Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges (2012)

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10 This chapter provides a brief summary of the approach used in addressing the objectives of the NCHRP Project 12-79 research. The primary project tasks were: •• Task 1. Review and evaluation of pertinent research, •• Task 2. Synthesis of owner/agency policies, •• Task 3. Identification of existing bridges, •• Task 4. Identification of geometric factors, •• Task 5. Selection of range and levels of geometric factors, •• Task 6. Selection of existing and parametric design bridges, •• Task 7. Analytical studies, •• Task 8A. Data reduction and assessment of analysis procedures, •• Task 8B. Development of improvements to simplified methods, and •• Task 9. Development of guidelines. The following descriptions are organized and arranged in the order of these tasks. 2.1 Review and Evaluation of Pertinent Research The first task of the research was to review and evaluate pertinent domestic and international research on the basis of applicability, conclusiveness of findings, and usefulness for the development of guidance for selecting analytical methods for the construction engineering of curved and/or skewed steel girder bridges. An extensive bibliography of the pertinent research was developed, including abstract summaries of research in progress, conference and workshop presentation slides, research reports, and archival journal papers. The references were scanned, indexed, and loaded into an internal database for ease of document access. The bibliography was focused primarily on references since 1993. Since Zureick et al. (1994) developed a comprehensive bibliography of the published literature on curved I- and box-girder bridges before 1994, the bibliography focused only on references not identified by the earlier bibliography for any citations prior to 1994. 2.2 Synthesis of Owner/Agency Policies and Practices The second project task was to synthesize current owner/agency policies and practices related to the construction engineering, construction plan preparation, and construction plan review for the above structure types. During this task, the project team coordinated its work with the C H A P T E R 2 Research Approach

Research Approach 11 AASHTO/NSBA Steel Bridge Collaboration Task Group 13, which conducted a “Survey of Current Practice in Steel Girder Design” during the early stages of the NCHRP Project 12-79 research. The project team also conducted its own survey, which was sent to the 50 state bridge engineers and bridge engineering contacts as well as the Commonwealth of Puerto Rico, the District of Columbia, and the bridge engineering contacts of various other owner agencies. The mailing, (see Appendix F of the contractors’ final report), included a short slide presentation summariz- ing the focus of Project 12-79, requested pertinent bridge cases (descriptions and plans) encoun- tered in the recipient’s practice that fit the criteria highlighted in the slides (summarized in the third project task below), and asked for input on state policies and practices regarding analysis methods and construction engineering of curved and/or skewed steel girder bridges. Thirty-one responses were received. Of these, 20 provided one or more bridges that fit the criteria provided with the mailing, 12 states provided specific input regarding their poli- cies and practices, and 9 states responded but indicated that they did not have any relevant information to provide. In addition to the specific request regarding state policies and prac- tices, the project team researched various state policies and practices available via the Web. Appendix G of the contractors’ final report contains a summary of the policies and practices from several representative states. The results of the AASHTO/NSBA Steel Bridge Collabo- ration Group Task Group 13 (TG13) Survey of Current Practice also are discussed in this appendix. The TG13 and Project 12-79 efforts were complementary to one another, with the TG13 efforts focusing on synthesis of current practices and practical recommendations con- cerning analysis methods, while the Project 12-79 focus was directed at identifying specific representative bridges and specific state policies and practices. 2.3 Identification of Existing Bridges During Task 3, the project collected more than 130 representative curved and/or skewed steel girder bridges based on a specific set of selection criteria. These included the bridges provided by the states as well as bridges from the professional practice of the project team members and various consultants contacted by the project team. The primary criteria posed for the collection of existing bridges were: •• Availability of quality field instrumentation data, or at least field observations, particularly during intermediate stages of construction, •• Availability of detailed construction and erection plans, and •• Successful construction but with significant challenges or concerns about the state of stress, etc. Cases involving generally acknowledged poor practices, such as the inappropriate use of oversize holes or inadequate attachment of cross-frames leading to loss of control of the structural geometry, were specifically ruled out from consideration. One of the key existing bridges identified for the NCHRP Project 12-79 studies was an eight-span curved I-girder fly-over ramp in Nashville, Tennessee, in which the Tennessee Department of Transportation gave the Georgia Institute of Technology researchers the opportunity to instrument and monitor the girders throughout the erection of the steel and the placement of the concrete deck. The results of this research are documented in Dykas (2012). The collected bridges, which are documented in the project’s Task 8 report (Appendix C of the contractors’ final report), showed a wide diversity in span arrangements, span lengths, span- to-width ratios, horizontal curvature, skew angles, and skew patterns (i.e., radial, non-radial, parallel, and non-parallel supports). In the Task 8 report, the collected bridges are summarized

