Proposed Modification to AASHTO Cross-Frame Analysis and Design (2021)

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12 Research Approach To address the major objectives of NCHRP Project 12-113, four distinct phases were system- atically carried out by the research team. The four phases, as identified in the project RFP, are as follows: (I) Planning, (II) Part 1 of the Analytical Program and Field Experiment, (III) Part 2 of the Analytical Program and Proposed AASHTO Specifications, and (IV) Final Products. Each successive phase built on the findings and lessons learned from the previous phase. This is demonstrated schematically with a workflow diagram in Figure 2-1. As noted in Section 1.4, comprehensive summaries of Phases I, II, and III are provided in the respective Appendices D, E, and F. The information provided in this chapter is intended only to provide a general overview of the research approach and methodologies. In Phase I, an extensive literature review and industry survey was conducted, and preparations were made for the Phase II field experiments. Phase I consisted of six tasks, which are briefly summarized below: • Task 1: Conduct a literature review, document current design practices, and examine how design software incorporates these design practices. • Task 2: Synthesize the results of the literature review to identify the knowledge gaps related to the research objectives. • Task 3: Propose an analytical program to be executed in two parts. Part 1 (executed in Phase II) includes preliminary analysis to aid in the development of the experimental program, and Part 2 (executed in Phase III) includes comprehensive parametric studies to achieve the research objectives outlined in Chapter 1. • Task 4: Propose field experiments (executed in Phase II) that include one straight bridge, one skewed bridge, and one horizontally curved bridge. Measurements include (i) cross- frame member fatigue force ranges under controlled application of live load and (ii) in-service effective and maximum stress ranges for the same cross-frame members. • Task 5: Identify existing articles of AASHTO LRFD that require modification. • Task 6: Prepare Interim Report No. 1. Note that several of these tasks were previously summarized in Chapter 1, including the review of pertinent research and AASHTO LRFD articles. Section 2.1 briefly details the industry survey performed by the research team that facilitated the planning of the experimental and analytical studies and also aided in the development of the research scope. Appendix D also outlines this material in a more thorough manner. The experimental and analytical programs are covered under the Phase II and III tasks discussed in the following paragraphs. In Phase II, field experimental studies and model validation were performed, which allowed the analytical studies in Phase III to commence. In total, Phase II consisted of four tasks, which are briefly summarized below: C H A P T E R 2

Research Approach 13 • Task 7: Execute Part 1 of analytical program related to the specific bridges evaluated in the field experiments. • Task 8: Execute the field experiments, as introduced in Task 4. • Task 9: Validate the analytical modeling approach based on the results of the field experiments. • Task 10: Prepare Interim Report No. 2. The methodology used to accomplish Tasks 7 through 9 is briefly outlined in Section 2.2. Appendix E provides a more detailed overview of these Phase II tasks. Phase III expanded on the work completed in Phase II. In order to properly evaluate the behavior and design criteria of cross-frames in common highway I-girder bridge systems, a much larger representative sample than the three unique bridge geometries and traffic condi- tions from the field studies was necessary. As such, the validated modeling approach established in Phase II was used to conduct extensive analytical studies that ultimately improve the breadth and depth of knowledge pertaining to cross-frame behavior. Phase III included three major tasks, as summarized with the following: • Task 11: Execute Part 2 of the analytical program. Perform parametric studies related to live load-induced fatigue performance and stability bracing characteristics of cross-frames. • Task 12: Based on the analytical and experimental investigations, develop specification and commentary language for proposed changes to AASHTO LRFD. • Task 13: Prepare Interim Report No. 3. Sections 2.3 through 2.6 describe the methodology used to carry out Task 11 and the analy- tical studies. The project culminated in Phase IV, for which the proposed specification and commentary language was finalized for potential balloting by AASHTO. These proposed modi- fications are summarized in Chapter 4 and provided in full in Appendix A. As outlined in Section 1.2, there were five major objectives of the NCHRP Project 12-113 investigation. Task 11 was structured in such a way to address these objectives through a series Conduct field experiments and obtain measured data Phase II Validate FEA modeling approach Phase II Conduct parametric FEA studies Phase III Develop specification and commentary language Phase III Finalize report and prepare AASHTO ballot items Phase IV Perform background review and industry survey Phase I Figure 2-1. General flow of work related to NCHRP Project 12-113 (FEA = finite element analysis).

14 Proposed Modification to AASHTO Cross-Frame Analysis and Design of four independent computational studies. Table 2-1 summarizes these independent studies as well as the various software platforms used. The studies related to Objectives (a) and (b) in Table 2-1 were performed together via a parametric study in Abaqus and are abbreviated as the “Fatigue Loading Study” herein. The Fatigue Loading Study addresses the first of the three major research focuses: (i) fatigue loading criteria. Objective (c), related to cross-frame stiffness and eccentric connections, was also accomplished in a standalone parametric study in Abaqus/CAE (2017). This is referred to as the “R-Factor Study.” The studies related to Objective (d), abbreviated as the Commercial Design Software Study,” draw comparisons between commercially available software programs (Software A and Software B) and validated models in Abaqus. Both the R-Factor Study and the Commercial Design Software Study address the second of the three major research focuses: (ii) analysis techniques. Before addressing Objective (e), there are two important notes to make with respect to the Commercial Design Software Study. First, no references to the specific commercial software packages are made throughout this report to avoid critiquing or endorsing a specific brand. As such, the two software packages studied extensively are generically referred to as Software A and B. Second, the term “commercial software program” must be clearly defined. The term “commercial” can refer to any program available to the public with a one-time or annual fee. However, the focus of the Commercial Design Software Study is on those commercially avail- able programs specifically designed for and marketed as a bridge design tool. Abaqus, although a commercial program, is a general purpose finite element analysis (FEA) software program and is not specifically suited for bridge design, whereas Software A and B are. Abaqus is one of many general purpose FEA programs commonly used in research. As noted in Section 2.2, the modeling approaches in Abaqus were adjusted based upon validation data from the field monitoring. With that in mind, the research team could have also used a variety of other general purpose FEA packages and completed the same validation. The study related to Objective (e), which involves the stability bracing component to the project, was performed independently of the others and is identified as the “Stability Study” in this report. The Stability Study investigates the stiffness and strength requirements for cross- frames, which act as torsional braces for I-girders. This study addresses the final of the three major research focuses: (iii) stability bracing requirements. The methodology used for the Fatigue Loading Study is summarized in Section 2.3, whereas the methodology for the R-Factor, Commercial Design Software, and Stability Studies are summarized in Sections 2.4, 2.5, and 2.6, respectively. Objective Overview Model Software a Fatigue loading studies Fatigue Loading Study Abaqus b Influence of bridge geometry and cross-frame details c Cross-frame stiffness reduction R-Factor Study Abaqus d Commercial software evaluation Commercial Design Software Study Software A, Software B e Stability bracing requirements Stability Study Abaqus Table 2-1. Overview of Task 11 work breakdown.

Research Approach 15 2.1 Industry Survey An industry survey was developed and distributed by the research team to gather informa- tion from DOTs and consultants in the following areas: (i) commonly used software packages, (ii) fatigue design of cross-frames, (iii) load-induced fatigue issues with cross-frames encoun- tered in existing bridges, and (iv) standard cross-frame details. In total, there were 57 responses to the distributed survey representing a variety of states and consultants. Given the high response rate, especially from DOTs, the results of the survey served as a reliable tool with respect to focusing research efforts in the remaining phases of the project. This section summarizes many of the key observations obtained from the industry survey, which further emphasize the research motivations outlined in Section 1.1. For a detailed review of the survey and responses, refer to Appendix D. In terms of commercial design software use and fatigue design procedures, the following observations were made: • A wide range of commercial design software packages are used for bridge design in the United States. Software B (2D program) was the most popular among bridge owners and consultants, and Software A (3D program) received the third most votes. Given their wide- spread use, Software A and B were selected by the research team for further investigation in Phase II and III of the project. • In many cases, respondents indicated that their software of choice (i.e., not limited to just Software A and B) could not perform fatigue design checks or that they were unsure if it had such capabilities. • Several respondents indicated that they have encountered issues or had concerns related to the fatigue checks made by their software of choice. This was largely attributed to the “black box” nature of many software programs, as it was not transparent how the design checks were performed. • As a spot check for the analysis program, many respondents indicated that they also use hand calculations to perform fatigue design checks for cross-frames. In terms of load-induced fatigue issues encountered in existing bridges, the following observations were made: • Only 28% of the respondents indicated that their organization has previously experienced load-induced fatigue cracking in existing bridges and cross-frames. • In many of those cases, the respondents noted that load-induced fatigue often worked in tandem with corrosion-related section loss and distortion-induced fatigue. • Multiple responses pertaining to distortion-induced fatigue were recorded. However, as noted in Chapter 1, the focus of the present study is on load-induced fatigue effects. In terms of standard cross-frame details used across the country, the following observations were made: • Since many states provide multiple standard details depending on bridge geometry and force demands, a total of 58 details (including cross-frames at supports) were compiled and reviewed. The following characteristics were of interest when reviewing typical cross-frame details: cross-frame configuration/layout (X-frame, K-frame, or Z-frame), member sections, member end connections (bolted or welded), and use of gusset plates. • Single-angle sections are the most common structural shape specified for the cross-frame members, ranging from L3×3×5/16 to L6×6×5/8. In some instances, DOTs do not provide typical sizes. For these cases, it is the responsibility of the designer to conduct a rational analysis and determine the required size.

16 Proposed Modification to AASHTO Cross-Frame Analysis and Design • Cross-frame connection details vary substantially from state to state. Figure 2-2 graphically summarizes the range of cross-frame configurations used in the United States. • In almost all instances where gusset plates are used to attach cross-frame members to the connection plates, the gusset plates are bolted to the connection plates and welded to the member. The lone exception to this is the Texas DOT (TxDOT) cross-frame details, where cross-frames often utilize erection bolts and the final connections are welded prior to deck construction. • In instances where gusset plates are not used to connect members to the connection plates, the diagonals are connected in one of two ways. They are either bolted directly to the connec- tion plate or they are fillet welded to the top and bottom chords. As indicated above, many DOTs specify bolted connections between the cross-frame member and the gusset plate with TxDOT being a key exception. In these instances, slip-critical (SC) connections are commonly used. Given that the present study is generally focused on the cross-frame response within the elastic range (i.e., the fatigue limit state), it is assumed that SC bolted connections do not slip at these service-level loads, such that the response of a cross- frame is relatively unaffected by the connection method. Therefore, the studies presented herein do not differentiate between bolted and welded connections, as cross-frame behavior is more influenced by its member size and overall geometry. As noted in Section 1.3, the industry survey allowed the research team to develop a manageable scope for the analytical studies. Based on these results, only single-angle sections in traditional X-type and K-type cross-frames were assessed [i.e., cross-frames (a) and (d) in Figure 2-2]. It should also be noted that cross-frames without top and/or bottom struts can potentially lead to stability issues during construction. Although zero-force members in many stability applica- tions (i.e., construction condition), struts are important since they prevent buckling mode in which the girders rotate out-of-plane in opposite directions. Thus, it is always recommended to include top and bottom struts for both X- and K-type frames, particularly in horizontally curved bridges and/or bridges with significant skews. 2.2 Field Experimental Program and Model Validation For Phase II of the project, three different bridges were instrumented with strain gages and experimentally tested with two different methods: (i) a controlled live load test and (ii) in-service monitoring of effective and maximum stress ranges. As noted previously, three different bridge geometries were explored as part of this field study, including a straight bridge with normal (a) X-Type (b) X-Type; No Top Strut (c) X-Type; No Struts (d) K-Type (e) K-Type; No Top Strut (f) Inverted K-Type (g) Z -Type Figure 2-2. Various cross-frame configurations used across the United States.

Research Approach 17 supports, a straight bridge with skewed supports, and a horizontally curved bridge. The research team assessed the pros and cons for a variety of existing bridge structures throughout Texas but ultimately elected to study three in the greater Houston area. This section provides a brief overview of the field experimental program and the general procedures used to validate the FEA modeling approach. Section 2.2.1 succinctly describes the key geometric parameters and traffic conditions of each instrumented bridge. Section 2.2.2 outlines the two experimental tests performed with additional guidance on how the data was processed. Note that the commentary provided herein is not intended to cover every aspect of the field tests. For a more detailed synopsis, refer to Appendix E. 2.2.1 Instrumented Bridge Overview The straight bridge with normal supports (referred to as instrumented Bridge 1 hereafter) serves as an off-ramp for traffic traveling northbound (NB) on IH-45. The bridge contains 13 spans, including 10 spans of prestressed concrete girders and a three-span continuous unit of built-up steel plate girders. In the steel spans, five nonprismatic girders across the width support three lanes of traffic. The respective lengths of the steel I-girder spans are 194 feet, 240 feet, and 194 feet. X-type cross-frames are oriented normal to the longitudinal axis of the bridge at a 19-foot typical spacing. According to the 2007 construction drawings, the estimated average daily traffic (ADT) for this bridge is approximately 3,300. Only the southernmost end span (194-foot) was instrumented with strain gages due to favorable ground access in the field. The skewed bridge (referred to as instrumented Bridge 2 hereafter) serves five lanes of south- bound (SB) IH-45 traffic, which is a major arterial road through Houston. Instrumented Bridge 2 is a three-span continuous, steel I-girder bridge with a total length of 465 feet (125-foot, 215-foot, and 125-foot spans). A 42-degree support skew relative to the centerline of the bridge was used to accommodate the roadway below. The five lanes of traffic are supported by 12 non- prismatic steel I-girders; thus, the superstructure is highly redundant. X-type cross-frames are oriented normal to the longitudinal axis of the bridge at a 17-foot typical spacing. Near the skewed supports, the designers selectively eliminated cross-frame panels in accordance with AASHTO LRFD Article 6.7.4.2 guidance. In these cases, lean-on braces were used to mitigate load-induced force effects. According to Texas DOT traffic maps (2020), the ADT for this bridge is 120,510, which provided excellent loading conditions in terms of the in-service moni- toring study. Only the northernmost end span (125-foot) was instrumented with strain gages due to favorable ground access and traffic control requirements. The horizontally curved bridge (referred to as instrumented Bridge 3 hereafter) is a direct connector along the SH-146 corridor that primarily serves large trucks traveling to nearby shipping ports. The bridge contains 14 spans, including 10 straight spans of prestressed concrete girders and a four-span continuous unit of curved steel I-girders. The radius of curvature of instrumented Bridge 3 is 800 feet, and the supports are normal to the centerline of the bridge. The respective lengths of the steel spans are 255 feet, 310 feet, 238 feet, and 216 feet. The four girders across the width support one lane of striped traffic. X-type cross-frames are oriented normal to the longitudinal axis of the bridge at a 12.5-foot typical spacing. According to the 2009 construction drawings, the ADT is approximately 40,000. Only the 216-foot end span was instrumented with strain gages due to favorable ground access. For additional information on the instrumented bridge geometries and cross-frame details, refer to Appendix E. 2.2.2 Experimental Testing Program Overview For each bridge, several cross-frame members and the bottom flanges of neighboring girders were selected for strain gage instrumentation. The locations of these gages were based

