collinear pattern, utilized constant cross frame spacing for the outermost girder and cross frames were rotated relative to a normal to this girder s web to create larger unbraced lengths for other girders. These skewed cross frames were placed along a continuous line between all three girders. At a 20 skew relative to the girder web, the collinear pattern resulted in mid-span unbraced lengths for the innermost girder that were significantly larger than AASHTO (2012) allowable limits. Therefore, to address this issue, a staggered bracing pattern was created that resulted in more reasonable brace spacings for the interior girder. 166

185 In general, the staggered bracing pattern replicated all the benefits of the collinear bracing pattern by reducing maximum girder deformations (approximately 4%) and girder rotations (approximately 16%) for the outermost girder and resulted in more uniform load sharing between structural members in the bridge under construction dead loads. In addition, as opposed to the collinear bracing pattern, the staggered bracing produced small increases in innermost girder web rotations and flange stresses. Therefore, this arrangement appeared to be preferred over the collinear bracing pattern for the bridge models that were studied. Case 4: Four-girder bridge All the previous cases considered three-girder bridge models. The purpose of the study in Case 4 was to examine the performance of skewed bracing arrangements in a bridge with a differing, and more realistic, number of girders, during construction. Therefore, a single-span bridge model was designed that contained four girders. The bridge plan geometry for this bridge is shown in Figure The optimized skewed bracing arrangement examined for this bridge used a collinear pattern for the skewed braces with 20 skew angles between the three exterior girders and, to create approximately uniform brace spacings for the innermost girder, a staggered bracing pattern was created between the two interior girders. In addition, to further optimize the bracing plan arrangement in the four-girder bridge, braces between the two interior girders near the end supports, that were deemed unnecessary, were removed (see Figure 5-54). Similar to the previous three-girder bridges, the skewed bracing arrangement, in general, improved the construction behavior of the four-girder bridge model studied in this case, although a smaller number of intermediate braces were used between the girders. This configuration produced more uniform load sharing between the structural members, and 167

186 subsequently reduced maximum deformations and stresses for critical members in the bridge. It is believed that the skewed bracing arrangement examined here can be extended to curved bridges having more than four girders, where, in similar fashion to the bridge examined here, a combination of collinear skewed bracing for the exterior girders and staggered bracing for the interior girders can create uniform brace spacings and possibly reduce the number of required braces. Case 5: Two-span bridge All the previous cases examined skewed bracing influence on the behavior of single-span bridges. The performance of skewed bracing arrangements within a two-span bridge during construction was examined for Case 5. The bridge was formed by reflecting the plan geometry of the base, single-span bridge model, for each of the two spans, as shown in Figure For the skewed bracing arrangements, the unbraced length was increased for the interior girders by changing the orientation of cross frames relative to a normal to web of the outermost girder, using 20 skew angle, and then removing cross frames between the two interior girders near their supports at the abutments and pier (see Figure 5-61). Similar to the single-span bridges, sequential analyses were performed to investigate the behavior at critical construction stages. The following list summarizes conclusions from these analyses. The skewed bracing arrangement produced smaller girder rotations at splice locations (approximately 25% smaller for G3) at the completion of the critical construction steps. This could reduce the possibility of fit-up problems during erection of girder segments in curved, multi-span bridges. The largest girder rotations occurred after placement of the first segment of concrete deck. For this critical stage, skewed bracing produced smaller deformations and 168

187 rotations (approximately 20% smaller for G3 maximum rotation) when compared against normal-to-web bracing. The reduction in girder deformations and rotations during deck placement reduces the possibility of girder misalignments that could produce web-out-of plumbness and undesirable final deck elevations. Similar to results for the single-span bridges, the skewed bracing arrangement resulted in more uniform load sharing between structural members at the critical stages of construction in the two-span bridge model. The findings from the study for this case could be extended to curved bridges having a larger number of spans, because critical stages of construction for multi-span bridges are when the girders in the first constructed span were erected, or when the concrete deck was placed in the first erected span. 169