12 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges succinctly in the form of sketches of their overall plan geometry, along with a title block listing specific bridge geometric parameters. 2.4 Identification of Geometric Factors In its fourth task, the project team developed a list of various geometric factors that potentially could have a significant impact on the accuracy of simplified methods of analysis. It was clear that if NCHRP Project 12-79 was to consider analysis accuracy for curved and/or skewed steel I- and tub-girder bridges, then the project would need to consider the following factors in the design of its parametric studies: •• Some measure of the horizontal curvature and •• Some quantification of the skew magnitude and pattern. Furthermore, it was apparent that the bridge responses, and hence the analysis accuracy, can be affected significantly by the magnitude of the span lengths as well as the span length-to-width ratios. Longer span bridges tend to be affected more substantially by dead load effects, potentially resulting in more significant stability considerations during construction. In addition, beyond a certain span length, I-girder bridges are more likely to need partial or full-span horizontal flange-level bracing systems to ensure adequate stability and sufficient resistance to lateral loads during construction. Flange lateral bracing systems cause corresponding portions of the structure to act as “pseudo-box girders,” fundamentally changing the behavior of the structural system. Furthermore, longer span bridges generally exhibit larger overall deflections. These larger overall deflections can lead to larger relative deflections at certain locations in the structural system, which can sometimes be problematic during construction. Longer span bridges often have a smaller ratio of the girder spacing relative to the girder depths, and typically have larger girder depth-to-flange-width ratios. These attributes can fundamentally affect various relative deflections in the structure as well as the local and overall behavior and analysis accuracy at the different stages of construction. In addition, the bridge span length-to-width ratios can significantly impact the influence of skew. Skewed bridges with smaller span length-to-width ratios tend to have more significant load transfer to the bearing lines across the width of the structure and hence more significant “nuisance stiffness” effects that need to be addressed in the design. Furthermore, relatively narrow horizontally curved bridges experience a greater torsional “overturning component” of the reactions, which tends to increase the vertical reactions on the girders farther from the center of curvature and decrease the vertical reactions on the girders closer to the center of curvature. Of equal or greater importance, these types of bridges potentially can experience significant global second-order amplification of their displacements. In addition, relatively wide horizontally curved bridges can have more substantial concerns related to overturning at intermediate stages of the steel erection, prior to assembly of the girders across the full width of the bridge cross- section. These spans become more stable as additional girders are erected and connected by cross-frames across the width of the bridge. Wide horizontally curved bridges also can cause greater concerns associated with overturning forces during deck placement. Lastly, it was apparent that the bridge responses (and the analysis accuracy) can be significantly affected by whether the spans are simply supported or continuous. Simple-span bridges tend to have larger deflections for a given geometry and potentially can be more difficult to handle during construction. Although simple-span girders can see negative bending during erection (due to lifting or temporary support from holding cranes, etc.), continuous spans have more significant negative bending considerations. Furthermore, particularly in I-girder bridges,

Research Approach 13 continuous-span bridges can have significant interactions between adjacent spans with respect to both major-axis bending as well as the overall torsional response. All of the above factors can have a substantial influence on the many detailed structural attributes of steel I-girder and tub-girder bridges. Also, there can be significant interactions between these factors in terms of their influence on the bridge responses, as well as the accuracy of different bridge analysis methods. If one considers the many detailed attributes of steel I- and tub-girder bridge structural systems and their members and components addressed subsequently, the combinations and permutations of potential bridge designs become endless. Hence, it was decided that the most practical way of covering the design space of curved and/or skewed I-girder and tub-girder bridges was to consider a range of practical combinations and permutations of the following primary factors: •• Span length of the bridge centerline, Ls, •• Deck width normal to the girders, w, (in phased construction projects, w is determined separately for each bridge unit), •• Horizontal curvature, of which the most appropriate characterization is discussed below, •• Skew angle of the bearing lines relative to the bridge centerline, q (equal to zero for bridges in which the bearing lines are not skewed), •• Skew pattern of the bearing lines, of which the most appropriate characterization is discussed below, and •• Span type, simple and various types of continuous spans. 2.5 Selection of Range and Levels of Geometric Factors As part of its fifth task, the project team compiled a summary of the range of values encountered for the above primary factors, as well as for various other geometric factors, considering the existing bridges collected in Task 3. This summary is documented in the project’s Task 8 research report (see Appendix C of the contractors’ final report). Given, this summary and the project team’s knowledge of maximum practical limits on the values, the primary factor ranges and levels shown in Table 2-1 were selected. Several nomenclature terms for categorizing the collected existing bridges as well as the bridges studied analytically in the project research appear in Table 2-1. These are the terms ICCR, TCCR, ICSS, TCSS, ICCS, and TCCS. The complete categories and their designations, which are used extensively throughout the remainder of this report, are as follows for the I-girder bridges: •• Simple-span, straight, with skewed supports (ISSS), •• Continuous-span, straight, with skewed supports (ICSS), •• Simple-span, curved, with radial supports (ISCR), •• Continuous-span, curved, with radial supports (ICCR), •• Simple-span, curved, with skewed supports (ISCS), and •• Continuous-span, curved, with skewed supports (ICCS). The same designations are used for the tub-girder bridges, except the first letter in the designation starts with a “T” rather than an “I.” A specific geometric factor used to characterize the bridge horizontal curvature is introduced in Table 2-1. This is the bridge torsion index: I s s s T ci ci co = + Eq. 1