18 Proposed Modification to AASHTO Cross-Frame Analysis and Design on numerous factors including field access, traffic control considerations, the capacity of the data acquisition system (DAQ), and preliminary analysis. The preliminary analysis, which was conducted prior to the field studies, identified the cross-frames likely to experience the largest stress ranges, as well as the critical truck positions to evaluate in the controlled live load test. The results of these preliminary studies are further detailed in Appendix E. A wireless monitoring system was designed to collect continuous strain time-histories, as well as simultaneously calculate stress cycle counts in accordance with commonly used rainflow- counting algorithms (Downing and Socie 1982). For each instrumented cross-frame member, four quarter-bridge strain gages were installed at various locations along a cross-section. By measuring the strain at various points on a cross-section, the axial component of stress could be isolated from the biaxial bending effects in the member via the linear regression method developed by Helwig and Fan (2000). This procedure was validated by laboratory studies prior to implementation in the field. Given that the Category E′ designation established for these welded connection details was explicitly based on axial stresses (i.e., P/A), reliably measuring axial stress was critical. It is important to note, however, that significant bending stresses were measured as part of this study to verify the impact of eccentric end connections. Two strain gages were also installed at each edge of the girder bottom flange, such that longitudinal bending stress and warping stress components could be independently evaluated. Upon completing the strain gage installation, two different experimental tests were performed for each bridge. The controlled live load test consisted of four dump trucks loaded with sand (of known axle configuration and weight) positioned at various longitudinal and transverse points on the concrete deck. The in-service monitoring consisted of recording live load stress range cycles in the instrumented elements over a one-month period. The subsections herein briefly address each test separately. 2.2.2.1 Controlled Live Load Test In total, seven different static load cases were considered for the controlled live load test, as well as a set of pseudo-static moving loads (i.e., trucks driven at slow speeds) to generate influence lines for each instrumented component. Cross-frame stresses, girder flange stresses, and select girder displacements were measured during each successive load case. Refer to Appendix E for full instrumentation plan and results, including sketches of strain gage locations and static load cases performed on each bridge. As noted in Chapter 1, the accuracy of even the most sophisticated FEA analysis is limited by the modeling assumptions. Thus, the field-measured data obtained from the controlled live load tests were vital in validating the FEA models used in the Phase II parametric studies. The load cases from the field studies were effectively replicated in a 3D FEA model, which allowed the research team to fine-tune the parameters and assumptions to achieve a close match of the actual conditions of the bridge system. The modifications required to obtain good agreement with the measured data are detailed in Section 3.1. Parameters such as boundary conditions and the presence of concrete barriers were explored. 2.2.2.2 In-Service Monitoring For the in-service monitoring phase of the field tests, continuous strain history over a four- week period was converted into histograms of stress cycle counts (i.e., a spectrum of stress range measurements). Two key performance characteristics of the instrumented cross-frame members and girder flanges can be inferred from these spectra: (i) the number of truck events causing a significant stress cycle and (ii) the stress cycle magnitudes that a structural component typically

Research Approach 19 experiences. Note that the one-month monitoring period is sufficiently long to capture stabi- lized data as well as hourly, daily, and weekly trends that may deviate from normal or average conditions (Connor and Fisher 2006; Fasl 2013). In addition, three supplementary metrics derived from the histogram plots were used by the research team to compare data between the various instrumented components of the different bridges. These three metrics are effective stress range (Sre), maximum stress range (Srm), and index stress range (Sri). All three, which can be derived from simple postprocessing of the measured data, are effective tools for evaluating the fatigue performance of structural components and details. Effective and maximum stress ranges are defined in the current 9th Edition AASHTO LRFD Specifications (2020). Index stress range was a metric developed by Fasl (2013) but is not currently adopted by standard codes. These metrics are further explained in the context of measured data in Section 3.1 and Appendix E. 2.3 Fatigue Loading Study The Fatigue Loading Study is intended to investigate two major questions concerning load-induced fatigue design of cross-frame systems: (i) the influence of bridge geometry (e.g., support skew, horizontal curvature) and cross-frame details (e.g., cross-frame layout) on cross-frame fatigue force effects and (ii) the appropriate fatigue stress ranges for the evaluation of cross-frames in right, skewed, and horizontally curved bridges. These questions represent Objectives (a) and (b) established in Section 1.2. Preliminary answers to these questions were obtained from the field experiments conducted during Phase II. However, the instrumented bridges represent only three unique framing geometries subjected to local traffic conditions in the greater Houston area. In order to adequately assess the AASHTO fatigue loading criteria, a more expansive study was required. Based on the lessons learned from the modeling validation process in Phase II, the research team developed and conducted a robust, finite element parametric study. The parametric study effectively improved the depth of knowledge by expanding the breadth of the data set. In total, 4,104 unique bridge geometries were studied including various girder and cross-frame layouts, girder cross-sections, and cross-frame details. The 4,104-model data set is intended to reason- ably represent a wide variety of steel I-girder bridges currently in service in the United States. The details of the 4,104-model matrix are briefly discussed in the subsequent sections. To address the major objectives of the Fatigue Loading Study outlined above, the research team focused its efforts on three major tasks. First, a unified modeling approach was developed to analyze the various bridge structures (i.e., right, skewed, and horizontally curved bridges) in a consistent and repeatable manner. Second, the research team implemented the current fatigue design criteria on the full set of bridges. The cross-frames of the representative bridges were effectively designed for AASHTO LRFD 9th Edition fatigue provisions. Lastly, measured high-resolution WIM data from different U.S. states were implemented on a subset of the bridges considered in the study. The observations and findings of each successive task were used to address Objectives (a) and (b) outlined in Section 1.2. By comparing the load-induced force response of cross-frames in a variety of bridge types, the influence of bridge geometry and cross-frame details can be evaluated. By then comparing the fatigue design forces with the force effects due to real traffic patterns (i.e., via WIM records), a realistic approximation of actual fatigue stress ranges can be assessed. The appropriateness of current AASHTO LRFD fatigue design criteria such as critical

20 Proposed Modification to AASHTO Cross-Frame Analysis and Design load position and Fatigue I and II load factors [which have been calibrated for longitudinal girders only through the research efforts documented in SHRP 2 R19B (Modjeski and Masters 2015)] can also be evaluated with respect to cross-frame systems. Following this introduction, the modeling approach adopted in the Fatigue Loading Study is outlined (Section 2.3.1). The procedures used to evaluate the bridges for current fatigue design loading are presented (Section 2.3.2), as well as the procedures related to the WIM study (Sec- tion 2.3.3). Major results and outcomes related to the studies are summarized in Section 3.3. 2.3.1 General Methodology Preliminary work and preparation are critical for any comprehensive analytical parametric study. It was especially important for the Fatigue Loading Study, which parametrically consid- ered entire bridge structures in refined 3D analyses. As such, the research team dedicated its efforts in the early stages of Phase III to prepare for and expedite the large-scale analytical study. Due to the wide range of parameters considered, an ordered process that systematically conducts the analyses and processes the results was developed. The balance between compu- tational efforts and modeling accuracy was of particular interest. For a parametric study of this size, it was not feasible to evaluate every permutation of every critical parameter given limitations to computational resources and time. With computational constraints in the forefront of the discussion, the research team conducted preliminary analyses to establish expectations and develop a reasonable scope. In particular, the seven tasks below were performed prior to conducting the Fatigue Loading Study to help narrow the focus and limit the required computational time: • Adopt a consistent modeling technique for cross-frame elements; • Develop a reasonable analytical testing matrix for purposes of the Fatigue Loading Study; • Refine the applied loading approach; • Conduct mesh sensitivity studies to optimize computational efforts; • Streamline postprocessing efforts by evaluating only critical cross-frames; • Develop scripts to automate the modeling process; and • Develop Excel-based macros for processing the FEA results. Each of these preliminary tasks was important in developing a robust study. For purposes of this report, each task is cursorily addressed herein with the following subsections; for a more detailed overview of the methodology used, refer to Appendix F. 2.3.1.1 Modeling Assumptions As documented in subsequent sections of the report, there are numerous ways that designers commonly model bridge superstructures, ranging from sophisticated (3D) to simplified (2D) approaches. For each approach, there are several widely regarded methods specifically directed at modeling cross-frames. This section does not address and compare every modeling tech- nique. Rather, a unified modeling approach that is both repeatable and justifiable is explored for use in the Fatigue Loading Study. For additional information, Sections 2.4 and 2.5 provide an overview on the most common modeling techniques. Because they generally produce more reliable cross-frame results, only 3D models were considered for use in the Fatigue Loading Study. Most 3D models, either from commercial design software packages or high-fidelity FEA software, employ the same basic modeling strategies for the concrete deck and girders. The strategies for cross-frames in 3D models, though, often vary. While the most rigorous and accurate representation of a cross-frame makes use of shell elements, most common 3D models do not explicitly model the cross-frames with

Research Approach 21 shells but instead use line elements. While the present study utilized line elements with R-factors, additional studies were also carried out making use of shell-element models. A discussion of these modeling techniques is provided in the following paragraphs. In terms of the more rigorous approach, each component of the cross-frame panel is explicitly modeled with a shell-element: connection plates, gusset plates, cross-frame members (typically single angles), and fill plates commonly found at the intersection of diagonals in X-frames. The eccentric load path is also explicitly considered. These models are simply referred to as shell-element models herein. Although generally more accurate, this refined technique (especially with a fine mesh) rapidly increases the computational demands and run time. The second alternative, which is more common in design practice and even in many research studies, simplifies the cross-frames as pin-ended truss elements. Therefore, the eccentric load path and the stiffness contributions of connection and gusset plates are neglected. The actual stiffness of the cross-frame system, though, can be approximated with a simple modifica- tion factor (R-factor) assigned to the cross-sectional area of the truss element or the elastic modulus of the material. These types of models are referred to as truss-element models in this section. Given the scope of the study, the shell-element modeling approach was deemed too computationally intensive. Thus, the research team elected to perform the Fatigue Loading Study with truss-element models to balance computational speed and modeling accuracy. To better understand the behavior, preliminary studies were performed using the general purpose FEA program, Abaqus, to determine which variation of truss-element models produces the most accurate results relative to the validated, shell-element models. Based on these preliminary studies, the following observations were made, which influenced the approach adopted for the Fatigue Loading Study (Figure 2-3): • Explicitly modeling the offset dimension between cross-frame working line and the girder flanges better represents the stiffness of the panel and the inclination angle of the diagonal members (as opposed to connecting the cross-frame member into a shared node at the web-flange juncture); • Without the connection plates, cross-frame forces are greatly affected by web distortion caused by the concentrated forces acting on an unstiffened, flexible web (note that distortion effects would not be impactful if cross-frames were connected directly into the girder flange); and • Terminating the cross-frame truss member at the edge of or in the middle of the connection plate affected the force distribution in the cross-frame members. Figure 2-3. Final selected modeling technique for truss-element cross-frames.

22 Proposed Modification to AASHTO Cross-Frame Analysis and Design Note that the vertical offset dimension and the presence of connection plates are further discussed in Sections 2.4 and 2.5, which focus heavily on the approaches traditionally adopted by 3D commercial design software programs. Prior to conducting the parametric study, the accuracy of the assumed truss-element modeling approach and its simplifications outlined above was validated with the field-measured data. This procedure is detailed in Appendix F. Lastly, it was determined that a first-order, elastic analysis was suitable for the fatigue load- ing conditions that are representative of service-level load magnitudes. This approach is also consistent with common design practice and AASHTO LRFD recommendations. To verify this assumption, supplementary FEA studies that considered second-order effects and initial imperfections in cross-frame members were performed, and their influence on the cross-frame force demands was found to be negligible. 2.3.1.2 Analytical Testing Matrix As previously stated, the bridges considered in the study were intended to represent a common range of steel I-girder bridges currently in service. Developing a testing matrix that represents thousands of steel I-girder bridges in the United States is challenging, considering there are many different parameters that potentially impact the behavior and response of cross-frames. To establish a reasonable scope, 13 parameters were identified as independent variables (i.e., the parameters to be evaluated throughout the study), and the remaining parameters were considered constants. Table 2-2 describes the range of values used for the 13 independent variables. Note that this list assumes that the truss-element modeling approach described in the previous sub- section is implemented. Thus, parameters related to cross-frame details such as gusset plate thickness are not considered in the Fatigue Loading Study but are instead considered in the R-Factor Study. To facilitate the understanding of these parameters and the bridge geometries evaluated, several sketches are provided in Appendix F (e.g., a sketch demonstrating how stag- gered cross-frame layouts were considered). Based on the parameters outlined in Table 2-2, it is apparent that conducting every permu- tation of every parameter was not feasible given computational constraints. Thus, the matrix Parameter Range of Values Number of spans {1, 2, 3} L/d ratio {25, 30, 35} Girder spacing [ft] {6, 8, 10} Number of girders {3, 5, 7} Support skew {0, 30, 60};{Parallel, Trapezoidal} Radius of curvature [ft] {Infinite, 1,500, 750} Cross-frame spacing [ft] {20, 30} Cross-frame layout {Contiguous, Staggered} Web depth [in] {72, 96} Deck thickness [in] {8, 10} Cross-frame type {X, K} Cross-frame area [in2] {2.86, 4.79};{L4x4x3/8, L5x5x1/2} Concrete modulus [ksi] {3,600, 5,000} Table 2-2. Range of values used for independent variables in Fatigue Loading Study (L/d = span-to-depth ratio, ksi = kips per square inch).

Research Approach 23 was filtered to generally provide only the cases that are likely most critical. Rules were estab- lished to eliminate permutations deemed unnecessary or less important than others. Many of these rules were based on current AASHTO provisions related to geometric limits. For example, AASHTO Article 2.5.2.6.3 recommendations were implemented to establish reason- able bounds on the span-to-depth-ratios considered in the study. To further clarify Table 2-2, the following bulleted items provide key discussion points. The following list represents an abbreviated version; refer to Appendix F for a more thorough overview and a comprehensive list of every bridge parameter studied. • The differentiation between independent variables and constants is based on prioritization of the parameters, the information gained from the validation studies, data collected from the industry survey, and common bridge design practice. For example, the presence of a concrete barrier was deemed a pertinent variable based on the validation studies in Phase II, as is documented in Section 3.1.1. However, for purposes of the Phase III parametric study, the research team elected to simply maintain constant dimensions and parameters for barriers rather than evaluate variable dimensions. • In the same vein, overhang dimensions were taken as a uniform 3 feet (measured from the centerline of the fascia girder to the outer edge of the deck) to simplify the calculations. For bridges with a wider girder spacing, overhang lengths are commonly greater than 3 feet. As is discussed later in the report, fatigue forces in cross-frames are often maximized by truck passages where a wheel line rides along the overhang portion of the deck. It was found, however, that force effects in a cross-frame member due to an “overhang” live load are gener- ally linearly dependent on the outer wheel distance to the centerline of the fascia girders. • Staggered cross-frame layouts were considered in the study, but skewed cross-frames were not. For skewed bridges, cross-frames were either: (i) contiguous lines normal to the longitu- dinal girders or (ii) discontinuous lines that run parallel to the support skew angle but placed at a line normal to the longitudinal girders. Because both support skew angles considered in the study (30 and 60 degrees) exceed 20 degrees, skewed cross-frames parallel to the support skew were not considered in accordance with AASHTO LRFD Article 6.7.4.2. • In general, the research team ensured AASHTO Article 6.7.4.2 was satisfied for the layout of cross-frames. In particular, the cross-frame depth was selected to exceed 75% of the girder depth to preclude any significant web-distortional effects. Additionally, the cross-frame layout recommendations near obtuse and acute corners of skewed bridges given in AASHTO Article C6.7.4.2 were followed. • A constant stiffness modification factor (R-factor) is assumed for all cross-frame systems, regardless of the assumed geometry and connection details. Because the focus of the Fatigue Loading Study is not on stiffness modification factors, the research team elected to normalize the effects of eccentric load paths for all cross-frame systems in the 4,104-model parametric study. That way, any potential variability associated with the R-factors is eliminated, and the relative impact is then consistent across all bridge models. A uniform factor of 0.60 was selected, as it was shown to be a reasonable representation of the “average” value in preliminary R-factor studies. • The decision to use R = 0.6 was made early in Phase III, several months before the final results were obtained from the R-Factor Study (i.e., proposed R = 0.75). Clearly, consistently larger cross-frame force effects would have been determined had the parametric study been conducted with the larger R-factor. As noted in Section 3.3, an increase in R-factor from 0.60 to 0.75 would only increase those predicted force effects about 5% to 7%. As such, it was justified that this deviation did not warrant a repeat of the parametric study since no major conclusions outlined herein would be affected. • The girder sections of each bridge were evaluated with respect to proportion limits (AASHTO Article 6.10.2) and strength limit state resistance requirements (AASHTO Article 6.10.6).