188 Chapter 6. Skewed Bracing Performance in In-Service, Horizontally Curved, I- Girder Bridges As explained in Chapter 5, it is generally understood that bracing systems have the largest influence on the behavior of steel girders in horizontally curved bridges during their construction, in the absence of a hardened concrete deck. Once in-service, on the other hand, these bracing systems still play important roles with respect to transferring loads and maintaining acceptable deformation and stress levels in the girders and deck. Therefore, it was necessary to extend the studies completed in Chapter 5 to examine the effects of different bracing systems on the in-service behavior of composite curved bridges. It was shown in Chapter 5 that the optimized skewed bracing arrangement, one that used a 20 skew angle relative to the girder web, created more reasonable brace spacings for different girders in horizontally curved bridges and, concurrently, improved their construction behavior by reducing deformations and stresses in the most critical members, while generally requiring fewer intermediate braces. In addition, the use of skewed bracing increased the stresses and deformations in the girders and cross frames that had small stresses and deformation with normal-to-web bracing without exceeding permissible values for the AASHTO Constructability Limit State (AASHTO 2012). Therefore, this bracing system was shown to produce more uniform load sharing between structural members in the bridges during construction. Given the discussion above, it was of interest to examine what influence, if any, the recommended skewed bracing systems would have on the behavior of in-service horizontally curved bridges via the continued examination of stress and deformation levels in critical components under live loads. Here, those components included the girders, cross frames and the concrete deck. As explained in Chapter 5, studied curved bridges were designed assuming a conventional normal- 170

189 to-web bracing arrangement. Given that, this chapter investigated in-service performance of the recommend skewed bracing systems. It was also of interest to evaluate appropriate AASHTO (2012) design limits for those systems, especially for interior girders that were found to experience larger stresses and deformations in Chapter 5. To complete this examination, bridge behavior under unfactored live loading was examined, and the resulting stresses and deformations were compared against corresponding AASHTO limits (AASHTO 2012) for Service and Strength Limit States, for both the skewed bracing and the normal-to-web bracing arrangements. Furthermore, the effect of changing bracing arrangement on inelastic behavior of the curved girders and concrete deck was examined under an elevated live load that approached the structure s ultimate load. Tedesco et al. (1995) and Stallings et al. (1999) have shown that increasing brace spacing or removing intermediate braces for in-service bridges increased differential deflections between adjacent girders and, therefore, caused large bending stresses and strains in the concrete deck that could eventually result in cracking. As a result of these studies, one important issue of concern, that was examined, was the effect that skewed bracing, which created larger unbraced lengths for the interior girders, would have on the behavior of concrete deck at an ultimate load level. 6.1 Studied bridge The bridge that was considered in this chapter was the same single-span bridge model examined in Section 5.1. The framing plan and the girder cross sections for this bridge are shown in Figure 5-3 and are listed in Table 5-1. This structure was selected because it had the same global geometry as the bridge tested during Phase 3 of the CSBRP project, which examined the behavior of composite curved bridges under live loads (Jung 2006). It was 171

190 possible for the current study to take advantage of that research by using similar assumptions to those made when testing the CSBRP structure at ultimate loads. In particular, simplified schemes that were developed for the CSBRP structure for the application of ultimate live loads on the bridge deck were used here. These loading schemes are explained in detail later in this section. Figure 6-1 shows the normal-to-web bracing and the skewed bracing arrangements that were considered for the bridge. The skewed bracing arrangement, similar to that examined in Section 5.1.3, is the one that utilized a 20 skew angle for the braces and had two braces removed between interior girders near the end supports to create nearly uniform brace spacing for the innermost girder in the bridge. This case is referred to as Skew=20-Removed CF in this chapter. 172

191 a) Normal-to-web bracing b) Skewed bracing Figure 6-1 Single-span bridge bracing arrangements The concrete deck considered in the composite single-span bridge had a thickness of 20.3 cm (8 in) and an overhang of 0.91 (3 ft), similar dimensions to those assumed in Chapter 5 for the calculation of wet concrete dead load. Reinforcement plan details for the deck are shown in Figure 6-2. This concrete deck was used by Jung (2006) for the structure tested for Phase 3 of the CSBRP project. The design was checked to ensure that it complied with corresponding requirements in the current edition of AASHTO (2012). Accordingly, four layers of reinforcement were provided in the longitudinal and transverse directions at the top and bottom of the deck. For the top reinforcing layers, #4 bars at 30.5 cm (12.0 in) center-to-center were considered at both directions. To provide minimum concrete cover at the top, as specified in AASHTO (2012), the centerline of the transverse reinforcing bars was assumed to be 6.9 cm 173

