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

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24 3.1 Evaluation of Conventional Simplified Analysis Methods A substantial number of studies were conducted as part of NCHRP Project 12-79 to determine the ability of approximate 1D and 2D methods of analysis to capture the behavior predicted by refined 3D finite element models. This chapter summarizes the findings and applications from the above research. Section 3.1.1 first addresses procedures for checking of (and in many cases, preventing) large second-order amplifications. Once these considerations are addressed, attention can be focused on selecting a suitable method of analysis for estimating the primary (i.e., first-order) forces, stresses, and displacements. Section 3.1.2 presents an overall scoring matrix for use in selecting the appropriate analysis type for I-girder bridges. Sections 3.1.3 and 3.1.4 provide examples illustrating how the scoring matrix should be used. Sections 3.1.5 and 3.1.6 parallel the above sections and focus on tub-girder bridges. Sections 3.1.2 through 3.1.6 focus on the evaluation of conventional methods of 1D line-girder and 2D-grid analysis (i.e., methods of 1D line-girder and 2D-grid analysis repre- sentative of the current standards of care in the bridge design profession). However, as noted in the statement of the objectives and scope of this research (Section 1.3) and in the summary of Task 8B of the project, development of improvements to simplified methods (see Section 2.9), substantial research effort was devoted to identifying the major causes of shortcomings in the conventional methods and to the development of easily implemented, low-cost solutions that provide substantial improvements to these methods. Sections 3.2 through 3.4 describe these improvements. The ultimate goal of the NCHRP Project 12-79 research is to provide substantive recommenda- tions on the level of construction analysis, plan detail, and submittals for curved and skewed steel girder bridges. The project’s Task 9 report, “Recommendations for Construction Plan Details and Level of Construction Analysis,” included as Appendix B of this document, addresses this goal. Section 3.6 of this chapter provides an overview of this guidelines document. 3.1.1 Checking for (and Preventing) Large Second-Order Amplification 3.1.1.1 Global Second-Order Amplification In certain situations, steel I-girder bridges can be vulnerable to overall (i.e., global) stability- related failures during their construction. The noncomposite dead loads must be resisted predominantly by the steel structure prior to hardening of the concrete deck. Relatively narrow C H A P T E R 3 Findings and Applications

Findings and Applications 25 I-girder bridge units (i.e., units with large span-to-width ratios) may be susceptible to global stability problems rather than cross-section or individual unbraced length strength limit states (Yura et al., 2008). Furthermore, due to second-order lateral-torsional amplification of the displacements and stresses, the limit of the structural resistance may be reached well before the theoretical elastic buckling load. Therefore, in curved and/or skewed bridge structures sensitive to second-order effects, simply ensuring that the loads for a given configuration are below an estimated global elastic buckling load is not sufficient. Large displacement amplifications can make it difficult to predict and control the structure’s geometry during construction well before the theoretical elastic buckling load is reached. Possible situations with the above characteristics include widening projects on existing bridges, pedestrian bridges with twin girders, phased construction involving narrow units, and erection stages where only a few girders of a bridge unit are in place. In all of these cases, the problem unit is relatively long and narrow. The NCHRP Project 12-79 research recommends a simple method that can be used to alert the engineer to undesired response amplifications due to global second-order effects. The linear response prediction obtained from any of the first-order analyses can be multiplied by the following amplification factor: AF M M G G crG = − 1 1 max Eq. 2 where MmaxG is the maximum total moment supported by the bridge unit for the loading under consideration, equal to the sum of all the girder moments, and M C sE L I IcrG b s ye x= pi2 2 Eq. 3 is the elastic global buckling moment of the bridge unit (Yura et al., 2008). In Equation (2.25), Cb is the moment gradient modification factor applied to the full bridge cross-section moment diagram, s is the spacing between the two outside girders of the unit, E is the modulus of elastic- ity of steel, I I b c Iye yc yt= + Eq. 4 is the effective moment of inertia of the individual I-girders about their weak axis, where Iyc and Iyt are the moments of inertia of the compression and tension flanges about the weak-axis of the girder cross-section respectively, b and c are the distances from the mid-thickness of the ten- sion and compression flanges to the centroidal axis of the cross-section, and Ix is the moment of inertia of the individual girders about their major-axis of bending (i.e., the moment of inertia of a single girder). Yura et al. (2008) developed Equation 3 considering multiple girder systems with up to four girders in the cross-section of the bridge unit. The individual girders were assumed to be prismatic and all the girders were assumed to have the same cross-section. The engineer must exercise judgment in applying this equation to general I-girder bridge units with stepped or other non-prismatic cross-sections, as well as cases where the different I-girders have different cross-sections.

26 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges In addition to providing an estimate of the second-order effects on the overall girder displacements, Equation 2 also can be used to predict potential increases in the girder stresses. Hence, to address potential second-order amplification concerns with narrow structural units, the results of an approximate 1D or 2D analysis should be amplified, using Equation 2, prior to conducting the constructability checks required by AASHTO LRFD Article 6.10.3. The limit states in Article 6.10.3 are: •• Nominal initial yielding due to combined major-axis bending and flange lateral bending, •• Strength under combined major-axis and flange lateral bending, •• Bend buckling or shear buckling of the girder webs, •• Reaching a flange lateral bending stress of 0.6Fy, and •• Reaching the factored tensile modulus of rupture of the concrete deck in regions not adequately reinforced to control the concrete crack size. Section 2.9 of the NCHRP Project 12-79 Task 8 report provides a detailed example showing the results of these calculations for an example narrow bridge unit that experienced construction difficulties (over-rotation of the bridge cross-section) during the deck placement. The NCHRP Project 12-79 research suggests that Equation 2 should be used to detect possible large response amplifications during preliminary construction engineering. If the amplifier shows that a structure will exhibit significant nonlinear behavior during the deck placement, then in many cases, the scheme adopted for the construction should be revisited. In these cases, by conducting a detailed 3D FEA of the suspect stages, one often may find that the physical second-order amplification is somewhat smaller than predicted by the above simple estimate. If the second-order amplification is still relatively large in the more refined model, one should consider reducing the system response amplification by providing shoring or by bracing off of adjacent units. If AFG from Equation 2 is less than approximately 1.1, it is recommended that the influence of global second-order effects may be neglected. If it is found necessary to construct a structure that has potentially large response amplification during the deck placement, the engineer should perform a final detailed check of the suspect stages using a second-order (geometric nonlinear) 3D FEA. (It is recommended that this scenario with an AFG larger than approximately 1.25 should be considered as requiring an accurate second-order 3D FEA.) In addition, it will be necessary to ensure that the deck placement does not deviate from the assumptions of the analysis in any way that would increase the second- order effects. Obviously, in most cases, it is best to stay away from these issues. Substantial second-order effects during the steel erection may be a concern in some situations; however, particularly during the earliest stages of the steel erection, if the steel stresses are small and if the influence of the displacements on fit-up is not a factor, large second-order amplification of the deformations typically does not present a problem. Steel tub girders generally have as much as 100 to more than 1,000 times the torsional stiffness of a comparable I-girder section. Therefore, when steel tub girders are fabricated with proper internal cross-frames to restrain their cross-section distortions as well as a proper top flange lateral bracing (TFLB) system, which acts as an effective top flange plate creating a pseudo- closed cross-section with the commensurate large torsional stiffness, lateral-torsional buckling is rarely a concern. Furthermore, second-order amplification in bridge tub girders is rarely of any significance even during lifting operations and early stages of the steel erection. However, overturning stability of curved tub girders, or tub-girder bridge units, can be a significant issue if it is not properly identified and addressed. Overturning stability considerations are addressed in Section 3.1.1.3.

Findings and Applications 27 3.1.1.2 Second-Order Amplification of Flange Lateral Bending between Cross-Frame Locations Design-analysis compression flange lateral bending estimates usually are based on a first- order analysis. They do not consider any potential amplification of the bending between cross- frame locations due to second-order effects. That is, they do not consider equilibrium on the deflected geometry of the structure in the evaluation of the stresses. The corresponding “local” second-order flange lateral bending stresses (local to a given unbraced length between cross- frames) can be estimated by multiplying the first order f values by the following amplification factor discussed in Article 6.10.1.6 of the AASHTO LRFD Specifications: AF f Fb cr = − ≥0 85 1 1 0 . . Eq. 5 where Fcr is the elastic lateral-torsional buckling stress for the compression flange, based on the unbraced length Lb between the cross-frames, and fb is the maximum major-axis bending stress in the compression flange within the targeted unbraced length. It should be noted that when Equation 5 gives a value less than 1.0, AF must be taken equal to 1.0; in this case, the second-order amplification of the flange lateral bending is considered negligible. When determining the amplification of f in horizontally curved I-girders, White et al. (2001) indicate that for girders with Lb/R ≥ 0.05, Fcr in Equation 5 may be determined using KLb = 0.5Lb. For girders with Lb/R < 0.05, they recommend using the actual unsupported length Lb in Equation 5. The use of KLb = 0.5Lb for Lb/R ≥ 0.05 gives a better estimate of the amplification of the bending deformations associated with the approximate symmetry boundary conditions for the flange lateral bending at the intermediate cross-frame locations and assumes that an unwinding stability failure of the compression flange is unlikely for this magnitude of the girder horizontal curvature. Figure 3-1 illustrates the flange lateral deflections associated with the horizontal curvature effects, as well as the unwinding stability failure mode for a straight elastic member. 3.1.1.3 Overturning Stability Two straight dashed lines are drawn along the length direction of the plan sketches in Figure 3-2. One of the dashed lines is the chord between the fascia girder bearings on the outside of the curve. The other is the chord between the fascia girder bearings on the inside of the curve. Also shown on the plan sketches is the symbol “x,” which indicates the centroid of the deck area (and hence the approximate centroid of dead weight of the structure). For bridges that are more highly curved (smaller R), the centroid (x) is closer to the outside chord line. If the curvature is (a) Flange lateral deflection mode associated with horizontal curvature effects (b) “Unwinding” elastic stability failure mode for straight members Cross-frame position (TYP.) Figure 3-1. Second-order elastic deflection of a horizontally curved flange versus the unwinding stability failure mode of the compression flange in a straight member.

28 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges such that the centroid is positioned directly over the outside chord line, then all the bridge reactions have to be zero except for the reactions at the outside bearings. That is, the bridge unit is at the verge of tipping about its outside bearings (assuming a single span, simply supported ends, and no hold-downs at the other bearings). This is obviously an extreme condition. Even a bridge with a much smaller curvature (larger radius of curvature) would require hold-downs at bearings closer to the center of curvature to handle uplift and equilibrate (or balance) the structure weight. The more common practice is to avoid uplift at any of the bearings. As noted previously in Section 2.5, the bridge torsion index IT provides a rough indication of the tendency for uplift at the bridge bearings. IT is equal to 1.0 for the extreme hypothetical case where the deck centroid is located on the chord between the bearings on the outside of the curve, as discussed previously. It is equal to 0.5 for a straight bridge with zero skew. The NCHRP Project 12-79 research studies identified that simply supported I-girder bridges with IT ≥ 0.65 are often susceptible to uplift at some of the bearings under the nominal (unfactored) dead plus live loads. 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. Continuous-span bridges can tolerate larger IT values due to the continuity with adjacent spans. It should be emphasized that IT is only a rough indicator of uplift or overturning problems. It is relatively easy to calculate, but it is based on the idealization that the structure weight is uniformly distributed over the slab area. Also, when considering intermediate stages of the steel erection, it should be noted that until all the girders are erected and connected together sufficiently with cross-frames, the width of the bridge cross-section is only equal to the perpendicular distance between the connected girders on the inside and the outside of the curve. (IT can be determined using this approximation for intermediate stages of the steel erection.) In addition, it should be noted that individual spans of continuous-span bridges may be supported essentially in a simple-span condition during some of the intermediate steel erection stages. Lastly, it should be noted that on highly curved bridge units, it may be useful to start the placement of the deck concrete on the inside of the curve to avoid a potential bearing uplift or overturning stability issue. 3.1.2 Selection of Analysis Methods for I-Girder Bridges A quantitative assessment of the accuracy of conventional 1D line-girder and 2D-grid analysis methods was obtained in the NCHRP Project 12-79 research by identifying several error measures Ls = 300 ft, w = 30 ft, wg = 24 ft, R = 1000 ft, Ls/R = 0.30 Ls = 300 ft, w = 80 ft, wg = 74 ft, R = 353 ft, Ls/R = 0.85 Fascia Girder (Typ.) Deck Centroid Deck Centerline Figure 3-2. Plan geometries of two representative simple-span horizontally curved bridges with Ls = 300 ft.

Findings and Applications 29 that compared the conventional approximate (1D and 2D method) solutions to 3D FEA bench- mark solutions. Using these quantitative assessments, the simplified methods of analysis were graded based on a scoring system developed to provide a comparative evaluation of the accuracy of each analysis method with regard to its ability to predict various structural responses. Table 3-1 summarizes the results for the various methods and responses monitored for I-girder bridges. The grading rubric was as follows: •• A grade of A is assigned when the normalized mean error is less than or equal to 6 percent, reflecting excellent accuracy of the analysis predictions. Traditional 2D-Grid 1D-Line Girder Traditional 2D-Grid 1D-Line Girder C (I C < 1) B B A B C (I C > 1) D C B C S (I S < 0.30) B B A A S (0.30 < I S < 0.65) B C B B S (I S > 0.65) D D C C C&S (I C > 0.5 & I S > 0.1) D F B C C (I C < 1) B C A B C (I C > 1) F D F C S (I S < 0.30) B A A A S (0.30 < I S < 0.65) B B A B S (I S > 0.65) D D C C C&S (I C > 0.5 & I S > 0.1) F F F C C (I C < 1) C C B B C (I C > 1) F D C C S (I S < 0.30) NAa NAa NAa NAa S (0.30 < I S < 0.65) Fb Fc Fb Fc S ( I S > 0.65) Fb Fc Fb Fc C&S (I C > 0.5 & I S > 0.1) Fb Fc Fb Fc C (I C < 1) C C B B C (I C > 1) F D C C S (I S < 0.30) NAd NAd NAd NAd S (0.30 < I S < 0.65) Fb Fe Fb Fe S (I S > 0.65) Fb Fe Fb Fe C&S (I C > 0.5 & I S > 0.1) Fb Fe Fb Fe C (I C < 1) NAf NAf NAf NAf C (I C > 1) NAf NAf NAf NAf S (I S < 0.30) B A A A S (0.30 < I S < 0.65) B B A B S (I S > 0.65) D D C C C&S (I C > 0.5 & I S > 0.1) F F F C Response Geometry Worst-Case Scores Mode of Scores Major-Axis Bending Stresses Vertical Displacements Cross-Frame Forces Flange Lateral Bending Stresses Girder Layover at Bearings a Magnitudes should be negligible for bridges that are properly designed & detailed. The cross-frame design is likely to be controlled by considerations other than gravity-load forces. b Results are highly inaccurate due to modeling deficiencies addressed in Ch. 6 of the NCHRP 12-79 Task 8 report. The improved 2D-grid method discussed in this Ch. 6 provides an accurate estimate of these forces. c Line-girder analysis provides no estimate of cross-frame forces associated with skew. d The flange lateral bending stresses tend to be small. AASHTO Article C6.10.1 may be used as a conservative estimate of the flange lateral bending stresses due to skew. e Line-girder analysis provides no estimate of girder flange lateral bending stresses associated with skew. f Magnitudes should be negligible for bridges that are properly designed & detailed. Table 3-1. Matrix for recommended level of analysis—I-girder bridges.

30 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges •• A grade of B is assigned when the normalized mean error is between 7 percent and 12 percent, reflecting a case where the analysis predictions are in “reasonable agreement” with the bench- mark analysis results. •• A grade of C is assigned when the normalized mean error is between 13 percent and 20 percent, reflecting a case where the analysis predictions start to deviate “significantly” from the benchmark analysis results. •• A grade of D is assigned when the normalized mean error is between 21 percent and 30 percent, indicating a case where the analysis predictions are poor, but may be considered acceptable in some cases. •• A score of F is assigned if the normalized mean errors are above the 30 percent limit. At this level of deviation from the benchmark analysis results, the subject approximate analysis method is considered unreliable and inadequate for design. The normalized mean error used in the assessment of the above grades is calculated as µe FEA ii N N R e= = ∑1 1 • max Eq. 6 where N is the total number of sampling points along the bridge length in the approximate model, RFEAmax is the absolute value of the maximum response obtained from the FEA, and ei is the absolute value of the error relative to the 3D FEA benchmark solution at point i: e R Ri approx FEA= − Eq. 7 The summation in Equation 6 is computed for each girder line along the full length of the bridge. The largest resulting value is reported as the normalized mean error for the bridge. The error measure µe is useful for the overall assessment of the analysis accuracy since this measure is insensitive to local discrepancies, which can be due to minor shifting of the response predictions, etc. The normalized local maximum errors, ei/RFEAmax, generally are somewhat larger than the normalized mean error. Also, in many situations, unconservative error at one location in the bridge leads to comparable conservative error at another location. Hence, it is simpler to not consider the sign of the error as part of the overall assessment of the analysis accuracy. In Table 3-1, the scoring for the various measured responses is subdivided into six categories based on the bridge geometry. These categories are defined as follows: •• Curved bridges with no skew are identified in the geometry column by the letter “C.” •• The curved bridges are further divided into two subcategories, based on the connectivity index, defined as: I R n m C cf = +( ) 15000 1 Eq. 8 where R is the minimum radius of curvature at the centerline of the bridge cross-section in feet throughout the length of the bridge, ncf is the number of intermediate cross-frames in the span, and m is a constant taken equal to 1 for simple-span bridges and 2 for continuous-span bridges. In bridges with multiple spans, IC is taken as the largest value obtained from any of the spans. •• Straight skewed bridges are identified in the geometry column by the letter “S.” •• The straight skewed bridges are further divided into three subcategories, based on the skew index: I w L S g s = tanθ Eq. 9 where wg is the width of the bridge measured between fascia girders, q is the skew angle mea- sured from a line perpendicular to the tangent of the bridge centerline, and Ls is the span

Findings and Applications 31 length at the bridge centerline. In bridges with unequal skew of their bearing lines, q is taken as the angle of the bearing line with the largest skew. •• Bridges that are both curved and skewed are identified in the geometry column by the letters “C&S.” Two letter grades are indicated for each of the cells in Table 3-1. The first grade corresponds to the worst-case results encountered for the bridges studied by NCHRP Project 12-79 within the specified category. The second grade indicates the mode of the letter grades for that category (i.e., the letter grade encountered most often for that category). It is useful to understand the qualifier indicated on the “C&S” bridges, i.e., “(IC > 0.5 & IS > 0.1)” in Table 3-1. If a bridge has an IC < 0.5 and an IS > 0.1, it can be considered as a straight-skewed bridge for the purposes of assessing the expected analysis accuracy. Furthermore, if a bridge has an IC > 0.5 and an IS ≤ 0.1, it can be considered as a curved radially supported bridge for these purposes. Table 3-1 can be used to assess when a certain analysis method can be expected to give acceptable results. The following examples illustrate how this table should be used. 3.1.3 I-Girder Bridge Level of Analysis Example 1 Consider a horizontally curved steel I-girder bridge with radial supports, “very regular” geometry (constant girder spacing, constant deck width, relatively uniform cross-frame spacing, etc.), and IC < 1, for which the engineer wants to perform a traditional 2D-grid analysis to determine the forces and displacements during critical stages of the erection sequence. (It should be noted that if IC is calculated for an intermediate stage of the steel erection in which some of the cross-frames have not yet been placed, the number of intermediate cross-frames ncf in Equation 8 should be taken as the number installed in the erection stage that is being checked. In addition, the radius of curvature R and the constant m should correspond to the specific intermediate stage of construction being evaluated, not the bridge in its final erected configuration.) For the girder major-axis bending stresses and vertical displacements (fb and D), the results are expected to deviate somewhat from those of a 3D analysis in general, since a worst-case score of B is assigned in Table 3-1 for these response quantities. The worst-case normalized mean error in these results from the 2D-grid analysis will typically range from 7 percent to 12 percent, compared to the results from a refined geometric nonlinear benchmark 3D FEA. However, one can expect that for most bridges, the errors will be less than or equal to 6 percent, based on the mode score of A for both of these responses. Therefore, in this example, if the major-axis bending stresses and vertical displacements are of prime interest, a 2D-grid model should be sufficient if worst-case errors of approximately 12 percent are acceptable. Given that the bridge has “very regular” geometry, it is likely that the fb and D errors are less than or equal to 6 percent. (The worst-case score is considered as the appropriate one to consider when designing a bridge with complicating features such as a poor span balance, or other “less regular” geometry characteristics.) It is important to note that the engineer can compensate for potential unconservative major-axis bending stress errors in the design by adjusting the performance ratios targeted for the construction engineering design checks. For example, for the above bridge, the engineer may require that the performance ratios be less than or equal to 1/1.12 = 0.89 or 1/1.06 = 0.94 for the girder flexural resistance checks to gain further confidence in the adequacy of the resulting design. Conversely, over-prediction or under-prediction of the vertical displacements can be equally bad. Nevertheless, 12 percent or 6 percent displacement error may be of little consequence if the magnitude of the displacements is relatively small, or if the deflections are being calculated at an early stage of

32 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges the steel erection and it is expected that any resulting displacement incompatibilities or loss of geometry control can be subsequently resolved. However, if the magnitude of the displacements is large, or if it is expected that the resulting errors or displacement incompatibilities may be difficult to resolve, the engineer should consider conducting a 3D FEA of the subject construction stage to gain further confidence in the calculated displacements. This step in the application of Table 3-1 is where the bridge span length enters as an important factor, since longer-span bridges tend to have larger displacements. It should be noted that compared to the creation of 3D FEA models for overall bridge design, including the calculation of live-load effects, the development of a 3D FEA model for several specific construction stages of potential concern involves a relatively small amount of effort. This is particularly the case with many of the modern software interfaces that facilitate the definition of the overall bridge geometry. For calculation of the girder flange lateral bending stresses and the cross-frame forces in the above example bridge, the worst-case errors are expected to be larger, on the order of 13 percent to 20 percent (corresponding to a grade of C for both of these responses). However, the mode score is B, and since the bridge has a very regular geometry, it is likely that the normalized mean error in the flange lateral bending stresses and cross-frame forces is less than 12 percent. If these errors are acceptable in the engineer’s judgment, then the 2D-grid analysis should be acceptable for the construction engineering calculations. As noted above, the engineer can compensate for these potential errors by reducing the target performance indices. In addition, with respect to the flange lateral bending stress, it should be noted that the f values are multiplied by 1⁄3 in the AASHTO 1⁄3 rule equations. Therefore, the errors in f have less of an influence on the performance ratio than errors in fb when considering the strength limit state. When checking the AASHTO flange yielding limit for constructability, both f and fb have equal weights though. Based on these considerations, the simplest way to compensate for different potential unconservative errors in the f and fb values is to multiply the calculated stresses from the 2D-grid analysis by 1.20 and 1.12 (or 1.12 and 1.06) respectively prior to checking the performance ratios. 3.1.4 I-Girder Bridge Level of Analysis Example 2 Consider a straight steel I-girder bridge, with skewed supports and a skew index Is = 0.35 (corresponding to the intermediate erection stage being evaluated), for which the engineer wants to perform a traditional 2D-grid analysis to determine the forces and displacements. After reviewing Table 3-1, it is observed that for the major axis bending stresses and vertical deflections, a worst-case score of B is shown for straight skewed I-girder bridges with 0.30 < IS ≤ 0.65. Furthermore, it can be observed that the mode of the scores for these bridge types is a B for the major-axis bending stresses and an A for the vertical displacements. Therefore, a properly prepared conventional 2D-grid analysis would be expected to produce major-axis bending stress and vertical deflection results that compare reasonably well with the results of a second-order elastic 3D FEA, such that the normalized mean error would be expected to be less than or equal to 12 percent. If the layout of the cross-frames in the skewed bridge is such that overly stiff (nuisance) transverse load paths are alleviated, the engineer may expect that the error in the displacement calculations may be close to 6 percent or less. In this case, the engineer should be reasonably confident in the 2D-grid results for the calculation of the displacements. As noted in the previous example, the potential unconservative errors in the stresses can be compensated for in the construction engineering design checks; however, positive or negative displacement errors are equally bad.