24 Proposed Modification to AASHTO Cross-Frame Analysis and Design In other words, it was ensured that the bridges studied represent realistic structures that have been properly designed and detailed. • Aside from the parameters listed in the tables, the research team also conducted spot checks to examine the influence of several parameters including different flange dimensions and the removal of bridge barriers. 2.3.1.3 Load Application Current AASHTO fatigue loading criteria require that the design fatigue truck be positioned in all transverse and longitudinal positions across the bridge deck to maximize the stress ranges in the detail under consideration (Article 3.6.1.4.3). To implement these criteria, the research team generated an influence-surface analysis in Abaqus by applying a unit point load across a defined grid of load positions. The loading grid extended over the entire width (between the inside faces of the barriers) and length of the bridge to capture the full influence. To then capture the effects of a realistic truck passage instead of a static unit load, an Excel program and script were developed to simulate these effects, as discussed in Section 2.3.1.6. In terms of accuracy, it is important that the critical loading position is captured by the specified loading grid. Preliminary studies were performed to optimize the loading grid dis- cretization (i.e., how frequently is the unit load applied in the longitudinal and transverse directions). Grids ranging from 1-foot to 5-foot longitudinal and transverse increments were considered. From these analyses, it was determined that a relatively coarse discretization still produced reasonable levels of accuracy in terms of predicting cross-frame force effects. The applied loading grid was accordingly optimized by assigning load points at every 5 feet longitu- dinally and at the following transverse locations: at each girder line, midpoint of every cross- frame bay, and at a point 1 foot outboard from the centerline of each fascia girder. 2.3.1.4 Mesh Sensitivity In addition to the sensitivity of the loading grid mesh, the sensitivity of the finite element model mesh was also extensively studied. Fine element meshes potentially offer more accurate solutions but at the cost of increased computation time and storage. Acknowledging that com- putation efficiency was of the utmost importance for the Fatigue Loading Study, the research team studied the effects of a coarser mesh on the predicted response of cross-frame members. From preliminary studies, it was observed that cross-frame force effects were not overly sensitive to the specified mesh size. Since localized stress concentrations or behaviors are not important to the study at hand, a coarse mesh was adopted. The girder shells were meshed at 24 inches (i.e., three to four shells along the depth of the web), and the deck shells were meshed at 48 inches (in addition to the loading grid). Quadrilateral meshes were typically used, except at extreme cases near skewed ends. In those cases, special care was taken to ensure reasonable aspect ratios (i.e., within 3 to 5). 2.3.1.5 Postprocessing Critical Cross-Frames Rather than evaluating the forces in every cross-frame from every model in the Fatigue Loading Study, it was more reasonable to evaluate only the most critical members. Knowing that cross-frame force effects are likely dependent on load position, bridge geometry, and location on the span, identifying the critical cross-frame for all 4,104 unique bridges was challenging. Thus, the research team took a more systematic approach to predicting the most likely “critical” cross-frames for any given bridge. Prior to the Fatigue Loading Study, a prelimi- nary parametric study was conducted to gain a general understanding of (i) the typical location of the critical cross-frame members and (ii) the influence of the transverse and longitudinal load position on the force effects in these critical members.

Research Approach 25 Based on the findings of this preliminary study, the research team preselected cross-frames to evaluate prior to running the full parametric study. Given the unique geometries and cross-frame layouts, the selected cross-frames are not necessarily in the exact same location for each bridge. Rather, general locations/regions that are consistent across all 4,104 models were selected. These four general locations/regions, which apply equally to both straight and horizontally curved bridges, included: • Edge cross-frame bay closest to the point of maximum positive dead load moment (i.e., midspan for single-span bridges and approximately 0.35L measured from the end support in end spans of continuous units); • Interior cross-frame bay closest to the point of maximum positive dead load moment; • Interior cross-frame bay nearest to the end support (skewed or non-skewed); and • Interior cross-frame bay nearest to the intermediate support (skewed or non-skewed). For these regions, every cross-frame member in the panel was examined (i.e., top strut, bottom strut, and diagonals). These cross-frame panels were selected given the propensity for more significant differential girder displacements in the surrounding regions. For a more detailed evaluation of these assumptions, refer to Appendix F. To illustrate these general locations, a representative sketch is provided in Figure 2-4. The figure presents the critical cross-frame panels evaluated for representative single-span, two- span continuous, and three-span continuous straight bridges. Note that these figures are simply schematic and not drawn to scale. In addition, only intermediate cross-frames are illustrated; cross-frames at end and intermediate supports are hidden for clarity, as these are not the focus of the study. 2.3.1.6 Automation and Scripts To expedite the analysis and postprocessing, the research team developed Python scripts to automate the calculations. The Python scripts were designed to perform the following Key: Critical cross-frame evaluated Figure 2-4. Critical cross-frames evaluated for sample bridges: (top) single-span with skewed supports, (middle) two-span continuous with skewed supports, and (bottom) three-span continuous with normal supports.

26 Proposed Modification to AASHTO Cross-Frame Analysis and Design tasks: (i) develop the 3D FEA models for all geometries, (ii) run the simulated influence- surface analysis, (iii) output the axial-force and displacement response of the preselected cross-frame members of interest, and (iv) create summarized output files of all pertinent data (i.e., influence-surface results for every critical cross-frame member). These extensive calcula- tions were performed using the Texas Advanced Computing Center (TACC) supercomputers. Once the summarized data files were obtained, further postprocessing was performed in an external Excel program, which is documented in the next subsection. 2.3.1.7 Data Visualization Due to the tremendous amount of data produced from the parametric study, the research team identified a need to view and evaluate data in an organized manner. Considering that 4,104 bridges were analyzed, the total number of influence-surface results exceeded 65,000, which is obviously not practical from a publication perspective. Thus, an Excel-based “data visualization” workbook was developed as a centralized tool to view all influence-surface data obtained from the finite element models. An example showing two of the many influence-surface plots is presented in Figure 2-5. These two plots correspond to a two-span, straight bridge with normal supports and an other- wise identical bridge (e.g., same deck thickness, girder spacing, etc.) with 60-degree support skews. Thus, the only discernible differences between the bridges are the support skew and the cross-frame layout near the skewed supports. As depicted in the figure, influence-surface results are best displayed as color contour plots where the x (longitudinal)- and y (transverse)-coordinates represent the spatial coordinates of the applied load on the deck surface and the z-axis (color intensity) represents the corresponding magnitude of the force effect. To clarify the contour data, a framing plan for each bridge is also overlaid. Girder lines and cross-frames are depicted with thin green lines; supports are represented as bolded and dashed black lines. The cross-frame of interest, for which the influence-surface results are presented, are bolded. Note that the x- and y-coordinates are not to scale; the transverse (y-axis) is scaled up for clearer viewing. The contour plot in Figure 2-5 demonstrates the axial-force response of a diagonal cross- frame member near the intermediate support. The diagonal that frames into the top flange of the first interior girder from the left (i.e., “left” corresponding to negative values about the y-axis) is displayed. The color intensity on the plots corresponds to the axial force induced in the cross-frame member due to a 1-kip load positioned at a given spatial coordinate on the deck surface. Thus, the color intensity is specified in units of kips of axial force per kips of applied load on the deck (i.e., kips/kip). For loads applied in the “red” area on the plot, the compression force in the diagonal is maximized. For loads applied in the “blue” area, the tension force in the diagonal is maximized. Load positions applied in the grey area have negligible influence on the axial-force response in that cross-frame member. As an example, a value of +0.1 kips/kip at a given location on the deck (representing a critical tensile load position) means that a 1-kip load applied there results in 0.1 kips of axial tension in the cross-frame diagonal of interest. From the sample results in Figure 2-5, it is obvious that the longitudinal load influence of the skewed bridge is broadened relative to the straight bridge. In fact, there is strong tensile load influence along the outer “right” edge of the bridge deck (i.e., a blue area), which is not present for the straight bridge. The overall magnitudes at the critical load positions are also slightly higher in the skewed system. For additional influence-surface results and commentary, refer to Appendix F.

Research Approach 27 In total, the research team produced thousands of influence-surface plots similar to those in Figure 2-5. Influence-surface plots are useful visual tools and provide important insights on the load-induced response of cross-frames, particularly in the case of this study where evaluating over 65,000 plots in an efficient and meaningful way is virtually impossible given the number of parameters investigated and the variability in the response. While the contour graphs show the response for a 1-kip force applied anywhere on the deck, the actual force generated in the cross-frame due to a truck on the bridge is the sum of the wheel loads multiplied by the corresponding influence-surface values at the wheel locations. In order to assess the correla- tion between truck placement, fatigue stresses, and the various bridge geometries, additional processing of the data was necessary. Thus, Sections 2.3.2 and 2.3.3 introduce realistic loading conditions for these 4,104 bridge models. 0 25 50 75 10 0 12 5 15 0 17 5 20 0 22 5 25 0 27 5 30 0 32 5 35 0 Lo ng itu di na l P os iti on (f t) Transverse Position (ft) 0. 2 (T ) -0 .2 (C ) -1 3 130 - 25 0 25 50 75 10 0 12 5 15 0 17 5 20 0 22 5 25 0 27 5 30 0 32 5 35 0 37 5 Lo ng itu di na l P os iti on (f t) 0. 2 (T ) -0 .2 (C ) -1 3 130 Transverse Position (ft) Influence (kip/kip) Influence (kip/kip) Figure 2-5. Comparison of influence-surface plots for axial-force response of cross-frame diagonal near intermediate support with (left) no skew and (right) 60-degree skew (T = axial tension force, C = axial compression force).

28 Proposed Modification to AASHTO Cross-Frame Analysis and Design More specifically, Section 2.3.2, utilizes the over 65,000 influence-surface plots to assess current AASHTO fatigue loading criteria. Section 2.3.3 expands on that analysis and inves- tigates the effects of measured WIM traffic streams. 2.3.2 AASHTO Design Loads Although the unit loads outlined in Section 2.3.1.7 highlight key trends in the behavioral response of cross-frame systems, the Fatigue Loading Study objectives can only be met by evalu- ating the response under realistic traffic loads. This section focuses on the approach used for current AASHTO design loads, whereas Section 2.3.3 focuses on the use of recent WIM records. Results of these studies are subsequently presented in Chapter 3. Implementing AASHTO fatigue design criteria involves an understanding of both the load response and the corresponding resistance of a given detail. In terms of the organization of this section, the methodology used to apply AASHTO fatigue loading on the influence-surface results is first outlined. Load-induced fatigue resistance is covered later in the subsection. To consider realistic traffic conditions on the numerous influence-surface results obtained, an Excel program was developed to perform a series of bi-linear interpolation calculations. Thus, truck passages were simulated over the various influence-surface plots to develop influence-line plots, for which the axial-force response of a cross-frame due to the passing truck is illustrated. AASHTO Article 3.6.1.3.4 states that “a single design truck shall be positioned transversely and longitudinally to maximize stress range at the detail under consideration, regardless of the position of traffic or design lanes on the deck.” To satisfy this requirement, a single AASHTO fatigue truck is analytically traversed along the entire length of the influence-surface in a speci- fied lane (i.e., the transverse position was held fixed for the entire longitudinal passage, such that “zig-zag” loads were not considered). Different lane passages were considered by repeating the process in 1-foot transverse increments in both the forward and backward directions (i.e., truck traversing upstation and downstation). At deck overhangs, a 1-foot clear distance between the centerline of the outermost wheel line and the inside face of the barrier was maintained. Although consideration of trucks driving in close proximity to the barrier (within 1 foot) is not required (particularly for fatigue evaluation) because it is an infrequent occurrence in most cases, these extreme loading conditions were considered in the study for completeness. An example of this procedure is presented in Figure 2-6. The sample influence-surface plot corresponds to a cross-frame diagonal near the intermediate support of a heavily skewed bridge (i.e., one of the 4,104 bridges studied). The AASHTO fatigue truck was analytically traversed across the influence surface in the upstation direction for three distinct lane positions: left wheel line 1-foot from the left barrier, centerline of bridge, and right wheel line 1-foot from the right barrier. Note that in the actual analyses of this particular influence surface, this process was completed many more times to consider all possible lane passages between the barriers. The x-axis of the influence-line plots corresponds to the longitudinal position of the front truck axle relative to the coordinate system established in the color contour plot. Therefore, at skewed ends, the truck is introduced to the deck surface at different x-coordinates, as is reflected in the figures. As presented in Figure 2-6, the axial-force response of the cross-frame varies significantly for each truck passage. When traversing the bridge along the left barrier, the cross-frame member of interest experiences a force reversal, resulting in a substantial force cycle (11.8 kips). For the fatigue truck passing along the right barrier, a large force reversal is also observed, except in the opposite order (9.4 kips). For the truck passing along the centerline of the bridge, the force

Research Approach 29 magnitudes are smaller, but the response is more complex. Whereas the other two truck passages resulted in one primary force cycle, the centerline passage results in additional secondary cycles of lesser magnitude. The primary cycle (3.7 kips) and the two secondary cycles (0.5 kips and 0.2 kips) are illustrated in the figure. Given that the permanent stress states in the cross-frames of these representative bridges are unknown (e.g., locked-in fit-up forces, dead loads, residual stresses), only live load force/stress cycles entirely in tension or subject to reversal, regardless of how small the tension component is, were considered. It is recognized that, in actuality, this assumption would only be valid when the Fatigue I factored tensile stress component exceeds the compressive stress due to unfactored permanent loads or the permanent loads are tensile in nature. To simplify the analysis, the tensile force component was always considered to be large enough to propagate a crack. Thus, truck passages that induce a purely compressive force/stress cycle with no tensile force component are disregarded from the evaluation, as they are not a fatigue-sensitive loading condition. Also note that, since the primary focus of this study is related to the fatigue limit -10 -5 0 5 10 -30 20 70 120 170 220 270 320 370 420 )spik( ecroF laix A Longitudinal Position of Truck (ft) -10 -5 0 5 10 -30 20 70 120 170 220 270 320 370 420 )spik( ecroF laix A Longitudinal Position of Truck (ft) -10 -5 0 5 10 -30 20 70 120 170 220 270 320 370 420 )spik( ecroF laix A Longitudinal Position of Truck (ft) -3 0 20 70 12 0 17 0 22 0 27 0 32 0 37 0 42 0 Lo ng itu di na l P os iti on (f t) Transverse Position (ft) -17 170 11.8 kips 9.4 kips 3.7 kips 0.2 kips 0.5 kips Figure 2-6. Sample data showing the development of influence-line plots from an influence surface.