14 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Factor I-girder bridges Tub-girder bridges Type of span Simple, 2-span continuous, and 3-span continuous with one balanced end span and one end span equal in length to the main center span. Use the above 3-span continuous bridges as base ICCR and TCCR cases. Consider both 2- and 3-span continuous bridges for the ICSS and TCSS cases. Consider only 2-span continuous cases for the ICCS and TCCS designs. Consider at least one 2-span continuous bridge with a significant unbalance between the span lengths. Max imum span length of bridge centerline, L s 150, 225, and 300 ft. for simple spans 150, 250, and 350 ft. for continuous spans (measured along the curve) Deck w id th , w 30 ft. (1 to 2 traffic lanes + shoulders and barriers) 80 ft. (4 to 5 traffic lanes + shoulders and barriers 30 ft. (1 to 2 traffic lanes + shoulders and barriers) Torsion Index, I T 0.58 to 0.71 for ISCR bridges 0.66 to 0.88 for ICCR bridges 0.72 to 0.87 for TSCR bridges 0.69 to 1.14 for TCCR bridges Sk ew angle relative to the bridge centerline, 20 o , 3 5 o , 50 o , and 7 0 0 but with at the inside edge of the deck < 7 0 o in curved spans 15 o and 3 0 o , plus additional sensitivity s tudies with variations up to ±15° from zero skew Sk ew p attern Consider the + com binations of skew angles shown in Figure 2-1 (for straight bridges) and Figure 2-2 (for curved bridges), but using = 35 and 7 0 o for I-girder bridges and = 15 and 30 o for tub-girder bridges. Limit the ratio of the span lengths along the edges of the deck, L 2 / L 1 , to a maximum value of 2.0 in all cases. Limit the difference in orientation of adjacent bearing lines to a maximum of 9 0 o in all cases. Give preference to ty pical (i.e., non-exceptional) bridge geometries. Table 2-1. Primary factor ranges and levels for the NCHRP Project 12-79 main analytical study. The terms in this equation, illustrated in Figure 2-3, are: •• sci, the distance between the centroid of the deck and the chord between the inside fascia girder bearings, measured at the bridge mid-span perpendicular to a chord between the intersections of the deck centerline with the bearing lines, and •• sco, the distance between the centroid of the deck and the chord between the outside fascia girder bearings, measured at the bridge mid-span perpendicular to a chord between the intersections of the deck centerline with the bearing lines. The torsion index IT is an indicator of the overall magnitude of the torsion within a span. It is a strong indicator of the tendency for uplift at the bearings under the nominal (unfactored) dead loads. This parameter was selected over various other factors that could be used to characterize the horizontal curvature effects on the bridge behavior and analysis accuracy, because it can be used to set minimum practical values for the radius of curvature of a span for a given deck width. A value of IT = 0.5 means that the centroid of the deck area is mid-way between the chords intersecting the outside and inside bearings. This is the ideal case where the radius of curvature

Research Approach 15 is equal to infinity and the skew is zero, (i.e., a straight tangent bridge). A value of IT = 1.0 means that the centroid of the deck area is located at the chord line between the outside bearings. This implies that the bridge is at incipient overturning instability, by rocking about its outside bearings under uniform self-weight. For a curved radially supported span, the denominator in Equation 1, sci + sco, is equal to wg cos(Ls/2R), where wg is the perpendicular width between the fascia girders. The NCHRP Project 12-79 research identified that simple-span I-girder bridges with IT ≥ 0.65 are often susceptible to uplift at the bearings under nominal (unfactored) dead plus live load. Similarly, for simple-span tub-girder bridges with single bearings on each tub, IT = 0.87 was identified as a limit beyond which bearing uplift problems are likely. The maximum values of 0.71 and 0.87 for the ISCR and TSCR bridges shown in Table 2-1 are similar to, and the same as, these values respectively. Continuous-span bridges can tolerate larger IT values due to the continuity with the adjacent spans. Therefore, the maximum IT values shown in Table 2-1 are larger for the ICCR and TCCR bridges. Figures 2-1 and 2-2 are referenced in Table 2-1 for the consideration of the skew pattern in straight and curved bridges, respectively. 2.6 Selection of Existing and Parametric Design Bridges Task 6 of the NCHRP Project 12-79 research involved the selection of various existing and parametric design bridges for detailed analytical study. The project’s Task 8 research report provides a detailed discussion of the considerations in the selection of the study bridges. An initial preliminary selection of these bridges was conducted at the start of Task 7 of the Case 1 - Parallel Skew, θ = 20° Scale in feet 0 20 50 100 Case 2 - Parallel Skew, θ = 35° Case 3 - Parallel Skew, θ = 50° Case 4 - Parallel Skew, θ = 70° Case 5 - Skewed at One Bearing Line, θ = 35° Case 6 - Skewed at One Bearing Line, θ = 50° Case 10 - Unequal Skew, θ = 60° & -30° Case 7 - Equal and Opposite Skew, θ = ±35° Case 8 - Skewed at One Bearing Line, θ = 70° Case 9 - Unequal Skew, θ = 70° & 35° Figure 2-1. Potential skew combinations for straight I-girder bridge spans with w = 80 ft. and Ls = 250 ft. (sketches with a dashed border are considered unusual; unshaded sketches with a grey border are considered exceptional).