Findings and Applications 33 The girder layover (i.e., the relative lateral deflection of the flanges) at the skewed bearing lines is often of key interest in skewed I-girder bridges. Table 3-1 shows that the girder layover calculations essentially have the same magnitude of error (i.e., the same resulting grades, as the girder vertical displacements). This is because properly designed and detailed skewed bearing line cross-frames are relatively rigid in their own planes compared to the lateral stiffness of the girders. Hence, the girder layovers are essentially proportional to the girder major-axis bending rotations at the skewed bearing lines. For the calculation of the cross-frame forces and/or the girder flange lateral bending stresses in the above example, one should observe that the conventional 2D-grid procedures are entirely unreliable. That is, the scores in Table 3-1 are uniformly an F. The reason for this poor performance of the traditional 2D-grid methods is the ordinary modeling of the girder torsional properties using only the St. Venant torsional stiffness GJ/L. The physical girder torsional stiffnesses are generally much larger due to restraint of warping (i.e., flange lateral bending) effects. In addition, for wide skewed bridges and/or for skewed bridges containing specific overly stiff (nuisance) transverse load paths, the limited accuracy of the cross-frame equivalent beam stiffness models used in conventional 2D-grid methods may lead to a dramatic loss of accuracy in the cross-frame forces. Lastly, conventional 2D-grid methods generally do not include any calculations of the girder flange lateral bending stresses due to skew. Hence, the score for the calculation of the flange lateral bending stresses is also an F in Table 3-1. 3.1.5 Selection of Analysis Methods for Tub-Girder Bridges Similar to the I-girder bridges, a quantitative assessment of the analysis accuracy of tub-girder bridges was obtained in the NCHRP Project 12-79 research by focusing first on the normalized mean errors in the approximate (1D and 2D method) solutions for the girder major-axis bending stresses, internal torques and vertical displacements, compared to benchmark 3D FEA results. Using the quantitative assessments, the various methods of analysis were assigned scores in the same manner as the scoring for the I-girder bridge responses. Table 3-2 summarizes the scores for the above responses in tub-girder bridges. 2D-P1 1D-Line Girder 2D-P1 1D-Line Girder S B B A B C B C A B C&S B C B B S F F D F C D D A B C&S F F A B S B B A A C A B A A C&S B B A A S B B A A C NAa NAa NAa NAa C&S B B A A Worst-Case Scores Mode of Scores Major-Axis Bending Stresses Vertical Displacements Girder Layover at Bearing Lines Response Geometry Girder Torques a Magnitudes should be negligible where properly designed and detailed diaphragms or cross- frames are present. Table 3-2. Matrix 1 for recommended level of analysis—tub-girder bridges.

34 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges It is interesting that the Table 3-2 scores for the major-axis bending stresses and vertical displacements are relatively good. However, the worst-case scores for the internal torques are generally quite low. These low scores are largely due to the fact that the internal torques in tub-girder bridges can be sensitive to various details of the framing, such as the use and location of external intermediate cross-frames or diaphragms, the relative flexibility of these diaphragms as well as the adjacent internal cross-frames within the tub girders, skewed interior piers without external cross-frames between the piers at the corresponding bearing line, incidental torques introduced into the girders due to the specific orientation of the top flange lateral bracing system members (particularly for Pratt-type TFLB systems), etc. Jimenez Chong (2012) provides a detailed evaluation and assessment of the causes for the errors in the girder internal torques for the tub-girder bridges considered in the NCHRP Project 12-79 research. Similar to the considerations for I-girder bridges, the external diaphragms and/or cross-frames typically respond relatively rigidly in their own plane compared to the torsional stiffness of the girders (even though the tub-girder torsional stiffnesses are significantly larger than those of comparable I-girders). Therefore, the girder layovers at skewed bearing lines tend to be proportional to the major-axis bending rotation of the girders at these locations. As a result, the errors in the girder layover calculations obtained from the approximate methods tend to be similar to the errors in the major-axis bending displacements. The connectivity index, IC, does not apply to tub-girder bridges. This index is primarily a measure of the loss of accuracy in I-girder bridges due to the poor modeling of the I-girder torsion properties. For tub-girder bridges, the conventional St. Venant torsion model generally works well as a characterization of the response of the pseudo-closed section tub girders. Hence, IC is not used for characterization of tub-girder bridges in Table 3-2. Furthermore, there is only a weak correlation between the accuracy of the simplified analysis calculations and the skew index IS for tub-girder bridges. Therefore, the skew index is not used to characterize tub-girder bridges in Table 3-2 either. Important differences in the simplified analysis predictions do exist, however, as a function of whether the bridge is curved, “C,” straight and skewed, “S,” or curved and skewed, “C&S.” Therefore, these characterizations are shown in the table. It should be noted that there was a measureable decrease in the accuracy of the 2D-grid solutions for the tub-girder bridges obtained with Program P2 compared to Program P1. Since the research team had greater control over the calculations, as well as more detailed information regarding the specifics of the procedures in Program P1, the P1 results are pre- sented in Table 3-2 as being the most representative of the results achievable with a 2D-grid procedure. In addition to the above assessments, the accuracy of the bracing component force calculations in tub-girder bridges is assessed separately in Table 3-3. It is useful to address the accuracy of these response calculations separately from the ones shown in Table 3-2 since the simplified bracing component force calculations take the girder major-axis bending moments, torques, and applied transverse loads as inputs and then apply various useful mechanics of materials approximations to obtain the force estimates. That is, there are two distinct sources of error in the bracing component forces relative to the 3D FEA benchmark solutions: 1. The error in the calculation of the input quantities obtained from the 1D line-girder or 2D-grid analysis, and 2. The error introduced by approximations in the bracing component force equations. Chapter 2 of the NCHRP Project 12-79 Task 8 report provides an overview of the bracing component force equations evaluated here, which are used frequently in current professional practice. It should be noted that the calculation of the top flange lateral bending stresses in tub girders is included with the bracing component force calculations. This is because these stresses

Findings and Applications 35 are influenced significantly by the interaction of the top flanges with the tub-girder bracing systems. The NCHRP Project 12-79 research observed that in many situations, the bracing component force estimates are conservative relative to the 3D FEA benchmark solutions. Therefore, it is useful to consider a signed error measure for the bracing component force calculations. In addition, the bracing component dimensions and section sizes often are repeated to a substantial degree throughout a tub-girder bridge for the different types of components. Therefore, it is useful to 2D-P1 1D-Line Girder 2D-P1 1D-Line Girder S D D D C C D F B F C&S Da F B F Pratt TFLB System C F A F S Fb C C -- c -- C&S -- -- Pratt TFLB System -- -- S C C C F F C&S F Fd Pratt TFLB System F F S C C C -- A C&S -- C Pratt TFLB System D D S NAe NAe C F F C&S F F Pratt TFLB System -- Ff S NAe NAe C -- -- C&S -- D Pratt TFLB System B -- S C C C F F C&S F Fd S C C C -- A C&S -- C c The symbol "--" indicates that no cases were encountered with this score. d Modified from a B to an F considering the grade for the C bridges. e For straight-skewed bridges, the internal intermediate cross-frame diagonal forces tend to be negligible. f Modified from an A to an F considering the grade for the C and C&S bridges. TFLB & Top Internal CF Strut Force Internal CF Diagonal Force Positive (Conservative) Negative (Unconservative) Positive (Conservative) Negative (Unconservative) Positive (Conservative) Negative (Unconservative) Top Flange Lateral Bending Stress (Warren TFLB Systems) Positive (Conservative) Negative (Unconservative) b Large unconservative error obtained for bridge ETSSS2 due to complex framing. If this bridge is considered as an exceptional case, the next worst-case unconservative error is -15 % for NTSSS2 (grade = C). a Modified from a C to a D considering the grade for the C and the S bridges. Response Sign of Error Geometry Worst-Case Scores Mode of Scores TFLB Diagonal Force Table 3-3. Matrix 2 for recommended level of analysis—tub-girder bridges.

36 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges quantify the analysis error as the difference between the maximum of the component forces determined by the approximate analysis minus the corresponding estimate from the 3D FEA benchmark, that is: e R R Rapprox FEA FEAmax . . .= −( )max max max Eq. 10 for a given type of component. The grades for these responses were assigned based on the same scoring system used for the assessments based on the normalized mean error with one exception: Separate grades were assigned for the positive (conservative) errors and for the neg- ative (unconservative) errors in Table 3-3. In situations where no negative (unconservative) errors were observed in all of the bridges considered in a given category, the symbol “—” is shown in the cells of the matrix, and the cells are unshaded. The mode of the grades is shown only for the top flange diagonal bracing forces in Table 3-3. The mode of the grades for the other component force types is not shown because of substantial positive and negative errors in the calculations that were encountered in general for the tub-girder bridges and because, in cases where a clear mode for the grades existed, the mode of the grades was the same as the worst-case grade. In addition to the above considerations, it should be noted that current simplified estimates of the tub-girder bridge bracing component forces are generally less accurate for bridges with Pratt-type top flange lateral bracing (TFLB) systems compared to Warren and X-type systems. A small number of tub-girder bridges with Pratt-type TFLB systems were considered in the NCHRP Project 12-79 research. Therefore, the composite scores for these bridges are reported separately in Table 3-3. 3.1.6 Tub-Girder Bridge Level of Analysis Example Consider a horizontally curved steel tub-girder bridge with a Warren top flange lateral bracing system and skewed supports for which the engineer wants to perform a traditional 2D-grid analysis to determine the forces and displacements during critical stages of the erection sequence. The bridge has a “very regular” geometry (constant girder spacing, constant deck width, a relatively uniform top flange lateral bracing [TFLB] system and internal cross-frame spacing, solid plate end diaphragms, single bearings for each girder, etc.). A properly prepared 2D-grid analysis would be expected to produce major axis bending stresses and vertical deflections with mean errors less than 12 percent relative to a rigorous 3D FEA solution, since the worst-case score assigned for both of these quantities is a B in Table 3-2 for the subject “C&S” category. Furthermore, it can be observed that the mode of the scores for the vertical displacements is an A; hence, given the “very regular” geometry of the above bridge, it is expected that the vertical displacements most likely would be accurate to within 6 percent. Unfortunately, the worst-case score is an F for the 2D-grid estimates of the internal torques in the C&S bridges. As noted previously, this low score is due to the fact that the internal torques in tub-girder bridges can be very sensitive to various details of the framing, such as the use and location of external intermediate cross-frames or diaphragms, the relative flexibility of these diaphragms as well as the adjacent internal cross-frames within the tub girders, skewed interior piers without external cross-frames between the piers at the corresponding bearing line, incidental torques induced in the girders due to the specific orientation of the top flange lateral bracing system members (particularly for Pratt-type TFLB systems), etc. Fortunately though, the web and bottom flange shear forces due to the internal torques are often relatively small compared to the normal stresses associated with the major-axis bending response of the

Findings and Applications 37 girders. Furthermore, the mode of the scores for the internal torques is an A from Table 3-2. Therefore, the engineer must exercise substantial judgment in estimating what the expected error may be for the internal torque from a 2D-grid analysis and in assessing the impact of this error on the bridge design. As noted for I-girder bridges, one can compensate for any antici- pated potential unconservative error in the internal force or stress response quantities by scal- ing up the corresponding responses by the anticipated error, or by adjusting the target values of the performance ratios. Based on Table 3-3, the worst-case score for the positive (conservative) error in the calculation of the TFLB diagonal forces in the above example is a D, whereas the mode of the scores is a B. The table shows that no unconservative errors were encountered in this calculation for the tub-girder bridges studied in NCHRP Project 12-79. Since the example bridge is “very regular,” the engineer may assume that the TFLB diagonal force calculations are conservative, but reasonably accurate, relative to the refined 3D FEA benchmark values. For both the TFLB and top internal cross-frame strut forces and the internal cross-frame diagonal forces in C&S bridges, Table 3-3 shows a grade of F for the conservative error. Also, the table shows that no unconservative errors were encountered in the NCHRP Project 12-79 calculations for these responses. Therefore, the engineer can assume that the forces for these components, as determined from a 2D-grid analysis plus the bracing component force equations, are highly conservative. It should be noted that the forces in the top struts of the internal cross-frames near exterior diaphragm or exterior cross-frame locations can be very sensitive to the interaction of the external diaphragm or cross-frame with the girders. These forces should be determined based on consideration of statics at these locations, given the forces transmitted to the girders from the external diaphragm or cross-frame components. NCHRP Project 12-79 did not consider these component forces in its error assessments. Lastly, Table 3-3 shows that the tub-girder top flange lateral bending stresses tend to be estimated with a high degree of conservatism by 2D-grid methods combined with the bracing component force equations. In addition, no unconservative errors were encountered in the tub-girder bridges studied by NCHRP Project 12-79 for the top flange lateral bending stresses. Therefore, the engineer can also assume that these stress estimates are highly conservative. 3.2 Improvements to Conventional Analysis Methods Various essential improvements to conventional methods of analysis were developed during the course of the NCHRP Project 12-79 research. In all cases, the project team strived to identify specific sources of errors relative to 3D FEA benchmark solutions and then to develop solutions to these errors by addressing the inadequacies in the conventional models at a fundamental structural mechanics level. In addition, solutions were sought that provided substantial benefits, yet involved little computational expense and were relatively easy to implement in software. The following sections highlight these major improvements. First, Section 3.2.1 introduces a basic simply supported I-girder bridge used for illustration purposes in a number of the subsequent sections. This bridge was designed and tested at the FHWA Turner-Fairbank Highway Research Center in prior FHWA research (Jung, 2006; Jung and White, 2008). The bridge has substantial horizontal curvature and zero skew. Furthermore, the bridge was designed at, or slightly above, a number of limits in the AASHTO LRFD Design Specifications. Therefore, this structure is particularly sensitive to a number of parameters that influence the accuracy of simplified analysis methods. Because of the fact that the bridge is relatively basic and easily modeled in a short amount of time, due to the sensitivity of the structure to attributes influencing the analysis accuracy, and since the calculations are backed up by a large

38 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges number of experimental measurements, this bridge is an excellent case for discussion of analysis error sources as well as analysis improvements. Section 3.2.1 is followed by four sections that describe four key improvements to conventional 2D-grid analysis methods recommended by the NCHRP Project 12-79 research for I-girder bridges: 1. Use of an equivalent St. Venant torsion constant for the I-girders that accounts approximately for the contribution of warping (i.e., flange lateral bending) to the girder torsional stiffnesses, 2. Use of equivalent beam models for the cross-frames that better capture the true bending and shear racking stiffnesses of various types of cross-frames used in I-girder bridges, 3. Direct calculation of flange lateral bending stresses in skewed or curved and skewed I-girder bridges based on the cross-frame forces calculated from the above improved 2D-grid procedures, and 4. 2D-grid (or 3D FEA) calculation of locked-in forces due to steel dead load fit (SDLF) or total dead load fit (TDLF) detailing of the cross-frames in curved and/or skewed I-girder bridges. The first three of the above sections focus on the recommended calculations and their imple- mentation, as well as the resulting improvement of the analysis results. However, the fourth of these sections focuses just on the recommended calculations and their implementation. A longer discussion is necessary to convey the detailed characteristics of the locked-in force effects from SDLF or TDLF detailing. These considerations are addressed in depth in Section 3.3. 3.2.1 The FHWA Test Bridge Figure 3-3 shows key particulars of the geometry for a 90 ft. span curved I-girder bridge tested in 2005 at the FHWA Turner-Fairbank Highway Research Center (Jung, 2006; Jung and White, 2008). This bridge is introduced here because it is used for illustration purposes in a number of the subsequent sections. The reader is referred to the Task 8 report (Appendix C of the con- tractors’ final report), and to Sanchez (2011), Ozgur (2011) and Jimenez Chong (2012) for similar results to those discussed in Sections 3.2.2 through 3.2.6 for a wide range of bridges. The radius of curvature of the centerline of the FHWA Test Bridge was 200 ft. and the bridge cross-section contained three I-girders spaced at 8.75 ft. Figure 3-4 shows a photo of the bridge after the girders G3 and G2 were placed on the supports and the cross-frames were installed. The fascia girder on the outside of the curve (G1) is blocked on the laboratory floor toward the right-hand side of the photo. It should be noted that the naming of the outside girder as G1 and the inside girder as G3 follows the naming convention adopted within the NCHRP Project 12-79 project research. The above referenced research reports refer to the outside girder as G3 and the inside girder as G1. The total width of the 8-in. concrete deck was 23.5 ft., with 3.0-ft. overhangs outside the fascia girders. The bridge was constructed using V-type cross-frames composed of circular tube section members. These HSS 5 × ¼-in. members had areas comparable to the member areas that would have been required for this bridge with other more common cross-frame section types. However, the tube-member cross-frames facilitated the measurement of the cross-frame forces, since the tubes were essentially instrumented as load cells. The FHWA Test Bridge was designed with a number of characteristics that pushed or slightly exceeded the limits of prior AASHTO curved I-girder bridge specifications, as well as some of the limits in the current AASHTO (2010) LRFD Specifications, as follows: •• Intermediate cross-frames were employed at only three cross-sections within the bridge span, resulting in a subtended angle between the cross-frames of Lb/R = 0.1125. This is slightly larger than the maximum limit of Lb/R = 0.10 specified in AASHTO (2010) Article 6.7.4.2.

Findings and Applications 39 (a) G1 cross-section (b) G2 cross-section (b) G3 cross-section 48.062 in 0 24.185 in 24.185 in Ls = 90 ft., R = 2000 ft., w = 23.5 ft. .362 in PL 1.389 in PL 1.001 in PL G1 (d) Bridge cross-section (e) Bridge plan view 48.057 in 22.168 in 14.188 in 0.323 in PL 1.002 in PL 0.877 in PL 48.102 in 0.331 in PL 17.266 in 0.886 in PL 0.883 in PL 12.188 in Figure 3-3. FHWA Test Bridge (EISCR1) geometry. Figure 3-4. FHWA Test Bridge (EISCR1) during the steel erection, with cross-frames attached between girders G2 and G3 (Jung, 2006; Jung and White, 2008).

40 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges •• The fascia girder on the outside of the curve (G1) utilized a hybrid HPS 70W bottom flange. Hybrid curved girders were not permitted in the AASHTO Specifications at the time of the FHWA research. The use of grade 70 steel allowed the bottom flange thickness for Girder G1 to be reduced from approximately 2 in. if grade 50 steel had been used. •• Due to the grade 70 bottom flange on G1, the 48-in. web depth for this girder slightly violates the arc-span length-to-depth requirements in AASHTO (2010) Article 2.5.2.6.3. However, this bridge satisfies the Span/800 deflection limit of AASHTO Article 2.5.2.6.2. •• The web slenderness D/tw of all the girders was close to the AASHTO (2010) limit of 150 for straight and curved transversely stiffened web panels. •• Transverse stiffening of the girder webs varied from a maximum of close to do/D = 3 in all the girders near the mid-span of the bridge to do/D < 1 near the supports for Girder G1. Prior AASHTO Specifications have required a much tighter spacing of web transverse stiffeners in curved I-girder webs. •• The top compression flange slenderness bfc/2tfc was slightly larger than 12, which is the maximum limit on the flange slenderness specified in AASHTO LRFD Article 6.10.2.2. •• Both girders G1 and G2 were sized close to the AASHTO (2010) strength limits. The tight radius of curvature (R = 200 ft.) combined with the use of only three intermediate cross-frames (ncf = 3) results in a value of IC of 18.75 from Equation 8 for this bridge in its final constructed condition. Therefore, this bridge significantly exceeds the IC ≤ 1 limit utilized for scoring the accuracy of the simplified analysis methods in Table 3-1. As noted previously, the NCHRP Project 12-79 research found that the connectivity index, IC, tended to correlate well with the magnitude of the errors exhibited by conventional 2D-grid methods. Figures 3-5 and 3-6 show several photos of the test bridge during its construction. 3.2.2 Improved I-Girder Torsion Model for 2D-Grid Analysis As noted previously, the conventional use of just the St. Venant term (GJ/L) in characterizing the torsional stiffness of I-girders results in a dramatic underestimation of the true girder Figure 3-5. FHWA Test Bridge, overhang brackets attached to the fascia girder on the outside of the curve (Girder G1 per the NCHRP Project 12-79 naming convention, Girder G3 in reports and papers on the FHWA research) (Jung, 2006; Jung and White, 2008).