30 Proposed Modification to AASHTO Cross-Frame Analysis and Design state, permanent (dead) loads and locked-in stresses are not explicitly addressed herein, except as described above. The computationally intensive procedure introduced in Figure 2-6 was performed for all influence-surface plots obtained from the 4,104 bridge models analyzed in Abaqus, which resulted in over 3 million influence-line plots similar to the three examples presented in the figure. A script was developed to automate these moving-load simulations. The script was designed to perform rainflow counting on the various influence-line plots (i.e., axial-force time-histories), and the following output was recorded for each cross-frame member evaluated: (i) the transverse position of the AASHTO fatigue truck that maximizes the force range, (ii) the magnitude of the primary force cycle caused by the critical lane passage, and (iii) the number and magnitude of any secondary cycles caused by that same critical lane passage. The maximum force range for a given cross-frame member represents the unfactored design force for which the engineer evaluates the Fatigue I or II limit state. Note that many of the results presented in Chapter 3 are based on the unfactored design force ranges obtained from this specified procedure. The unfactored fatigue force ranges were then factored in accordance with current 9th Edition AASHTO LRFD load factors (as well as the cross-frame-specific load factors proposed and outlined in Section 3.2) and converted into an axial stress with consideration of shear lag effects [i.e., the U factor specified in AASHTO Table 6.6.1.2.3-1 (Condition 7.2)]. Dynamic load allowance (1.15 for fatigue limit state) was also subsequently applied to produce factored axial- stress ranges for fatigue design and evaluation. The resistance side of the AASHTO fatigue design criteria (i.e., load-induced fatigue) was then introduced into the computational studies. Based on the guidance provided in Article 6.6.1.2.3, it was assumed that the critical Category E′ welded cross-frame details were all governed by the finite-life calculations and Fatigue II limit state, which is a reasonable assumption for Category E′ details and practical traffic conditions. Given that finite fatigue life is inherently a function of both stress magnitude and frequency of load occurrence, the anticipated number of stress cycle counts over the service life of these “fictional” bridges must be considered in some capacity. The 4,104 bridges evaluated in this study, however, are not representative of any particular location or traffic conditions. As such, the research team elected to investigate different repre- sentative traffic conditions to bound the problem. In other words, these “fictional” bridges were effectively constructed along different highway corridors in Texas, both in rural areas with low traffic volumes and congested urban areas with high traffic volumes. Rather than utilize measured WIM data (Section 2.3.3) at this stage, the research team utilized ADT maps readily available on the TxDOT website (2020), as well as simplifying assumptions recommended by AASHTO Article C3.6.1.4.2. A state highway system in rural Llano, Texas, and a heavily trafficked corridor in Houston, Texas, were selected as the extreme conditions, but the general process would be identical for other highway systems in other states. Although not outlined in the report herein, AASHTO LRFD guidance was followed to obtain an estimate of realistic single lane, average daily truck traffic (ADTTSL) values for the purposes of computing the Fatigue II resistance stress ranges for the two extreme traffic conditions considered. A detailed overview of these calculations is provided in Appendix F. As noted previously in this section, the design stress ranges and resistances (as computed by the procedures outlined above) are used throughout Chapter 3 to establish generalized observations about load-induced cross-frame behavior in composite systems. More specifically, these analytical data are processed and investigated in several different ways, as listed below.

Research Approach 31 Note that many of these concepts were previously introduced in Section 1.2 as the major questions to be answered in NCHRP Project 12-113. • Maximum stress ranges are compared for the various bridge geometries, such that the impact of each parameter can be evaluated with respect to cross-frame behavior. • Maximum stress ranges are compared for the various cross-frame members, such that generalized observations about which members typically govern fatigue design can be made. • The governing lane positions corresponding to those maximum stress ranges are compiled for each representative bridge, such that the AASHTO fatigue loading model can be assessed in terms of truck positioning. • Factored stress ranges (based on current AASHTO LRFD fatigue loading criteria) are compared to factored resistances, such that the overall efficiency of cross-frame design can be assessed. • Factored stress ranges, determined by analytical methods, are compared to effective and maximum stress ranges obtained from WIM records and the field experiments, such that the accuracy and appropriateness of the current Fatigue I and II load criteria for application to cross-frames can be assessed. 2.3.3 WIM Records While Section 2.3.2 discussed the use of the AASHTO design load (i.e., the fatigue design truck) and its design implications related to cross-frame force effects, this section discusses the application of measured WIM data obtained from sites throughout the United States to a subset of the 4,104-model data set. These analyses provided the opportunity to understand the behavior of critical cross-frames for a representative set of bridge systems subjected to a variety of realistic traffic conditions. This study of WIM data is intended to provide an indication on the appropriateness of the current AASHTO LRFD Fatigue I and II load factors for the design of cross-frames, as these load factors were developed primarily for girder response to truck traffic per Modjeski and Masters (2015). In addition, this study investigates whether it is necessary to consider multiple presence effects in the fatigue evaluation of cross-frames. The research team obtained high-resolution WIM records from the Federal Highway Administration (FHWA) for 16 specific pavement study (SPS) sites across the United States. The records, collected in 2014, generally include a full year’s worth of measurements. The record timestamps are reported with 0.01-second measurement resolution. In total, the unfiltered records include approximately 46 million vehicle records from 16 sites over 15 states. Since some sites have multiple lanes, the records include 23 one-lane records. Consistent with SHRP 2 Project R19B (Modjeski and Masters 2015), the research team applied a set of filtering techniques to this data set in an attempt to eliminate questionable records (i.e., unrealistic geometry or erroneous data), apparent permit vehicles or illegally loaded vehicles, and lightweight vehicles [i.e., gross vehicle weights (GVWs) less than 20 kips]. Many of these filters were based on NCHRP Research Report 683 (Sivakumar, Ghosn and Moses 2011). Appendix F summarizes these filters. The lightweight vehicle entries were elimi- nated, since previous research has indicated that these vehicles have a negligible effect on the accumulated fatigue damage in a member or detail (Connor and Fisher 2006). White (2020) conducted multiple sensitivity studies using the same records studied by the research team, in which the lightweight vehicles were retained in order to study the impact on force effects in cross-frames. The results of these studies indicated the inclusion of lightweight vehicle records contributed negligibly to the accumulation of fatigue damage. Table 2-3 provides a summary of WIM records prior to any filtering, as well as the total number of vehicle records and ADT counts for each SPS site before and after removing lightweight

32 Proposed Modification to AASHTO Cross-Frame Analysis and Design vehicles. The records used for the fatigue study contain approximately 11 million truck measure- ments after all appropriate filters are applied. Figure 2-7 shows the cumulative distribution functions (CDFs) of GVWs captured by the WIM sensors for all SPS sites. The GVWs are shown on a normal probability plot, in which the horizontal axis is GVW (in kips), and the vertical axis is the standard normal variable (i.e., axis values are “Z-values” indicating the number of standard deviations the GVW value is from the mean of the distribution). A normal probability plot can be used to determine how well the data represents a normal distribution; nonlinear data sets indicate departures from normality. As clearly shown, the GVW populations are not normally distributed. This likely is the result of natural groupings of different vehicle types and payload. The mean GVWs for the 11 million truck records range from 40 to 62 kips, with a maximum GVW of 220 kips. As documented in Appendix F, the shape of the CDFs appears to be generally consistent with the CDFs of WIM data used in the SHRP 2 R19B project (Modjeski and Masters 2015), which were recreated by the research team with the same filters described above. Thus, the 2014 data obtained from FHWA that were used on this project were deemed acceptable given their good agreement with the SHRP 2 R19B data. State Initial Number of Records (Before Filtering) Records Including Light Vehicles Records After Removing Light Vehicles Total Number of Records Lane ADT Total Number of Truck Records Lane ADTT AR 3,529,952 3,414,934 9,356 1,704,481 4,670 AZ 2,711,532 2,626,954 7,704 1,227,567 3,600 CA 4,873,640 4,779,602 13,167 1,380,075 3,802 CO 1,675,744 1,645,722 4,509 352,198 965 IL 2,807,183 2,707,469 7,584 798,935 2,238 IN (Lane 1) 1,886,428 1,865,543 5,111 370,241 1,014 IN (Lane 2) 500,621 471,291 1,302 21,340 59 IN (Lane 3) 1,843,395 1,802,053 4,937 360,458 988 IN (Lane 4) 548,382 524,003 1,460 20,158 56 KS 2,312,975 2,262,526 6,199 436,913 1,197 LA 1,734,519 1,722,311 4,719 76,547 210 MD 3,040,831 3,005,933 8,564 108,881 310 MN 864,803 857,522 2,349 52,757 145 NM1 1,018,250 1,005,887 2,786 147,077 407 NM2 1,675,090 1,609,947 4,485 892,295 2,486 PA 2,292,235 2,251,316 8,187 873,903 3,178 TN (Lane 1) 2,792,715 2,120,750 7,913 550,858 2,055 TN (Lane 2) 1,945,926 1,842,295 6,874 118,000 440 TN (Lane 3) 2,042,591 1,942,114 7,247 191,330 714 TN (Lane 4) 2,757,569 2,690,197 10,038 1,182,136 4,411 VA (Lane 1) 1,462,016 1,445,614 3,961 224,928 616 VA (Lane 2) 438,126 430,438 1,179 19,300 53 WI 1,468,798 1,358,660 5,435 120,079 480 Total 46,223,321 44,383,081 11,230,457 Table 2-3. SPS sites from which WIM data was obtained by FHWA.

Research Approach 33 2.3.3.1 Application of WIM Records In order to estimate real live load force effects on cross-frame members, the research team applied the filtered WIM traffic streams to a subset of the analytical testing matrix discussed in Section 2.3.1.2. As the computational effort is significant (discussed in the next subsection), approximately 20 models were selected from the 4,104-model data set. While the models were somewhat arbitrarily selected, bridge models that would likely represent extreme cross-frame force effects based on preliminary studies were chosen. Note that these same representative bridges were subsequently evaluated in the parametric studies related to the R-Factor Study outlined in Section 2.4 and the Commercial Design Software Study outlined in Section 2.5. 2.3.3.2 Processing of WIM Data Using MATLAB, a script was developed to read, format, and filter the entire WIM record database. Since the axle tracks (i.e., the transverse distance between the centerline of two wheels on the same axle) are not recorded in the WIM records, the research team set all axle tracks to 6 feet and assumed that the weight of each axle is evenly distributed between the driver and passenger side wheels. The research team created scripts to perform the following basic load configuration routines: • Load Configuration 1 – Single Traffic Stream: This routine steps a stream of user-defined WIM traffic on a bridge deck along an influence surface at 1-foot longitudinal increments in any defined transverse position (also 1-foot increments). The script uses a cluster analysis to include the effects of groups of vehicles in the same traffic stream, provided that any wheel of the following truck is on the bridge during the time window in which the leading vehicle is still on the bridge. Time windows are calculated based on the respective vehicle speeds. 5 -5 4 3 2 1 0 -1 -2 -3 -4 0 50 100 150 200 250 GVW (kips) St an da rd N or m al V ar ia bl e Figure 2-7. CDF of GVWs from FHWA 2014 data (excluding light vehicles).

34 Proposed Modification to AASHTO Cross-Frame Analysis and Design • Load Configuration 2 – Two Traffic Streams: Similar to Load Configuration 1, a WIM traffic stream is stepped along a bridge deck in a defined transverse position. The script uses a cluster analysis to include the effects of groups of vehicles in any adjacent, user-defined transverse position, provided any tire on the following truck is on the bridge during the time window in which the leading vehicle is still on the bridge. Time windows are calculated based on the respective vehicle speeds. • Load Configuration 3 – Realistic Meandering Traffic Stream: This routine systemically steps a stream of WIM traffic along a defined influence surface at 1-foot longitudinal increments in any defined 12-foot-wide lane position. The routine randomly selects a transverse position (within the 12-foot-wide lane) for each vehicle record based on an assumed distribution, such that effects of lane meandering are considered due to cross-frames being highly sensitive to transverse position. This script uses a cluster analysis to include the effects of groups of vehicles, provided any of a following vehicle tires are on the bridge during the time window that a leading vehicle is still on the bridge. Time windows are calculated based on the respec- tive vehicle speeds. Recalling that the influence-surface output from Abaqus provides results that are relatively sparsely gridded (Section 2.3.1.4), the scripts use bi-linear interpolation methods to re-mesh the influence surfaces to a 1-foot by 1-foot grid. This procedure is similar to that described in Section 2.3.1 for the single AASHTO fatigue truck. The output of each load configuration routine above includes the following: • A sample load event history (see Figure 2-8); • The total number of stress cycles (using rainflow-counting techniques discussed in Appendix F); • The average number of cycles per passage; • The maximum stress and stress cycle recorded; • The equivalent stress range using the Palmgren-Miner damage accumulation model (i.e., stress range corresponding to the Fatigue II limit state); and • The lowest stress range of the top 0.01% of all stress ranges (i.e., the 99.99th percentile criteria corresponding to the Fatigue I limit state). Past research has indicated that eliminating smaller stress cycles of a variable-amplitude loading source has negligible effect on the damage accumulated in a fatigue detail. Connor and Fisher (2006) showed that stress cycle magnitudes less than 25% of the constant ampli- tude fatigue limit (CAFL) generally have little impact on the long-term fatigue performance, 2.0 0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 2000 2200 2400 2600 2800 3000 3200 Load Step (1-ft Increment) C ro ss -F ra m e A xi al S tr es s ( ks i) Figure 2-8. Sample load event history showing three vehicles back to back.

Research Approach 35 which is outlined in Appendix E. To study the effects of this lower stress range truncation on cross-frames, the research team compiled the results of each loading iteration (as listed above) for two different conditions: (i) including the effects of stress magnitudes less than 25% of the CAFL and (ii) filtering out those effects. For the purposes of this study, a detail Category E′ is assumed based on the results of McDonald and Frank (2009) and AASHTO LRFD Table 6.6.1.2.3-1 (Condition 7.2). The corresponding CAFL value for this detail is 2.6 ksi; thus, all stress cycles less than 0.65 ksi are truncated when the filter is applied. For the remainder of this report, the data discussed refers to the unfiltered data (i.e., no stress ranges are truncated). A more detailed discussion of the effects of stress truncation is provided in Appendix F. The routines discussed in the previous section systematically and sequentially apply a defined traffic stream in a specific load configuration over the influence surface corresponding to one cross-frame member in one of the 20 representative bridges. Considering the computational time for each iteration (depending on the total number of vehicle entries and bridge length) the number of potential iterations becomes unmanageable. For this reason, the studies presented in this report were obtained by positioning the WIM traffic stream in a realistic drive lane, rather than a worst-case drive lane. The worst lane position is often when the truck is positioned at the edge of the deck (e.g., the right tire of a truck is positioned along the deck edge where a traffic barrier would be placed). As a worst-case lane position would lead to unnecessary conservatism, the research team opted for defining a realistic lane position based on the actual bridge width and recommendations from AASHTO Article 3.6.1.1.1. As illustrated in Figure 2-9, this realistic truck position does not produce the largest tensile stress range for the selected cross-frame; however, this position would be consistent with realistic loading and lane striping on the bridge (and thus consistent with the intent of the fatigue limit state). With that in mind, the research team deemed Loading Configuration 3 introduced above to be more pertinent with respect to fatigue design. As such, the focus of the study presented herein is on realistic traffic streams as opposed to considering all possible lane positions (i.e., overhang loads). Figure 2-9. Illustration of realistic truck position in drive lane.