16 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Case 11 θ = 30, 30 Case 8 θ = -30, -15 Case 4 θ = -30, 0 Case 13 θ = -15, 30 Case 14 θ = -30, 30 Case 9 θ = 30, 8.5 Case 7 θ = -15, -15 Case 3 θ = -15, 0 Case 10 θ = -15, 15 Case 2 θ = 21.5, 0 Case 1 θ = 15, 0 Case 6 θ = 30, -15 Case 5 θ = 10.7, -10.7 Case 12 θ = 30, -30 Parallel Parallel Parallel Figure 2-2. Example potential skew and horizontal curvature combinations for curved tub-girder bridge spans with w = 30 ft., Ls = 150 ft., and R = 400 ft. (sketches with a dashed border are considered unusual). Fascia Girder (Typ.) Deck Centroid Deck Centerline sci sco wg Figure 2-3. Illustration of parameters used in calculating IT.

Research Approach 17 research (i.e., the specific analytical studies discussed in the next section). These selections were revisited and revised at various subsequent stages, based on information learned during the analytical studies. A total of 58 I-girder and 18 tub-girder bridges were considered by the project at the completion of its analytical studies. In addition, another 10 tub-girder bridges were studied that involved taking several of the above tub-girder bridges and varying the skew angle at one of the bearing lines to study the sensitivity of the bridge response and the analysis accuracy to skew effects. Of the 86 total bridges studied, 16 were existing I-girder bridges and 5 were existing tub-girder bridges. In addition, three of the I-girder bridges and two of the tub-girder bridges studied were detailed example designs taken from prior AISI, NSBA, NCHRP, and NHI developments. Throughout the project documentation, the various bridges are referred to by their category (e.g., ISCR, ICCS, TCCS, etc.), preceded by the letters: •• E if the structure is an “existing” bridge, •• X if the structure is an AISI, NSBA, NCHRP, or NHI “example” bridge, and •• N if the structure is a “new” parametric bridge design. A unique number is appended to the end of the designation to arrive at the specific bridge name. Therefore, for example, the 8-span continuous I-girder ramp flyover in Nashville, Tennessee, studied by the NCHRP project team, has the designation EICCR22a (the number “22a” was selected in this case to group this bridge with other Tennessee EICCR bridges considered within the project research without modifying the numbers that had already been assigned to the other EICCR bridges). For all of the above bridges, the erection sequences used in the bridge construction were considered, or hypothetical erection sequences were developed where the specific erection sequences were not known. Various critical stages of the construction were then selected for study. In general, from 4 to 10 construction stages were selected for analysis with each bridge. As a result, more than 500 construction stages were considered in total, including the execution of multiple analysis methods for each stage. For the 58 - 16 - 3 = 39 additional “new” I-girder bridges and the 28 - 10 - 5 - 2 = 11 “new” tub-girder bridges, hypothetical parametric designs were developed by the practic- ing design members of the project team. These 39 + 11 = 50 bridges were complete designs satisfying the current AASHTO LRFD Specifications requirements. Specific supplementary criteria used for the design of these parametric bridges are explained in Appendix H of the contractors’ final report. It is important to note that the results of simply varying design parameters without checking Specification requirements can be misleading. The AASHTO requirements were satisfied for the parametric study bridges such that the research could establish appropriate relationships between bridge design variables and recommended levels of analysis and construction engineering effort. It should be noted that the study of 86 different bridges, as well as more than 500 construction stages, is not enough to develop a relevant data set for valid statistical assessment of analysis accuracy, given the vast range of potential situations that can be encountered during construction. However, the evaluations of the accuracy are certainly a large representative sample of the results that can be encountered in professional practice. Furthermore, a major focus of the project research, in Task 8 below, was the identification of mechanistic causes of the errors observed in the simplified analysis calculations, as well as the development of specific improvements to the simplified methods. By adopting this approach in the project research, various improvements were identified that are relatively easy to implement and lead to substantial gains in the general accuracy of the simplified methods.