Findings and Applications 41 torsional stiffness. This is due to the neglect of the contributions from flange lateral bending, that is, warping of the flanges, to the torsional properties. Even for intermediate steel erection stages where some of the cross-frames are not yet installed, the typical torsional contribution from the girder warping rigidity (ECw) is substantial compared to the contribution from the St. Venant torsional rigidity (GJ). It is somewhat odd that structural engineers commonly would never check the lateral-torsional buckling capacity of a bridge I-girder by neglecting the term ECw and using only the term GJ. Yet, it is common practice in conventional 2D-grid methods to neglect the warping torsion contribution coming from the lateral bending of the flanges. The NCHRP Project 12-79 Task 8B research observed that an equivalent torsion constant, Jeq, based on equating the stiffness GJeq/Lb with the analytical torsional stiffness associated with assuming warping fixity at the intermediate cross-frame locations and warping free conditions at the simply supported ends of a bridge girder, potentially could result in significant improvements to the accuracy of 2D-grid models for I-girder bridges. This observation was based in part on the prior research developments by Ahmed and Weisgerber (1996), as well as the commercial implementation of this type of capability within the software RISA-3D (RISA, 2011). The term Lb in the stiffness GJeq/Lb is the unbraced length between the cross-frames. When implementing this approach, a different value of the equivalent torsional constant Jeq must be calculated for each unbraced length having a different Lb or any difference in the girder cross-sectional properties. Furthermore, it is important to recognize that the use of a length less than Lb typically will result in a substantial over-estimation of the torsional stiffness. Therefore, when a given unbraced length is modeled using multiple elements, it is essential that the unbraced length Lb be used in the equations for Jeq, not the individual element lengths. By equating GJeq/Lb to the torsional stiffness (T/f) for the open-section thin-walled beam associated with warping fixity at each end of a given unbraced length Lb, where T is the applied end torque and f is corresponding relative end rotation, the equivalent torsion constant is obtained as: J J pL pL pL pL eq fx fx b b b −( ) = − ( ) + ( )−[ ] 1 1 2 sinh cosh b bpLsinh( )     −1 Eq. 11 Figure 3-6. FHWA Test Bridge placement of the slab concrete (Jung, 2006; Jung and White, 2008).

42 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Similarly, by equating GJeq/Lb to the torsional stiffness (T/f) for the open-section thin-walled beam associated with warping fixity at one end and warping free boundary conditions at the opposite end of a given unbraced length, one obtains: J J pL pL pL eq s fx b b b −( ) − = − ( ) ( )    1 1 sinh cosh Eq. 12 Section 6.1.2 of the Task 8 report shows a complete derivation of these equivalent torsion constants. The assumption of warping fixity at all of the intermediate cross-frame locations is certainly a gross approximation. TWOS 3D-frame analysis (see Section 2.7.3 for a description of this terminology) generally shows that some flange warping (i.e., cross-bending) rotations occur at the cross-frame locations. However, the assumption of warping fixity at the intermediate cross-frame locations leads to a reasonably accurate characterization of the girder torsional stiffnesses pertaining to the overall deformations of a bridge unit as long as: •• There are at least two I-girders connected together, and •• They are connected by enough cross-frames such that the connectivity index IC is less than 20 (IC ≤ 20). Therefore, the FHWA Test Bridge in its final constructed condition represents essentially the maximum limit at which the above approach provides a sufficient solution. 3.2.2.1 Comparison of the Vertical Displacement Results from Various Approaches for the FHWA Test Bridge Figure 3-7 shows representative results for the vertical displacement of Girder G1 of the FHWA Test Bridge under the nominal (unfactored) total dead load, i.e., the self-weight of the structural steel plus the weight of the concrete deck, with all the loads being resisted by the noncomposite structure. The benchmark 3D FEA prediction, 4.49 in. downward deflection of the centerline of G1 at its mid-span, matches closely with the results from the physical test (Jung, 2006; Jung and White, 2008). Figure 3-8 shows a rendering of the magnified bridge vertical deflections from the 3D FEA solution. Figure 3-7. FHWA Test Bridge (EISCR1) vertical displacements in fascia girder on the outside of the curve under total dead load (unfactored).

Findings and Applications 43 The 1D-line girder solution in Figure 3-7 is obtained using the V-load method, applying the primary loads as well as the V-loads to Girder G1 on the outside of the horizontal curve, and ana- lyzing the uniaxial bending deformations of the member subjected to these loads. Unfortunately, this solution under-predicts the vertical displacement of Girder G1 by 33.4 percent. The actual displacements are larger due to the coupling between the girder mid-span vertical displacements and the twisting deformations, particularly the twisting deformations of the girder near the supports. That is, the twisting of the girder near the supports produces corresponding additional vertical displacements at the mid-span. In the 2D-grid analyses, the girders are modeled by four elements within each of their unbraced lengths, with the nodes being positioned along the circular arc between the cross-frames. Only one conventional 2D-grid solution is shown in the plot. However, essentially the same results are obtained by models built in MDX and LARSA-4D, as well as one other 2D-grid model created using a third independent code for this problem. The improved method of modeling the cross-frames discussed in Section 3.2.3 is employed for all the 2D-grid solutions discussed in this section. One can observe that the improved 2D-grid solution, based on the use of Jeq, predicts a slightly larger mid-span displacement of Girder G1 than obtained in the 3D FEA solution (4.73 in. versus 4.49 in.). Furthermore, it should be noted that the benchmark 3D FEA solution shown here includes geometric nonlinearity. However, if the 3D FEA simulation model is run as a geometrically linear (first-order analysis) solution, the mid-span displacements reduce to only 4.40 in. Therefore, the second-order effects on the vertical displacements are only about 2 percent for this structure and loading. The improved 2D-grid solution over-estimates the 3D FEA linear elastic solution by 7.5 percent and over-predicts the 3D FEA geometric nonlinear benchmark solution by 5.3 percent. Conversely, the conventional 2D-grid solutions predict a displacement of 15.37 in. at the mid-span of G1, 342 percent larger than the benchmark result. Obviously, this discrepancy between the 2D-grid prediction and the physical result is some cause for concern. Table 3-4 summarizes the above numerical results and presents a number of additional solutions for the vertical displacement of Girder G1. The only solution that is dramatically in error is the conventional 2D-grid solution discussed above. Interestingly, if the girders are modeled using only one straight conventional element between each of the cross-frames, the predicted dis- placement is 4.35 in. (3.1 percent smaller than the benchmark solution). Furthermore, the same Figure 3-8. Magnified deflected geometry of EISCR1 under total dead load, from 3D FEA (displacements scaled 20x, initial vertical camber not scaled).

44 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges displacement solution is obtained if the girders are modeled with four elements positioned along the straight chord between each of the cross-frames. If the improved 2D-grid model is used with only one straight element between each of the cross-frames, or with four elements positioned along the straight chord between each of the cross-frames, the predicted displacement is 4.28 in. (4.7 percent smaller than the benchmark solution). One solution to the above problem that some engineers might consider is to simply never represent any unbraced length with the nodes positioned along the curved arc of an I-girder member when using conventional methods. However, this can lead to an awkward handling of situations where the same model is used to analyze girders with different numbers of cross-frames inserted in the structure at an early intermediate stage of construction. In addition, a common practice for modeling of staged deck placement in 2D-grid programs such as MDX is to use conventional frame elements to model portions of the bridge that are not yet composite, but then to connect these elements to a plate representation of the slab once the slab has been activated for a given stage. Usually, it is desirable to use more than one plate element within each unbraced length for modeling of the structure in its composite condition. Furthermore, it is desirable to model the slab with nodes along arcs about the center of curvature (assuming a circular arc). Therefore, it is desirable to also position the I-girder element nodes along the circular arcs between the cross-frames. Several additional 2D-grid solutions are provided at the bottom of Table 3-4 using the TWOS frame element in MASTAN2 (MASTAN2, 2011; McGuire et al., 2000). The reader is referred to Section 2.7.2 for a discussion of the meaning of a TWOS 2D-grid analysis. One can observe that this element predicts a displacement of 4.42 in. (1.6 percent smaller than the benchmark solution) when four elements are used between each of the cross-frames and the nodes are positioned along the circular arc, whereas a displacement of 4.34 in. (3.3 percent smaller than the benchmark solution) is obtained when four elements are used with the nodes positioned along a straight chord between the cross-frames. Another interesting solution is one obtained using the TWOS frame element if the warping is artificially fully fixed at the intermediate cross-frame locations rather than being modeled as a 2D-grid discretization and idealization G1 (in.) Four conventional e le ments within each L b , node s located on the circular arc 15 .3 7 One convent ional element withi n each L b , straight between each CF 4.35 Four conventional e le ments within each L b , straight between each CF 4.35 Four elements within each L b , nodes lo cated on the circular arc, using J e q 4.73 Four elements within each L b , strai ght between each CF, using J e q 4.28 One element within each L b , straight between each CF, usi ng J e q 4.28 Four TWOS b eam elements withi n each L b , nodes located on the circular arc 4.42 Four TWOS b eam elements withi n each L b , straight between each CF 4.34 Four TWOS b eam elements withi n each L b , nodes located on the circular arc, wa rping fixed at the intermed ia te CF locations 4.32 Four TWOS b eam elements withi n each L b , straight between each CF, warping fi xed at the intermediate CF locations 4.28 Four TWOS b eam elements withi n each L b , nodes located on the circular arc, all girder J values taken equal to zero 4.43 One TWOS beam element within each L b , straight between each CF 4.38 One TWOS beam element within each L b , straight between each CF, warpi ng fi xed at the intermediate CF locations 4.32 Table 3-4. FHWA Test Bridge (EISCR1) mid-span vertical displacement of Girder G1 (DG1) under total dead load (unfactored) for different 2D-grid girder discretizations and idealizations, cross-frames modeled using shear deformable beam element (DG1 = 4.49 in., second-order 3D FEA; DG1 = 4.40 in., first-order 3D FEA).

Findings and Applications 45 continuous function along the girder lengths. In this case, the displacement prediction is reduced to 4.28 in. (4.5 percent smaller than the benchmark displacement). This demonstrates the accuracy of assuming full fixity at these locations, subject to the limitations discussed in the development of the improved 2D-grid procedure (i.e., at least two I-girders connected together, and IC ≤ 20). The accuracy of these solutions is not influenced significantly if the girders are modeled using only one element between the cross-frames for the evaluation of the non-composite behavior of this bridge. Lastly, it is interesting to investigate the influence of completely neglecting the St. Venant torsional contribution to the stiffness within the Mastan TWOS 2D-grid analysis. In this case, the solution with four elements modeled along the circular arcs between the cross-frames increases from 4.42 in. to 4.43 in. The torsional stiffness of the I-girders is dominated by the restraint of flange warping, once the girders are sufficiently connected together such that IC is less than approximately 20. It should be noted that if the improved 2D-grid model is used to predict the vertical displacements for the girder pair G2-G3, connected together by the bearing-line cross-frames and only a single intermediate cross-frame, the results are very poor. In this case, the connectivity index IC is equal to 38. However, if G2 and G3 are connected together by three intermediate cross-frames, as shown in Figure 3-4, the accuracy of the improved 2D-grid prediction is comparable to that demonstrated above. One question that could be asked relative to the implementation of the improved 2D-grid model is the following: Will the improved 2D-grid model work properly when used to model composite conditions with a plate-eccentric beam approach? Separate studies conducted in the NCHRP Project 12-79 research indicate that the improved 2D-grid model works sufficiently with a plate representation of the slab as long as one handles the calculation of the girder bottom flange lateral bending stresses properly. Chang and White (2008) have shown in previous research that, if TWOS 3D-frame elements are used to model the steel I-girders, and if these elements are constrained by rigid offsets to a shell representation of the slab, the bottom flange lateral bending stresses predicted by the TWOS element are drastically underestimated. This is because the slab bending stiffness, along with the rigid link of the TWOS element to the slab, essentially prevents any lateral bending of the bottom flange in the TWOS model (unless special procedures are invoked to release the torsional constraint of the TWOS element by the slab model). As noted above, these issues are not encountered for the improved 2D-grid element with the use of Jeq, as long as the flange lateral bending stresses (f) are calculated properly. The calculation of f is addressed subsequently in Section 3.2.4. 3.2.2.2 Mechanical Explanation of the Large Error in the Conventional 2D-Grid Procedure with the Nodes Positioned along a Circular Arc Consider the idealized 2D-frame representation of an I-girder unbraced length between two cross-frames shown in Figure 3-9. All the degrees of freedom at the end nodes are constrained with M, θM, θ β1 β2 M cos β1 M cos β1 M cos β2 M cos β2 M sin β1 M sin θ1 M sin β2 M sin β2 LL Figure 3-9. Behavior for a chorded representation of a curved girder using four straight elements.

46 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges the exception of the rotational dof corresponding to the applied end moments. In addition, all of the dofs are free at the interior nodes in this model. The reason for the dramatic over-prediction of the vertical displacements by the conventional 2D-grid procedure shown in the previous section is due to the fact that statics requires that a portion of the bending moment transmitted to the elements must be taken by element torsion (when the elements are modeled along a circular arc). However, the torsional stiffness of the elements is drastically underestimated by the St. Venant torsional stiffness GJ/L, where L is the length of each of the individual elements. The bending end rotations in the idealized problem shown in Figure 3-9 can be calculated with relative ease, using the principle of virtual work, as: 1 1 11 1 1 1 • • •θ β β β β= ( ) ( ) + ( ) ( )sin sin cos cosM L GJ M L EI M L GJ M Eq. 13 + ( ) ( ) + ( )1 12 2 2• •sin sin cos coβ β β sβ2( )L EI After some algebra, Equation 13 may be expressed as: θ β β= +  ∑ ML EI EI GJ i i i sin cos2 2 Eq. 14 Clearly, one can obtain a significant contribution from the torsional term in this equation, i.e., the first term inside the brackets. If one substitutes the relevant parameters for Girder G1 of the previous problem into Equation 14, i.e., I = 37,600 in4, J = 29.4 in4, L = 70.45 in., b1 = 0.04218 radians and b2 = 0.01462 radians, along with the yield moment of the G1 cross-section, M = MyG1 = 5,564 ft-kips = 66,768 in-kips, one obtains q = 0.0369 radians. This prediction matches precisely with the first-order elastic MASTAN2 solution for this problem. If the improved 2D-grid procedure with Jeq(fx-fx) = 688.3 in.4 is employed, the predicted value for q is 0.00983 radians. Finally, if the more rigorous TWOS frame element solution is employed, where Cw = 1,662,000 in. 6 for Girder G1, equal end rotations of q = 0.00889 radians are obtained. The rotations predicted by the recommended improved 2D-grid model are 10.6 percent larger than the more rigorous predictions from the TWOS frame element. Correspondingly, the conventional 2D-grid solution over-predicts the end rotations by 315 percent. 3.2.2.3 Comparison of the Major-Axis Bending Stresses from Various Approaches for the FHWA Test Bridge Figures 3-10 and 3-11 show the results from the different methods of analysis for the major-axis bending stresses in the FHWA Test Bridge Girder G1 (on the outside of the horizontal curve) and Girder G3 (on the inside of the horizontal curve) respectively. It should be noted that these results are shown at the factored load level, i.e., 1.5 of the total dead load, associated with the Strength IV loading condition. One can observe that the major-axis bending stress at the mid-span of Girder G1 is under- predicted by 12.3 percent in the 1D line-girder solution conducted using the V-load method. All of the other solutions are very comparable. Therefore, it can be concluded that the poor vertical displacement estimate for Girder G1 does not impact the accuracy of the conventional 2D-grid estimate of the major-axis bending stress in this problem. The percentage accuracy of the results is not as good for Girder G3. However one should notice that the scale on the vertical axis of the plot in Figure 3-11 is highly magnified compared to the

Findings and Applications 47 scale in Figure 3-10. It may be more useful to consider the differences between the maximum predicted stresses when considering the errors for Girder G3 in this bridge. The maximum major- axis bending stress in the 3D FEA simulation model is 8.1 ksi. The corresponding maximum predicted by the improved 2D-grid method is 9.9 ksi versus 10.8 ksi by the conventional 2D-grid solution. The 1D line-girder (V-load) solution exhibits the largest error in this problem, predicting a maximum major-axis bending stress of only 4.1 ksi. Furthermore, the 1D solution does not capture any semblance of the shape of the stress diagram from the benchmark. 3.2.3 Improved Equivalent Beam Cross-Frame Models Figure 3-12 shows the geometry of the V-type cross-frames used in the FHWA Test Bridge. The cross-frames are 34 in. deep and L = 8.75 ft. = 105 in. wide between the work points at the girder centerlines. The areas of all the tube members are A = 3.73 in.2 In this section, various Figure 3-10. FHWA Test Bridge (EISCR1) top flange major-axis bending stresses in the fascia Girder G1 on the outside of the curve under the Strength IV load combination (1.5  total dead load). Figure 3-11. FHWA Test Bridge (EISCR1) top flange major-axis bending stresses in the fascia Girder G3 on the inside of the curve under the Strength IV load combination (1.5  total dead load).

48 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges idealized beam solutions are compared to the “exact” equivalent beam stiffnesses of this cross- frame, where the “exact” solutions are taken as the stiffnesses from an explicit truss representa- tion of the cross-frames in their own plane. The most appropriate simplified equivalent beam modeling of the cross-frames becomes apparent by evaluating these results. 3.2.3.1 Equivalent Beam Stiffness Based on the Flexural Analogy Approach Figure 3-13 illustrates the calculation of the equivalent moment of inertia for the cross- frames in the FHWA Test Bridge using the “flexural analogy” approach discussed as one of two commonly used options in the AASHTO/NSBA (2011) G13.1 document “Guidelines for Steel Girder Bridge Analysis.” This is the default option for calculation of the cross-frame equivalent beam stiffness in the MDX software. The lighter arrows in the figure represent displacement constraints at the corner nodes of the cross-frame. The truss support reactions corresponding to the loading applied in the figure are shown with these arrows. The cross-frame is effectively supported as a propped cantilever and is loaded by an end moment at its simply supported end in the flexural analogy approach. It is fixed against rotation and vertical displacement at its left-hand side and restrained against vertical movement at its right-hand side in the figure. A couple composed of equal and opposite unit loads is applied to the top and bottom joints on the right-hand side. The associated horizontal displacements of the truss are determined via a 34 in 8.75 ft = 105 in All members, A = 3.73 in2 Figure 3-12. Cross-frame configuration, FHWA Test Bridge. 1 kip 1 kip 0.0005568 in 0.0005568 in -0.1473 kip 0.1473 kip -0.2761 kip 0.2761 kip θ = 2(0.0005568)/34 = 0.00003275 radians = ML/4EIeq = (34)(105)/4(29000)Ieq Ieq = 940 in4 Figure 3-13. Calculation of equivalent moment of inertia based on the flexural analogy method.

Findings and Applications 49 structural analysis. The equivalent beam moment of inertia, Ieq, is then calculated by equating the corresponding rotation at the right-hand side to the Euler-Bernoulli beam rotation M/(4EIeq/L), as shown in the figure. Figure 3-14 compares the physical cross-frame end shears and moments to the nodal shears and moments in the equivalent Euler-Bernoulli beam. One can observe that the moment induced at the left-hand side of the physical cross-frame is much smaller than the “carry-over moment” of one-half of the applied end moment in the equivalent Euler-Bernoulli beam element. In fact, it is even of the opposite sign. Correspondingly, the vertical shear forces induced in the physical cross-frame are much smaller than the ones associated with the equivalent beam based on the flexural analogy. These smaller internal forces are due to the shear raking deformations in the physical truss system. The equivalent Euler-Bernoulli beam does not consider any beam shear deformations. 3.2.3.2 Equivalent Beam Stiffness Based on the Shear Analogy Approach Figure 3-15 illustrates the second common method of determining an equivalent beam stiffness discussed by the AASHTO/NSBA (2011) G13.1 document. This approach is termed the shear analogy method. In this approach, the cross-frame is supported as a fixed-fixed beam subjected to a transverse shear force. In the figure, all of the truss dofs are fixed on the left-hand side, 34 in-kip 5.008 in-kip 0.2761 kip 0.2761 kip 34 in-kip 17 in-kip 0.4857 kip 0.4857 kip Cross-Frame Nodal Shears and Moments Equivalent Euler-Bernoulli Beam Nodal Shears and Moments Figure 3-14. Cross-frame nodal shears and moments and equivalent Euler-Bernoulli beam shears and moments based on flexural analogy method. 1 kip 1.544 kip -1.544 kip 1.544 kip -1.544 kip -1 kip 0.006543 in ∆ = 0.0006543 in = PL3/12EIeq = (1)(105)3/12(29000)Ieq Ieq = 508 in3 Figure 3-15. Calculation of equivalent moment of inertia based on the shear analogy method.

50 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges the horizontal dofs are constrained on the right-hand side, and the truss is subjected to a unit vertical load on the right-hand side. It should be noted that the vertical members at the sides of the cross-frames represent the stiffness of the girder webs and connection plates, which typically involves a larger effective area than the cross-frame members themselves. The unit load is applied on the right-hand side and the truss is supported on the left-hand side in Figure 3-15 such that no deformations of the end vertical elements come into play. The equivalent beam cross-frame stiffness is obtained by equating the relative end deflection to the Euler-Bernoulli beam solution P/(12EIeq/L 3). Figure 3-16 shows the nodal shears and moments for both the physical cross-frame and the equivalent Euler-Bernoulli beam in this problem. That is, the nodal shears and moments are identical for the equivalent beam idealization and the physical truss in this case. However, it should be noted that a large portion of the vertical displacement in the physical truss is due to shearing type deformations whereas the Euler-Bernoulli beam does not include any consideration of shear deformations. Therefore, the equivalent moment of inertia is in essence “artificially reduced” to account for these large shearing deformations in the shear analogy approach. 3.2.3.3 Equivalent Beam Stiffness for a Timoshenko Beam Element Figure 3-17 illustrates the first step of a more accurate approach for the calculation of the cross-frame equivalent beam stiffnesses. This approach simply involves the calculation of an equivalent moment of inertia, Ieq, as well as an equivalent shear area Aseq for a shear-deformable (Timoshenko) beam element representation of the cross-frame. In this approach, the equivalent 52.50 in-kip 52.50 in-kip 1 kip 1 kip Figure 3-16. Cross-frame nodal shears and moments and equivalent Euler-Bernoulli beam shears and moments based on shear analogy. 1 kip 1 kip 0.0009707 in 0.0009707 in 0.002998 in θ = 2(0.0009707)/34 = 0.0000571 = ML/EIeq = 34(105)/29000Ieq Ieq = 2156 in4 -1 kip 1 kip Figure 3-17. Calculation of equivalent moment of inertia based on pure bending.