36 Proposed Modification to AASHTO Cross-Frame Analysis and Design For the actual location of each 6-foot vehicle track width (i.e., transverse distance between left and right wheel lines) within the realistic lane, the research team assumed a distribution for which the vehicle is located in the center of the lane 55% of the time; 30% of the time, the vehicle is located plus or minus 1 foot of the lane centerline; 10% of the time, the vehicle is located plus or minus 2 feet of the lane centerline; and 5% of the time, the vehicle is riding along one of the lane edges. This is illustrated in Figure 2-10 for a sample 30-foot wide bridge, although the procedure is similar for different bridge widths and lane configurations. The intent of this assumed distribution, although not based on past precedents or measurements, is to consider the inherent variability in driving ability and consistency. 2.3.3.3 WIM Drive Lane Truck Traffic Positioned in Realistic Drive Lanes The traffic streams discussed herein exclude vehicles with GVW less than 20 kips and utilized a drive lane defined by AASHTO Article 3.6.1.1.1, along with an assumed vehicle position distribution discussed in Section 2.3.3.2. For Fatigue I, the maximum stress ranges produced by the WIM traffic streams were normal- ized to the maximum stress range produced by applying the unfactored AASHTO design load (i.e., HS-20) to the same bridge in all possible transverse positions within the clear distance of the barriers. An impact factor was not included since the WIM stations attempt to correct for dynamic effects and relate measured drive-by weights to static weights. Note that it may be prudent to include a portion of the typical 0.15 impact factor (AASHTO Article 3.6.2.1) in subsequent studies, since the effects of bridge dynamics are not accounted for explicitly in this FEA. The governing Fatigue I maximum stress ranges calculated for each individual WIM site and bridge combination were determined based on a 99.99th percentile criteria. This stress range corresponds to the lowest magnitude of the top 0.01% of all stress ranges recorded (for the governing cross-frame for each bridge given the defined lane position). In other words, one critical cross-frame produced WIM stress ranges that exceeded the remainder of the cross- frames in the bridge and thus served as the governing case. Similarly, the governing Fatigue II stress ranges calculated for each WIM site and bridge combination were determined based on the maximum effective stress range. Recall that the Fatigue II limit state is intended to represent the average effect of the traffic spectrum. These effects are generally characterized by the effective stress range (or equivalent stress range) of the variable-amplitude response. Appendix F discusses how the effective stress range is Figure 2-10. Illustration showing distribution of vehicle transverse location within 12-foot design lane for a 30-foot wide bridge.

Research Approach 37 obtained from a variable-amplitude spectrum via rainflow-counting techniques and the use of Palmgren-Miner’s rule. In order to compare the WIM force effects to the unfactored AASHTO design truck force effects, it is more appropriate to compare the accumulated fatigue damage, rather than just the effective stress ranges. The accumulated fatigue damage metric inherently considers both the variable stress range magnitudes and the number of cycles. Thus, the total damage accumulated by the various WIM streams on the critical, governing cross-frame members is compared directly to the damage caused by the AASHTO fatigue truck. 2.3.3.4 Reliability Study Using Proposed Load Factors A reliability study was performed to investigate the implications of using the load statistics developed in Section 2.3.3.3 for fatigue of cross-frames in steel I-girder bridges in the context of a reliability-based design. Using the statistical characterizations of the total load effect, it is possible to compare this to the statistical characterization of fatigue resistance provided in the literature (Modjeski and Masters 2015). Stochastic models can be implemented that utilize these characterizations of load and resistance to provide an indication of reliability, given a specific choice of design parameters. The reliability study is performed via Monte Carlo simulation, in which values of load and resistance are obtained through a randomly determined process using distribution parameters for load and resistance. For this reliability study, the research team adopted the resistance models developed in the SHRP 2 R19B report (Modjeski and Masters 2015). Appendix F provides the step-by-step procedure used in applying the Monte Carlo simulation. Results of the Monte Carlo simulations are subsequently summarized in Section 3.2.2.3. 2.3.3.5 Multiple Presence Study The WIM data set used in this study and outlined in Section 2.3.3 includes multi-lane records for three sites. As two of the sites (Indiana US-31 and Tennessee IH-40) include two lanes of traffic in two directions, the multiple presence data set includes a total of five 2-lane records. In order to determine if a passing truck is in the same or an adjacent lane as another truck, it is necessary that the time stamps have high resolution. For example, many available WIM records have a time resolution of 1 second. At 70 miles per hour, a time difference of plus or minus a half-second is equivalent to a range in truck position of approximately plus or minus 50 feet, making assessment of multiple presence for cross-frame fatigue highly unreliable. Higher reso- lution data is essential to evaluate the frequency of occurrence of multiple presence. At 70 miles per hour, the 0.01 second resolution data that the research team obtained from the FHWA generally provides a resolution of plus or minus 0.5 feet. With the availability of this higher-resolution, multi-lane data, the team performed a study on the WIM records to better understand the statistical parameters surrounding multiple presence. This was performed using a cluster analysis to consider if a bridge may be loaded with other simultaneous truck traffic during a primary drive lane load event (i.e., passage of one or more axles of a vehicle). The cluster analysis is performed based on the time stamps of the individual truck events, the lengths of the individual trucks, and the speed of the individual trucks. Since bridge lengths vary, the following study incorporates multiple presence load events that occur within a plus or minus 1,000-foot window of the primary drive lane event. The research team developed a script to determine how many times a second truck is in a lane adjacent to the primary drive lane truck (i.e., a second truck is passing or is being passed by the truck in the drive lane, anywhere along an arbitrary length of 1,000 feet). This scenario is illustrated schematically in Figure 2-11. Clear distances are measured from the rear axle of the drive lane truck to the front axle of the passing lane truck. Positive values indicate front axles of the passing truck are “behind” the rear axles of the drive-lane truck, and negative values

38 Proposed Modification to AASHTO Cross-Frame Analysis and Design indicate the front axles of the passing truck are “ahead of ” the rear axles of the drive-lane truck. This provides a smooth, continuous function of clear distances, with increasing clear distances (positive or negative) indicating a larger separation between vehicles. Note that a clear distance of zero corresponds to a staggered configuration, which was deemed critical to cross-frame force effects in the 7th Edition AASHTO LRFD Specifications. This provision was removed in the 2016 Interims to the 7th Edition Specifications, citing too infrequent occurrences. This study herein evaluates the frequency of occurrence in the context of cross-frame response. Results of the multiple presence study are subsequently summarized in Section 3.2.3. 2.4 R-Factor Study (3D Analysis) The R-Factor Study, as noted previously, addresses Objective (c) of this project (Section 1.2): to provide quantitatively based guidance on the influence of end connections on cross-frame member stiffness. With that in mind, the primary focus of this study is the 3D modeling techniques commonly employed by commercial design software programs. While the term “3D models” are often synonymous with highly accurate representations of the framing system, the level of detail in modeling the cross-frames can have a dramatic impact on the accuracy of the brace forces estimated. Commercial 3D programs generally simplify cross-frame members and their connections as pin-ended truss elements, thereby misrepresenting the stiffness of the panel and potentially producing unreliable design force estimates. As noted in Section 1.1.3, significant research has been conducted over the past decade to approximate the softening effects of the eccentric load path through unsymmetrical cross- frame members (Wang 2013; Battistini et al. 2016). Although the previous work developed equations for evaluating the reduction in stiffness, the recommendations incorporated into the design provisions consist of a fixed stiffness modification factor of 0.65 (AASHTO LRFD Article C4.6.3.3.4). However, these past studies were primarily limited to the response of cross-frames in noncomposite systems during construction. As such, the stiffness response of cross-frames in composite in-service bridges has not been properly investigated. This section provides an overview on the computational studies performed to evaluate the limitations of the traditional Drive Lane Truck Passing Truck Drive Lane Truck Passing Truck Positive Clear Distance Negative Clear Distance Figure 2-11. Illustration of an adjacent lanes (truck-passing-truck) scenario; negative and positive clear distances indicate the front axle of the passing truck is ahead or behind the rear axle of the drive lane truck, respectively.

Research Approach 39 truss-element modeling approach for cross-frames, appropriate stiffness modifications for composite conditions, and a proposed alternative approach to model cross-frames in refined 3D analyses (i.e., an eccentric-beam model). Section 2.4.1 outlines the general load-induced behavior of cross-frames in composite sys- tems to provide context to the analysis procedures and results presented herein. Section 2.4.2 highlights key aspects of the stiffness modification approach, and Section 2.4.3 outlines the proposed eccentric-beam modeling approach. Lastly, Section 2.4.4 summarizes the parametric studies performed to evaluate each of these methodologies. 2.4.1 General Load-Induced Cross-Frame Behavior Prior to examining the various 3D modeling techniques, it is important to understand how cross-frames deform under applied live loads in composite systems. By comparing “real” deformations with the deformations assumed in common 3D analyses, the remaining discus- sions of this section can be contextualized. Intuitively, the relative movement of the corners of a cross-frame dictates the axial-force demands in each member. During this study, the observation was made that any load-induced deformed shape in a composite system is a combination of the following responses between adjacent girders framing into a common cross-frame: (i) equal rotation, (ii) differential vertical displacement, and (iii) differential rotation. These deformations are presented schematically in Figure 2-12. The cross-frame also undergoes rigid-body motion as the entire superstructure displaces under applied load. However, rigid-body motion does not induce strains and stresses on cross-frame members and is therefore not relevant to the discussion herein. The previous work reported in Battistini et al. (2016) and Wang (2013) focused only on the case of equal rotation, which resulted in the fixed reduction factor of 0.65 recommended in AASHTO LRFD Article C4.6.3.3.4. Even in Figure 2-12, the deformation of the cross-frame members is idealized (i.e., only axial deformations are represented). In reality, the entire panel (members, gusset plates, and connection plates) undergoes some degree of in-plane and out-of-plane rotation under an applied load. Consequently, cross-frame members experience biaxial bending stresses in addi- tion to axial stresses. A schematic of this behavior is presented in Figure 2-13 for a scenario in which the girders sustain purely differential vertical displacements. A cross-sectional view and a plan view are provided to examine the in-plane and out-of-plane behavior, respectively. The column depicting the shell-element models indicates that FEA models comprised of shell elements accurately capture these rotational effects, compared to the column depicting truss-element models. Equal Rotation Differential Vertical Displacement Differential Rotation θx θx θy θy θx1 θx2 Figure 2-12. Types of cross-frame deformation caused by relative movement of adjacent girders.

40 Proposed Modification to AASHTO Cross-Frame Analysis and Design Figure 2-13 divides the deformations into in-plane and out-of-plane deformations. The in-plane rotations depicted in Figure 2-13 are attributed to the in-plane rigidity typically provided by the connection and gusset plates. In order to accommodate the girder displacements and compatibility of the system, the cross-frame members have to flex given the rotational restraint provided by the end connections. Given that the in-plane rotational stiffness of the connection plate and welds is generally large (i.e., full-depth web stiffener welded along three sides), this behavior is likely observed in all practical steel I-girder systems in service. This behavior, however, is typically ignored in the truss-element modeling approach. Out-of-plane rotations, on the other hand, are attributed to the eccentric nature of the connec- tions. As internal force is transmitted from the girder webs into the cross-frame members, the load must pass several “eccentric jumps.” These “jumps” of potentially several inches in magnitude can have a considerable impact on the stiffness response of cross-frame members and systems. The stiffness modification factors (R-factors), detailed in the next subsection, were primarily developed to account for this modeling oversight in truss-element models. 2.4.2 Stiffness Modification Approach As indicated above, most 3D commercial design software programs implement a truss- element approach for modeling cross-frames. Rather than explicitly modeling each component and the load eccentricity of the cross-frame with shell elements, commercial design software programs typically simplify them as pin-ended truss elements capable of only resisting axial forces. It should be noted that even 3D models utilized in many past research investigations have idealized the cross-frames using truss elements. In this simplified approach, the eccentric load path and the stiffness contributions of the gusset and connection plates are neglected, as are the biaxial bending stresses induced. In general terms, there are three inherent shortcomings with this modeling approach when compared to realistic cross-frame systems or a shell-element modeling approach for cross-frames (two of which were illustrated in Figure 2-13). These shortcomings are summarized below: • The truss-element modeling approach neglects the out-of-plane bending effects caused by the eccentric end connections. At the most fundamental level, the eccentric load path in an unsymmetrical cross-frame member is introduced by the connections. The stiffness modi- fication factors established by Battistini et al. (2016) were directly developed to account View Shell-Element Models Truss-Element Models In- Plane Out-of- Plane Figure 2-13. Schematic showing the in-plane and out-of-plane flexural deformations commonly observed in shell-element models compared to simplified truss-element models.

Research Approach 41 for the softening effect of eccentric end connections (i.e., the R-factor is often assigned to reduce the cross-sectional area of the truss element or the elastic modulus of the material). As such, this aspect is a well-researched behavior. • The truss-element modeling approach neglects the in-plane bending effects caused by the connec- tion rigidity. Unlike the out-of-plane effects, the stiffness modification factors formulated by Battistini et al. (2016) did not explicitly address these effects. Instead, in-plane rotational behavior is implicitly considered in those previous studies. • The truss-element modeling approach neglects the axial stiffness of the connection and gusset plates in favor of the cross-frame member axial rigidity. In other words, truss-element models replace the stiffer connection and gusset plates with an additional length of an often, more flexible cross-frame member. Again, this behavior was not explicitly considered in the development of the R-factors but was implicitly considered in the corresponding laboratory and analytical studies. To expand on the previous work, two generalized solutions were explored in NCHRP Project 12-113. First, stiffness modification factors similar to those derived by Battistini et al. (2016) were analytically developed in the context of composite bridge systems. To accomplish this, a series of 3D analyses in Abaqus and approximate hand calculations were conducted at three different model-scale conditions: (i) member-level, (ii) panel-level, and (iii) system-level studies. The second generalized solution is a proposed alternative to the truss-element modeling approach that makes use of eccentrically loaded beam elements for the cross-frame members. These procedures are briefly outlined herein. 2.4.3 Proposed Eccentric-Beam Approach As noted previously, the truss-element modeling approach has three major limitations with regard to representing the in-plane flexural, out-of-plane flexural, and axial rigidity of cross- frame systems. The stiffness modification (R-factor) approach corrects for those limitations by approximately adjusting the axial rigidity of a pin-ended truss element such that the overall cross-frame stiffness in the analysis model reasonably matches its actual stiffness. As with many specifications and design guides around the world though, it is often beneficial to provide designers with multiple solutions to a problem with varying levels of refinement. As an alternative, the research team sought an approach more refined than using pin- ended truss elements, yet simpler to employ than the shell-element approach. The proposed eccentric-beam modeling approach addresses the major limitations of the truss-element approach, similar to assigning R-factors but in a more explicit and less circuitous manner. The proposed methodology represents the cross-frame members (and connection and gusset plates) as beam elements with flexural degrees of freedom. Note that the eccentric-beam approach does not necessarily correspond to one specific technique but rather encompasses a wide range of modeling assumptions with varying levels of accuracy and modeling simplicity, as demonstrated below. The eccentric-beam approach is shown schematically in Figure 2-14 along with a representa- tion of the shell-element and truss-element approaches for cross-frames in 3D models. Unlike the truss-element approach, in which the cross-frame stiffness is represented as only the axial rigidity of the cross-frame member modified by an R-factor (i.e., RAa), the eccentric-beam model is represented by a larger set of parameters, including but not limited to y–, the out-of-plane eccentricity, and Aa, the cross-sectional area of the cross-frame member (not modified by R). The biggest challenge with respect to the eccentric-beam approach is deciding how/if to handle the axial and flexural stiffness of the connection plate, the gusset plate, and the over- lapped portions. As such, three different variations of the approach were explored, ranging

42 Proposed Modification to AASHTO Cross-Frame Analysis and Design from more refined to relatively simple. Each variation (i.e., the refined, simplified, and angle- only models) is presented schematically in Figure 2-15. This figure schematically depicts these different modeling methods for a sample cross-frame panel. Note that the “eccentric jumps” only apply to the out-of-plane direction, as the behavior is assumed concentric in the in-plane axis. To illustrate some of the assumptions used in the development of these models, an abbre- viated set of notes is provided below. This discussion specifically highlights key aspects of the “refined” model, as defined by Figure 2-15, but are also directly applicable to the “simplified” and “angle-only” approaches, where appropriate. For a more detailed overview, as well as sample calculations demonstrating the proposed methodologies, refer to Appendix F. • In the “refined” model, the cross-frame member, gusset plate, connection plate, and individual overlapped sections are treated independently when developing the eccentric- beam model. 8″ 4″ 1″ 7¼″ L4x4x3/8 9/16″ PL. ½″ PL. 16¼″ 96″ 60″ Girder web shell (typ.) L4x4x3/8 Overlap 2 Gusset plate Overlap 1 Connection plate Refined Model L4x4x3/8 Connection plate Simplified Model L4x4x3/8 L4x4x3/8 Angle-Only Model Pinned or fixed Note: Out-of-plane view shown; all “offset” links are rigid elements Figure 2-15. Sample cross-frame panel and the corresponding eccentric-beam models. Shell-Element Pin-Ended Truss-Element with R-Factor Eccentric-Beam RAa A y a – Figure 2-14. Various cross-frame modeling approaches considered in system-level studies.