192 (2.75 in) below top surface and the centerline of the longitudinal bars was considered to be 8.3 cm (3.25 in) below the top surface. For the bottom reinforcing, #4 bars at 20.3 cm (8.0 in) center-to-center were used in both directions. Again, to meet AASHTO requirements (AASHTO 2012) for minimum concrete cover at the bottom, the centerline of the transverse bars was 3.2 cm (1.25 in) above the bottom surface and the centerline of the longitudinal bars was 4.4 cm (1.75 in) above the bottom surface. Concrete deck plan Concrete deck section Figure 6-2 Single-span bridge concrete deck reinforcement Finite element modeling techniques discussed in Section 4.2 were used here. As explained in that section, all bridge members and components were modeled using ABAQUS S4R shell elements. Steel components were assembled using rigid ties. Composite action between the girder top flanges and the concrete deck was created by coupling nodal degrees of freedom of 174

193 those elements together using beam type MPC elements. Nonlinear material behavior was assumed for all members in the composite bridge model using the material models discussed in Section For the steel superstructure, following the assumption made by Jung (2006) for the girders in the CSBRP Phase 3 structure, ASTM A572 Grade 50 steel (see Figure 4-20) was used for all members and components except for the G3 bottom flange for which ASTM A709 HPS 70W (see Figure 4-21) was used to help optimize the design. For the concrete deck elements, a material model based on the Concrete Damage Plasticity formulation was used. The four layers of steel reinforcement (see Figure 6-2) were embedded in the concrete deck elements using a smeared modeling approach. Elastic-perfectly plastic material behavior was assumed for the reinforcement, with nominal yield strength of 414 MPa (60 ksi). As stated in Section 4.2, all assumptions behind the fully nonlinear composite bridge model were adopted from finite element modeling by Jung (2006). Two cases were considered to examine the effects of live loads on the composite bridges. For the first case, an AASHTO design live load that consists of a single design truck at midspan and a single lane load was placed near the inside radius of the bridge deck and, subsequently, over interior girders G1 and G2. This load case was found, during design, to produce maximum stresses on the G1 flanges. The second considered load case represented application of two AASHTO design trucks aligned side by side at mid-span plus two design lane loads that were applied on the entire bridge deck. This loading case was found during design to cause maximum flange stresses in G2 and G3 of the composite bridge. As explained earlier, simplified schemes were used for the application of AASHTO live loads onto the concrete deck. These schemes are shown in Figure 6-3 and they were adopted from loading setups used for the composite structure tested for Phase 3 of CSBRP project (Jung 175

194 2006). These loading schemes were developed such that they created approximately the same load effects (maximum stresses and deformations) in the bridge girders when compared to those resulting from AASHTO design live load for the application of one or two trucks. As shown in Figure 6-3a, for load Case-1 a group of six loads, representing both the wheel and lane loads, were positioned on the bridge deck directly over G1 and G2. The total live load for this loading case was found by considering loads from a single design truck load and a single lane load from AASHTO (2012) and was equally distributed between the six load points as reported by Jung (2006). For load Case-2, shown in Figure 6-3b, a group of nine load points were applied to the bridge deck directly over the girders to represent load effects from the two design truck loads and the two lane loads. Similar to load Case-1 the point loads were applied on the bridge deck directly over the top of the three girders in the model. The total live load for this case was obtained considering loads from the two design trucks loads and the two lane loads (AASHTO 2012). Following Jung (2006), this total load was equally distributed between the three points above each girder. Also, the magnitude of the loads above G2 was set to be 2.24 times the magnitude of those the loads above G1 and G3. For both loading cases, each load was applied to the concrete deck over an area approximately equal to AASHTO (2012) wheel load patch dimensions (25.4x50.8 cm or 10x20 in). 176