Findings and Applications 51 moment of inertia is determined first based on pure flexural deformation of the cross-frame (zero shear). The cross-frame is supported as a cantilever at one end and is subjected to a force couple applied at the corner joints at the other end, producing constant bending moment. The associated horizontal displacements are determined at the free end of the cantilever, and the corresponding end rotation is equated to the value from the beam pure flexure solution M/(EIeq/L). One can observe that this results in a substantially larger Ieq and that this EIeq represents the “true” flexural rigidity of the cross-frame. In the second step of the improved calculation, using an equivalent Timoshenko beam element rather than a Euler-Bernoulli element, the cross-frame is still supported as a cantilever but is subjected to a unit transverse shear at its tip. Figure 3-18 shows the corresponding displacements and reactions for this model, as well as the Timoshenko beam equation for the transverse displacement and the solution for the Aseq of the FHWA Test Bridge cross-frame. It should be noted that the end rotation of the equivalent beam in Figure 3-18 is θ = − = ( ) ( )( )−( ) VL EI V GAeq seq2 2 2 1 105 2 29000 2156 1 2 6 29000 2 008 0 00004352. . .( ) ( )( ) = radians However, from the deflected shape in Figure 3-18, q = 2(0.001499)/34 = 0.00008818 radians. Therefore, it can be observed that the shear-deformable Timoshenko beam element is not able to match the “exact” kinematics of the cross-frame. Figure 3-19 compares the physical cross-frame end shears and moments to the nodal shears and moments for the equivalent Timoshenko beam for the case of a propped cantilever subjected to end moment. One can observe that the Timoshenko beam comes much closer to fitting the force response of the cross-frame, compared to the earlier result with the Euler-Bernoulli beam element in Figure 3-14. However, it can be seen that the Timoshenko beam shear forces are still 2.9 percent smaller than those of the physical truss, and the left-hand end moment is 16.5 percent larger than the “actual” left end moment. The left-hand moment is in the correct direction though in Figure 3-19, whereas in the previous Figure 3-14, the left-hand end moment is not even in the correct direction. 1 kip 0.001499 in 0.01086 in 0.001499 in ∆ = 0.01086 in = VL3/3EIeq + VL/GAeq = 1(105)3/3(29000)(2156) + (1)(105)(2.6)/29000Aseq Aseq = 2.008 in2 -3.088 kip 3.088 kip -1 kip Figure 3-18. Calculation of equivalent shear area based on tip loading of the cross-frame supported as a cantilever.

52 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 3.2.3.4 Overall Comparison of Cross-Frame Models Table 3-5 provides a detailed comparison of the force and displacement results for the three different equivalent beam elements described in the above compared to the “exact” results for the physical truss model of the cross-frame from Figure 3-12. All the “exact” solutions are shown in bold. It can be observed that the equivalent Euler-Bernoulli beams are able to fit the exact solution for only one response, whereas the Timoshenko beam is able to fit the exact solution for two responses. Furthermore, the Timoshenko beam provides a closer approximation to the physical truss results in the cases where the fit is not exact. This is due to the fact that the Timoshenko element is able to represent both flexure and shear deformations. The approximations are due in part to the fact that the Timoshenko beam formulation considered here is a close representation of prismatic solid web members. The truss-type cross-frame deformations generally lead to different stiffness results than provided by a prismatic solid web member though. It can be shown that the Timoshenko beam element provides a closer approximation of the physical cross-frame behavior compared to the Euler-Bernoulli beam for all other types of cross- frames typically used in I-girder bridges as well, including X and inverted V cross-frames with top and bottom chords, as well as X and V cross-frames without top chords. However, similar to the above demonstrations, the Timoshenko beam model is only able to provide an exact fit for two of the five responses listed across the rows of Table 3-5. Given the “exact” equivalent beam stiffness values developed in the above solutions, the next logical refinement is to develop generic X, V, inverted-V, X without top chord, and V without top chord models with variable width and height and variable cross-section area for the cross-frame members (including different cross-section areas for the different members). Section 6.2.2 of the Task 8 report describes the development of one “exact” equivalent beam element of this form as well as a rather easy implementation of this element as a user-defined element within the LARSA 4D software system. Sanchez (2011) provides detailed developments of this form for all of the above cross-frame types. 3.2.3.5 Influence of the Cross-Frame Equivalent Beam Stiffness Model on the Vertical Displacement Results in the FHWA Test Bridge Table 3-6 shows the influence of the different equivalent beam stiffness models considered in the above developments on the vertical displacement at the mid-span of the fascia girder (G1) on the outside of the horizontal curve in the FHWA Test Bridge. It can be observed that the 2D-grid model using the Timoshenko equivalent beam element generally provides the best estimate of the models developed in the above section. The results provided by the “exact” equivalent beam model of the test bridge cross-frame are essentially the same as those obtained using the 34 in-kip 5.008 in-kip 34 in-kip 5.834 in-kip 0.2761 kip 0.2761 kip 0.2682 kip 0.2682 kip Cross-Frame Nodal Shears and Moments Equivalent Shear-Deformable Element Nodal Shears and Moments Figure 3-19. Cross-frame nodal shears and moments and equivalent shear-deformable beam shears and moments.

Findings and Applications 53 Timoshenko beam formulation. The predicted mid-span displacement is 8.7 percent larger using the model with the Euler-Bernoulli element based on the shear analogy. This demonstrates that the cross-frame model can have a measurable influence on the prediction of the constructed geometry. The absolute difference in the displacements is relatively small for the test bridge; however, for a longer span, the difference could be more consequential. The last row of Table 3-6 gives the solution obtained if the cross-frame torsional stiffness is neglected (i.e., J = 0) for the cross-frame equivalent beam element. One can observe that this results in little change in the bridge vertical displacement. 3.2.3.6 FHWA Test Bridge Cross-Frame Forces Predicted by Different Methods Figure 3-20 shows the forces calculated in the two cross-frames at the mid-span of the FHWA Test Bridge from the various methods of analysis. One can observe that, of the various solutions, CF Model Ieq (in4) Aseq (in2) Mfar/Mnear propped cantilever subjected to end moment Transverse deflection of cantilever in pure bending (M = 34 in-k) Transverse deflection of tip-loaded fixed-fixed member (V = 1 k) End rotation of propped cantilever (M = 34 in-k ) Cantilever in pure bending, end rotation (M = 34 in-k) Euler-Bernoulli element with Ieq based on flexural analogy 940 NAa +0.5 6.88E-3 inches 3.54E-3 inches 3.27E-5 radiansb 13.1E-5 Euler-Bernoulli element with Ieq based on shear analogy 508 NAa +0.5 12.7E-3 inches 6.55E-3 inches 6.06E-5 radians 24.9E-5 Timoshenko beam element 2156 2.01 -0.172 3.00E-3 inches 6.23E-3 inches 3.28E-5 radians 5.71E-5 radians Physical truss model -- -- -0.147 3.00E-3 inches 6.55E-3 inches 3.27E-5 radians 5.71-5 radians a The shear area is effectively ∞ for the Euler-Bernoulli beam element. b Exact values are shown in bold font. Table 3-5. Comparison of equivalent beam responses to the physical truss cross-frame model responses for the V-type cross-frame of Figure 3-12. Cross-frame idealization G1 (in.) Sh ear-deformable (Timoshenko) be am element – I = 2156 in 4 , A s = 2.01 in 2 , J = 39.8 in 4 4.73 Equ ivalent Euler-Bernoulli beam element based on flexural analogy – I = 940 i n 4 , A s = , J = 39.8 in 4 4.87 Equ ivalent Euler-Bernoulli beam element based on shear analogy – I = 508 in 4 , A s = , J = 39.8 in 4 5.14 Sh ear deformab le (Timoshenko) beam el ement – I = 2156 in 4 , A s = 2.01 in 2 , J = 0 in 4 4.74 Table 3-6. FHWA Test Bridge (EISCR1) mid-span vertical displacement of Girder G1 (DG1) under total dead load (unfactored) for different 2D-grid cross-frame idealizations, girders modeled using Jeq and four elements in each Lb, nodes located on the circular arc (DG1 = 4.49 in., second-order 3D FEA; DG1 = 4.40 in., first-order 3D FEA).

54 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 274 33.4 133 23.2 +19.1 -46.8 +8.2 +32.7 -32 .3 17.8 17.6 +54.5 -96.6 -11.8 +49.9 -50 .2 27.1 27.3 (a) 3D FEA benchmark 249 60.0 179 10.0 +29.8 -63.2 +3.5 +39.7 -39 .7 21.6 21.6 +54.5 -87.9 -21.2 +39.7 -39 .7 21.6 21.6 (b) V-Load 259 28.0 108 30.1 +13.8 -38.2 +10.6 +29.0 -29 .0 15.8 15.8 +50.6 -91.3 -9.9 +48.5 -48 .5 26.3 26.3 (c) Conventional 2D-grid, cross-frame equivalent beam stiffness based on flexural analogy 236 31.6 118 24.4 +16.6 -41.7 +8.6 +30.0 -30 .0 16.3 16.3 +47.2 -83.1 -11.2 +42.9 -42 .9 23.3 23.3 (d) Improved 2D-grid, cross-frame equivalent beam stiffness based on Timoshenko beam model 273 41.4 143 20.7 +21.0 -50.5 +7.3 +34.7 -34 .0 18.9 18.5 +56.0 -96.5 -14.6 +49.5 -48 .0 26.9 26.1 (f) TWOS 3D-frame, explicit modeling of cross-frames by beam and truss elements G1 G2 G3 G1 G2 G3 G1 G2 G3 G1 G2 G3 G G2 G3 265 39.6 138 20.1 +20.7 -48.5 +7.1 +33.1 -33 .1 18.0 18.0 +53.9 -93.7 -14.0 +47.5 -47 .5 25.8 25.8 (e) TWOS 2D-grid, cross-frame equivalent beam stiffness based on Timoshenko beam model G1 G2 G3 Figure 3-20. FHWA Test Bridge (EISCR1) unfactored (nominal) cross-frame dead load forces calculated at mid-span by different methods (units: ft-kip moments, kip forces).

Findings and Applications 55 only the TWOS results shown in Figure 3-20e and f give results that never deviate more than 3 kips from the 3D FEA benchmark solution. (The percentage errors can be large even for these solutions in cases where the cross-frame forces are small. However, these percentage errors are not of any consequence when the cross-frame member sizes are repeated throughout the structure and sized for the most critical demand.) The TWOS 3D-frame solution shown in Figure 3-20f gives the best correlation with the 3D FEA benchmark. This is because this is the only “simplified” solution that accounts for: 1. The second-order effects in the calculation of the cross-frame forces (although the second- order effects are only a few percent for this structure and loading, as discussed previously in Section 3.2.2.1), and 2. The location of the various components and entities through the depth of the structure (i.e., the girder centroids and shear centers, the cross-frame depths and locations through the depth of the girders, the load height of the concrete slab, and the location of the bearings relative to the girder centroidal and shear center axes). Nevertheless, all of the 2D-grid solutions as well as the V-load solution give reasonable results for this bridge. The maximum error in the prediction of the maximum cross-frame bottom chord force is -14.0 percent, corresponding to the improved 2D-grid solution shown in Figure 3-20d, while the maximum error in the maximum cross-frame diagonal force is -20.4 percent, correspond- ing to the V-load method solution shown in Figure 3-20b. It is clear that the V-load method gives the greatest misrepresentation of the true cross-frame vertical shear forces. This is due to the fact that the V-load method is based on an assumption of “equal vertical stiffness” across all the girders at each of the intermediate cross-frame locations. The concept of equal vertical stiffness in the above means that, if each girder were considered in isolation, and if a unit load were applied at each girder in succession, the same vertical dis- placement would be obtained. However, the girders can never be physically isolated from each other in any meaningful way for calculation of these so-called vertical stiffnesses. Isolating the girders requires the application of artificial boundary conditions to them, which changes the way they respond to the load. An implicit assumption of an equal vertical stiffness from each girder is invoked in the derivation of the coefficient C used in calculating the V-load as a function of the number of girders in the bridge system (NHI, 2011). Since the “equal vertical stiffnesses” can never be calculated in any meaningful way, they can never be checked. Conceptually, the girder vertical stiffnesses can be thought of typically as being very different though, even in radially supported bridges with a “very regular” geometry such as the FHWA Test Bridge, at least when the structure is highly curved. This is because the outside girder generally must resist substantially more load. It should be noted that in the calculation of the results shown for the simplified methods in Figure 3-20b through e, part of the solution involves the calculation of the contribution from the overhang eccentric bracket loads. These load effects are approximated based on the AASHTO LRFD Equation (C-6.10.3.4-2) and are shown in Figure 3-21. 13 13 13 13 +4.6 -4.6 -4.6 +4.6 -4.6 -4.6 G1 G2 G3 Figure 3-21. FHWA Test Bridge (EISCR1) unfactored (nominal) cross-frame dead load forces due to eccentric bracket loads (units: ft-kip moments, kip forces).

56 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 3.2.3.7 Improved Prediction of Cross-Frame Forces in Skewed I-Girder Bridges Figure 3-22 shows a sketch of the framing plan for Bridge NICSS16 from the NCHRP Project 12-79 Task 7 analytical studies. This is a continuous-span structure with an extreme parallel skew of its bearing lines of 70o, combined with an 80-ft.-wide deck (w = 80 ft.), a perpendicular distance between its fascia girders of wg = 74 ft., and 120-, 150- and 150-ft. span lengths. As a result, its skew index IS is 1.69 from Equation 9. The skew index captures the tendency for the development of substantial transverse load paths in I-girder bridge structures and is used as a key term in scoring the accuracy of the simplified methods of analysis in Table 3-1. This bridge is framed with staggered cross-frame lines, which reduces the large forces developed particularly in the transverse direction between the obtuse corners of each span. However, these forces still are significant. Figures 3-23 through 3-25 show the cross-frame forces calculated in Bay 3 (between Girders G2 and G3), Bay 6 (between Girders G3 and G4, and Bay 8 (between Girders G6 and G7) Bay 3 Bay 6 Bay 8 G1 G9 Figure 3-22. Framing plan of Bridge NICSS16. Figure 3-23. Bridge NICSS16 cross-frame forces in Bay 3 (between Girders G3 and G4) under total dead load (unfactored) from conventional 2D-grid analysis (M1), improved 2D-grid analysis (M2), and 3D FEA. - 500.0 - 400.0 - 300.0 - 200.0 - 100.0 0.0 100.0 200.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 M om en t M i (k ip- ft) Sh ea r F or ce , V i (k ips ) Cross-Frame Number 2D M1 2D M2 3D FEA - 20.0 - 15.0 - 10.0 - 5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Cross-Frame Number 2D M1 2D M2 3D FEA

Findings and Applications 57 in the Bridge NICSS16 using the conventional 2D-grid approach, the improved 2D-grid method, and the 3D FEA benchmark simulation. The improved 2D-grid solution implemented here uses the Jeq girder torsion model discussed in Section 3.2.2 as well as the “exact” equivalent beam element discussed Section 3.2.3 and described in detail in the Task 8 report (Appendix C of the contractors’ final report). The first plot in each of the figures cited above shows the nodal moment at the ends of the cross-frames toward the bottom of the plan view shown in Figure 3-22. The second plot shows the vertical shear transferred by each cross-frame. The horizontal axis shows the cross-frame number within each of the bays, starting from the left-hand end of the bridge in Figure 3-22 and progressing to the right-hand end. The forces are calculated assuming no-load fit detailing of the cross-frames for simplicity of the discussion. Steel dead load fit (SDLF) and total dead load fit (TDLF) detailing effects are addressed subsequently in Sections 3.2.5 and 3.3. From the above plots, it is apparent that the conventional 2D-grid solution predicts essentially zero cross-frame forces throughout the NICSS16 bridge structure. The primary reason for this behavior is the dramatic under-estimation of the girder torsional stiffnesses due to using only the St. Venant torsional stiffness term (GJ/L) in the 2D-grid idealization. Conversely, the improved 2D-grid method provides a reasonably good estimate of the cross-frame forces in this extreme structure, compared to the benchmark 3D FEA solutions. The results in Figures 3-23 through 3-25 for the above skewed I-girder bridge, combined with the results in Figure 3-7 for the FHWA Test Bridge highlight the importance of using a - 500.0 - 400.0 - 300.0 - 200.0 - 100.0 0.0 100.0 200.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Cross-Frame Number 2D M1 2D M2 3D FEA - 40.0 - 30.0 - 20.0 - 10.0 0.0 10.0 20.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Cross-Frame Number 2D M1 2D M2 3D FEA M om en t M i (k ip- ft) Sh ea r F or ce , V i (k ips ) Figure 3-24. Bridge NICSS16 cross-frame forces in Bay 6 (between Girders G6 and G7) under total dead load (unfactored) from conventional 2D-grid analysis (M1), improved 2D-grid analysis (M2), and 3D FEA.

58 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges better I-girder torsion model than the simplistic one commonly used in conventional 2D-grid methods. Furthermore, the results in Table 3-6 clearly show the importance of also using a better representation of the cross-frame stiffnesses in bridges where the cross-frame deformations start to have some influence on the overall structure response. Of major importance is the fact that these improvements require little additional computational expense, and their software implementation is relatively straightforward. However, professional software implementation of these methods is essential for them to be used efficiently in practice. Manual calculation and input of the corresponding improvements into the software is too laborious to be workable given common professional time constraints. 3.2.4 Improved Calculation of I-Girder Flange Lateral Bending Stresses from 2D-Grid Analysis Given the above improvements in the I-girder and cross-frame stiffness representations, it is still essential to address the calculation of the flange lateral bending stresses in curved and/or skewed I-girder bridges. This section recommends specific improvements in these calculations. Figure 3-26a shows the plan view of Bridge NISSS16 considered in the NCHRP Project 12-79 Task 7 analytical studies. This is a 150-ft. simple-span straight bridge with an 80-ft.-wide deck (w = 80 ft.), a perpendicular distance between its fascia girders of 74 ft., and a skew of 50 degrees at its left-hand abutment. These geometry factors produce a skew index of IS = 0.59, placing this - 400.0 - 300.0 - 200.0 - 100.0 0.0 100.0 200.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Cross-Frame Number 2D M1 2D M2 3D FEA - 40.0 - 30.0 - 20.0 - 10.0 0.0 10.0 20.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Cross-Frame Number 2D M1 2D M2 3D FEA M om en t M i (k ip- ft) Sh ea r F or ce , V i (k ips ) Figure 3-25. Bridge NICSS16 cross-frame forces in Bay 8 (between Girders G8 and G9) under total dead load (unfactored) from conventional 2D-grid analysis (M1), improved 2D-grid analysis (M2), and 3D FEA.

Findings and Applications 59 bridge just inside the second category of straight-skewed bridge structures in the scoring system of Table 3-1. Figure 3-26b illustrates the forces in cross-frame 2 (CF2) of Bay 6 in this structure and the corresponding statically equivalent nodal horizontal and vertical forces transferred to the I-girders at the cross-frame chord levels. These horizontal forces can be transformed to statically equivalent lateral forces applied at the flange levels of the I-girders by determining the couple associated with these horizontal forces and then multiplying the chord-level couple forces by the ratio of the cross-frame depth to the girder depth between the flange centroids, dCF/h. In typical 2D-grid solutions, Cx = -Dx and Bx = -Ax, and thus the forces shown in Figure 3-26b are the couple forces. Figure 3-26c shows the top flange forces applied to Girder G6 in this bridge, determined from the improved 2D-grid method discussed in the previous sections. The forces are still labeled “Cx” and “Bx,” for simplicity of the presentation. It should be noted that the chord-level couple forces shown in Figure 3-26b are multiplied by (dCF/h) to determine the flange-level forces. Given a general statical free-body diagram of a girder flange, such as the one shown for Girder G6 in the figure, one would expect that the subsequent determination of the flange lateral bending stresses is an easy strength of materials calculation. If the girder is also horizontally curved, the equivalent radial lateral loads corresponding to the horizontal curvature can be included in the free-body diagram. Furthermore, eccentric bracket loads from the overhangs can be included on fascia girders. Unfortunately, the solution for the flange lateral bending stresses is not this simple. The prob- lem is that the girder torsional stiffnesses, upon which the above calculation of the cross-frame (a) Plan view of Bridge NISSS16 (b) Forces transferred from cross-frame B6-CF2 to Girders G6 and G7 (c) Top flange of Girder G6 subject to the horizontal components of the nodal forces C F C,5 B,x 8 )77.0( C F C,5 B,x 7 (- )82.3 C F C,5 B,x 6 )31.0( C F C,5 B,x 5 )55.01( C F C,5 B,x 4 )76.62( C F C,5 B,x 3 )61.4( C F C,5 B,x 2 )71.9( B B,x 6 F C, 9 )56.0( B B,x 6 F C, 8 )82.3( B B,x 6 F C, 7 )21.0( B B,x 6 F C, 6 (- )27.01 B B,x 6 F C, 5 (- )63.52 B B,x 6 F C, 4 (- )38.1 B B,x 6 F C, 3 (- )06.8 B B,x 6 F C, 2 (- )31.31 C B,x 5 F C, 1 )77.1( B B,x 6 F C, 1 )87.5( G1 G9 G6 B6-CF2... Figure 3-26. Calculation of lateral bending stresses in the top flange of Girder G6, in Bridge NISSS16 under total dead load (unfactored).

60 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges forces is based, include a contribution both from the girder warping torsion as well as the girder St. Venant torsion. As such, a portion of the above forces is transferred (by the interaction of the flange with the girder web) into the internal St. Venant torsion in the girders. More specifically, corresponding small but undetermined distributed lateral forces are transferred to the flange from the web in Figure 3-26c. Because of this effect, if the statical free-body diagram shown in Figure 3-26c is used to calculate the girder flange lateral bending stresses, slight errors accumulate as one moves along the girder length. Solutions to this problem include: 1. Use the girder torsional rotations and displacements along with the detailed open-section thin-walled beam stiffness model associated with Jeq to directly determine the flange lateral bending stresses. This results in an imbalance in the flange lateral bending moments on each side of the intermediate cross-frames (since Jeq is based on the assumption of warping fixity at the cross-frame locations). This moment imbalance could be re-distributed along the girder flange to determine accurate flange lateral bending moments. A procedure analogous to elastic moment distribution could be utilized for this calculation. Although this approach is a viable one, it is relatively complex. Therefore, it was not pursued in the NCHRP Project 12-79 research. 2. Focus on an approximate local calculation in the vicinity of each cross-frame, utilizing the forces delivered to the flanges from the cross-frames as shown in Figure 3-26c. Because of its relative simplicity, this approach was selected in the NCHRP Project 12-79 research. It should be noted that the girder flange lateral bending stresses are calculated directly and explicitly from the element displacements and stiffnesses in the TWOS 2D-grid and TWOS 3D-frame solutions. Therefore, these methods provide the best combination of accuracy and simplicity for the grid or frame element calculation of the flange lateral bending stresses. However, the disadvantage of this approach is the additional complexity of the element formulation and the requirement that an additional warping degree of freedom has to be included in the global structural analysis. Figure 3-27 illustrates the simplified approach adopted in the NCHRP Project 12-79 research for calculating the I-girder flange lateral bending moments given the statically equivalent lateral loads transferred at the flange level from the cross-frames. The calculation focuses on a given cross-frame location and the unbraced lengths, a and b, on each side of this location. For simplicity of the discussion, only the force delivered from the cross-frame under consideration a b L Pab/L Pa2b/L2Pab 2/L2 Pa2b2/L3 = Mmax if a > b = Mmax if b > a P P Pab (1+ab/L2) /2L Pa2b/2L2Pab 2/2L2 Averaged Moments Figure 3-27. Lateral bending moment, M in a flange segment under simply supported and fixed-end conditions.