Research Approach 43 • The out-of-plane “jumps” in the neutral axis, or the eccentricities, are represented by rigid offsets, which imply that the welded or bolted connections between the plates and angles perfectly constrain these elements together (i.e., no slip is assumed in the bolted connections). • The lengths of the rigid elements are based on the distances between the neutral axes of the connected components. • The length of each individual beam component is assigned assuming there are no shear lag effects. In other words, the length of the angle member starts precisely at the termination of the gusset plate and ends at the gusset plate edge on the opposite side of the angle. • Section properties (i.e., A, out-of-plane I, in-plane I, and length L) of the connection and gusset plates are based on a Whitmore approach. Section properties of the equivalent overlapped portions are taken from an assumed composite section. A detailed calculation demonstrating this process is provided in Appendix F. • Unlike the pin-ended truss models, rotations are not released at both ends of the eccentric- beam model. The distortional stiffness of the girder web is assumed substantial enough to warrant in-plane and out-of-plane rotational restraint of the cross-frame system. • In X-type cross-frames, the diagonal members are connected on opposite faces of the cor- responding gusset plates to avoid interference at the crossover point. These different eccentric effects were considered accordingly. • Although not shown here, a similar procedure was also used to develop equivalent beam properties for K-type cross-frames. This proposed 3D modeling technique was evaluated for a variety of bridge geometries, as outlined in the next subsection. 2.4.4 Parametric Study Overview To evaluate the various simplified 3D cross-frame modeling techniques, three different model-scale conditions were identified and studied including: (i) member-level, (ii) panel-level, and (iii) system-level studies. The work of each successive study built on the previous one. Member-level studies computationally investigated the fundamental behavior of axial-loaded sections with eccentric end connections, similar to the experimental approach by McDonald and Frank (2009). Panel-level studies analyzed isolated cross-frame panels consisting of various geometries, as well as various gusset and connection plates. The objective of the system- level study was to demonstrate how the behavior of cross-frames (i.e., load-induced force demands) is affected when serving as a small part of a larger composite system. A parametric study consisting of full 3D bridge models was conducted to assess the accuracy of the stiffness modification factors, as well as the eccentric-beam approach. To elaborate on these studies, two subsections are provided herein. An overview of the panel-level computational studies is provided in Section 2.4.4.1, while the system-level studies are discussed in Section 2.4.4.2. Note that the member-level studies are not explicitly addressed in this report, but the pertinent background and results are provided in Appendix F. 2.4.4.1 Panel-Level Studies (Noncomposite) The panel-level studies, as noted in the preceding section, examined the elastic stiffness response of individual cross-frame panels. First-order finite element models were developed to duplicate the laboratory experiments and analytical studies previously conducted by Wang (2013). However, the research team expanded on those past investigations by subjecting the panels to a variety of deformation patterns, as illustrated in Figure 2-12. Note that the previous studies, which focused on stability bracing applications, were limited to the equal rotation deformation pattern to simulate the buckling and torsional response of straight and horizontally curved noncomposite bridge girders during construction.

44 Proposed Modification to AASHTO Cross-Frame Analysis and Design In general, the behavior of a shell-element model is compared to the equivalent truss- element model. The ratio of the corresponding stiffness of a shell-element model to that of a truss-element model represents the R-factor. The shell-element model explicitly considers the reduction in stiffness due to eccentric end connections and the effects of additional in-plane rotational restraint, while the truss-element model considers only axial stiffness. The elastic stiffness of each model iteration was taken as the torsional stiffness of the entire cross-frame panel, or the rotational displacements due to an applied moment or twist on the entire panel. Prior to parametrically studying the effects of different cross-frame geometries, member sizes, and connection details, the research team conducted preliminary studies to verify the modeling approach and to achieve good agreement with measured results from past experi- mental tests. For instance, considerable effort was spent on connection restraints (i.e., how welded connections were represented computationally), which were ultimately found to have only a small influence on the cross-frame response. In the parametric studies, various cross-frame parameters were systematically evaluated through a series of FEAs. The variables considered in the study included girder spacing, cross-frame height, gusset/connection plate thickness, and angle sizes. The range of values considered for each parameter represented practical values as well as additional extreme condi- tions to bound the solutions. Note that the previous analytical studies by Wang (2013) and Battistini et al. (2013) did not explicitly consider the effects of connection plate thickness. A thicker connection plate implies two contrasting behaviors: a larger eccentricity but also a larger out-of-plane rotational stiffness. Therefore, this study examined how the stiffness modification factor is influenced by these two factors. In total, approximately 2,000 unique cross-frame geometries were developed as part of this parametric study. The general-purpose 3D FEA program Abaqus was used for the parametric studies. Refer to Appendix F for a detailed overview of the parameters considered. The researchers found that a key factor in the behavior of the panel-level analytical studies was related to the orientation of the cross-frame member, specifically the diagonals, relative to the gusset and connection plates. For X-type cross-frames, the diagonal members must be connected to opposite faces of the gusset plates to avoid interference at the crossover point. The position of the struts can also vary and depends on the standard DOT details and fabri- cator preference. Based on preliminary analyses, it was discovered that the orientation of the member connection had substantial impact on the stiffness response of the panel, depending on the deformation pattern and the distribution of forces. This is depicted schematically in Figure 2-16, which presents a detailed close-up of the top strut and diagonal connection for different loading conditions. In the right side loading condition, the top strut and the grey diagonal member are directly engaged and provide the primary resistance to the applied force. These two members are connected to the same face of the gusset plate. In this case, the corresponding eccentricity of this connection is additive with respect to the gusset-to-connection plate connection. Conversely in the left side loading condition, the top strut and the yellow diagonal member are directly engaged and provide resistance to the applied force. The diagonal in this scenario is fastened to the opposite face of the gusset plate as the top strut. Therefore, the individual member eccentricities tend to counterbalance each other. This, in turn, often results in a stiffer response by reducing the net eccentricity. This particular behavior is quantified in Section 3.3. 2.4.4.2 System-Level Studies (Composite) Whereas the panel-level studies evaluated the global stiffness of an isolated cross-frame, the system-level study assessed the accuracy of the simplified modeling approaches for 3D composite

Research Approach 45 bridge structures with various gusset plate thicknesses. More specifically, the sensitivity of the load-induced cross-frame response due to the assigned R-factor was examined. To make those assessments, a variety of analytical studies were conducted as part of the system-level study. A parametric study (consisting of only straight and normal bridges) was performed to emphasize the influence of connection and gusset plate dimensions on this behavior. Additionally, a second study was conducted to further investigate the effects of support skew and horizontal curvature on the R-factor and eccentric-beam approaches. For each individual study, the research team developed and compared multiple versions of the same 3D bridge model. Each iteration investigated a different approach to modeling the cross-frame elements, while the remainder of the 3D model, as outlined in Section 2.3.1, remained unchanged. In general, the rigorous shell-element model served as the benchmark model to which the rest of the iterations were compared. Truss-element models were also developed where the assigned R-factor was varied. Lastly, the proposed eccentric-beam modeling approach was also implemented for the various bridge models. Thus, for each individual study, the following model iterations were considered: • Shell-element model (representing the most accurate solution), • Truss-element model with different stiffness modification factors {R = 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, and • Eccentric-beam model. For the shell-element and eccentric-beam modeling approaches, different values of the gusset plate thickness were also considered as a variable. Note that the gusset plate thickness and the corresponding effects on the load eccentricity were explicitly considered in the shell-element model. In the eccentric-beam model, the eccentric offset dimension, as illustrated in Figure 2-15, and the equivalent beam properties were adjusted accordingly. Gusset plates were not explicitly represented in the truss-element models. Ultimately, the effectiveness of the simplified 3D model (i.e., the truss-element model with an assigned R-factor or the eccentric-beam model) is evaluated as a ratio between the predicted axial force in select cross-frame members from the simplified models to the predicted force in the same members from the shell-element model. This is described algebraically as Fsimplified/Fshell. Figure 2-16. Assumed orientation of cross-frame members for differential- rotation loading.

46 Proposed Modification to AASHTO Cross-Frame Analysis and Design Rather than compare the stiffness of the models indirectly, it is more appropriate to evaluate the ability of the simplified model to accurately obtain design forces, which is of most importance to designers. A ratio of unity represents perfect agreement between the shell-element model and the simplified models. This implies that the approximated stiffness is identical to the “true” stiffness of the panel. Values below unity indicate that either: (i) the assigned R-factor is likely too low for the truss-element model or (ii) the equivalent section properties assigned for the eccentric-beam model are too low (i.e., the cross-frame attracts less force). In other words, the simplified models underpredict the cross-frame force when compared to the more accurate shell-element model (i.e., unconservative estimates), and an increase in the modification factor or equivalent beam properties is needed. The opposite is true for force ratios above unity. In all cases, the applied load on these composite bridge systems represented fatigue design load conditions. Thus, the cross-frame force effects compared between the various modeling approaches were based on the same corresponding load cases (i.e., same truck configuration, weight, and lane position). For more information on the parameters considered and the perti- nent loading conditions, refer to Appendix F. The results of this computational study are sub- sequently summarized in Section 3.3. 2.5 Commercial Design Software Study (2D Analysis) As discussed extensively throughout this report, analysis refinement generally improves accuracy but at the cost of increased modeling complexity and computational effort. As such, many designers and commercial design software programs elect to use simplified methods to conduct refined analysis and obtain design forces. These simplified methods range from high-fidelity 3D FEA solutions, which were introduced in Section 2.4, to 1D line-girder analyses. One-dimensional line-girder analyses can be useful tools for girder analysis and design of simple structures but generally provide no information related to cross-frame behavior. With that in mind, outlining the methodology behind the 2D analysis methods prevalently used in the bridge design industry is the primary focus of this section [i.e., Objective (d) of NCHRP Project 12-113]. Two-dimensional analysis methods have been shown to produce relatively accurate results for girder forces (White et al. 2012), but that performance is less understood and quantified for cross-frame forces, particularly in composite bridge systems. In general, the Commercial Design Software Study examined the limitations of these common 2D methods with regards to cross-frame force effects in composite bridge systems through a series of analytical studies. As noted in Section 2.1, the research team polled bridge owners and consultants across the country on their preferred commercial design software programs. Based on these results, two different programs were identified for further investigation—one 2D program and one 3D program. Since the specific analysis techniques are of more importance to the scope of the project than the software brand itself, it was decided to conduct all studies related to this study in Software A, a widely used 3D analysis program. Although specifically developed and marketed as a 3D bridge design tool, Software A is also capable of developing essentially the identical 2D models associated with Software B. By using Software A to conduct all 2D and 3D commercial design software analyses, two major benefits were realized. First, rather than being restricted to the inherent “black box” assumptions adopted by many 2D-specific software packages, manually developing 2D models in a general- use program provided more flexibility, especially in terms of implementing several modeling improvement methods (Section 2.5.3). Secondly, this general approach shifted the focus away

Research Approach 47 from comparing specific commercial design software packages and instead highlighted the modeling approach, which is of more general value to designers and software developers. In total, three distinct types of analyses were performed in Software A as part of this study, including 3D models (as a direct comparison with the 3D truss-element models produced in Abaqus and documented in Section 2.4), 2D plate and eccentric-beam (PEB) models, and 2D grid/grillage models. The following subsections outline key attributes related to these 2D and 3D modeling procedures. Section 2.5.1 briefly introduces the major assumptions and features related to PEB and grillage models, as well as explains the subtle differences between the 3D models developed in Software A and Abaqus (Section 2.4). Section 2.5.2 then expounds on the equiva- lent beam approach for modeling cross-frames that is inherent with any 2D analysis. Finally, Sec- tion 2.5.3 offers techniques commonly used to improve the 2D equivalent beam approach, and Section 2.5.4 outlines the parametric studies used to evaluate these analytical techniques. Note that the general approach for these simplified methods is largely based on the guidelines docu- mented in AASHTO G13.1 Guidelines for Steel Bridge Analysis (2019) and White et al. (2012). 2.5.1 General Modeling Assumptions The inherent modeling assumptions associated with 3D, 2D PEB, and 2D grillage models in Software A (and many other commercial programs) are briefly outlined in this section. Individual subsections are provided to discuss each approach. 2.5.1.1 3D Models Three-dimensional modeling of bridge systems in any commercial design software (e.g., Software A) can vary from package-to-package and engineer-to-engineer. The accuracy of the model as it pertains to cross-frame forces can be sensitive to the assumptions made by the engi- neer. With this in mind, the intent was to provide a modeling approach that is representative of most 3D software packages and assumptions likely utilized by bridge engineers. However, it is important to note that variability in the results should be anticipated for a different software package or a different set of assumptions. The 3D models in Software A were developed similarly to the 3D Abaqus model (Section 2.3.1) with a few notable exceptions. First, girder flanges were modeled as beam elements rather than shells. Second, rather than framing cross-frame members into the stiffened girder web (i.e., how cross-frames are fabricated and erected in practice), cross-frames were framed into the shared node at the girder web-to-flange juncture to maintain consistency with most 3D bridge-related software programs. Upon developing the model, the built-in influence-surface feature of Soft- ware A was utilized to move a 1-kip load along the length and width of the bridge deck, similar to the procedures outlined previously. 2.5.1.2 2D PEB Models The 2D PEB models were developed in accordance with AASHTO LRFD Article 4.6.3.3.1. The concrete deck was explicitly modeled as a shell element. Girders were modeled as beam elements offset from the deck shell elements to represent the height difference between the centroids of the respective components. The shells representing the concrete deck and the beam elements representing the girders were constrained together to simulate composite action. Given that the depth component of the girders is neglected in the PEB modeling approach, cross-frames cannot be explicitly modeled. Instead, they are represented as equivalent beams, as outlined in more detail in Section 2.5.2. Similar to the 3D approach, bearings were represented as linear springs, except that they acted at the centroid of the girder section. Additionally, at least one intermediate node was placed on

48 Proposed Modification to AASHTO Cross-Frame Analysis and Design the beam element representing the girder between two adjacent cross-frame intersections on that beam. This is particularly important for horizontally curved bridge systems, which generally model the girders as a series of chorded, straight-line segments (White et al. 2012). The influence-surface loads were applied directly to the deck shell elements, in the same manner as the 3D model. 2.5.1.3 2D Grillage Models For 2D grillage models, the deck is not explicitly modeled with shell elements, which drama- tically impacts the modeling assumptions related to live load application and overall stiffness. There are, however, procedures to approximately consider the contributions of the concrete deck as a transverse load distribution mechanism, which are discussed in Section 2.5.3. In terms of load application in grillage models, live load force distribution is typically based on the simpli- fied distribution factors specified in AASHTO LRFD. In this case, the lever rule was utilized to assign a percentage of the load to the neighboring girders. In terms of stiffness, the girders, much like the 2D PEB approach, were modeled as beam elements. However, the section properties of the beam elements in grillage models consider the composite section. Thus, the grillage models inherently reflect the longitudinal stiffness of the deck but neglect its transverse stiffness. Similar to the PEB approach, at least one inter mediate node was placed along the girder beam element between adjacent cross-frame intersections. Cross-frames and bearings in grillage models were handled the same way as with the PEB models. Table 2-4 summarizes the general techniques used by the various analysis methods con- sidered in the study. The table serves to highlight the major differences in how the key load- distributing elements were represented, namely the deck, girders, and cross-frames. Note that Table 2.4 does not consider any modifications that potentially improve the predicted response of the bridge; however, these concepts are introduced in subsequent sections. Based on the descriptions of each analysis method, it is apparent that the 2D approaches rely heavily on simplifications and thus have a greater potential for error. This is particu- larly true for cross-frame force predictions in 2D models because cross-frames are modeled as equivalent beams. In the context of Table 2-4, it is expected that the performance of the analysis method moving from left to right will decrease in accuracy with respect to predicting cross-frame forces. 2.5.2 Equivalent Beam Approach for Cross-Frames For both the 2D grillage and PEB modeling approaches discussed in the preceding sub- section, cross-frames are commonly converted into equivalent beams for analysis purposes Element Analysis Method Control 3D 2D PEB 2D Grillage Concrete deck Shells Shells Shells --a Girders Shells Shells/beams Beam elementsb Beam elementsb Cross-frames Truss elements Truss elements Equivalent beamsc Equivalent beamsc Notes: aGrillage models do not explicitly consider the concrete deck. bIn the 2D PEB model, beam elements represent the steel section alone; in the 2D grillage model, beam elements represent the effective composite section. cThere are several methods by which the equivalent beam section properties are computed, as discussed in Section 2.5.2. Table 2-4. Parameters considered in the analytical-model parametric study.