18 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 2.7 Analytical Studies Task 7 of the NCHRP Project 12-79 research involved the development and execution of a large number of analysis studies aimed at identifying when simplified 1D and 2D analysis methods are sufficient for the evaluation of the constructability and the prediction of the constructed geometry of curved and/or skewed steel girder bridges. Results from the Task 7 research are provided in written report form for three benchmark cases extracted from the larger studies. This report is included as Appendix D of the contractors’ final report. Three main levels of design analysis were considered in the NCHRP Project 12-79 research: 1D or line-girder analysis, 2D-grid (or grillage) analysis, and general 3D finite element analysis (FEA). The specifics of the methods evaluated in each of these categories are summarized in the sections below. Chapter 2 of the Task 8 project report, Appendix C to the contractors’ final report, provides a more detailed description of each of the methods. 2.7.1 1D Line-Girder Analysis The first level of analysis targeted in the NCHRP Project 12-79 research was a conventional line-girder analysis including approximations such as the V-load (Richardson, Gordon & Associates, 1976; USS, 1980; Grubb, 1984; Poellot, 1987) and M/R (Tung and Fountain, 1970) methods to account for horizontal curvature effects. For these 1D solutions, a commonly available commercial line-girder analysis program, STLBRIDGE (Bridgesoft, 2010) was used to analyze the behavior for straight skewed I- and tub-girder bridges. The 1D analysis of curved, and curved and skewed, I-girder bridges was based on the V-load method using the software VANCK (NSBA, 1996). The 1D analysis of curved and skewed tub-girder bridges was based on a line-girder analysis coupled with supplementary calculations implemented by the project team based on the M/R Method. In addition, a useful method developed in the NCHRP Project 12-79 research for estimating the internal torque due to skew was implemented within the calculations for skewed tub-girder bridges. The recommended procedure is summarized subsequently in Section 3.2.6 of this report. Furthermore, for the estimation of the flange lateral bending stresses and the bracing forces in tub-girder bridges, the component force equations developed originally by Fan and Helwig (1999 and 2002), supplemented by additional equations presented in Helwig et al.( 2007) for the calculation of external intermediate cross-frame forces and the top flange lateral bracing (TFLB) system strut forces in Pratt systems, are used. Section 2.7 of the Task 8 report provides a detailed summary of these equations. Lastly, one additional improvement developed in the NCHRP Project 12-79 research is included in the calculation of the top flange average longitudinal normal stresses in tub-girder bridges. An additional local “saw-tooth” contribution to these stresses that comes from the longitudinal component of the TFLB diagonal forces is included in the project calculations. These additional “saw-tooth” stresses are discussed in Section 3.2.6 of this report. 2.7.2 2D-Grid Analysis To evaluate conventional 2D-grid methods of analysis, two commercially available software packages, employed by many bridge designers, were used to analyze the behavior of the same bridges considered with the above 1D methods: the software MDX (MDX, 2011) for analysis using a conventional 2D-grid approach, and a subset of the capabilities of the general-purpose LARSA-4D (LARSA, 2010) software for analysis using a conventional 2D-frame approach. In

Research Approach 19 the subsequent presentations in this report, the LARSA-4D software is referred to as Program P1 and the MDX software is referred to as Program P2. The 2D-frame model is referred to as such, even though the nodes in this model have 6 degrees of freedom (dofs) (3 translations and 3 rotations), because the entire structural model is cre- ated in a single horizontal plane. As discussed in Section 2.3 of the NCHRP Project 12-79 Task 8 report (see Appendix C to the contractors’ final report), if the structural model is constructed all in one plane with no depth information being represented, and if the element formulation does not include any coupling between the traditional 2D-grid dofs and the other dofs (which is practically always the case), 2D-frame models do not provide any additional forces or displace- ments beyond those provided by ordinary 2D-grid solutions. Assuming gravity loading normal to the plane of the structure, all the displacements at the three additional nodal dofs in the 2D-frame solution are zero. All of these conditions are satisfied by the LARSA-4D models developed in the NCHRP Project 12-79 research. Therefore, the 2D-frame and 2D-grid procedures are conceptu- ally and theoretically synonymous. Unfortunately, the programs typically do not provide identi- cal results for various reasons, some of which are addressed in the subsequent discussions. For the estimation of the flange lateral bending stresses and the bracing component forces in tub-girder bridges, NCHRP Project 12-79 used the same component force equations described above for the 1D-methods in its 2D-grid solutions. In a limited number of cases, 2D-grid calculations for the staged placement of the concrete deck were evaluated in the NCHRP Project 12-79 research. These calculations were conducted using a refinement on the basic 2D-grid modeling approach implemented in the MDX software system. For these calculations, once the deck was made composite with the girders in a staged construction analysis, the composite deck was modeled using a flat shell finite element model and the girders were represented by 6 dof per node frame elements with an offset relative to the slab. This modeling procedure is