Findings and Applications 61 is shown in the figure, and the cross-frame is assumed to be non-adjacent to a simply supported end of the girder. In general, the lateral forces from horizontal curvature effects and/or from eccentric bracket loads on fascia girders also would be included. Two flange lateral bending moment diagrams are calculated as shown in the figure, one based on simply supported end conditions and one based on fixed-end conditions at the opposite ends of the unbraced lengths. For unbraced lengths adjacent to simply supported girder ends, similar moment diagrams are calculated, but the boundary conditions are always pinned at the simply supported end. The cross-frame under consideration is located at the position of the load P in the sketches. In many situations, the moments at the position of the load are the controlling ones in the procedure specified below. Given the moment diagrams for the above cases, the project Task 8B research determined that an accurate-to-conservative solution for the flange lateral bending moments and stresses is obtained generally by: 1. Averaging the above moment diagrams, and 2. Taking the largest averaged internal moment in each of the unbraced lengths as the flange lateral bending moment for that length. This solution is repeated cross-frame location by cross-frame location along the length of the girders and the largest moment from the two solutions obtained for each unbraced length is taken as the estimate of the flange lateral bending moment in that unbraced length. (For the unbraced lengths at girder simply supported ends, only one solution is performed.) The above procedure recognizes that the true flange lateral bending moment is bounded by the “pinned” and “fixed” moment diagrams (neglecting the small St. Venant torsional contributions from the interaction with the web) and ensures that the flange lateral bending moments required for static equilibrium are never underestimated. Also, the average of the pinned and fixed moment diagrams is analogous to the use of the approximation qLb 2/10 rather than qLb 2/12 when estimating the flange lateral bending moments due to horizontal curvature, where q is the equivalent flange radial load. In addition, the above solution is insensitive to any inaccuracies in the calculation of the cross-frame forces as described in Sections 3.2.3.6 and 3.2.3.7. Figure 3-28 illustrates the accuracy associated with using the procedure from Figure 3-27 for the NISSS16 Bridge. One can observe that the flange lateral bending stresses from the 3D FEA simulation model are predicted quite well. The recommended procedure of using the maximum of the internal moments from the calculations for the two adjacent cross-frames for each unbraced (a) Girder G3 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 f (k si) Normalized Length 2D-Pinned 2D-Avg. 2D-Fixed 3D FEA (b) Girder G6 f (k si) -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Normalized Length 2D-Pinned 2D-Avg. 2D-Fixed 3D FEA Figure 3-28. Bridge NISSS16 flange lateral bending stresses under total dead load (unfactored).

62 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-29. FHWA Test Bridge (EISCR1) flange lateral bending stresses in Girder G1 under Service IV load combination (1.5  total dead load). length as the flange lateral bending moment value tends to be somewhat conservative in extreme cases where the dimensions a and b are substantially different. Figure 3-29 compares the results of simplified calculations of the maximum flange lateral bending stresses for the different unbraced lengths to the 3D FEA benchmark solution for the fascia girder (G1) on the outside of the curve in the FHWA Test Bridge. For curved radially supported I-girder bridges with relatively regular geometry, the basic “conventional” estimate from the AASHTO LRFD equation (C4.6.1.2.4b-1), using a coefficient of N = 12 rather than 10, works quite well. In the FHWA Test Bridge calculations, the results from the above improved calculations give essentially the same results as those obtained from the AASHTO equation. However, the AASHTO equation is obviously much simpler. Nevertheless, for bridges having non-zero skew, the improved method is able to account in a rational manner for the skew effects. The net result is a significantly improved estimate of the girder flange lateral bending stresses compared to the coarse values recommended in AASHTO (2010) Article C6.10.1. It should be emphasized that the AASHTO LRFD equation (C4.6.1.2.4b-1) gives an estimate of the maximum flange lateral bending moment in a given unbraced length. Therefore, in Figure 3-29, the simplified solution is shown just as a constant value within each unbraced length. Lastly, in Figure 3-29, the TWOS 3D-frame geometric nonlinear solution is provided along with the 3D FEA benchmark result to illustrate the high accuracy achievable with this TWOS solution. However, as stated in the Task 8 report (Appendix C of the contractors’ final report), the TWOS approach was not pursued as an improved simplified solution in the NCHRP Project 12-79 research due to the additional complexities associated with its implementation. 3.2.5 Calculation of Locked-In Forces Due to Cross-Frame Detailing This section addresses the fourth major improvement recommended by the NCHRP Project 12-79 research for the simplified 2D-grid analysis of curved and/or skewed I-girder bridges. However, it is important to note that this improvement also applies to 3D FE design analysis. This section addresses the calculation of locked-in forces due to steel dead load fit (SDLF) or total dead load fit (TDLF) cross-frame detailing. The emphasis here is predominantly on the calculation aspects. Section 3.3 addresses the broader attributes of the behavior and the ques- tion of when the locked-in forces due to the detailing of the cross-frames should be considered in the design. Appendix A provides summary definitions of key terms pertaining to cross-frame

Findings and Applications 63 detailing. It is essential that the reader understand these definitions to facilitate study and inter- pretation of the corresponding results and discussions throughout the report. As noted previously, regardless of whether the analysis is a 2D-grid or a 3D FEA method, it can only give the bridge internal forces associated with no-load fit (NLF) detailing if it is conducted without the modeling of initial lack-of-fit effects. 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 solutions. Fortunately, with some qualifications (discussed subsequently in Section 3.3), the SDLF or TDLF detailing effects tend to be opposite in sign to the internal forces due to the dead loads in straight-skewed bridges. Therefore, the 2D-grid or 3D FEA solutions for the cross-frame forces and the flange lateral bending stresses are conservative when they neglect the SDLF or TDLF initial lack-of-fit effects. Unfortunately, in some cases, these solutions can be prohibitively conservative. In addi- tion, unfortunately, for curved radially supported structures, the cross-frame forces and girder maximum flange lateral bending stresses tend to be increased by the SDLF or TDLF detailing effects (see the subsequent discussions in Section 3.3). For generally curved and skewed bridges, the effects can go both ways. Technically, it is relatively easy to include the influence of locked-in forces in either 2D-grid or 3D FEA calculations. Basically, the calculation amounts simply to the inclusion of an initial stress or initial strain effect. This is similar to the handling of thermal strains and deflections. Therefore, for cases where the initial lack-of-fit effects are important, including them in the analysis should not provide any significant hardship in terms of modeling effort or computational expense. Of course, as emphasized with the other key improvements recommended in the pre- vious sections, the implementation of the calculations into professional software is essential for the methods to be used efficiently by the design engineer. In addition, it is essential for engineers to understand the methods, calculations, and potential issues; therefore, the software methods need to be well documented. 3.2.5.1 Key Conceptual Configurations Associated with SDLF and TDLF Detailing To understand the calculation of the locked-in forces due to SDLF or TDLF detailing of the cross-frames, it is essential to first understand the basic geometry calculations associated with these methods. These calculations do not require any structural analysis, but rather, they utilize the specified girder camber profiles to determine the fabricated geometry of the cross-frames. Figure 3-30 illustrates four different configurations associated with SDLF or TDLF detailing. Geometric factors such as cross-slope, super-elevation, and profile grade line are not shown in the figure for clarity. The cross-frame shown in the figure is assumed to be an arbitrary one within the bridge span (considerations at bearing line cross-frames are addressed subsequently). The two configurations used by structural detailers are Configurations 1 and 4. In Configuration 1, the girders are assumed to be blocked and under zero load with their webs vertical in their initially fabricated (cambered and plumb) geometry. If either TDLF or SDLF detailing is employed, the cross-frame, if connected to the girder on one side, will not fit up with the connection on the other side. This is because the cross-frame geometry is detailed to fit between the girder connection work points, assuming that the girder webs remain vertical while the corresponding camber values are taken out of the girders at the cross-frame location. If TDLF detailing is employed, Configuration 4 is the idealized girder geometry, with plumb webs and with the total dead load camber taken out of both of the girders. Correspondingly, if SDLF detailing is used, Configuration 4 is the idealized plumb girder geometry with the steel dead load camber taken out of both of the girders. Therefore, given the total dead load or steel dead load camber profiles, the TDLF or SDLF calculation is simply a geometrical one for the detailer and fabricator.

64 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges (a) Configuration 1 – No-load geometry before connecting the cross-frames (b) Configuration 2 – Girders “locked” in the initial no-load, plumb and cambered geometry, cross-frames subjected to initial strains and initial stresses to connect them to the girders Drop due to differential camber between girders = initial lack of fit Figure 3-30. Important conceptual configurations associated with total (or steel) dead load fit detailing (geometric factors such as cross-slope, super-elevation, and profile grade line are not shown for clarity). (continued) In order to include the initial lack-of-fit effects due to the above procedures in the structural analysis, Configuration 2 needs to be considered. It should be emphasized that Configuration 2 is never experienced in the physical bridge. However, this configuration is very convenient for setting up the analysis of the SDLF or TDLF effects. In this configuration, the girders are conceptually “locked” into position in their no-load ideally plumb geometry, and the cross-frames are conceptually deformed (i.e., forced) into the position where they fit up with the corresponding points on the girder connection plates. In many cases, the drops due to the differential camber, labeled in Configuration 1, are sufficiently large such that substantial initial strains need to be induced into the cross-frames in order for the connection points to fit up. This is not a problem, since the girders have been artificially locked in their no-load plumb position in this configuration. This is similar to the conceptual model used in the calculation of thermal loading effects, where the structure “nodes” are initially locked into position, the temperature changes are applied

Findings and Applications 65 to the model, producing initial stresses, and then the nodes are “released” and the structure is allowed to deform due to the “fixed-end” forces induced at the nodes when everything was initially “locked up.” Configuration 3 represents the geometry achieved by the structure once the girders are “unlocked” and allowed to deform under the fixed-end forces induced from the cross-frames at the connection points in Configuration 2. It should be emphasized that, conceptually, the dead loads (i.e., the self-weight of the steel and the dead weight of the concrete deck) have not been applied to the structure yet in Configuration 3. Therefore, similar to Configuration 2, this configuration also is never directly experienced by the bridge. However, the internal forces and stresses induced in Configuration 3 are the locked-in values due to the SDLF or TDLF detailing effects. When the corresponding steel or total dead load is added to this configuration, Configuration 4 (the state of the bridge under the combined dead load and locked-in force effects) is achieved. The goal of TDLF or SDLF detailing is to achieve approximately plumb girder webs under the total dead load or the steel dead load respectively. Once the girders are released from their locked (c) Configuration 3 – Theoretical geometry under no-load (dead load not yet applied), after resolving the initial lack of fit by connecting the cross-frames to the girders, then “releasing” the girders to deflect under the lack-of-fit effects from the cross-frames (d) Configuration 4 – Geometry under the combined effects of the total (or steel) dead load plus the locked-in internal forces due to the dead load fit detailing Figure 3-30. (Continued).

66 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges positions in Configuration 2, the cross-frames tend to “spring back” or “elastically rebound.” Since the cross-frames tend to be relatively stiff in their own planes compared to the resistance of the individual girders to lateral bending and twisting, the cross-frames tend to snap-back close to their original undeformed geometry. However, this cannot occur without the twisting of the girders, since compatibility must be maintained between the cross-frame connection points and the corresponding points on the girder connection plates. As a result, the girders are twisted into the position shown in Configuration 3. When the dead load (i.e., the total dead load or the steel dead load) is applied conceptually to the bridge, starting in Configuration 3, the structure tends to bend and twist under the load such that the geometry shown in Configuration 4 is achieved. The sketch of Configuration 4 shown in Figure 3-30d implies TDLF detailing, since the bridge cross-slopes, etc., are not shown in the figure and the drawing indicates that both of the girders have deflected to the same final elevation. For TDLF detailing, the girder webs are approximately plumb in this condition under the total dead load. If SDLF detailing is employed, the additional camber associated with the concrete dead load (plus any additional camber for dead load from appurtenances, etc.) still remains in the girders in Configuration 4. However, in this case, the girder webs are approximately plumb under the steel dead load. It should be noted that the twisting induced into the girders in Configuration 3 is largely due to the differential camber between the girders in Configuration 1. Furthermore, the differential camber in Configuration 1 is due to the different vertical displacements that occur in the girders due to the bending and twisting of the structure under the applied loads. Therefore, the displace- ments that the cross-frames tend to “pull” into the girders in Configuration 3 are approximately equal and opposite to the displacements at these locations under the corresponding total or steel dead load. 3.2.5.2 Calculation of the Initial Strains, Initial Stresses or Initial Forces Associated with SDLF or TDLF Detailing of the Cross-Frames The calculation of the initial strains generated in the cross-frames in Configuration 2 of Figure 3-30b simply involves the identification of the nodal positions of the girder connection work points in the desired “final” Configuration 4, as well as the corresponding nodal positions of the girder connection work points in Configuration 2. (Note that the Configuration 2 girder nodal positions are the same as the nodal positions in Configuration 1 since the girders are in their undisplaced no-load plumb-web geometry in both of these configurations.) The difference between the nodal positions in Configurations 2 and 4 gives the displacements that the cross- frame is subjected to in order to connect it with the girders in Configuration 2. •• Calculation of the initial strains, initial stresses, or initial forces in 3D FEA software. Figure 3-31 shows a spatial representation of Configurations 2 and 4 for a hypothetical location within a bridge span. It should be noted that, if the individual cross-frame members are represented explicitly by truss and/or beam elements, the calculated initial strain is simply the axial extension of the individual members associated with the above displacements. If beam elements are employed for the individual cross-frame members, it is generally sufficient to assume that these elements are “pinned” to the girder connection work points at their ends, such that only axial deformation is produced by the displacements from Configuration 4 to Configuration 2. The engineer may wish to insert rotational releases explicitly in the model at the end of the cross-frame members in many situations where they are modeled by beam elements. However, the bending rigidity of the individual cross-frame members is typically sufficiently small such that including or not including the rotational releases is not of any significance.

Findings and Applications 67 Once the cross-frame element initial (axial) strains are calculated, the corresponding initial stresses are determined simply by multiplying the strains by the elastic modulus of the material. The initial strains and initial stresses are simply a computational device to determine the locked- in force effects. Therefore, even if the initial stress is larger than the material yield strength, the material behavior should be assumed to be linear elastic. The initial cross-frame member forces are determined simply by multiplying the initial stress by the cross-frame area. It should be noted that the implementation of the above calculations requires that the soft- ware, and the structural elements used in the software, must have either initial stress or initial strain capabilities. Any software that already is capable of modeling thermal loading has these capabilities. Calculation of the initial strains and initial forces in cross-frame equivalent beam elements. If the cross-frames are represented by equivalent beam elements, the calculations are exactly the same as in the above discussion. However, the displacements at the two cross-frame end connec- tion work points are resolved into element end displacements and end rotations. These element end displacements and rotations are then applied to the equivalent beam element. Assuming the use of a structural element for the equivalent beam, the best approach is to calculate the initial forces induced by the above displacements from Configuration 4 to Configuration 2. These forces are then handled as initial fixed-end forces in the equivalent beam element. This procedure requires that the beam element implementation must be able to handle initial fixed- end forces (e.g., fixed-end forces due to thermal loading, fixed-end forces due to internal ele- ment loads, etc.). If this is the case, the implementation of the “initial force” effects is relatively straightforward. As noted above, elements that are able to handle thermal loading already include these effects. In addition, elements that incorporate the calculation of fixed-end forces from internal loading between the nodes already include this type of effect. (a) Girders in the final geometry (Configuration 4) (b) Girders “locked” in their initially plumb cambered geometry, cross-frames subjected to initial strains to connect to the girders (Configuration 2) Girder Flanges Connection Plates Figure 3-31. Configurations used for calculation of initial strains in cross-frame members due to initial lack of fit.

68 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 3.2.5.3 Handling of Cross-Frame Initial Strains and Initial Stresses (or Initial Forces) at Skewed Bearing Line Cross-Frames The computational handling of the initial lack-of-fit or locked-in force effects at bearing line cross-frames is essentially no different than described in the above section. However, the behavior is somewhat different since the girders cannot displace vertically at the bearings and because the skewed cross-frames impose a twist into the girders associated with the girder major-axis bending rotation at the bearings. Figure 3-32 illustrates the rotations, due to applied loads within the bridge span, at the end of a girder connected to a skewed bearing line cross-frame. A fixed bearing is assumed at this position to simplify the discussion. The girder web and the bearing line cross-frame are assumed to be plumb in the current configuration shown in the figure. The double arrow perpendicular to the girder web represents the major-axis bending rotation of the girder, fx, about the fixed point. This rotation induces the longitudinal displacement Dz at the top flange of the girder. However, since the girder is attached to the skewed bearing line cross-frame, the top flange can only displace significantly in the direction normal to the plane of the cross-frame. This is indicated by the arrow labeled D. The cross-frame deflects essentially only by rotating about its longitudinal axis through the fixed point. This is shown by the double-arrow vector f. In order to maintain compatibility between the girder and the cross-frame, the top flange of the girder must deflect by the vector component labeled Dx in the figure. Therefore, the girder web lays over by the deflection Dx relative to the fixed point. This deflection, divided by the height h, gives the girder twist rotation fz. Figure 3-33 shows an alternate plan view of the behavior illustrated in Figure 3-32, except that the rotations are in the opposite direction to the rotations associated with the structure’s dead loads. If one considers the “deflections” of the girders due to the camber, the typical upward displacement in the spans induces a major-axis bending rotation at the bearing line shown by the double arrows normal to the girders in Figure 3-33 (using the right-hand rule). That is, if the bearing stiffener/connection plate at the bearing is placed normal to the flanges, this stiffener is rotated to a non-vertical position in the initial cambered, no-load, plumb geometry of the girder. This is comparable to Configuration 1 in Figure 3-30a. In order to fit-up with the girders in Configuration 2, the bearing line cross-frames have to be rotated about their Figure 3-32. Illustration of the girder major-axis bending and twist rotations required for compatibility at a skewed bearing line cross-frame.

Findings and Applications 69 longitudinal axis, and then (because of the skewed geometry), strained into position to connect them with the rotated connection plates in the initial cambered no-load, plumb geometry of the girders (i.e., assuming no drops between the girders at the bearing line, the bearing line cross-frames have to be deformed from their rectangular geometry in Configuration 1 into a parallelogram geometry in Configuration 2, assuming equal f at both ends of the cross-frame). When the girders are then “unlocked” and “released,” the cross-frames elastically rebound approximately to their initial rectangular geometry and force a twist into the girders opposite to the direction that they twist under the dead loads. This corresponds to Configuration 3 in Figure 3-30c. However, the girders only lay over at the bearing lines. They cannot displace vertically. It should be noted that skewed intermediate cross-frames involve a combination of the two effects shown in the above for the intermediate cross-frames in Figure 3-30 and for the skewed bearing line cross-frames in Figures 3-32 and 3-33. That is, at skewed intermediate cross-frames, the girders are subjected to twisting due to the differential vertical displacements between the cross-frame connection points on the girders as well as the compatibility of the rotations between the girders and the skewed cross-frames at the connection points. 3.2.6 Simplified Analysis Improvements for Tub-Girder Bridges Significant improvements also were developed for the simplified analysis of curved and skewed tub-girder bridges in the NCHRP Project 12-79 Task 8B research. These improvements are of a somewhat different nature though, since tub-girder bridges are fundamentally different from I-girder bridges. The key improvements for tub girders were: 1. The development of an improved method for estimating the influence of skew on tub-girder internal torques using basic 1D analysis procedures, 2. The investigation of the influence of skew (and torsion due to skew) on the cross-section distortion of box-girders, and 3. The calculation of local effects from the longitudinal components of the axial forces in the diagonals of the top flange lateral bracing (TFLB) system, which result in “saw-tooth” type local spikes in the longitudinal normal stresses in tub-girder top flanges. These improvements are described briefly in the following subsections. The NCHRP Project 12-79 Task 8 report (Appendix C of the contractors’ final report) and Jimenez Chong (2012) provide more detailed discussions of these improvements. Skewed end cross-frame Girder major-axis bending camber rotation Component of girder major-axis bending camber rotation causing cross-frame flexure Girder Girder Component of girder major-axis bending camber rotation causing cross-frame torsional rotation Figure 3-33. Flexural and torsional rotation components at the ends of a skewed end cross-frame due to the girder major-axis bending camber rotations.

70 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 3.2.6.1 Improved Estimation of Tub-Girder Internal Torques in 1D Line-Girder Analysis Methods Figure 3-34 shows a plan view of Bridge NTSCS29 analyzed in the NCHRP Project 12-79 Task 7 studies. This is a 225-ft. span simply supported curved and skewed tub-girder bridge with a horizontal radius of curvature of 820 ft., a deck width of 30 ft., and a skew angle of 15.7o at the left-hand abutment. The bearing line at the right-hand abutment is radial. The girders are each supported on single bearings at their ends, and the structure is built with two intermediate external cross-frames. The top flange lateral bracing (TFLB) system in this bridge is a Warren-type truss system. Figure 3-35 compares the internal torques calculated with two different line-girder analyses of this bridge (including the use of the M/R method for estimating the effects of the horizontal curvature), to two different 3D FEA benchmark simulations. The lighter dotted curve in the figure shows the results for the internal torque calculated solely by using the conventional M/R method without any accounting for the skew effects at the left-hand end. The bold dotted curve shows the combination of this conventional calculation with a separate additional estimate of the internal torque due to the left-hand end skew. The two 3D FEA solutions for the internal torque shown in the plot are: 1. A 3D FEA solution of the bridge as shown in Figure 3-34, indicated by the dark solid line, and 2. A 3D FEA solution of the bridge constructed without any intermediate external cross-frames, indicated by the light dashed line. One can observe that the M/R solution, combined with the improved method of estimating the internal torque, gives a close representation of the response of the bridge if it did not have any intermediate external cross-frames tying the girders together along the span length. Furthermore, the left-most intermediate external cross-frame appears to cause a shift in the internal torque on Figure 3-34. Plan view of Bridge NTSCS29. -2000 -1000 0 1000 2000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 To rs io na l M om en t ( kip -ft ) Normalized length M/R torque 3D FEA M/R+T 3D FEA no Ext. Interm. CF Figure 3-35. Comparison of total dead load torsional moments (unfactored) in the girder on the outside of the horizontal curve of Bridge NTSCS29 predicted by 1D analysis assuming rigid end diaphragms versus 3D FEA.