Research Approach 49 and converted back into an idealized truss system for obtaining internal member forces. With respect to this transformation, there are two major questions that often arise when imple- menting these procedures into 2D models. The first is related to the section properties of the equivalent beam elements. Several different methods are commonly used to compute the equivalent moment of inertia, torsional constant, and shear area as outlined in Section 2.5.3. The second question is related to the postprocessing procedures once the shear forces and end moments in the equivalent cross-frame beams are obtained from the 2D models. Typically, these end moments and shears are applied as external loads on an idealized truss model, from which internal axial forces in struts and diagonals are computed. The end moments are often resolved as a force couple between the top and bottom nodes of the truss, resulting in equal- and-opposite forces in the top and bottom struts. For X-type cross-frames, the end shears are equally distributed between top and bottom nodes resulting in equal-and-opposite diagonal member forces. In many cases, these behaviors are seldom observed. For K-type cross-frames, the vertical shear component is resisted entirely by the single diagonal member framing into a given end, such that no distribution assumption is required. Aside from the general procedures, this equivalent beam approach is also thought to produce cross-frame force results with varying levels of accuracy, particularly for heavily skewed and/or curved bridges. With the background information outlined, the next subsection offers and explores specific improvements on these simplified 2D methodologies. 2.5.3 Approaches for Improving 2D Analyses In conjunction with the 2D models outlined previously, several improvement techniques based on NCHRP Research Report 725 (White et al. 2012) and past experience were explored in the Commercial Design Software Study. These improvement techniques range from analysis to postprocessing assumptions. These techniques, along with their designation as analysis or postprocessing modifications, are briefly summarized as follows: 1. Improve the equivalent cross-frame beam properties [Analysis]. As outlined in NCHRP Research Report 725 (White et al. 2012), there are three approaches by which cross-frames can be transformed into an equivalent beam for use in 2D models: the flexural-analogy approach, the shear-analogy approach, and the Timoshenko beam approach. Per NCHRP Research Report 725, the Timoshenko beam approach provides the most realistic estimate of the cross-frame stiffness because it considers both flexural and shear deformations. In this study, this assertion is verified in the context of composite systems. For the sake of brevity, refer to Section 3.2.3 of NCHRP Research Report 725 for a detailed description of each method. 2. Use an equivalent torsion constant for the beam elements representing girders to account for the neglected warping stiffness [Analysis]. As documented in NCHRP Research Report 725, many software programs, including Software A and Software B, neglect the warping stiffness term when idealizing girders as beam elements in 2D analyses. This results in a significant underestimation of the torsional stiffness of the system. Significant errors in predicted bridge response (e.g., deflections and cross-frame forces) have been shown to occur in non- composite systems, particularly in skewed and horizontally curved bridges. Given that the current study is focused on composite systems subjected to live loads, the equivalent torsion constant (represented as Jeq) proposed by White et al. (2012) was explored, but it was found that this modification is less impactful for composite systems. In a composite system, the torsional stiffness of the concrete deck is substantial compared to the added contribution of the warping stiffness in the girders. Therefore, implementing an equivalent torsional constant for girders was shown to have little effect on the cross-frame

50 Proposed Modification to AASHTO Cross-Frame Analysis and Design force predictions. For completeness, the models and corresponding results presented later in this report include equivalent torsional properties. 3. Consider the transverse stiffness of the concrete deck [Analysis]. Given that 2D PEB models represent the deck as a shell element, this technique is only applicable to grillage models. In this modification, the transverse stiffness of the deck can be simulated using (i) equivalent section properties or (ii) notional beams. For the first approach, the section properties of the equivalent cross-frame beam can be modified by simply summing the contributions from both the cross-frame (as previously determined by the Timoshenko beam approach) and the deck, whose effective width is taken as half the distance to the nearest cross-frame on each side. Thus, the additional moment of inertia, shear area, and torsional constant contributed by the applicable concrete deck strip (as modified by the appropriate modular ratio) is added to the equivalent cross-frame beam properties. In addition to the effective section property approach, consideration of the transverse deck stiffness can also be implemented with separate, notional transverse beam elements in the grillage model per AASHTO Article C4.6.3.3.4. 4. Reconsider the distribution of shear forces and moments when postprocessing results [Post- processing]. As introduced previously, the end moments obtained from 2D analyses are often resolved as equal-and-opposite force couples to the top and bottom nodes of the truss, and end shears are assumed equally distributed to top and bottom nodes for X-type cross-frames. These assumptions, although simple to implement, can produce significant errors for com- posite bridge systems as is shown by the results herein. Three-dimensional FEA results and measured data have shown that top and bottom struts generally do not have equal-and- opposite forces; similarly, diagonal members generally do not have equal-and-opposite forces in X-frames. As such, a unique postprocessing tool is proposed for both 2D grillage and 2D PEB models, as illustrated in Figure 2-17. The grillage improvement technique (left side of Figure 2-17.) is related to the equivalent section properties approach outlined in item #3 above (i.e., considering the contributions of the transverse deck stiffness). By effectively increasing the stiffness of the equivalent beam to account for the contributions of the concrete deck, considerations for the deck must also be made when processing 2D analysis results. Otherwise, the force demands on the cross-frame elements will be artificially amplified due to the increase in assigned stiffness. In contrast, the methodology below is not applicable to the notional beam approach introduced in item #3. For that case, the end moments and shears obtained for the equivalent cross-frame beam can be resolved directly to the cross-frame truss model given that the contributions of the concrete deck are explicitly and independently considered in analysis. As such, rather than applying the end moments and shear forces on the cross-frame panel alone, the applied loads are resolved on a pseudo-composite system including the cross-frame and deck. The end moment is resolved between the centroid of the deck and the centroid of the bottom strut, which results in a moment arm of H (or the distance M1 / H M1 / H VCF H FTS FBS Vdeck M1 / hb M1 / hb nVCF hb FTS FBS (1-n)VCF Figure 2-17. Postprocessing improvement methods for 2D grillage models (left) and 2D PEB models (right).

Research Approach 51 between the respective centroids). This modification is intended to remedy the problem of having equal-and-opposite strut forces in composite systems. It should be noted that in noncomposite systems, equal-and-opposite force distribution in the struts is anticipated. Thus, it is apparent from this simple discussion that the behavior of cross-frames in non- composite and composite systems is different and should be treated as such in the cor- responding analysis. This discussion is advanced further in subsequent sections. Additionally, in this proposed technique for 2D grillage models, the shear force acting on the equivalent beam must be distributed to the concrete deck and cross-frame panel. The research team utilized a rational approach in which the shear force is distributed based on the relative stiffness of the deck and cross-frame to determine Vdeck (sheer force component resisted by the concrete deck) and VCF (shear force component resisted by the cross-frame), although other methodologies are possible. These expressions can be found in Appendix F. The distribution of VCF to the top and bottom nodes of the cross-frame truss is discussed below. The PEB postprocessing technique (right side of Figure 2-17) differs from the grillage technique in that the internal moments and shears obtained from the analysis (via equiva- lent beams) only consider the contributions from the cross-frame panel. Recall that the stiffness of the concrete deck is explicitly considered by shell elements in PEB models. Thus, in the process of converting end moments and shears into cross-frame force effects, the added discussion about the moment arm H above is not relevant. With that in mind, the remaining discussion relates to how the VCF acting on the cross- frame panel is distributed to the top and bottom nodes. This significantly impacts the assumed force effects in the cross-frame members, particularly for the diagonals since the force effects in these members are directly related to the vertical component of the nodal forces. The research team explored several solutions to this problem and compared the behavior of noncomposite and composite systems. Ultimately, the equal distribution “50-50” assumption (i.e., 0.5VCF to the top node and 0.5VCF to the bottom node) is compared to a “100-0” assumption (i.e., conservatively and independently evaluate the cases in which 100% of VCF is resisted by the top node and 100% of VCF is resisted by the bottom node). This is demonstrated schematically in Figure 2-17, where the percentage of shear force distribution is represented by the variable n. Table 2-5 summarizes the analysis-related improvement techniques outlined above and the relative impact on deck, girder, and cross-frame elements. Blank table entries indicate that no modifications are made to that particular element. Element 2D Analysis Improvement Techniques Improve Equivalent Cross-Frame Beamsa Equivalent Torsional Constant for Girdersa Equivalent Cross-Frame Beamsb Concrete deck --c -- c Consider with equivalent cross-frame beam Girders -- c Assign Jeq to girder properties -- c Cross-frames Use Timoshenko beam approach -- c Adjust equivalent section properties to include deck Notes: aApplies to both grillage and PEB models. bApplies to grillage models only. cBlank cells indicate that the improvement technique does not directly apply to that structural element. Table 2-5. Summary of improved analysis techniques for 2D models.

52 Proposed Modification to AASHTO Cross-Frame Analysis and Design 2.5.4 Parametric Study Overview To assess the limitations of these various 2D modeling techniques and modifications, a series of analyses were conducted as part of this Commercial Design Software Study. The results of 3D truss-element models and various 2D models were compared for a variety of bridge geometries, where the 3D models served as the basis for comparisons. More specifically, the research team compared 3D truss-element models developed in Abaqus, which were used in the Fatigue Loading Study, with the following analysis methods in the general-use software package, Software A: • 3D model (cross-frame modeled with truss elements; stiffness modification, R, taken as 0.6 to maintain consistency with Section 2.3.1.1 of this report), • 2D PEB model (cross-frames idealized as equivalent beams; modifications outlined in Section 2.5.3 and R = 0.6 inherently considered in equivalent beam properties), and • 2D grillage or grid model (cross-frames idealized as equivalent beams; modifications outlined in Section 2.5.3 and R = 0.6 inherently considered in equivalent beam properties). As noted above, the stiffness modification factor (R = 0.6) was directly applied in the development of the equivalent beams in the 2D models, thereby reducing the equivalent beam section properties. The various models outlined above were developed for a scaled version of the 4,104-model matrix outlined in Section 2.3.1.2. Ultimately, the same 20 representative bridges of the 4,104 total introduced in Section 2.3.3 were evaluated. These representative bridges sampled key parameters from the full matrix that most often affect cross-frame force predictions in 2D simplified analyses (e.g., support skew and horizontal curvature). Thus, information or knowledge gained from this abbreviated study is directly applicable to a broader range of bridges. For each of the 20 representative models, the research team conducted an influence- surface analysis (or equivalent) for the various iterations. For reference, the pertinent variables that describe the overall geometry and cross-frame layout of these sample bridges can be found in Appendix F. The results of the Commercial Design Software Study are subsequently presented in Section 3.4. 2.6 Stability Study The Stability Study focused on investigating two major design issues related to Objective (e) introduced in Section 1.2: (i) development of stability bracing requirements for steel I-girders extending the available solutions to include negative moment regions and (ii) combination of stability bracing strength requirements with consideration of force effects generated during construction and throughout the service life of the bridge. Although the major stability force effects likely come from gravity loading during deck casting, the impact of forces induced from other sources such as wind and overhang construction loads needs to be considered. This section addresses the methodology and background associated with item (i), whereas item (ii) is largely addressed in Section 3.5.4 and the design examples (Appendices B and C). As noted previously, AASHTO LRFD currently has no guidance for stability bracing requirements. As such, the primary goal of this study was to develop design guidance in the context of steel I-girder bridge systems, using the bracing provisions in the AISC Specifications as the template. Consequently, the AISC provisions were reviewed and modified accordingly based on a series of finite element studies introduced in this section. First, a cursory overview of the AISC guidelines for torsional braces is provided below for context. Effective stability bracing of beams can be provided by either restraining twist of the cross- section (torsional brace) or restraining lateral movement of the compression flange (lateral brace). Cross-frames generally restrain twist of the section and are therefore categorized as

Research Approach 53 torsional bracing. Regardless of whether a system is categorized as torsional or lateral bracing, adequate stability bracing must satisfy both stiffness and strength requirements. The stiff- ness requirement, in general terms, is instituted to limit the twist and/or lateral displacement of a girder at the brace point. The recommended brace stiffness is often given as an integer multiplier of the “ideal” brace stiffness (βi), which corresponds to the brace stiffness required for a perfectly straight member to reach a specified capacity or load level. Since “real” structural members are not perfectly straight (i.e., they possess some initial imperfection or out-of- straightness over the length), the current AISC design provisions assume that twice the ideal stiffness adequately controls deformations and brace forces. Based on the work conducted by Winter (1960), it was found that providing twice the ideal stiffness (2βi) limits the out-of-plane deformations for columns to a value equal to the initial imperfection as the applied load reaches the critical buckling load. This observation, coupled with simplifications covered in the commentary (Yura 2001), is the basis for the required torsional brace stiffness adopted in the AISC Specifications (2016): 2.4 2.1, 2 2 LM nEI C T req r yeff b =b f where bT,req = required system torsional brace stiffness, L = span length, Mr = maximum factored moment within the critical unbraced segment, n = number of intermediate braces within the span, E = the modulus of elasticity of steel (29,000 ksi), Iyeff = effective moment of inertia, f = resistance factor, taken as 0.75, and Cb = moment gradient factor assuming the beam buckles between the brace points. These variables, in the context of single- and reverse-curvature conditions, are discussed in Section 3.5. Although this “twice the ideal stiffness” assumption works well for columns, studies on beam torsional bracing have shown a larger value may be warranted. In response, the research team conducted a parametric study that investigated the effects of girder cross-section, loading conditions, intermediate bracing schemes, girder spacing, and number of girders on the required stiffness of a torsional beam brace (i.e., a cross-frame). A brief overview of the research methodology used in this study is provided in Section 2.6.3. While a reasonable view of stiffness requirements might focus solely on the stiffness of the cross-frame, in reality, the total torsional brace stiffness is a function of several components. The total torsional stiffness of a brace is a combination of three main components including: (i) brace stiffness, bb, (ii) cross-sectional distortion stiffness, bsec, and (iii) in-plane girder stiff- ness of the beams, bg. The AISC bracing provisions do not include the expression for in-plane girder stiffness since that component is not a major factor in most building applications. However, for bridge applications, bg can significantly impact the behavior, particularly for relatively narrow girder systems (Yura et al. 2008; Han and Helwig 2016). The individual stiffness components tend to follow the expression for springs in series, as demonstrated mathematically with the following expression: 1 1 1 1 2.2 T b sec g = + + b b b b