195 a) Case-1 b) Case-2 Figure 6-3 Single-span bridge live load cases As also explained earlier, in addition to examining behavior of the composite, single-span, bridge model under unfactored live loads, its behavior was also studied under a higher, ultimate loading case. The ultimate loading considered here was taken from the ultimate load test completed for Phase 3 of the CSBRP project (Jung, 2006). Accordingly, load Case-2 in Figure 6-3b was used for this ultimate loading and it represented the application of two, side by side, AASHTO design truck loads at mid-span and two design lane loads. Following what is reported by Jung (2006), the total load was incrementally increased in the models to 5783 kn (1300 kips), a load that was approximately equal to four times the total unfactored live load for 177

196 Case-2. This total load was distributed over the bridge deck between the nine point loads in Figure 6-3b. 6.2 Analysis results and discussions To examine the performance of different bracing arrangements on the behavior of the composite single-span bridge model, a series of analyses were first completed under unfactored live loads and response for the skewed bracing bridge were compared against those from the normal-to-web bracing bridge. Preliminary analyses showed that stresses and deformations in the composite bridge under the unfactored live loads remained in the elastic range. Therefore, it was possible for these stresses and deformations, to be checked against the AASHTO for Service and Strength Limit States (AASHTO 2012). To further examine the effect of different bracing arrangements in-service response in key bridge members, a second series of inelastic analyses were completed under the increased load discussed previously; one that produced behavior approximating the ultimate level for the bridge components, for skewed and normalto-web bracing designs Bridge behavior under unfactored live loads The initial response quantity examined here under unfactored loads was girder vertical deformation, a serviceability criterion that is usually restricted in different bridge design codes and specifications. Article of AASHTO (2012) provides optional criteria for girder elastic live load deflections in different bridge systems. According to this article, for steel bridge girders deflections under vehicular loads may be limited to Span/800. These deflections are to be calculated under unfactored design truck loads. The AASHTO optional deflection limit was not considered for the design of the single-span bridge model, so it was expected that some girders would not meet this limit. It was decided to examine girder deflection in relation 178

197 to this limit, to provide another assessment tool related to the influence of the optimized skewed bracing arrangement on in-service behavior. Figure 6-4 compares girder live load deflections for the skewed and normal-to-web bracing arrangements against the optional Service Limit State AASHTO deflection limits. Analysis deflections are obtained for load Case-2 (Figure 6-3b), which created the largest vertical deflection for all girders. It should be noted that, although the simplified loading scheme considered for the load Case-2 differed from the AASHTO design live loading scheme, it was arranged to cause approximately the same maximum load effects as the AASHTO design loads. As expected, the AASHTO deflection limit was violated for the outermost girder, G3, for both bracing arrangements in Figure 6-4. However, in similar fashion to construction behavior, skewed bracing, produced slightly smaller (approximately 5%) vertical deflection for G3 when compared against the normal-to-web bracing arrangement and, subsequently, closer to the AASHTO optional limit. Vertical deflections in G2 remained almost unchanged when using the skewed bracing arrangement and vertical deflection for G1 slightly increased (approximately 7%). It appears that using a skewed bracing arrangement resulted in slightly more balanced deflected shape between the girders in the composite single-span bridge model under unfactored live loads and produced girder deflections that are closer to the AASHTO limits. 179

198 Max deflection (cm) Normal to web Skew=20 Removed CF AASHTO optional limit Max deflection (in) 0 G1 G2 G3 0.0 Figure 6-4 Girder maximum vertical deflections, load Case-2 - unfactored live loads In addition to the optional elastic deflection limits, to avoid excessive permanent deformation in the girders AASHTO (2012) limits stresses in girder flanges under the Service load conditions. To examine effects of the optimized skewed bracing arrangement on girder design requirements in the composite bridge, flange stresses were compared against AASHTO Service Limit State criteria. Stress checks are provided here for the girder bottom flanges, which experienced larger stresses under live loads in the composite bridge. AASHTO Service Limit State requirements for bottom flange stresses in positive flexure (the tension flange) in a composite section are provided in Article and are reproduced in Equation 6-1 (AASHTO Equation ). The flange maximum lateral bending stress, f, and vertical bending stress, f, in this equation are obtained from the combined effects of dead loads and live loads using the Service II AASHTO load combination (dead load factor of 1.0, live load factor of 1.3) Where: = flange lateral bending stress. 180