Findings and Applications 71 one side, given by just the M/R method solution, and on the other side, given by the combination of the M/R method with the improved method of estimating the internal torque due to skew. The right-most external cross-frame does not appear to have any significant influence on the internal torque. The improved method of estimating the internal torque due to skew involves the relatively simple idealization that the bearing line diaphragms (or cross-frames) are rigid in their own plane with respect to torsional stiffness of the tub girders. Although tub girders generally have substantially larger torsional stiffness than I-girders, the bearing line diaphragms or cross-frames are relatively short in length. Therefore, particularly in relatively narrow tub-girder bridges, these components may be approximated reasonably well as acting rigidly in their own plane. As a result, once the major-axis bending rotations are estimated for the tub girders at a bearing line, the same type of rotational compatibility rules as discussed in Section 3.2.5.3 apply. The NCHRP Project 12-79 research has not addressed analysis of the effect of external inter- mediate cross-frames or diaphragms via a 1D analysis in the context of the above procedures. A number of the conceptual idealizations used in the development of the V-load method for I-girder bridges may be helpful for the development of such procedures. However, the tedious nature of the adjustments to the 1D solutions may outweigh the benefits of these procedures, given that the use of 2D-grid methods should be quite feasible in current practice (2012). 3.2.6.2 Investigation of the Influence of Skew (and Torsion Due to Skew) on the Cross-Section Distortion of Box-Girders AASHTO LRFD Article 6.7.4.3 generally requires the use of intermediate internal diaphragms or cross-frames in steel box girders to control cross-section distortion due to torsional loads. Cross-section distortion of box girders is caused by external torsional loads that are not distributed in proportion to the St. Venant shear flow. It is well known that the distortional behavior of a box girder is dependent on the manner in which the external torque is applied to the member. Fan and Helwig (2002) have developed equations for estimating the distortional bracing forces developed in internal diaphragm and cross-frame components by horizontal curvature effects and by eccentric vertical applied loads. However, to the knowledge of the NCHRP Project 12-79 research team, no prior studies have been conducted to understand and to estimate the influence of distortion due to skew. Evaluations of the tub-girder bridges studied in NCHRP Project 12-79 Tasks 7, 8, and 9 research have indicated that the tub-girder internal cross-frame forces tend to be negligible in straight-skewed tub-girder bridges and that these forces tend to be predicted conservatively by the Fan and Helwig (2002) equations in curved tub-girder bridges. It appears that the development of internal torsional moments in tub girders, due to support skew, is similar to the shear flow associated with St. Venant uniform torsion. This is largely because the support diaphragms restrain the distortion of the girder cross-sections at the skewed supports. As such, the discrete torque induced in the girders at the skewed supports is predominantly a St. Venant torque. Figure 3-36 shows two basic geometries that can be used to gain some further understanding of this problem: (1) a straight simply supported box girder with a square cross-section and 30o skew at its right-hand end, and (2) a horizontally curved, simply supported box girder having the same cross-section. The span length of these girders is Ls = 150 ft., and the girders are subjected to vertical loads representative of the weight from the placement of a concrete deck. The square box depth is set to D = Ls/25 = 72 in. and the web thickness is set to tw = 0.5 in. The top and bottom flange thickness is also set to tf = 0.5 in. for simplicity. The radius of curvature of the second girder is taken as R = 400 ft. Solid plate diaphragms with t = 1 in. are used at the ends of the boxes, but no intermediate internal diaphragms or cross-frames are employed along the spans. Both box girders are supported continuously across their bottom flange at the supports.

72 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges In Figures 3-37 and 3-38, the cross-section warping deformation is illustrated via a side view and the cross-section distortion is illustrated via a cross-section view from the 3D FEA at the four cross-sections labeled in Figure 3-36. Although the torsion is also smaller in the first case, the warping deformations, as well as the cross-sectional distortion deformations, are also small relative to the torsional deformations. Conversely, in the second case, the distortion of the box is quite evident. This is predominantly due to effective radial forces due to the horizontal curvature coming from the flanges. 3.2.6.3 Calculation of “Saw-Tooth” Longitudinal Normal Stresses in the Top Flanges of Tub Girders Figure 3-39 shows a simplified free-body diagram illustrating the forces Q and P delivered to one of the top flanges of a tub girder at the connection of the top flange lateral bracing (TFLB) Section C1 C2 C3 C4 Section S1 S2 S3 S4 (a) Straight and skewed box-girder (b) Curved box-girder Figure 3-36. Straight skewed and curved box girders used for study of distortion effects. (a) Side View S2 S3 S4S1Section (b) Cross-Section View Figure 3-37. Deformed cross-sections in the straight skewed box girder. (a) Side View C2 C3 C4C1Section (b) Cross-Section View Figure 3-38. Deformed cross-sections in the curved box girder.

Findings and Applications 73 system to the girder. For cases where the tub girder is resisting significant torsion, the diagonal forces often are dominated by the torsion and the forces in the diagonals alternate from tension to compression in the consecutive panels. In these cases, the effects of the tension and compression forces due to the torsion are additive in their contribution to P. Therefore, the tub-girder flanges are acted upon by a longitudinal concentrated load at the intersection of the diagonals with the flanges. Although the predominant flange stress is the major-axis bending stress, which is commonly estimated as fb = M/Sx.top, where M is the major-axis bending moment at the cross-section under consideration and Sx.top is the elastic section modulus to the top flange, neglecting the contribution of the TFLB system, the above axial load P has an important local effect on the flange stresses. Interestingly, the resulting top flange average normal stress tends to follow a saw-tooth pattern in which the saw-tooth “jump” in stress is essentially P/bftf. The saw-tooth effect appears as a + P/2bftf fluctuation about the “mean” value fb = M/Sx.top (see Figure 3-40). The researchers obtained the best accuracy of the simplified calculations relative to 3D FEA benchmark results when this saw-tooth effect is added to the stress fb with Sx.top determined as explained above. Figures 3-42 and 3-44 show example results comparing the top flange longitudinal normal stresses (labeled generally as fb) from 3D FEA simulation models to the “conventional” calculation of the top flange major-axis bending stress as fb = M/Sx.top from a 2D-grid model (neglecting the contribution of the TFLB system in determining Sx.top). These results correspond to the two simple-span tub-girder bridges shown in Figures 3-41 and 3-43. The first bridge (NTSSS2) is a straight-skewed 150-ft. span tub-girder bridge with 30o parallel skew, a 30-ft. wide deck, and cos cosTot i i Tot j jP D D sin sinTot i i Tot j j BendQ DQ P SBend DTot-i DTot-j α i αj D S Figure 3-39. Interaction of forces between top flange lateral bracing and girder top flange for Warren and X-type layouts. P/(bf tf) M/Sx,top (a) Upper Top flange axial stresses Torque Torque C T P/(bf tf)M/Sx,top (b) Lower Top flange axial stresses Figure 3-40. Top flange saw-tooth major-axis bending stresses due to interaction with the flange level lateral bracing system.

74 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-41. Plan view of Bridge NTSSS2. -35 -30 -25 -20 -15 -10 -5 0 5 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f (k si ) b Normalized Length 3D FEA-NL 2D-Grid Average 2D-Grid+Force P Figure 3-42. Bridge NTSSS2 exterior top flange normal stresses on Girder G1. Figure 3-43. Plan view of Bridge NTSCR1. -35 -30 -25 -20 -15 -10 -5 0 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f (k si ) b Normalized Length 3D FEA 2D-Grid Average 2D-Grid+Force P Figure 3-44. Bridge NTSCR1 exterior top flange normal stresses on Girder G1.

Findings and Applications 75 no intermediate external diaphragms within its span. The second case (NTSCR1) is a curved 150-ft. span, radially supported structure with a 30-ft. wide deck, and a horizontal radius of curvature R = 400 ft. The 2D-grid estimates of the top flange major-axis bending stresses are excellent in both of these examples. Correspondingly, this result is reflected in the mode grade of A for the calculation of the major-axis bending stresses in straight-skewed and curved radially supported tub-girder bridges in Table 3-2 of Section 3.1.5. Interestingly, the internal torque is constant (and solely due to the skew) in the straight-skewed bridge. Conversely, the internal torque is maximum at the supports and zero at the mid-span in the horizontally curved structure. These variations in the internal torque are reflected clearly in the saw-tooth patterns shown in Figures 3-42 and 3-44. The “jump” associated with the saw-tooth is constant throughout the length of the bridge in Figure 3-42, while this jump is maximum toward the ends of the bridge and relatively small near the mid-span in Figure 3-44. In cases such as the one in Figure 3-44, the saw-tooth stresses result in a significant local increase in stress above the conventionally calculated fb in the region of maximum moment. This effect can also occur in the negative moment region of continuous-span bridges. 3.3 Influence of Locked-In Forces Due to SDLF or TDLF Detailing of Cross-Frames This section provides a summary of the findings of the NCHRP Project 12-79 Task 8B research pertaining to the influence of steel dead load fit (SDLF) and total dead load fit (TDLF) detailing of the cross-frames in curved and/or skewed I-girder bridges. Two examples are extracted from the large suite of structures considered in the Task 7 studies for this purpose. The first example is a simple-span straight bridge with a substantial parallel skew; the second example is a horizontally curved, radially supported structure. The results presented show the impact of the above detailing methods on a relatively complete set of important responses including: •• Bridge displacements (i.e., the constructed geometry), •• Cross-frame forces, and •• Girder flange lateral bending stresses. This is followed by a broader discussion of key considerations, including the questions: •• When is it important or essential to consider locked-in force effects due to SDLF or TDLF detailing in the design? •• When can SDLF or TDLF initial lack-of-fit effects be considered as incidental? •• To what extent can standard connection tolerances relieve the locked-in internal forces induced by SDLF or TDLF detailing? Appendix C of the contractors’ final report provides a more detailed summary of results for the wide range of bridges studied in the NCHRP Project 12-79 research. Appendix A pro- vides summary definitions of key terms. It is essential that the reader understand these defini- tions to facilitate study and interpretation of corresponding results and discussions throughout this report. 3.3.1 Straight-Skewed Bridge Example Figure 3-45 shows the framing plan for a 300-ft. straight simple-span I-girder bridge with a 70o parallel skew of its bearing lines. The bridge has an 80-ft.-wide deck (i.e., w = 80 ft.) and 74-ft. spacing between its fascia girders. This geometry produces a skew index of IS = 0.68, which places

Figure 3-45. NISSS54 framing plan.

Findings and Applications 77 this structure in the third and most difficult “S” category of Table 3-1. Figure 3-46 and Table 3-7 give the girder dimensions, and Table 3-8 gives the cross-frame member sizes. The bridge uses staggered cross-frames to alleviate “nuisance” transverse stiffness effects due to the large skew. 3.3.1.1 Bridge Deflections It is useful to first consider how this example bridge deflects under its total construction dead load. This can be accomplished by conducting a 3D FEA of the structure assuming no-load fit (NLF) of the cross-frames. Figure 3-47 shows a plan view of the magnified deflected geometry. One can observe that the girders are subjected to substantial layover (i.e., twist rotations) at the bearing lines. This is due to the compatibility between the girders and the heavily skewed bearing line cross-frames, as discussed previously in Section 3.2.5.3. The bearing line cross-frame deflec- tions involve predominantly a rotation about the skewed bearing line, highlighted by the double arrows in the figure (right-hand rule). The large 70° skew induces girder end twists (denoted by fz in the previous Figure 3-32) approximately equal to fx tan(70°) = 2.75 fx, where fx is the girder end major-axis bending rotation. Twists of a similar but different magnitude are induced by the intermediate cross-frames due to the fact that they frame into the girders at different positions along the girder spans. The overall twisting of the girders is a rather complicated pattern, involv- ing twist rotations in opposite directions at the girder ends. 3.3.1.2 Girder Cambers and Camber Differences Based on the prior discussions in Section 3.2.5.1, it should be clear that the SDLF and TDLF detailing effects are driven largely by the girder camber profiles, or more specifically, by the dif- ferences between the camber profiles at each of the cross-frame positions. Figures 3-48 and 3-49 show two different camber profiles for this bridge, the first one based on a 1D line-girder analysis and the second based on a 3D FEA assuming NLF. Figure 3-50 shows the differential values for the 3D FEA girder cambers. Web 1"x144" A B C D E 1.25"x30"2.25"x30"1.25"x30" 2.75"x30" 2.25"x30" Bottom Flange G1-G9 A B C D E 1.25"x28"2"x28"1.25"x28" 2"x28" 2"x28" Top Flange G1-G9 Length Length Bearing Stiffener 1.5"x12" Figure 3-46. NISSS54 girder plate dimensions. Cross-Frame Type Top Chord Diagonals Bottom Chord Interior (X type) L6x6x1 L6x6x1 L6x6x1 End (Inverted V) WT6x53 WT6x60 WT9x38 Table 3-8. NISSS54, cross-frame member sizes. Girder A B C D E G1-G9 45 45 45 45 45 Table 3-7. NISSS54 girder plate lengths (ft.).

Bearing line rotations Bearing line rotations G1 G9 Figure 3-47. Bridge NISSS54 total dead load deflected geometry for the case of NLF detailing of the cross-frames (scale factor = 10x). 0 2 4 6 8 10 12 14 16 18 0 50 100 150 200 250 300 350 400 450 500 G ird er C am be rs (in .) G1 G2 G3 G4 G5 G6 G7 G8 G9 G1 G9 z (ft) z = 0 z = 505' z = 302' z = 203' z (ft) Figure 3-48. Bridge NISSS54 total dead load camber profiles from line-girder analysis. 0 2 4 6 8 10 12 14 16 18 0 50 100 150 200 250 300 350 400 450 500 G ird er C am be rs (in .) G1 G2 G3 G4 G5 G6 G7 G8 G9 G1 G9z = 0 z (ft) z (ft) z = 505' z = 302' z = 203' Figure 3-49. Bridge NISSS54 total dead load camber profiles from 3D FEA.

0.92.13.40.0 1.22.02.80.0 0.81.52.22.63.10.0 0.81.52.12.63.00.0 0.00.81.52.12.73.13.40.0 0.10.91.72.32.93.33.73.80.0 -0.30.61.42.12.73.23.63.94.10.0 0.21.01.82.43.03.43.84.14.34.40.0 0.0 -4.4-4.2-4.0-3.7-3.4-2.9-2.4-1.7-1.0-0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.1-3.9-3.6-3.2-2.7-2.1-1.4-0.60.3 -3.8-3.6-3.3-2.8-2.3-1.6-0.80.0 -3.4 -3.1-2.7-2.1-1.4-0.70.0 -3.0-2.6-2.1-1.5-0.8 -3.1-2.6-2.2-1.5-0.8 -2.8-2.0 -1.2 -3.4-2.0-0.8 G1 G9 Figure 3-50. Bridge NISSS54 total dead load camber differences (differential camber values) between girders, taken from the camber profiles based on the 3D FEA vertical deflections.

80 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Each of the curves in Figures 3-48 and 3-49 is the total dead load camber profile for a single girder. Only the total dead load cambers are shown to keep the discussion focused and brief. The focus of the subsequent discussions is on the TDLF detailing of the cross-frames and its effects. For TDLF detailing, the cross-frames are fabricated to fit to the girder connection work points in the conceptual geometry where the girder webs are still plumb but the total dead load cambers have been removed from the girders. The TDLF detailing induces twists in the girders in the opposite directions from those shown in Figure 3-47. The horizontal axis in Figures 3-48 and 3-49 is the longitudinal coordinate “z” along the length of the bridge. The origin for the z coordinate is located at the bearing on Girder G9 at the left-hand acute corner of the structure. Therefore, the left-most curve in the plots is the camber profile for Girder G9. Correspondingly, the right-most curve, ending at z = 505 ft., is the camber profile for Girder G1. One can observe that the 3D FEA camber profiles are substantially smaller for the girders near the center of the bridge width. This is due to the substantial transverse load path between the obtuse corners of the bridge, developed via the cross-frames (even though the cross-frames are staggered throughout the length of the bridge to reduce these effects). The differential cambers shown in Figure 3-50 are based on the girder cambers determined from the 3D FEA vertical deflections. The implications of using the line-girder analysis total dead load vertical displacements versus the 3D FEA vertical displacements for setting the total dead load cambers are discussed subsequently. 3.3.1.3 System Deflections Due to Initial Lack-of-Fit Effects Figure 3-51 shows the deflections of the NISSS54 Bridge after, first, the cross-frames conceptually are connected to the girders (Configuration 2 of the previous Figure 3-30b), then the girders are “unlocked” and “released” from their initial no-load plumb geometry such that they are deformed by the cross-frames into the Configuration 3 shown in Figure 3-30c. That is, Figure 3-51 shows the “final” deformed geometry due to the cross-frame locked-in force effects (from the TDLF detailing of the cross-frames) if, by some means, the bridge dead load were not yet applied to the structure. One can observe that the bridge deformations in Figure 3-51 are approximately the opposite of the deflections shown previously in Figure 3-47. Figure 3-52 shows the layover of the girders corresponding to the deflections in Figure 3-51, where the term “layover” is defined as the lateral deflection of the girder’s top flange relative to its bottom flange. The plot in Figure 3-52 is similar to the previous plots of the girder cambers in that (1) the horizontal axis is the horizontal z coordinate in the bridge plan view, measured from the bearing at the left-hand acute corner; and (2) each curve gives the layover of a different girder at the various positions along the length. Upon studying Figure 3-52 carefully, one can observe that the “curvature” of the fascia girder layover curves (i.e., the darkest solid curves in Figure 3-52) is largest near the acute corners Bearing line rotations Bearing line rotations Figure 3-51. Bridge NISSS54 “Configuration 3” deflected geometry under no-load due solely to the initial lack of fit associated with the TDLF detailing of the cross-frames (camber profiles based on 3D FEA vertical deflections).

Findings and Applications 81 of the span. This indicates that the TDLF detailing results in substantial “locked-in” flange lateral bending of the fascia girders at the acute corners. In addition, one can observe that the curvature of the layover curves for the interior girders is even more dramatic in the vicinity of the skewed bearing lines at each end of the bridge. Furthermore, if one looks carefully at the curves in the middle of the plot, it can be seen that the inner-most girders are subject to noticeable “back-and-forth” twisting actions. This is due to the use of the staggered cross-frames throughout the bridge, causing the load transfer between the obtuse corners to pass from cross-frame to cross-frame by twisting and flange lateral bending of the girders. 3.3.1.4 Approximate Canceling of Dead Load Layovers by Dead-Load Fit Effects Figure 3-53 shows the girder layovers in Bridge NISSS54 due solely to the total dead load. That is, these are the layovers associated with the deflected geometry illustrated previously in Figure 3-47. One should note that the girder values in Figure 3-53 are approximately equal and opposite the corresponding girder values in Figure 3-52. However, it should be emphasized that the values in these two plots are not exactly equal and opposite to one another. If one considers the application of the steel dead load to the bridge, resulting in the deflections of the girders from Configuration 3 to an intermediate configuration between 3 and 4 shown in the previous Figure 3-30c and d, the resulting girder layovers are the ones plotted in Figure 3-54. As one would expect (once the typical effect of TDLF on the girder layovers is understood), the girder webs are not plumb under the steel dead load. This is because TDLF detailing gives approximately plumb webs under the total dead load, but the total dead load has not been applied at the time of this graph. Next, if the concrete dead weight is added to the structure (assumed to be applied non- compositely to the bridge for simplicity of the example), the girders finally reach the conceptual Configuration 4 shown previously in Figure 3-30d. The resulting girder layovers corresponding to this configuration are obtained by summing the results from Figures 3-52 and 3-53 and are -6 -4 -2 0 2 4 6 0 100 200 300 400 500 La yo ve r ( in. ) Bridge Length (ft) G1 G2 G3 G4 G5 G6 G7 G8 G9 G1 G9 z (ft) z = 0 z = 505' z = 203' z = 302' Figure 3-52. Bridge NISSS54 girder “Configuration 3” layovers due to the initial lack of fit associated with the TDLF detailing of the cross-frames.

82 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges -6 -4 -2 0 2 4 6 0 100 200 300 400 500 La yo ve r ( in. ) G1 G2 G3 G4 G5 G6 G7 G8 G9 z (ft) G1 G9 z (ft) z = 0 z = 505' z = 302' z = 203' Figure 3-53. Bridge NISSS54 girder layovers solely due to the total dead load. -6 -4 -2 0 2 4 6 0 100 200 300 400 500 La yo ve r ( in. ) Bridge Length (ft) G1 G2 G3 G4 G5 G6 G7 G8 G9 Figure 3-54. Bridge NISSS54 girder layovers under steel dead load (due to the effects of TDLF detailing of the cross-frames plus the steel dead load effects) for the case where the cross-frames are detailed for TDLF.

Findings and Applications 83 shown in Figure 3-55. Many engineers would expect that the girder webs would be perfectly plumb in “Configuration 4” under the total dead load, since TDLF detailing was employed and the same analysis solutions were used consistently throughout the above developments. They would also expect that the girder flange lateral bending stresses would be perfectly zero in this “Configuration 4.” However, neither of these assumptions is correct. One can observe from Figure 3-55 that there is still a measurable amount of girder twisting (and corresponding flange lateral bending), particularly in the inner-most girders. Nevertheless, the layover of the webs is within the tolerance of D/96 = 144 in./96 = 1.5 in. Therefore, the webs may be considered as “approximately plumb.” The reason why the layovers are not zero in Figure 3-55, as well as why the corresponding girder flange lateral bending stresses discussed subsequently are not zero, is because of the following facts: •• The girders are twisted in the direction opposite to the total dead load displacements (in Figure 3-52) by concentrated lateral loads applied from the cross-frames. •• However, the torsional displacements of the girders under the total dead load, shown in Figure 3-53, are due to the distributed self-weight of the steel as well as the distributed concrete dead load. •• The concentrated lateral loads from the cross-frames, which induce the girder deflections due to the TDLF detailing, cannot possibly produce girder layovers exactly equal and opposite to the effects of the distributed dead loads acting on the girders. 3.3.1.5 Final Girder Vertical Deflections and Vertical Elevations Figure 3-56 shows the vertical deflections versus the normalized length along the fascia Girder G1 as well as the inner-most Girder G5 for the extreme example skewed I-girder bridge (NISSS54). (The girder normalized length coordinates vary from zero at the bearing at the left end of the bridge to 1.0 at the bearing on right end of the bridge.) -6 -4 -2 0 2 4 6 0 100 200 300 400 500 La yo ve r ( in. ) Bridge Length (ft) G1 G2 G3 G4 G5 G6 G7 G8 G9 D/96 Figure 3-55. Bridge NISSS54 girder “Configuration 4” layovers under total dead load (due to the combined effects of TDLF detailing of the cross-frames and the total dead load effects) for the case where the cross-frames are detailed for TDLF.