commonly referred to as a plate and eccentric beam (PEB) approach. In the PEB analyses of staged deck placement conducted by the project team, the concrete was assumed to be fully effective at the beginning of the stage just after the one in which it is placed. In addition, a limited number of additional “specialized” 2D-grid solutions were performed in the NCHRP Project 12-79 research using the first-order analysis capabilities of a thin-walled open-section (TWOS) frame element implemented in the educational program MASTAN2 (MASTAN2, 2011; McGuire et al., 2000). The TWOS frame element in MASTAN2 contains a seventh nodal warping degree of freedom, or a total of 14 nodal dofs per element. The specific element implemented in the MASTAN2 software, discussed in detail in McGuire et al. (2000), assumes a doubly symmetric cross-section such that the girder cross-section shear center is at the same position as the cross-section centroid. Therefore, the element is strictly not capable of representing the detailed response of singly symmetric bridge I-girders. However, in the 2D-grid models created with the MASTAN2 element, all the girder and cross-frame reference axes were modeled at the same planar elevation, and no depth information (e.g., bearing position relative to the reference axis of the girders, load height above the girder reference axis, etc.) was included in the model. As such, only the three conventional 2D-grid dofs plus the additional warping dof have non-zero displacement values and the influence of the shear center height relative to the height of the cross-section centroid does not enter into the first-order TWOS 2D-grid solutions. 2.7.3 3D Finite Element Analysis The ABAQUS software system was used to conduct linear elastic (first-order) design-analysis solutions as well as detailed geometric nonlinear (second-order) elastic “simulation” studies in the NCHRP Project 12-79 research. Furthermore, for selected cases from the full suite of 86

20 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges bridges considered in the NCHRP Project 12-79 analytical studies, ABAQUS was used to con- duct full nonlinear (material and geometric nonlinear) test simulations. Where possible, extant bridges were evaluated, and if those bridges had been instrumented, the test simulation results were validated against measured responses. The ABAQUS geometric nonlinear solutions were taken as the benchmarks to which all the simplified elastic analysis solutions were compared. Furthermore, the ABAQUS full nonlinear test simulation models were utilized as “virtual experiments” to evaluate questions such as the influence of different practices on the structural capacity of the physical bridges. Generally speaking, any matrix analysis software where the structure is modeled in three- dimensions may be referred to as a three-dimensional finite element analysis (3D FEA). The NCHRP Project 12-79 research adopts the more restrictive definition of 3D FEA stated by AASHTO/NSBA G13.1 (2011). According to G13.1, an analysis method is classified as 3D FEA if: 1. The superstructure is modeled fully in three dimensions, 2. The individual girder flanges are modeled using beam, shell, or solid type elements, 3. The girder webs are modeled using shell or solid type elements, 4. The cross-frames or diaphragms are modeled using truss, beam, shell, or solid type elements as appropriate, and 5. The concrete deck is modeled using shell or solid elements (when considering the response of the composite structure). Section 2.8 of the Project Task 8 report (Appendix C to the contractors’ final report) pro- vides a detailed description of the specific finite element modeling procedures employed for the elastic first- and second-order 3D FEA solutions as well as the full nonlinear test simulations conducted in the NCHRP Project 12-79 research. One additional 3D FEA solution (using a less restrictive definition of the term) is employed for limited additional checking and verification of the above linear elastic and geometric nonlinear 3D FEA solutions in the NCHRP Project 12-79 research. This approach involves a second TWOS frame element implemented in the GT-Sabre software (Chang, 2006; Chang and White, 2008). The GT-Sabre TWOS frame element formulation accommodates the geometrically nonlinear modeling of singly symmetric I-girders, where the cross-section shear center and centroid are located at different elevations. In addition, in the GT-Sabre software, all of the girder reference axes (taken as the shear-center axis) are modeled at their correct physical elevations, and all of the individual cross-frame members are modeled explicitly at their precise elevation in the physical bridge. The connection of these components to the girder reference axes is accomplished by the use of rigid offsets. Furthermore, the height of the girder reference axes above the bearings is modeled by rigid offsets, and the load height of the slab dead weight effects is included in the element formulation. Therefore, the GT-Sabre model captures all the essential three-dimensional attributes of the structure geometry. This approach is referred to as a TWOS 3D-frame method in the project’s Task 8 report. Specific comparisons of the geometric nonlinear results from GT-Sabre and ABAQUS are discussed subsequently in Section 3.2.4 of this report. 2.8 Data Reduction and Assessment of Analysis Procedures Task 8A of the NCHRP Project 12-79 research involved extensive data reduction and inter- pretation of the results from the various studies of Task 7. The detailed results of this research are documented in the Task 8 report, “Evaluation of Analytical Methods for Construction Engineer- ing of Curved and Skewed Steel Girder Bridges,” Appendix C to the contractors’ final report. Key results from this task are summarized in Chapter 3 of this report.