54 Proposed Modification to AASHTO Cross-Frame Analysis and Design A review of Eq. 2.2 demonstrates that the total stiffness of the system, bT, is always less than the smallest of the three individual terms on the right of the equation. For example, a stiff cross- frame with flexible connection details will severely limit the stiffness response of the system; or a flexible in-plane girder stiffness, which effectively equates to a narrow superstructure, will diminish the adequacy of a cross-frame. The previous research related to these topics is sum- marized by Helwig and Yura (2015). While Eq. 2.2 provides the total stiffness of the system, in design, the actual brace stiffness must exceed the required stiffness, as determined by Eq. 2.1 (i.e., bT ≥ bT,req). In addition to stiffness requirements, stability bracing must also satisfy strength require- ments. In general terms, the required brace strength is a function of the required stiffness and the assumed initial imperfection. A distinct change in bracing strength requirements, however, occurred between the 14th Edition AISC Specifications (2010) and the current 15th Edition (2016). Applying some simplifications which are documented in the commentary, the original 2010 version of the torsional brace strength requirement (based primarily on elastic buckling behavior) was as follows: 0.024 2.3M M L nC L br T o r b b = =b q where Mbr = required strength of a torsional brace, bT = required system torsional brace stiffness (based on twice the ideal stiffness), qo = initial imperfection in terms of a twist angle, and Lb = unbraced length of the critical segment. The remainder of the variables were previously introduced in Eq. 2.1. Note that the initial imperfection is based on the critical imperfection shape, which for torsional bracing consists of a lateral sweep of the compression flange equal to Lb/500 (sweep tolerance) while the tension flange remains straight (Wang and Helwig 2005). This produces an initial twist equal to Lb/500ho, where Lb and ho are the respective unbraced length and distance between flange centroids. In contrast, the latest AISC Specification (2016) introduced a change in the torsional brace moment equation based upon a study conducted by Prado and White (2015). The researchers carried out a detailed investigation on the stability bracing requirements with an emphasis on inelastic buckling of relatively short unbraced lengths. This research prompted the revision in the torsional brace moment equation, which is also given as Eq. A-6-9 in the current AISC Specifications: 0.02 2.4M Mbr r= This expression is simply a function of the internal factored moment in the critical unbraced segment. Although the simplicity of Eq. 2.4 is attractive, the applicability of the expression for general design situations is questionable when compared to the longstanding strength equations predicted by Eq. 2.3. To determine which strength design expression is more appropriate for implementation into AASHTO LRFD, an additional parametric study was conducted in tandem to NCHRP Project 12-113 (Liu and Helwig 2020). As noted above, LTB in bridge applications is most critical during girder erection and deck construction. Although the bending moments during construction are smaller in magnitude than the live load moments in the completed structure, the noncomposite girders are most susceptible to instability at this stage. During construction, the steel section alone generally supports the entire load, and all permanent bracing may not be installed. Additionally,

Research Approach 55 stay-in-place forms, commonly used to support wet concrete during deck construction, have connections to the girder that potentially can introduce significant flexibility and therefore are not considered bracing elements in bridge applications (Egilmez, Helwig and Herman 2016). While bracing demands are often considered most critical in positive moment regions, both the positive and negative moment regions in continuous girder systems need to be considered. In the finished structure, the composite deck provides significant continuous lateral and torsional restraint to the top flange. For simple spans, for which only the top flange is in compression, girders are not prone to LTB. In continuous spans, the stability behavior of continuous girders in the negative moment region (i.e., the bottom flange is in compression) is often questioned by designers. As a result, cross-frames are often provided in negative moment regions to control LTB, and the substantial restraint provided by the deck to the bottom flange is conservatively neglected in design. There are, however, a number of beneficial restraints that can be considered around interior supports of the finished structure. As discussed by Yura (2001), the composite deck not only continuously braces the top flange but also provides additional bracing benefits to the bottom compression flange in the negative moment regions, assuming web distortion is prevented. Additionally, the girder bearings themselves provide lateral restraint and some torsional restraint to the bottom compression flange in these regions, which further mitigates an LTB problem. Therefore, the stability bracing requirements for cross-frames in negative moment regions of composite systems are not critical for design. To demonstrate the beneficial effects of continuous top flange restraint, Figure 2-18 schematically depicts the results of an eigenvalue FEA buckling analysis on a bridge girder with stiffened webs subjected to reverse-curvature bending, which is representative of many continuous systems near the interior supports. In all three cases presented, twist of the cross- section is prevented at the girder ends. However, the level of top flange restraint is modified in each successive case. From the buckled shapes, it is evident that cases (i) and (ii), which are representative of the steel section alone and a noncomposite girder condition (i.e., a concrete deck without shear studs), respectively, tend to buckle in a traditional LTB (or similar) mode. In these cases, the unrestrained compression flange(s) displaces laterally in the out-of-plane direction along the (i) No continuous top flange restraint (ii) Continuous top flange lateral restraint (iii) Continuous top flange lateral & torsional restraint Figure 2-18. Typical buckled shape for beams subjected to reverse-curvature bending with various degrees of top flange restraint.

56 Proposed Modification to AASHTO Cross-Frame Analysis and Design full unbraced length. In case (iii), the buckled shape is indicative of a distortional mode in the web, which is not sensitive to the unbraced length but is sensitive to the web slenderness and transverse stiffener details. The buckling capacity of these cases generally increases from left to right, as presented in Figure 2-18. More specifically, the bucking capacity associated with case (iii) is typically substantially larger provided that transverse stiffeners are included and are properly detailed. As such, girder segments in the negative moment regions are often controlled by yielding or the distortional buckling mode, for which bracing demands in the cross-frames are much less significant than the construction condition. This phenomenon is supported with additional analyses outlined in Section 2.6.1. The critical stage, therefore, occurs during erection and deck construction with the non- composite steel girder system supporting the entire construction load. With that in mind, the studies outlined herein are focused primarily on noncomposite systems under various moment gradients including single- and reverse-curvature bending, as these conditions represent the most critical in terms of bracing requirements. Before outlining the stability bracing studies, a cursory overview of a computational study investigating buckling in composite conditions is provided in Section 2.6.1. An overview of the parametric studies conducted to address these bracing strength and stiffness requirements are then provided in the following subsections. Section 2.6.2 outlines the bracing strength study, and Section 2.6.3 outlines the bracing stiffness study. 2.6.1 Buckling in the Composite Condition To verify the assertion that buckling in the composite condition is not critical for design, several spot check computational studies were performed in the general purpose FEA software, Abaqus. Elastic eigenvalue buckling analyses were conducted on prismatic girder segments with unbraced length-to-depth ratios (i.e., Lb/d) of 5 and 10 and a slender flange-width-to- web-depth ratio (i.e., bf /d) of 1/6—the minimum value permitted in AASHTO LRFD. Cases with closely spaced transverse web stiffeners (i.e., stiffener spacing equivalent to the girder depth) and no transverse stiffeners along the length were also studied to examine the effects of these details on web distortion buckling. Rather than investigating entire girder systems with bracing elements as is done in the subsequent sections, the critical unbraced segment in the negative moment region is isolated and analyzed independently, while conservatively neglecting the additional warping restraint provided by the adjacent unbraced segments in the span. Moment gradients along the length of the critical segment are then simulated through a series of end moments. This procedure is consistent with past LTB studies that investigated concepts such as load-height effects and moment gradient factors for singly symmetric and nonprismatic sections (Helwig, Frank and Yura 1997; Reichenbach et al. 2020). With that in mind, various straight-line moment gradients were considered with varying degrees of negative flexure in the unbraced segment, ranging from uniform negative moment (i.e., constant compressive stress in the bottom flange) to equal-and-opposite end moments (i.e., straight-line moment diagram with an inflection point at mid-length of the segment). These parameters were selected to represent practical conditions commonly found in continuous, composite bridge girders near the interior supports. The primary intent of these studies was to evaluate the beneficial effects of continuous top flange restraint in the negative moment regions of composite systems. Thus, in addition to the torsional restraint provided by the cross-frames at the ends of the critical girder segments, three different types of additional top flange restraints were considered: (i) no additional top

Research Approach 57 flange restraint, which is representative of the steel section alone during deck construction, (ii) continuous lateral restraint to the top flange, which is representative of noncomposite finished bridges not utilizing shear connectors, and (iii) continuous lateral and torsional restraint to the top flange, which is representative of a composite bridge in its finished state. Case (iii) represents an upper-bound condition in terms of restraint fixity, as the composite deck does not supply a completely rigid torsional restraint to the top flange. For the various parameters and moment gradients, the critical buckling moment was evaluated and compared as a function of the top flange restraint provided. The critical eigen- vector either corresponded to an LTB mode or a web-distortional buckling mode, similar to the illustrations provided in Figure 2-18. The results of this study are subsequently presented in Section 3.5.1. 2.6.2 Bracing Strength Study To investigate torsional bracing strength behavior, a study was conducted in tandem with the NCHRP Project 12-113 study consisting of parametric 3D finite element analyses on twin I-girder systems using Abaqus (Liu and Helwig 2020). The results are directly applicable to the NCHRP Project 12-113 study, and therefore the following section provides a cursory over- view of these studies. For a more detailed outline of the modeling assumptions and procedures, refer to Appendix F. In general, a series of buckling analyses on twin-girder systems with an assumed initial imperfection were conducted. Two girders represent the fewest number of girders that are possible with torsional bracing provided by cross-frames or diaphragms; the bracing behavior is representative of systems with more than two girders. A variety of intermediate bracing configurations were considered, including one to five cross-frames between the girder supports. Both elastic and inelastic (i.e., assuming elastic-perfectly plastic behavior with yield strengths of 36, 50, and 70 ksi) material analyses were carried out to examine the influence of material nonlinearity on the response of the girders and cross-frames. In general, tailoring brace strength requirements around a specific material yield strength is not advisable, as it can potentially lead to unconservative estimates of the stability brace moments. Instead, provisions for brace strength should be applicable for a variety of yield strengths. Strain hardening and residual stresses, however, were not considered in the study since preliminary results demonstrated that elastic materials produced more critical results. Three different noncomposite, prismatic girder cross-sections (referred to as Cross-section 1, 2, and 3 herein) were examined, which primarily investigated different flange-width-to-web- depth ratios (bf/d) ranging from the extreme limit for built-up sections permitted in AASHTO LRFD (2020) to a value more consistent with typical rolled sections. These cross-sections are depicted schematically in Figure 2-19. The web depth and span length were subsequently held constant to achieve a span-to-depth ratio (L/d) of 25, which is representative of common bridge girders used in practice. In addition to various girder cross-sections, the research team also examined different loading conditions, including uniform moment and uniformly distributed loading (i.e., single- curvature bending). The distributed loads were either applied at the top flange or at mid-height of the sections to study the impact of load position on the cross-section. As part of NCHRP Project 12-113, additional spot checks were performed to examine bracing demands in bridge girders subjected to reverse-curvature bending, particularly for cross-frames in the negative moment region. As noted previously, only the noncomposite condition is repre- sented in these various loading conditions. Thus, no lateral and torsional restraint provided by a composite concrete deck is considered. For all loading scenarios, the girders were simply

58 Proposed Modification to AASHTO Cross-Frame Analysis and Design supported at the ends. For the cases involving reverse curvature, the girders were still simply supported, but girder continuity was simulated with applied end moments (positive and negative). Twist was also restrained at the girder ends, but the sections were free to warp. For the single-curvature bending cases, it was observed that the critical segment for buckling that consistently resulted in the largest brace forces was near midspan, where positive moment and out-of-plane girder displacements were maximized. As such, a critical asymmetric imper- fection consistent with Prado and White (2015) (i.e., the critical compression flange is dis- placed laterally in accordance with the discussion in the preceding section) was assumed in these critical areas. For reverse-curvature bending, it is not always as clear which segment is critical for buckling. In continuous girder systems, the negative moment regions generally have the largest moment magnitudes. However, these regions are aided by steeper moment gradients (i.e., larger Cb factors) and restraint provided by the nearby girder supports. In contrast, positive moment regions have smaller moment magnitudes but typically have Cb values close to unity, especially as additional intermediate braces are included. For example, in the interior span of a three-span continuous unit, the maximum negative moment is approximately double the maximum positive moment. However, the moment gradient factor in the negative moment region is often greater than 2.0 (depending on the unbraced length) based upon published values or AISC expressions, compared to a Cb factor near unity for the positive moment region. Thus, it was observed that the critical brace (i.e., maximum bracing forces) depended on the location of the critical imperfection. For instance, the midspan brace forces in the positive moment region were maximized when the imperfection was assumed along the compression top flange at that same location. In contrast, the brace forces at the first intermediate brace line in the negative moment region were maximized when the imperfection was assumed along the bottom (compression) flange at that location. Note that the imperfection in the negative moment case was not implemented directly at the support condition, where negative moment is actually largest, due to the presence of bearings that benefits the torsional bracing requirements. Permutations of different girder cross-sections, bracing configurations, and loading conditions were systematically analyzed by a series of independent eigenvalue and incremental analyses. Eigenvalue analyses were initially performed to obtain the ideal stiffness of the cross-frame required to buckle the girders between the brace points. The large-displacement incremental analysis, performed on an imperfect system and cross-frames with twice the predetermined Figure 2-19. Different cross-sections evaluated in the stability bracing parametric studies. Flanges: 8"×0.5" 12"×0.75" 16"×1" 0.75" 48" Cross-section 1 Cross-section 2 Cross-section 3 Extreme Limit in AASHTO Typical Built-Up Shapes Typical of Rolled W-Shapes

Research Approach 59 ideal stiffness, was subsequently performed to obtain the relationship between internal girder moments and critical bracing moments. An additional discussion on modeling convergence and inferring cross-frame brace forces from the results is provided in Appendix F for reference. The results of this stability bracing strength study are subsequently presented in Section 3.5.2. 2.6.3 Bracing Stiffness Study To investigate the torsional brace stiffness behavior, a similar parametric 3D FEA study was carried out in Abaqus. Many of the same features and modeling assumptions outlined in Section 2.6.1 were also examined in this study, including the various intermediate bracing configurations, girder cross-section proportions, load-height effects, and critical imperfections. There were two notable differences, though, between the brace strength study outlined previ- ously and the brace stiffness study. First, the effect of material inelasticity was not explicitly considered in the stiffness study. As noted in Section 2.6.1, it was demonstrated that elastic materials generally produce more critical demands for cross-frame stiffness and strength. As such, only elastic materials with no residual stresses were considered in these FEA studies. Second, the bracing stiffness study exam- ined redundant bridge systems, including those with two, three, four, and six girders across the width. Regardless of the number of girders, the applied loads and critical imperfections outlined above still applied. These variables were parametrically evaluated through a series of independent eigenvalue and incremental analyses. Eigenvalue analyses, as outlined in Section 2.6.1, were initially performed to obtain the ideal stiffness of the cross-frame required to buckle the girders between the brace points. An incremental analysis on the imperfect system was subsequently conducted for a variety of cases. More specifically, the stiffness of the cross-frames was examined for its effect on the out-of-plane girder twists at the brace points (relative to the initial imperfection) for increased load levels. To examine these effects, brace stiffness to ideal stiffness ratios, bb/b i, of {2, 2.5, 3, and 4} were studied. For each stiffness multiplier, girder twists were obtained as a function of increasing internal girder moments. Ultimately, the objective was to determine which ideal stiffness multiplier produced final girder twists equal in magnitude to the initial imperfection. Additional discussion on the overall methodology and procedures is provided in Appendix F. Results of this stability bracing stiffness study are summarized Section 3.5.3.