199 = flange vertical bending stress calculated without consideration flange lateral bending. = hybrid factor, specified in Article of AASHTO (2012) = nominal flange yield strength. Bottom flange normal stresses in the single-span bridge model with different bracing arrangements under non-composite dead loads were discussed in Section Here, the normal flange stresses were obtained from the composite model under unfactored live loads. Figure 6-5 and Figure 6-6 plot these normal stresses across the G3 and G1 bottom flanges at mid span. These results were obtained for load Case-2 for G3 and load Case-1 for G1, cases that created the largest stresses in the respective girders. As detailed in the figures, using a skewed instead of a normal-to-web bracing arrangement slightly reduced (approximately 20.7 MPa or 3ksi) the maximum normal stress in the G3 bottom flange and caused a relatively significant increase (approximately 41.4 MPa or 6 ksi) in G1 bottom flange maximum stress. No significant change was observed in the G2 flange stresses as a result of changes in the bracing arrangements. 181

200 Flange Width (in) Stress (MPa) Normal to web Skew=20 Removed CF North Stress (ksi) Flange Width (cm) Figure 6-5 G3 mid span, normal stress across the bottom flange, load Case-2 - unfactored live loads Stress (MPa) North Flange Width (in) Normal to web Skew=20 Removed CF Stress (ksi) Flange Width (cm) Figure 6-6 G1 mid span, normal stress across the bottom flange, load Case-1 - unfactored live loads To evaluate girder bottom flange stress levels against the AASHTO Service limit in Equation 6-1, the following steps were followed: Non-composite dead load vertical bending stresses,, for the G3 and G1 bottom flanges were found from the results in Figure 5-21 and Figure 5-22 by averaging 182

201 stresses at the flange tips. Vertical bending stresses under composite live loads were obtained using the same procedure from Figure 6-5 and Figure 6-6. Maximum non-composite dead load lateral bending stresses,, for the G3 and G1 bottom flanges were obtained from the results in Figure 5-10 and Figure 5-11 by determining the difference between the exterior flange tip normal stress and the previously calculated vertical bending stresses,. Again, similar composite live load stresses were obtained using the same procedure from Figure 6-5 and Figure 6-6. Composite live load stresses were multiplied by 1.3 (Service II live load factor) and then were combined with non-composite dead load stresses that were multiplied by a factor of 1.0 (Service II combination dead load factor), according to the relationship in Equation 6-1. They were then compared against the maximum allowable limit from Equation 6-1. Table 6-1 shows results for the Service Limit State checks in the G3 and G1 bottom flanges for bridges containing skewed or normal-to-web cross frames. Corresponding nondimensionalized ratios are also shown, with values less than one indicating acceptable stress levels. As explained in Chapter 5, girder design in the single-span bridge was governed by the AASHTO (2012) Strength Limit State. Therefore, nondimensionalized ratios in Table 6-1 were much less than one for both girder flanges for the initial normal-to-web bracing system. Also, from Table 6-1 it is understood that, although the maximum stress did increase in G1 bottom flange as a result of using skewed bracing, the corresponding nondimensionalized ratio for the Service Limit State check was still much less than one. 183

202 Table 6-1 Service Limit Sate checks for girder bottom flanges Bottom flange of girder Bracing arrangement (1) f 1 2 f [MPa (ksi)] (2) 0.95 [MPa (ksi)] (3) Nondimensionalized ratios (1)/(2) G3 G1 Normal-to-web (49.9) (65.8) 0.76 Skewed (47.1) (65.8) 0.72 Normal-to-web (24.6) (47.5) 0.52 Skewed (29.8) (47.5) 0.63 Similarly, the effect of skewed bracing on AASHTO (2012) Strength Limit State requirements for the girder flanges was examined. AASHTO requires using the non-compact section limits when designing curved girder segments. The Strength Limit State requirement for bottom flange stresses in positive flexure (the tension flange) in a non-compact section is provided in Article , and is also shown in Equation 6-2 (AASHTO Equation ). The flange maximum lateral bending stress,, and vertical bending stress,, are obtained from the combined effects of dead loads and live loads using the Strength I AASHTO load combination (dead load factor of 1.25, live loads load factor of 1.75). These flange stresses are then added using the one-third-rule Where: = flange lateral bending stress. = flange vertical bending stress calculated without consideration flange lateral bending. = resistance factor for flexure (1.0) 184