Research Approach 21 2.9 Development of Improvements to Simplified Methods Task 8B of the NCHRP Project 12-79 research involved the identification of various short- comings of the conventional simplified analysis methods studied in Tasks 7 and 8A and the development of specific improvements to these methods that lead to significantly better accuracy at little additional effort or cost. Specific calculations, as well as important considerations in the software implementation of these methods, were addressed. Several of the key improvements for the analysis of tub-girder bridges have already been outlined in Sections 2.7.1 and 2.7.2, and, with the exception of the saw-tooth top-flange major-axis bending stress effects, were included as part of the “conventional” analysis calculations evaluated in Tasks 7 and 8 of the research. This is because these improvements are all implemented as part of 1D line-girder calculations as well as “post-processing” calculations to determine top flange lateral bending stresses and bracing component forces given the internal major-axis bending moments and torques determined either from the 1D or 2D analysis procedures. These improvements are discussed in more detail in Section 3.2.6 of this report. The key improvements for the analysis of I-girder bridges require implementation within software if they are to be used efficiently in design practice. Furthermore, it is valuable to illustrate the critical inadequacies of the conventional methods to emphasize the importance of making the recommended improvements. Therefore, for I-girder bridges, the above Tasks 7 and 8 focus on evaluation of the accuracy of the simplified methods without the benefit of these improvements. The critical shortcomings of the conventional models and the essential improvements developed in the NCHRP Project 12-79 research are as follows: •• The conventional 2D-grid models used in current practice substantially underestimate the girder torsional stiffnesses in I-girder bridges. This is because the software considers only the St. Venant torsional stiffness of the girders. The contribution of warping torsion to the girder responses is generally neglected. It is interesting to note that competent structural engineers would never discount the girder warping rigidity ECw, and thus use only the girder St. Venant torsional stiffness GJ, when evaluating the lateral-torsional buckling (LTB) resistance of I-girders. Doing so would underestimate the girder LTB resistances in practical constructed geometries so drastically that the I-girders would become useless. Yet, it is common practice to completely discount the girder warping rigidity when conducting a structural analysis. This practice generally results in dramatic over-estimation of the structural displacements when curved I-girders are modeled with nodes along the arc between the cross-frame locations. Furthermore, it tends to discount the significant transverse load paths in highly skewed bridges, since the girders are so torsionally soft (in the structural model) that they are unable to accept any significant load from the cross-frames causing torsion in the girders. As such, the cross-frame forces can be under-estimated to a dramatic extent. In the Project 12-79 Task 8B research, this limitation is addressed by the development of an equivalent St. Venant torsion constant that accounts approximately for the girder stiffness from warping torsion. Section 3.2.2 of this report makes the case for this essential improvement. •• The conventional 2D-grid models commonly use an equivalent beam stiffness model for the cross-frames that substantially misrepresents the cross-frame responses. Fortunately, in many I-girder bridges, cross-frame deformations are small enough compared to the girder displacements such that the cross-frames perform essentially as rigid compo- nents in their own plane. However, in cases of significantly skewed I-girder bridges having “nuisance stiffness” transverse load paths (Krupicka and Poellot, 1993) and/or in general wide I-girder bridges, the deformations of the cross-frames can be a significant factor in the overall bridge response.