84 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges The results for the final (or total) dead load vertical displacements in these plots are shown considering each of the three main cross-frame detailing methods: NLF, SDLF, and TDLF. It can be observed that the mid-span displacements in Girder G5 are slightly smaller at the mid-span when NLF detailing is used. However, the vertical displacements are relatively insensitive to the type of cross-frame detailing. This is generally the case for all straight bridges. The vertical displacements of the fascia girders are essentially the same for all of the detailing methods. The above displacements can be added to the 3D FEA camber profiles of Figure 3-49 to obtain the final total dead load elevations of the girders. One can observe that the fascia girders are essentially at a “zero” elevation along their full length, whereas the interior Girder G5 has a final “flat” geometry for NLF detailing and is slightly less than 1 in. below the “zero” elevation for TDLF detailing. 3.3.1.6 Cross-Frame Forces The choice of NLF, SDLF, or TDLF detailing affects more than the girder displacements and stresses. It also can have a significant effect on the cross-frame forces. Figure 3-57 shows the Ve rt ic al D is pl ac em en t ( in. ) Ve rt ic al D is pl ac em en t ( in. ) Normalized Length NLF SDLF TDLF -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF SDLF TDLF Girder G1 Girder G5 Figure 3-56. Bridge NISSS54 “Configuration 4” vertical deflections under total dead load for different cross-frame detailing methods.

Figure 3-57. Bridge NISSS54 maximum amplitude of the component axial forces in each of the cross-frames under total dead load, NLF detailing of the cross-frames.

86 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges maximum magnitude of the total dead load member axial forces in each cross-frame throughout the NISSS54 Bridge for the ideal case where the cross-frames are fabricated NLF. Figure 3-58 parallels Figure 3-57, but shows the total dead load member axial forces in each of the cross- frames for the situation where the cross-frames are fabricated TDLF. One can observe that, in the NLF case, the cross-frame forces are relatively large in the vicinity of the short transverse load path between the obtuse corners of the bridge. The members having the largest axial force appear in bold in Figure 3-57. It should be noted that the figure also shows the mean of the cross-frame member force magnitudes for the cross-frame top chord, the cross-frame diagonals, and the cross-frame bottom chords. In addition, the sum of absolute value of all the cross-frame member forces, S|FNLF|, is shown in the upper right-hand corner of the figure. Conversely, in Figure 3-58, the cross-frame member axial forces along the short diagonal direction between the obtuse corners are relatively small (but not equal to zero). In this case, the maximum forces are in the diagonals of several “nuisance stiffness” cross-frames that frame into the girders very close to the skewed bearing lines. If these nuisance stiffness cross-frames are offset a sufficient distance from the bearing lines, as discussed subsequently, all of the final total dead load cross-frame forces are relatively small compared to the forces in the NLF case. However, the cross-frame forces generally are increased due to the TDLF detailing effects in the regions near the acute corners of the deck. One can observe that the chord forces are significantly smaller on average in Figure 3-58, but the average diagonal forces are larger compared to Figure 3-57. This is mainly due to the extremely large forces in the cross-frame diagonals at the acute corners, caused by the small offset distance of these cross-frames from the bearing line. These large forces are due to interactions between the first intermediate cross-frame near the acute corners with the bearing line cross-frames and the corresponding need to introduce a large force into the intermediate cross-frame to “pull” the fascia girders back to an approximately plumb position under the total dead load. Section 8.2.1 of the Task 8 report (Appendix C of the contractors’ final report) recommends that the first intermediate cross-frames should be placed a minimum distance of a D b= ( )max . , .1 5 0 4 Eq. 15 from the bearing line to alleviate the “nuisance stiffness” effects associated with the above spike in the cross-frame forces, where D is the girder depth and b is the second unbraced length within the span from the bearing line. Figures 3-59 and 3-60 show solutions comparable to the ones in Figures 3-57 and 3-58, but corresponding to the steel dead load condition. The values for the maximum cross-frame forces in these figures are somewhat representative of the difficulty the erector may encounter in fitting up the cross-frames with the girders in an unshored erection scenario. It is apparent from Figure 3-59 that, for the case with NLF detailing, the greatest difficulty may be encountered with the cross-frames located near the bearing lines and along the short diagonal direction. However, for the case of TDLF detailing, the largest cross-frame forces occur near the acute corners. In particular, it is apparent that some of the cross-frame diagonals near the acute corners may be particularly difficult to install. Again, these “nuisance stiffness” effects can be relieved by using a more appropriate offset distance from the bearing line for these cross-frames. This result demonstrates an important fact that fit-up problems tend to be exacerbated by TDLF detailing of the cross-frames. From Figures 3-57 through 3-60 as a whole, it is apparent that the locked-in force effects in the cross-frames tend to substantially relieve the cross-frame dead load forces in the short, stiff diagonal direction. However, the “locked-in” cross-frame forces in the vicinity of the acute corners tend to be additive with the dead load effects. Also, it is apparent that the TDLF

Figure 3-58. Bridge NISSS54 maximum amplitude of the “Configuration 4” component axial forces in each of the cross-frames under total dead load, TDLF detailing of the cross-frames.

Figure 3-59. Bridge NISSS54 maximum amplitude of the component axial forces in each of the cross-frames under steel dead load, NLF detailing of the cross-frames.

Figure 3-60. Bridge NISSS54 maximum amplitude of the component axial forces in each of the cross-frames under steel dead load, TDLF detailing of the cross-frames.

90 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges detailing of the cross-frames tends to exacerbate fit-up problems during the steel erection (due to the fact that the total dead load deflections have not yet been fully taken out of the girders by the application of the total dead loads). Lastly, it should be noted that for NLF detailing of the cross-frames, the forces needed to connect the structure together are theoretically zero if the girders are supported in their no-load position. Therefore, shoring of the girders is a good way to facilitate the erection when NLF detailing is used. 3.3.1.7 Girder Flange Major-Axis and Lateral Bending Stresses Figure 3-61 shows the top flange major-axis and lateral bending stresses for the fascia Girder G1 and for the inner-most Girder G5 of Bridge NISSS54. The plots in this figure again show the results for all the methods of detailing the cross-frames: NLF, SDLF, and TDLF. Similar to the results for the vertical deflections in G1 and G5, the major-axis bending stresses in these straight girders are relatively insensitive to the type of cross-frame detailing. However, the girder flange lateral bending stresses are substantially affected by whether the detailing of the cross- frames is NLF, SDLF, or TDLF. This should not be surprising given the above results for the girder layovers and cross-frame forces. -40 -30 -20 -10 0 10 20 30 Girder G1 Girder G5 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length σ (ks i) -40 -30 -20 -10 0 10 20 30 σ (ks i) NLF SDLF TDLF 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF fb f f fb SDLF TDLF Figure 3-61. Bridge NISSS54 “Configuration 4” top flange stresses under total dead load for different detailing methods.

Findings and Applications 91 The girder flange lateral bending stresses are the smallest when TDLF detailing is used. Many engineers expect that if TDLF detailing is used, the flange lateral bending stresses will be essentially zero (under the total dead load condition). This is not generally the case for the same reasons as explained in Section 3.3.1.4. In the fascia girder, the flange lateral bending stress is still approxi- mately 3 ksi near the left-hand end (see the top plot in Figure 3-61). This is related to the same nuisance stiffness effects of framing a number of the cross-frames in too close to the supports observed in Section 3.3.1.6. If the problem cross-frames are offset further from the bearing line, the flange lateral bending stresses in the fascia girder are essentially negligible. It should be noted that some small lateral bending stresses are induced in the top flange of the fascia girder due to the overhang loads. For the interior Girder G5, significant flange lateral bending stresses are still encountered even for the case of TDLF detailing. These stresses are due to the fact that the locked-in concentrated lateral forces acting on Girder G5 from the cross-frames are not able to cancel the torsional actions of this girder under the distributed total vertical dead load. The maximum flange lateral bending stresses, however, are actually reduced by more than a factor of two by the TDLF detailing effects in Girder G5. 3.3.2 Curved Radially Supported Bridge Example Figure 3-62 provides the framing plan for a 150-ft. simple-span curved radially supported I-girder bridge with a radius of curvature at its centerline of R = 438 ft., a deck width of w = 30 ft., and four I-girders spaced at 8 ft. apart. This bridge, named NISCR2, has a connectivity index of IC = 4.89, which places it in the second category of the “C” bridges of Table 3-1. It is expected that a conventional 2D grid analysis may have some difficulty in capturing all the responses of this structure. Figure 3-63 and Table 3-9 give this bridge’s plate girder dimensions, and Table 3-10 gives the sizes of its cross-frame members. A number of the attributes of this bridge are simpler than those of the previous example. However, this bridge is important to illustrate several key considerations with respect to SDLF and TDLF detailing effects on horizontally curved geometries. Figure 3-62. Bridge NISCR2 framing plan.

92 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges 3.3.2.1 Bridge Deflections Figure 3-64 shows a plan view of the Bridge NISCR2 magnified deformed geometry due to its total dead load. One can observe that there is an overall twisting of the bridge cross-section and all of the girders are laying over in the same direction. However, the layover at the radial supports is zero. 3.3.2.2 Girder Cambers and Camber Differences As noted in the previous example in Section 3.3.1, SDLF and TDLF detailing are driven by the girder camber profiles or, more specifically, by the differential camber at the cross-frame locations. Figure 3-65 shows the total dead load differential cambers for Bridge NISCR2. One can observe that all the cambers are negative values, indicating that in all cases, the girders with a smaller horizontal radius of curvature have smaller dead load deflections in this bridge. Similar to the previous example, the discussions are focused on the behavior for TDLF detailing of the cross- frames unless noted otherwise. 3.3.2.3 System Deflections Due to Initial Lack-of-Fit Effects Figure 3-66 shows the deflections of NISCR2 after the cross-frames are first connected to the girders in “Configuration 2” (see Figure 3-30b), second, the girders are “unlocked” and “released” from their initial no-load plumb geometry, and third, the girders are deformed by the cross-frames into Configuration 3 (described in Figure 3-30c). In other words, Figure 3-66 shows the “final” deformed geometry due to the locked-in forces caused by the TDLF detailing of the cross-frames. Web G1-G2 0.75"x84" A B C D E 1.25"x26"2"x26"1.25"x26" 2.75"x26" 2"x26" G1-G2 Bottom Flange A B C D E 1"x22"1.25"x22"1"x22" 2"x22" 1.25"x22" G1-G2 Top Flange Length Length Bearing Stiffener 1.5"x8" 1"x20"1"x20"1"x20" 1.5"x20" 1"x20" G3-G4 Top Flange 1"x24"1.25"x24"1"x24" 2"x24" 1.25"x24" G3-G4 Bottom Flange G3-G4 0.625"x84" Figure 3-63. NISCR2 girder plate dimensions. Girder A B C D E G1 G2 G3 G4 20.0 19.6 19.3 18.9 20.0 19.6 19.3 18.9 74.1 72.8 71.5 70.2 20.0 19.6 19.3 18.9 20.0 19.6 19.3 18.9 Table 3-9. NISCR2 girder plate lengths (ft.). Cross-Frame Type Top Chord Diagonals Bottom Chord Interior (X type) L6x6x0.75 L6x6x0.75 L6x6x0.75 End (Inverted V) L6x6x0.75 L6x6x0.75 L6x6x0.75 Table 3-10. NISCR2 cross-frame member sizes.

Findings and Applications 93 One can observe that the bridge deformations in Figure 3-66 are approximately the opposite of the deflections shown in Figure 3-64. Similar to the previous example, they are not exactly equal and opposite. 3.3.2.4 Approximate Canceling of Dead Load Layovers by Dead-Load Fit Effects Figure 3-67 shows the girder layovers in this bridge once the steel and concrete dead load effects have been added to deflect the structure conceptually from the previously discussed “Configuration 3” to “Configuration 4” (see Figure 3-30c and d). Results similar to those obtained in the previous skewed bridge example are observed in that each of the girder Figure 3-64. Bridge NISCR2 total dead load deflected geometry for the case of NLF detailing of the cross-frames (scale factor = 20x). 0 . 0 0.0 0 . 0 0 . 0 0 . 0 0 . 0 - 0 . 8 - 0 . 8 - 0 . 8 - 0 . 8 - 0 . 8 - 0 . 8 - 1 . 4 - 1 . 4 - 1 . 4 - 1 . 4 - 1 . 4 - 1 . 4 - 1 . 7 - 1 . 7 - 1 . 7 - 1 . 7 - 1 . 7 - 1 . 7 G 1 G 4 Figure 3-65. Bridge NISCR2 total dead load camber differences (differential camber values) between girders. Figure 3-66. Bridge NISCR2 “Configuration 3” deflected shape due to the initial lack-of-fit effects from TDLF detailing of the cross-frames (scale factor = 20x).

94 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges layovers is strictly non-zero; however, the final layovers are well within the typical tolerance of D/96 = 84 in./96 = 0.875 in. Also, as stated previously, the layovers cannot possibly be expected to be exactly equal to zero because the TDLF detailing effects are applied to the girders as con- centrated lateral loads from the cross-frames, whereas the total dead load layovers are caused by distributed vertical loads. Figure 3-68 illustrates the girder layovers in Bridge NISCR2 under the steel dead load when TDLF detailing of the cross-frames is used. Clearly, the girders are not plumb under the steel dead load. They are still rotated in the direction opposite to the direction that they twist under the action of the vertical loads. However, these layovers also satisfy the D/96 tolerance. Lastly, G1 G2 G3 G4 D/96 La yo ve r ( in. ) -2.0 -1.5 -1.0 -0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length Figure 3-67. Bridge NISCR2 “Configuration 4” girder layovers under total dead load for the case where the cross-frames are detailed for TDLF. G1 G2 G3 G4 La yo ve r (in .) 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length Figure 3-68. Bridge NISCR2 girder layovers under steel dead load for the case where the cross-frames are detailed for TDLF.

Findings and Applications 95 Figure 3-69 shows the girder layovers under the steel dead load for the case of NLF detailing of the cross-frames, while Figure 3-70 indicates the corresponding layovers under the total dead load. It can be observed that the layovers under the total dead load violate the above D/96 tolerance. Nevertheless, rigorous test simulation studies show that this layover does not have any measurable effect on the system capacity. 3.3.2.5 Final Girder Vertical Deflections Figure 3-71 shows the vertical deflections along the length of the fascia Girder G1 on the outside of the horizontal curve as well as the fascia Girder G4 on the inside of the curve of Bridge NISCR2. The results in these plots are shown for each of the three main cross-frame detailing G1 G4 G1 G2 G3 G4 La yo ve r ( in. ) -1.0 -0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length Figure 3-69. Bridge NISCR2 girder layovers under steel dead load for the case where the cross-frames are detailed for NLF. G1 G4 G1 G2 G3 G4 La yo ve r ( in. ) -2.0 -1.0 -1.5 -0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length Figure 3-70. Bridge NISCR2 girder layovers under total dead load for the case where the cross-frames are detailed for NLF.

96 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges methods: NLF, SDLF, and TDLF. One can observe that the percentage differences between these vertical displacement solutions are significantly larger than observed in the previous straight bridge example. Generally, the vertical displacements in horizontally curved bridges tend to be affected to a larger degree by the SDLF and TDLF detailing effects than in straight bridges. The mid-span vertical displacement of G1 in this specific example is 7 percent smaller than the solution for NLF when SDLF detailing is used. It is 17 percent smaller when TDLF detailing is employed. One other important observation that should be made from Figure 3-71 is that the influence on the vertical displacements from the SDLF detailing (i.e., the differences between the SDLF and NLF curves) are similar for all of the girders in the NISCR2 Bridge. The SDLF detailing reduces the displacements of all the girders equally by approximately 0.4 in. This statement also can be made regarding the influence of the TDLF detailing on the girder vertical displacements. The TDLF detailing reduces all the girder displacements by approximately 1.2 in. This is a general finding for all curved radially supported bridges and is demonstrated subsequently for several other bridges of this type. Ve rt ic al D is pl ac em en t ( in. ) Normalized Length NLF SDLF TDLF -8 -7 -6 -5 -4 -3 -2 -1 0 Ve rt ic al D is pl ac em en t ( in. ) -8 -7 -6 -5 -4 -3 -2 -1 0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF SDLF TDLF 0.0 0.2 0.4 0.6 0.8 1.0 Girder G1 Girder G4 Figure 3-71. Bridge NISCR2 vertical deflections under total dead load for different cross-frame detailing methods.

Findings and Applications 97 3.3.2.6 Cross-Frame Forces Figures 3-72 through 3-74 show the individual cross-frame member axial forces under the total dead load in the NISCR2 Bridge for the cases of NLF, SDLF, and TDLF detailing of the cross-frames, respectively. These figures indicate that the cross-frame chord forces are not significantly affected in this bridge by the type of cross-frame detailing. However, all the diagonal forces are significantly increased. The increase in the mean of the axial force in the diagonals is 35 percent for SDLF detailing versus NLF. The increase is 100 percent for TDLF detailing of the cross-frames. Correspondingly, Figures 3-75 through 3-77 show the individual cross-frame member axial forces under the steel dead load in the NISCR2 Bridge for NLF, SDLF, and TDLF. As discussed previously, the internal cross-frame forces in these solutions provide an indication of any potential difficulty regarding the fit-up of the cross-frames with the girders during the steel erection. One can observe again that the chord forces are not affected significantly Figure 3-72. Bridge NISCR2 maximum amplitude of the component axial forces in each of the cross-frames under total dead load, NLF detailing of the cross-frames. Figure 3-73. Bridge NISCR2 maximum amplitude of the component axial forces in each of the cross-frames under total dead load, SDLF detailing of the cross-frames. Figure 3-74. Bridge NISCR2 maximum amplitude of the component axial forces in each of the cross-frames under total dead load, TDLF detailing of the cross-frames.

98 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges by the cross-frame detailing type. However, the mean of the diagonal forces is increased by 100 percent from the NLF detailing to the SDLF case, and 283 percent from NLF to TDLF. Based on the results in Figure 3-77, one can conclude that the span length and radius of curvature for NISCR2 is such that the cross-frame fit-up forces are expected to be manageable for any of the methods. However, for a comparable bridge with a tighter radius curvature and/or a longer span length, the above percentage differences may be of greater significance. Based on the full set of bridge studies performed within NCHRP Project 12-79, the locked-in forces in the cross-frame diagonals always tend to be additive with the dead load effects when SDLF or TDLF detailing is used on curved radially supported bridge structures. Also, the chord forces tend to be increased, but not as much so. These are important findings. One can conclude that locked-in force effects generally should be considered when sizing the cross-frames in horizontally curved I-girder bridges. (These conclusions are independent of the specific sequence of girder erection, since assuming the structure remains elastic, and neglecting aspects such as Figure 3-75. Bridge NISCR2 maximum amplitude of the component axial forces in each of the cross-frames under steel dead load, NLF detailing of the cross-frames. Figure 3-76. Bridge NISCR2 maximum amplitude of the component axial forces in each of the cross-frames under steel dead load, SDLF detailing of the cross-frames. Figure 3-77. Bridge NISCR2 maximum amplitude of the component axial forces in each of the cross-frames under steel dead load, TDLF detailing of the cross-frames.

Findings and Applications 99 friction at the supports and non-zero connection tolerances, the bridge is a conservative elastic system for which the responses are path independent.) 3.3.2.7 Girder Flange Major-Axis and Lateral Bending Stresses Figure 3-78 gives the top flange major-axis and flange lateral bending stresses for Girders G1, G2, and G4 in the NISCR2 Bridge. The plots in this figure again show the results for all the methods of detailing the cross-frames: NLF, SDLF, and TDLF. One can observe that the major-axis bending stress in the girders is insensitive to the type of cross-frame detailing. This is a common result for horizontally curved structures, even though the girder vertical displacements exhibit some sensitivity to these attributes. This sensitivity is related to the coupling between the girder vertical displacements and the twist deformations in curved members. Conversely, the curved I-girder flange lateral bending stresses show some sensitivity to the cross-frame detailing type. This is due to the fact that, in a curved radially supported bridge, the locked-in cross-frame forces due to SDLF or TDLF detailing tend to displace the flanges laterally, and in a direction opposite to the direction the girders are tending to bend and twist under the vertical loads. These actions occur over the full span length of the girders. In the specific case of the NISCR2 Bridge, as well as other generally curved and radially supported structures, the flanges act effectively as continuous-span beams loaded effectively by uniformly distributed lateral loads coming from the horizontal curvature. The cross-frames are the supports for these effective continuous-span beams. The influence of SDLF or TDLF detailing on the effective continuous span beams is to essentially “pre-stress” the flanges by displacing their supports in the direction opposite to the equivalent horizontal curvature loading. On the inside Girder G4 of NISCR2, this “pre-stressing” is the dominant effect, essentially shifting the entire flange lateral bending moment diagram throughout the span. However in Girders G1 and G2, this pre-stressing effect is manifested predominantly in an increase in the flange “negative bending” moments and flexural stresses (using the above continuous-span beam analogy). These “negative bending stresses” are relatively small in this bridge, but they are increased by a maximum of approximately 20 percent due to SDLF detailing and 66 percent due to TDLF detailing. 3.3.3 General Considerations 3.3.3.1 Key Results from Studies of the Ford City Bridge (EICCR11) Because of its relative simplicity, Bridge NISCR2 considered in the previous section is useful to illustrate the influence of SDLF and TDLF cross-frame detailing effects on general horizontally curved bridges. Furthermore, this structure is illustrative of the behavior for reasonably “regular” curved I-girder bridges with relatively short-to-moderate span lengths. The Ford City Bridge (EICCR11) shown in Figures 3-79 through 3-81 represents the most extreme case encountered in the NCHRP Project 12-79 research regarding the influence of the cross-frame detailing method on the girder layovers, vertical displacements, and flange lateral bending stresses. This three-span continuous I-girder bridge has one straight end span of length 321 ft., a straight center span of 445 ft., and a highly curved end span of 292 ft. The minimum radius of curvature in the curved span is 511 ft. Furthermore, the bridge has a relatively narrow deck with w = 48.2 ft. given its span lengths. The torsion index on its curved span (Equation 1) is IT = 0.87. In addition, its four girders are 14 ft. deep and are spaced at 13.5 ft. apart. The circles in Figure 3-81 are highlighting a come-along beam that is being used to stabilize the curved girder during lifting. A cable goes up to a lifting beam from each end of the come-along beam.

100 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Girder G1 Girder G2 Girder G4 -30 -25 -20 -15 -5 -10 0 5 10 σ (ks i) 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF f f f fb fb fb SDLF TDLF -20 -15 -5 -10 0 5 10 σ (ks i) 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF SDLF TDLF -10 -5 0 5 10 σ (ks i) 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF SDLF TDLF Figure 3-78. Bridge NISCR2 top flange stresses under total dead load for different detailing methods.