203 = nominal flexural resistance of the tension flange = = hybrid factor, specified in Article of AASHTO (2012) = nominal yield strength in the tension flange. To evaluate girder bottom flange stress levels in the single-span bridges that contained both skewed and normal-to-web bracing arrangements against AASHTO Strength limits, flange vertical and lateral bending stresses ( and ) were obtained for the non-composite dead load and composite live loads in similar fashion to the Service Limit State checks. The noncomposite dead load stresses were multiplied by 1.25 (Strength I load combination dead load factor) and the composite live load stresses were multiplied by 1.75 (Strength I combination live load factor). These stresses were summed following the one-third rule and were compared against the maximum allowable limit from Equation 6-2. Table 6-2 shows results for the Strength Limit State checks in the G3 and G1 bottom flanges for bridges containing skewed or normal-to-web cross frames. Corresponding nondimensionalized ratios are also shown, with values less than one indicating acceptable stress levels. Table 6-2 Strength Limit State checks for girder bottom flanges Bottom flange of girder Bracing arrangement (1) f 1 3 f [MPa (ksi)] (2) φ F [MPa (ksi)] (3) Nondimensionalized ratios (1)/(2) G3 G1 Normal-to-web (64.5) (69.3) 0.92 Skewed (60.2) (69.3) 0.87 Normal-to-web (35.1) (50.0) 0.70 Skewed (40.2) (50.0)

204 As discussed in Chapter 5, girders in the single-span bridges were designed assuming that the cross frames were oriented normal to the girder webs. Therefore, the G3 bottom flange nondimensionalized ratio for the normal-to-web bracing arrangement is close to one in Table 6-2 (0.92). As expected, using skewed bracing arrangement, the G3 bottom flange nondimensionalized ratio reduced to 0.87 which indicates that it may be possible to use smaller flange sizes for this girder with skewed bracing. Sizing of the section for G1 in the single span bridge was governed by AASHTO proportion and shipping length limits rather than strength limits. This is evident in Table 6-2 by looking at the low nondimensionalized ratio for the G1 bottom flange for the normal-to-web bracing arrangement (0.70). However, the skewed bracing arrangement that was studied generated larger unbraced length for G1 and, subsequently, a more reasonable nondimensionalized ratio (0.81) was obtained for this girder. This indicates, again, that the skewed bracing not only reduced the number of required intermediate braces between the girders in this bridge, it also appeared to create a more efficient structure. The last item examined here is, the cross frame normal stresses. Figure 6-7 shows cross frame bottom cord member normal stresses for the normal-to-web or skewed bracing arrangements under unfactored live loads. These results were obtained for load Case-2 for cross frames between G3 and G2 and for load Case-1 for cross frames between G2 and G1, cases that created the largest normal stresses. Behavior similar to that observed for cross frame bottom cord normal stresses in the non-composite construction studies (see Figure 5-23), is also observed here for the composite bridge under live loads. Using skewed bracing resulted in slightly larger stresses in chord members that had small stresses for normal-to-web bracing arrangement and produced somewhat smaller stresses in the cross frame bottom cord members 186

205 at mid span (Station 4) that experienced the largest stresses. As expected, stress magnitude changes in the composite bridges under live loads were not as significant as those for the noncomposite construction studies (see Figure 5-23). Stress (MPa) Stress (MPa) Normal to web Skew=20 Removed CF 2L 3L 4 Between G2 and G3 (load Case-2) Normal to web Skew=20 Removed CF 2L 3L Stress (ksi) Stress (ksi) Between G1 and G2 (load Case-1) Figure 6-7 Cross frame maximum bottom chord normal stresses - unfactored live loads Both Figure 5-23 and Figure 6-7 indicate that the skewed bracing arrangement, although requiring a smaller number of intermediate braces between the girders in the bridge, created more uniform load sharing among the cross frame members and resulted in slightly smaller maximum cross frame design forces. As a result, it is clearly evident that design requirements 187