22 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges The Project 12-79 Task 8B research addressed this issue by the development of equivalent beam elements that capture the “exact” in-plane response for various cross-frame configurations. Section 3.2.3 of this report makes the case for this essential improvement. •• The conventional 2D-grid models do not address the calculation of girder flange lateral bending in skewed I-girder bridges. The current AASHTO LRFD Specifications Article C6.10.1 states: In the absence of calculated values of f from a refined analysis, a suggested estimate for the total f in a flange at a cross-frame or diaphragm due to the use of discontinuous cross-frame or diaphragm lines is 10.0 ksi for interior girders and 7.5 ksi for exterior girders. These estimates are based on a limited examination of refined analysis results for bridges with skews approaching 60 degrees from normal and an average D/bf ratio of approximately 4.0. In regions of the girders with contiguous cross-frames or diaphragms, these values need not be considered. Lateral flange bending in the exterior girders is substantially reduced when cross-frames or diaphragms are placed in discontinuous lines over the entire bridge due to the reduced cross-frame or diaphragm forces. A value of 2.0 ksi is suggested for f, for the exterior girders in such cases, with the suggested value of 10 ksi retained for the interior girders. In all cases, it is suggested that the recommended values of f be proportioned [apportioned] to dead and live load in the same proportion as the unfactored major-axis dead and live-load stresses at the section under consideration. An examination of cross-frame or diaphragm forces is also considered prudent in all bridges with skew angles exceeding 20 degrees. The above recommendations are intended as coarse estimates of the total unfactored stresses associated with the controlling strength load condition. Hence, for an example location in a straight skewed bridge governed by the STRENGTH I load combination, with discontinuous cross-frames over only a portion of the bridge and with a ratio of dead load stress to total stress (dead plus live load) of 1⁄3, the nominal total dead load flange lateral bending stress in the exterior girders may be taken as 7.5 ksi × 1⁄3 = 2.5 ksi. If discontinuous cross-frame lines are used throughout the entire bridge, then using this same example dead-to-live-load ratio, f may be taken equal to 2.0 ksi × 1⁄3 = 0.7 ksi. In both of these cases, the dead load f values may be taken as 10.0 × 1⁄3 = 3.3 ksi on the interior girders. In lieu of using a more rational method of determining the flange lateral bending effects, the NCHRP Project 12-79 research recommends that the value of f from the above AASHTO (2010) provisions should be combined additively with the results from other estimates for the effects of overhang bracket loads and horizontal curvature when using 1D (line-girder) and 2D-grid analysis methods. However, the variety of geometries and framing conditions in highway bridges is extensive, involving a large range of skew, length, width, number of spans, and curvature combinations. Therefore, the above recommendations are very coarse estimates. Section 3.2.4 describes a method to more closely predict the f stresses caused by skew effects within a 2D-grid analysis. •• None of the analysis calculations commonly employed in current bridge design practice address the calculation of internal locked-in forces due to cross-frame detailing. Yet, AASHTO (2010) Article C6.7.2 states that for curved I-girder bridges, “ . . . the Engineer may need to consider the potential for any problematic locked-in stresses in the girder flanges or the cross-frames or diaphragms . . . ” This article goes on to state, “The decision as to when these stresses should be evaluated is currently a matter of engineering judgment. It is anticipated that these stresses will be of little consequence in the vast majority of cases and that the resulting twist of the girders will be small enough that the cross-frames or diaphragms will easily pull the girders into their intended position and reverse any locked-in stresses as the dead load is applied.” This statement reflects a limited understanding of the detailed behavior associated with the locked-in forces due to steel dead load fit (SDLF) or total dead load fit (TDLF) detailing of the cross-frames. One major misconception in this statement is that these forces are canceled by the dead load effects calculated by the 2D-grid analysis or 3D FEA. This implicit assumption

Research Approach 23 is false. The 2D-grid and 3D FEA calculations, conducted without the modeling of initial lack-of-fit effects, only give the internal forces in the bridge associated with no-load fit (NLF) detailing. Any locked-in forces, due to the lack of fit of the cross-frames with the girders in the undeformed geometry, add to (or subtract from) the forces determined from the 2D-grid or 3D FEA design analysis. Fortunately, at many locations in a given bridge, the SDLF or TDLF detailing effects tend to be opposite in sign to the internal forces due to the dead loads. Therefore, the 2D-grid or 3D FEA solutions for the stresses at these locations are conservative (potentially, undesirably so). However, there are important locations where the SDLF or TDLF detailing effects and the dead load effects can be additive. These locations depend on the characteristics of the bridge geometry. Substantial effort was invested in the NCHRP Project 12-79 research to thoroughly evaluate the detailed behavior associated with the conceptually simple SDLF and TDLF detailing of the cross-frames in steel I-girder bridges. Sections 3.3 through 3.5 highlight the major findings and applications of this work. However, possibly the most important point related to the locked- in forces caused by SDLF and TDLF detailing is that they can be included in 2D-grid or 3D FEA calculations with relative ease and with little computational expense. Section 3.2.5 discusses how these locked-in force effects can be included in both of these types of analysis. •• Little guidance is available in the current literature on methods that can be used to estimate fit-up forces. In order to evaluate the potential for fit-up difficulties in the field for a given steel erection stage, generally, the engineer must conduct some evaluation of the corresponding fit-up forces. Better and more complete guidelines for conducting these types of analysis would be very useful. Section 3.3.5 of this report highlights major NCHRP Project 12-79 Task 8B findings that address this need. 2.10 Development of Guidelines for the Level of Construction Analysis, Plan Detail, and Submittals The tenth task of the NCHRP Project 12-79 studies involved the development of guidelines for the level of construction analysis, plan detail, and submittals for curved and skewed steel girder bridges. As noted previously, this major objective of the project is addressed by the Task 9 report “Recommendations for Construction Plan Details and Level of Construction Analysis,” which is included as Appendix B of this document. Section 3.6 of this report outlines the major recommendations from these guidelines.