206 for these members were not violated when the bracing orientations were skewed relative to the girder web Bridge behavior under ultimate loading In the previous section, behavior of the single-span composite bridge model under unfactored live load was examined mainly to compare the resulting deformations and stresses against the AASHTO permissible values for the Service and Strength limit states. It was shown that, under unfactored loading, the structural elements remained elastic. In this section, the effect of skewed bracing arrangements on the inelastic behavior of the bridge steel members and the concrete deck was studied by examining behavior of the composite bridges under the increased, ultimate loading case, discussed in Section 6.1. As explained in that section this ultimate loading was taken from the ultimate load test completed for Phase 3 of the CSBRP project (Jung, 2006). Under this ultimate load, in similar fashion to what was reported by Jung (2006), significant material nonlinearities occurred in the steel members and the concrete deck for the composite bridge that were examined. Figure 6-8 to Figure 6-10 detail mid-span vertical deflections for the skewed and normal-toweb bracing arrangements as a function of the total applied load. As indicated in these figures, the girders demonstrated nearly linear behavior with deformations being proportional to the total applied load until approximately 4000 kn (900 kips) of load was applied to the bridges. At this point, appreciable inelastic behavior occurred due to the girder flange and cross frame member stresses exceeding yield. Not surprisingly, this behavior is more significant for the outermost girder; G3. It was observed that the final girder deflection for G3 under the maximum applied load was slightly smaller (approximately 8%) for the skewed bracing arrangement than for the normal-to-web arrangement. Also, the maximum deflection for G1 188

207 was slightly larger (6%) when skewed bracing was used. Therefore, it can be surmised that behavior similar to that observed in the linear range under unfactored live loads in Section 6.2 was also observed here for inelastic girder deflections, with the skewed bracing producing more uniform deformations. This effect is also evident in Figure 6-11, which details vertical concrete deck deflections across the bridge at mid span at the ultimate load (5783 kn or 1300 kips) and shows reduced maximum vertical deflections (approximately 9%, compared to normal-to-web bracing) in the deck and a slightly more uniform profile across mid span with skewed cross frames. 6, Deflection (in) Total load (kn) 5,338 4,003 2,669 1, Normal to web 600 Skew=20 Removed CF Deflection (cm) Total load (kip) Figure 6-8 G3 mid span vertical deflection as a function of total applied load - ultimate loading 189

208 6, Deflection (in) Total load (kn) 5,338 4,003 2,669 1, Normal to web Skew=20 Removed CF Deflection (cm) Total load (kip) Figure 6-9 G2 mid span vertical deflection as a function of total applied load - ultimate loading 6, Deflection (in) Total load (kn) 5,338 4,003 2,669 1, Normal to web 600 Skew=20 Removed CF Deflection (cm) Total load (kip) Figure 6-10 G1 mid span vertical deflection as a function of total applied load - ultimate loading 190

209 Deflection (cm) Normal to web Skew=20 Removed CF North 1.8 Deck width (ft) Deck width (m) Figure 6-11 Mid-span deck vertical deflections, total applied load=5783 kn (1300 kips) - ultimate loading Performance of the differing bracing arrangements at ultimate loading was also studied by comparing girder bottom flange normal strains. Strain levels were generally considered in this section for comparison, instead of stresses, because under ultimate loading the girder flanges demonstrated extensive inelastic behavior, and stress levels were not as meaningful as strains to detect changes in the inelastic behavior as a result of changing bracing arrangements. Figure 6-12 to Figure 6-14 depict the normal strains across bottom flanges of the girders at mid span under the applied, ultimate load of 5783 kn (1300 kips). Yield strain limits in these figures are obtained from associated material stress-strain curves assumed for the girder flanges Deflection (in) 191

210 Normal to web Skew=20 Removed CF Flange Width (in) Strain (%) North Yield Strain = 0.26% Strain (%) Flange Width (cm) Figure 6-12 G3 mid span normal strains across the bottom flange, total applied load=5783 kn (1300 kips) - ultimate loading Normal to web Skew=20 Removed CF Flange Width (in) Strain (%) Yield Strain = 0.2% 0.5 North Strain (%) Flange Width (cm) Figure 6-13 G2 mid span normal strains across the bottom flange, total applied load=5783 kn (1300 kips) - ultimate loading 192

211 Normal to web Skew=20 Removed CF 6 4 Flange Width (in) Strain (%) North Yield Strain = 0.2% Flange Width (cm) Strain (%) Figure 6-14 G1 mid span normal strains across the bottom flange, total applied load=5783 kn (1300 kips) - ultimate loading As shown in these figures, with either normal-to-web bracing or skewed bracing, normal strains across the girder flanges for G3 and G2 exceeded yield limits at the ultimate load and no significant difference was observed for the effects of the two bracing arrangements on these strain distributions. For the innermost girder, however, skewed bracing produced a significantly different strain distribution across G1 bottom flange at ultimate load. When skewed bracing was used, normal strains exceeded the yield limit for G1 bottom flange across practically half the flange width while for the normal-to-web case strains were below yield. Given the understanding that, at this loads level, the G2 and G3 bottom flanges experienced significant yielding (see Figure 6-12 and Figure 6-13), increased strains in the G1 bottom flange again indicated that using the skewed bracing arrangement resulted in more uniformly distributed loads between the structural members. 193

212 As explained, earlier an important issue of concern with respect to changing bracing arrangement and removing intermediate braces is the effect that this change could have on the concrete deck strains and subsequent cracking pattern. Previous studies by Tedesco et al. (1995) and Stallings et al. (1999) on a straight, three-girder bridge indicated that removing intermediate cross frames increased differential deflections between adjacent girders and therefore caused large bending stresses and strains in the concrete deck that could eventually result in deck cracking. Therefore, it was of interest to examine the effects that changes in girder deformations pattern had on concrete deck inelastic behavior when skewed bracing was used. To study the effect of skewed bracing on concrete deck behavior under ultimate loading, Figure 6-15 plots mid span, compressive in-plane principal strains across the deck width for the bridges having skewed and normal-to-web bracing arrangements at the ultimate applied load of 5783 kn (1300 kips). As observed in this figure, except for some slight, local discontinuities, the two bracing arrangements resulted in approximately the same longitudinal strain levels and distributions at the ultimate loads. For both bracing arrangements, longitudinal strains varied almost linearly across the deck width and exceeded the nominal concrete compressive crushing capacity of 0.3% strain at the tip of the overhang, near G3. More details regarding the effects of the different bracing arrangements on compressive principal strains throughout deck is provided using Figure 6-16, which is a plan view of principal strain contours on the top surface of the concrete deck for normal-to-web and skewed bracing arrangements at the ultimate load level. Again, only some minor changes are observed between the two contours for the different bracing arrangements. For both bracing arrangements a relatively significant portion of the deck overhang outside G3 near mid span developed compressive strains larger than the nominal 194

213 concrete compressive capacity (0.3% strain). The crushed region in the deck was observed to be slightly smaller for the bridge with skewed bracing than that containing normal-to-web bracing. Normal to web Skew=20 Removed CF 12 6 Deck width (in) Strain (%) North Crushing Strain= 0.3 % Deck width (cm) Strain (%) Figure 6-15 Mid-span deck top surface compressive principal strains, total applied load=5783 kn (1300 kips) - ultimate loading 195

214 Normal to web Skew=20 - Removed CF Figure 6-16 Concrete deck compressive principal strain contours, top surface of the deck, total applied load=5783 kn (1300 kips) - ultimate loading In similar fashion, to investigate the effect of skewed bracing on the tensile cracking behavior of the concrete deck under ultimate loading conditions, Figure 6-17 provides contours for tensile in-plane principal strains on the top deck surface at the ultimate applied load of 5783 kn (1300 kips), for bridges having normal-to-web and skewed bracing arrangements. Again, 196

215 the two bracing arrangement gave similar tensile strain patterns and almost the entire deck developed tensile strains that were larger than the nominal tensile cracking strain of 0.02%. Figure 6-17 also indicates that localized high tensile principal strains occurred near the top corners of the concrete deck for both arrangements. In similar fashion to the compressive strains discussed earlier, under ultimate loading the skewed bracing arrangement produced slightly smaller localized tensile cracking regions in the deck when compared against the normal-to-web bracing. 197

216 Normal to web Skew=20 - Removed CF Figure 6-17 Concrete deck tensile principal strain contours, top surface of the deck, total applied load=5783 kn (1300 kips) - ultimate loading 6.3 Summary and conclusions The in-service performance of the conventional normal-to-web bracing arrangement and the optimized skewed bracing arrangement from Chapter 5 were compared in this chapter by examining a representative composite, single-span, horizontally curved, three-girder bridges 198