Findings and Applications 101 Figure 3-79. Ford City Bridge (EICCR11) (Chavel, 2008). Figure 3-80. Ford City Bridge (EICCR11) girder depth and spacing (Chavel, 2008). Figure 3-81. Ford City Bridge (EICCR11) installation of drop-in segment (Chavel, 2008).

102 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges The combination of the above attributes, along with various other factors, make the Ford City Bridge possibly one of the most challenging curved I-girder bridges that has ever been erected. This bridge was not originally designed with a top flange lateral bracing (TFLB) system, but one was provided as shown in Figure 3-79 to facilitate the steel erection and concrete deck placement. This bridge was studied without a TFLB system in the NCHRP Project 12-79 research so that legitimate comparisons could be made between the 3D FEA simulations and simplified 1D line-girder and 2D-grid methods (since the 1D line-girder and 2D-grid solutions are generally not sufficient for I-girder bridges with TFLB systems). Figure 3-82 provides a plan view of three magnified displaced shapes from the 3D FEA simulation model of the Ford City Bridge. These images illustrate the influence of the different methods of cross-frame detailing on the torsional and lateral bending response of the bridge. It is apparent that there are substantial layovers, lateral movements, and span interactions in the bridge if it is constructed with NLF detailing. SDLF detailing reduces these deformations substantially, whereas TDLF detailing gives effectively plumb webs under the total dead load. Unfortunately, because of the size and close spacing of the girders on this bridge, TDLF detailing results in prohibitive fit-up forces. Therefore, SDLF detailing, or possibly detailing for an intermediate condition between NLF and SDLF, is the best option for this bridge. Based on the analytical studies from the NCHRP Project 12-79, NLF detailing tends to minimize the cross-frame internal forces under the intermediate and final steel dead load conditions in curved radially supported bridges and also tends to minimize the forces required to fit-up the steel in these structure types. Figure 3-83 plots the girder layovers associated with the deflected geometries from Figure 3-82 and shows that the maximum layovers under the total dead load are 3.6 in. when SDLF detailing is used. The NCHRP Project 12-79 research studies show that these displacements do not have any significant impact on the strengths. Generally, if the stability (second-order amplification) checks of Sections 3.1.1.1 and 3.1.1.2 are satisfied and the cross-frames satisfy stability bracing requirements (Helwig, 2012), the influence of girder layover on the structural resistance is sufficiently addressed and does not need to be considered any further. (a) NLF (b) SDLF (c) TDLF G1 G4 Figure 3-82. Ford City Bridge (EICCR11) deflected shape under total dead load for different detailing methods (scale factor = 10x).

Findings and Applications 103 -8 -6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length G1 G2 G3 G4 G1 G4 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length G1 G2 G3 G4 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length G1 G2 G3 G4 D/96 D/96 NLF SDLF TDLF La yo ve r ( in. ) -8 -6 -4 -2 0 2 4 6 La yo ve r ( in. ) -8 -6 -4 -2 0 2 4 6 La yo ve r ( in. ) Figure 3-83. Ford City Bridge (EICCR11) girder layovers under total dead load for different detailing methods.

104 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-84 shows the vertical deflections from the above 3D FEA simulations of the Ford City Bridge. One can observe that the maximum vertical displacement on Girder G1 under the total dead load is 19.1 in. if NLF detailing were used, 16.9 in. if SDLF detailing were used, and 14.9 in. for TDLF detailing. Girder G4 experiences a more dramatic effect on its vertical deflections due to the span torsional interactions. The curved span on G4 sees a maximum downward displacement of 3.2 in. with NLF detailing of the cross-frames, a maximum upward displacement of 1.7 in. with SDLF detailing, and a maximum upward displacement of 3.4 in. with TDLF detailing. These differences are large enough such that it is clear that the influence of the type of cross-frame detailing would need to be considered in setting the girder cambers in this bridge. As noted previously for the NISCR2 Bridge (see Section 3.3.2.5 and Figure 3-71), SDLF or TDLF detailing tends to have a similar effect on all of the girder vertical displacements within a given bridge. In Figure 3-84, one can observe that the vertical displacements of all the girders are reduced by the SDLF and TDLF detailing effects in the right-hand curved end-span of the Ford City Bridge. Correspondingly, all the vertical displacements are increased by these effects in the Normalized Length NLF SDLF TDLF Ve rt ic al D is pl ac em en t ( in. ) -20 -15 -10 -5 5 0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Length NLF SDLF TDLF Ve rt ic al D is pl ac em en t ( in. ) -20 -15 -10 -5 5 0 0.0 0.2 0.4 0.6 0.8 1.0 Girder G1 Girder G4 Figure 3-84. Ford City Bridge (EICCR11) vertical deflections under total dead load for different detailing methods.

Findings and Applications 105 middle span. The SDLF and TDLF influences on the Girder G1 and G4 displacements, given by the differences between the SDLF and NLF and the TDLF and NLF curves, are somewhat different in the Ford City Bridge however. This is due to the interaction between the continuous spans. 3.3.3.2 Consideration of the Influence of Cross-Frame Connection Tolerances on the Development of Locked-in Forces Due to SDLF or TDLF Detailing One question that may be asked regarding the influence of SDLF or TDLF detailing is whether small connection tolerances can add up to relieve the locked-in forces to a substantial degree. This question can be evaluated in an informative but simplified fashion using the FHWA Test Bridge (EISCR1) considered in prior Section 3.2. Figure 3-85 shows an isometric view of the 3D FEA simulation model for this structure, illustrating the undeformed geometry as well as the “Configuration 3” geometry explained previously in Figure 3-30c. This is the conceptual con- figuration in which the girders are “unlocked” and “released” such that the cross-frames impose deformations on them due to the initial lack of fit. However, the vertical loads conceptually have not been applied to the structure at this stage. Figures 3-86 and 3-87 show the vertical displacements and the flange major-axis and lateral bending stresses under the total dead load (unfactored) for the cases where the cross-frames are detailed for NLF and for TDLF. These results closely parallel the results presented previously in Figures 3-71 and 3-78 for Bridge NISCR2. Basically, the method of detailing can have a significant influence on both of these quantities in horizontally curved I-girder bridge structures. Figure 3-88 shows the corresponding cross-frame forces in the test bridge associated with NLF and TDLF detailing of the cross-frames. These results show the same trends as illustrated earlier for the NISCR2 Bridge in Figures 3-72 and 3-74 (i.e., the cross-frame diagonal forces can be increased substantially by the use of TDLF detailing in horizontally curved bridges). Similar results are obtained for SDLF detailing, but the increase in the cross-frame forces is smaller. It is useful to plot the responses induced solely due to the TDLF detailing, to understand their magnitudes before addressing the connection tolerance question. Figure 3-89 shows the vertical displacements induced in Girder G1 on the outside of the horizontal curve and in Girder G3 on the inside of the horizontal curve corresponding to the deformed “Configuration 3” geometry from Figure 3-85. One can observe that comparable vertical displacements are induced in both girders due to the effects of the TDLF detailing (consistent with the previous results shown in Figures 3-71 and 3-84). Figure 3-90 shows the corresponding major-axis and flange lateral bending stresses in the girders due to the initial lack-of-fit effects from the TDLF detailing method, and Deformations due to lack-of-fit, TDLF (scale factor = 30x) Undeflected geometry G1 G3 Figure 3-85. FHWA Test Bridge (EISCR1) undeflected geometry and “Configuration 3” deflected geometry due solely to the initial lack of fit from TDLF detailing of the cross-frames.

106 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-91 shows the “Configuration 3” cross-frame forces (note that the cross-frame forces in Figure 3-91 are not exactly equal to the difference between the cross-frame forces in Figures 3-88a and 3-88b due to small second-order effects). Figure 3-92 illustrates the loading mechanism causing a reduction in the downward vertical displacements in all of the girders due to TDLF detailing in the FHWA Test Bridge. Basically, the locked-in forces in the intermediate cross-frames generate an internal couple that is applied to each of the girders at the intermediate cross-frame locations. These couples are balanced by couples in the opposite direction at the bridge bearing lines. The applied internal couples at the intermediate cross-frames twist the girders in the opposite direction from the direction they displace under the total dead load. However, because of the horizontal curvature, the girders cannot twist in this fashion without the girder vertical displacements also being reduced. In order to obtain a “representative upper-bound” estimate of “slip” at the connections due to small connection tolerances, two scenarios were considered as shown in Figure 3-93. In the first Figure 3-86. FHWA Test Bridge (EISCR1) vertical displacements under total dead load (unfactored) for NLF versus TDLF detailing of the cross-frames.

Findings and Applications 107 Figure 3-87. FHWA Test Bridge (EISCR1) major-axis bending and flange lateral bending stresses under total dead load (unfactored) for NLF versus TDLF detailing of the cross-frames. scenario, a 1⁄8-in. axial movement, or “slip,” was assumed in all the intermediate cross-frame diagonals of the test bridge. The bridge deformations due to these connection movements are shown in the top image of Figure 3-93. In the second scenario, a 1⁄8-in. “slip” was assumed in both chords of the three intermediate cross-frames in the direction of their dead load and TDLF axial forces. The deformations corresponding to these movements are shown in the bottom image of Figure 3-93. Figures 3-93 through 3-95 show plots of the corresponding induced girder vertical displacements, flange major-axis and lateral bending stresses, and cross-frame forces due to the first scenario above, and Figures 3-96 through 3-98 show plots of these responses due to the second scenario. The above 1⁄8-in. magnitude is obtained by assuming standard size bolt holes 1⁄16-in. larger than the fasteners and assuming that nominally (or on average) the bolts are in the center of the holes. Then, assuming two plies in every connection, the ideal “slip” that can occur at a given connection is 1⁄16-in. This value is then multiplied by two to account for the ideal influence of the connection slip at each end on the elongation or shortening of each member.

108 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges By comparing the values in the above plots, one can observe that the locked-in forces from the TDLF detailing can indeed be reduced to some extent by “joint slip” within standard connection tolerances. However, in this bridge, the TDLF effects are significantly larger than the effects of these “slip” displacements. The other key point that can be noted by considering the above influence of the “slip” displacements is that the use of oversize holes to allow adjustment in the structure basically amounts to giving up control of the geometry by the amount that the connections can move. In addition, the connections have to be engaged before the cross-frames can brace the girders. This example again shows that the locked-in forces in the cross-frames generally are additive with the dead load effects when SDLF or TDLF detailing is used on curved radially supported bridge structures. Therefore, one can conclude that locked-in forces generally should be considered when sizing the cross-frames in horizontally curved I-girder bridges. Figures 3-99 and 3-100 provide a combined summary of the above results in terms of the influences of TDLF detailing, as well as the upper-bound connection tolerance movements on the vertical displacements and the girder layovers. The dark solid curve in these figures shows the results for NLF detailing of the cross-frames, and the dashed curve shows the results with TDLF. The light solid curve illustrates the combined results of TDLF detailing along with the 1⁄8-in. slip in each of the cross-frame diagonals. Finally, the dot-dash curve shows the result of a 1⁄8-in. slip in all of the internal cross-frame members (diagonals and chords), in the direction of their axial forces, combined with the influence of the TDLF detailing. (a) NLF detailing (b) TDLF detailing 5 4 3 5 9 1 1 7 2 3 - 1 0 1 - 7 5 - 1 2 - 1 0 0 - 6 6 - 3 7 3 7 - 4 9 4 9 - 1 1 - 2 5 2 6 - 3 6 3 7 - 1 7 - 3 0 - 3 9 5 - 5 4 8 4 4 3 5 8 1 1 6 2 2 - 9 1 - 8 1 - 4 - 1 0 8 - 7 - 2 - 1 - 4 3 1 0 - 5 8 1 4 - 6 6 - 4 6 4 5 - 6 0 6 0 - 1 1 - 3 1 3 1 - 4 3 4 3 Figure 3-88. Cross-frame forces under total dead load (unfactored) for NLF versus TDLF detailing of the cross-frames.

Findings and Applications 109 One can conclude that “slip” due to standard connection tolerances within the cross-frames can indeed reduce the influence of TDLF detailing by a significant fraction. However, this is based on a representative upper-bound effect of standard connection tolerances. Indeed, if the TDLF detailing is successful at achieving its objective of approximately plumb webs under the total dead load, then the corresponding girder locked-in vertical deflections, internal stresses, and cross-frame forces would be induced. Similar results are obtained for SDLF detailing, but again, the SDLF effects are smaller (and hence the potential influence of connection tolerances tends to be larger with respect to these effects). In horizontally curved bridges, it can be concluded that TDLF and SDLF detailing effects generally should be included in the structural analysis, since they tend to produce an additive effect on the girder “negative” flange lateral bending stresses and on the cross-frame forces. 3.3.3.3 Potential Influence of Other Connection Tolerances The previous section considered several scenarios giving an indication of the influence of representative cross-frame connection tolerances on the final constructed geometry, the cross- frame forces, and the girder major-axis and flange lateral bending stresses in the FHWA Test Bridge (EISCR1). It was shown that these connection tolerances can indeed have a measurable effect. Figure 3-89. FHWA Test Bridge (EISCR1) “Configuration 3” vertical displacements solely due to the initial lack-of-fit effects from TDLF detailing of the cross-frames.

110 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-90. FHWA Test Bridge (EISCR1) “Configuration 3” top flange major-axis and lateral bending stresses solely due to the initial lack-of-fit effects from TDLF detailing of the cross-frames. -0.4 0.5 0.6 0.0 -0.1 -0.4 0.8 -0.2 -7.9 6.7 -10.9 9.9 0.2 -0.2 - 4.6 4.7 -6.0 6.7 0.6 -0.6 -8.7 8.6 -12.4 12.2 0.2 -0.2 -5.4 5.5 -7.5 7.6 Figure 3-91. FHWA Test Bridge (EISCR1) “Configuration 3” cross-frame forces solely due to the initial lack-of-fit effects from TDLF detailing of the cross-frames.

Findings and Applications 111 Deformations due to +1/8 inch “slip” in all intermediate cross-frame diagonals (scale factor = 100x) Deformations due to +1/8 inch “slip” in all intermediate cross-frame top and bottom chords (scale factor = 100x) G1 G3 Figure 3-93. FHWA Test Bridge (EISCR1) deformed geometry solely due to 1/8-in. “slip” in the direction of the internal load in all the intermediate cross-frame diagonals and due to 1/8-in. slip in the direction of the internal load in all the intermediate cross-frame chords. However, the influence on the overall displacements is not as large as erectors would commonly expect for more general structures based on experience. The discussion below addresses one example scenario where the connection tolerance effects can be significantly larger. Figure 3-102 illustrates the potential impact of a relative “slip” between the top and bottom of a girder splice. In this case, the impact of the displacement Dslip between the top and the bottom of the splice is multiplied by the length-to-depth ratio (L/D) of the girder field section. Basically, whenever there is a significant ratio of this sort, there is a lever effect that can have a substantial influence on the constructed geometry. This “lever effect” can also occur across the width of the bridge due to the types of cross-frame connection “slip” displacements discussed in the previous section. However, the FHWA Test Bridge is not wide enough relative to its length to exhibit a significant “lever effect” of the cross-frame connection slip displacements. 3.3.4 When Should Locked-in Forces from SDLF or TDLF Detailing be Considered in a Structural Design Analysis? Table 3-11 summarizes the recommendations and their corresponding rationale from Ozgur (2011) pertaining to the question: When must locked-in forces from SDLF or TDLF detailing be considered in the structural design analysis? Alternately, this question can be phrased as: When can a curved and/or skewed I-girder bridge be designed based on an analysis that assumes NLF detailing, but then the cross-frames are detailed subsequently for either SDLF or TDLF without any significant consequences? The answers are listed in terms of the key bridge responses and are all based on the assumption that the girder cambers are based on Figure 3-92. Loading mechanism associated with an increase in all the girder vertical displacements due to TDLF detailing in the FHWA Test Bridge (EISCR1).

112 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-94. FHWA Test Bridge (EISCR1) vertical displacements solely due to 1/8-in. slip in the direction of the internal load in all of the intermediate cross-frame diagonals. an accurate 2D-grid or 3D FE analysis (see Appendix A for a definition of these terms). These findings are derived from the detailed study of the various bridges analyzed in the NCHRP Project 12-79 Task 8A research. Based on the basic illustrative examples in Sections 3.3.1 to 3.3.3, one can observe clearly that the answer to the above questions is different for straight-skewed and horizontally curved, radially supported bridges. Furthermore, in short, it can be stated that the influence of SDLF and TDLF detailing on bridges that have both horizontal curvature and skew can involve any combination of the attributes shown for the above distinct bridge types. However, the requirements for when lack-of-fit effects need to be included in the analysis are the same for horizontally curved, radially supported bridges and horizontally curved bridges with skewed supports. Therefore, the rules in Table 3-11 are listed for straight-skewed and skewed and/or curved bridges. Influence of cross-frame connection tolerances. From the presentations in Sections 3.3.3.2 and 3.3.3.3, clearly connection “slip” displacements can have a substantial influence on the constructed geometry as well as the internal force state in the erected structure. However, this fact

Findings and Applications 113 Figure 3-95. FHWA Test Bridge (EISCR1) major-axis bending and flange lateral bending stresses solely due to 1/8-in. slip in the direction of the internal load in all of the intermediate cross-frame diagonals. - 0 . 8 0 . 3 0 . 9 0 . 0 - 0 . 3 0 . 2 - 0 . 5 0 . 1 4 . 2 - 2 . 5 1 . 9 - 2 . 6 - 0 . 1 0 . 2 2 . 9 - 2 . 2 1 . 6 - 1 . 5 - 0 . 4 0 . 4 3 . 7 - 4 . 2 2 . 3 - 3 . 0 - 0 . 2 0 . 2 2 . 8 - 3 . 2 1 . 7 - 2 . 1 Figure 3-96. FHWA Test Bridge (EISCR1) cross-frame forces solely due to 1/8-in. slip in the direction of the internal load in all of the intermediate cross-frame diagonals.

114 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-97. FHWA Test Bridge (EISCR1) vertical displacements solely due to 1/8-in. slip in the direction of the internal load in all of the intermediate cross-frame chords. should not be used as a justification for neglecting initial lack-of-fit effects and the calculation of locked-in forces due to SDLF or TDLF detailing in the structural analysis. Given the above examples and discussions, it is clear that the effects of SDLF and TDLF detailing on the cross- frame forces and girder flange lateral bending stresses are typically additive in curved radially supported I-girder bridges. When these types of structures have longer spans and/or tighter curvatures, the influence of SDLF and TDLF detailing on the girder vertical displacements can be significant. If the cross-frame detailing is indeed successful in controlling the girder layovers, as it is intended to do, these locked-in forces have to exist. However, it is evident that “standard” connection tolerances can nullify much of the SDLF or TDLF detailing effects for bridges with shorter spans or smaller horizontal curvature. Interestingly, when this is the case, the small initial lack of fit is an indication that the cross-frame detailing effects are sufficiently small such that NLF detailing may be a good option. Impact of using girder cambers from a line-girder analysis in cambering the girders and in SDLF or TDLF detailing of the cross-frames. Table 3-11 is based on the assumption that the girder cambers are determined from an accurate 2D-grid or 3D FE analysis. Some very interesting behavior occurs if the displacements from a line-girder analysis are used in setting the girder

Findings and Applications 115 Figure 3-98. FHWA Test Bridge (EISCR1) major-axis bending and flange lateral bending stresses solely due to 1/8-in. slip in the direction of the internal load in all of the intermediate cross-frame chords. 0 . 2 - 1 . 6 2 . 6 - 0 . 1 - 2 . 7 1 . 6 - 0 . 5 - 0 . 2 1 . 1 4 . 8 - 0 . 2 - 3 . 6 0 . 0 0.0 5 . 2 0 . 9 - 3 . 4 0 . 3 - 0 . 2 0 . 2 - 3 . 4 1 . 0 0 . 7 - 3 . 2 0 . 0 0 . 0 2 . 2 - 2 . 8 - 2 . 4 1 . 9 Figure 3-99. FHWA Test Bridge (EISCR1) cross-frame forces solely due to 1/8-in. slip in the direction of the internal load in all of the intermediate cross-frame chords.

116 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Figure 3-100. FHWA Test Bridge (EISCR1) vertical displacements under the total dead load, based on NLF detailing of the cross-frames, TDLF detailing of the cross-frames with zero tolerance in the fit-up of the connections, and TDLF detailing of the cross-frames with 1/8-in. slip in the direction of the internal load in the intermediate cross-frame diagonals or in all the intermediate cross-frame members. cambers and detailing the cross-frames for SDLF or TDLF. This behavior, and its implications on the analysis requirements, is detailed below. •• Steel dead load (SDL) response of straight-skewed bridges with the cross-frames fabricated for SDLF based on line-girder analysis cambers. For any straight-skewed bridge, if the steel dead load (SDL) cambers are obtained from a line-girder analysis, and if the cross-frames are detailed for SDLF based on these cambers, then theoretically there is zero lack of fit between the girders and the cross-frames in an idealized unshored SDL configuration where, prior to engaging the cross-frames, all the girders are placed on the supports and allowed to deflect under the self-weight of the steel. The girder webs are plumb in this configuration, since there is no interaction with the cross-frames. When the cross-frames are detailed for SDLF based on this configuration, they fit-up perfectly with the girders in this configuration without any forcing. Therefore, interestingly, the use of line-girder analysis for SDL gives the “optimum” SDL cambers in that the total cross-frame forces and total flange lateral bending stresses in the SDL condition will be zero. These statements are all predicated on including the correct tributary weights from the cross-frames in the above line-girder analysis. Very interestingly, but as should be expected based on a fundamental understanding of the analysis of initial lack-of-fit effects from Section 3.2.5, an accurate 2D-grid or 3D FE analysis

Findings and Applications 117 Figure 3-101. FHWA Test Bridge (EISCR1) girder layovers under total dead load, based on NLF detailing of the cross-frames, TDLF detailing of the cross-frames with zero tolerance in the fit-up of the connections, and TDLF detailing of the cross-frames with 1/8-in. slip in the direction of the internal load in the intermediate cross-frame diagonals or in all the intermediate cross-frame members. D L D L slip slip Figure 3-102. Influence of a relative slip between the top and bottom of a girder splice.

118 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Response Bridge Type Lack of Fit Required? Rationale Major-axis bending stress fb Straight-Skewed No Locked-in fb is negligible. Skewed &/or Curved General girder vertical displacements, layovers, and final elevations Straight-Skewed No The vertical displacements are insensitive to initial lack-of- fit effects. Skewed &/or Curved Yes Girder vertical displacements can be affected significantly by cross-frame detailing effects. Girder layovers in the DL condition corresponding to the type of cross- frame detailing Straight-Skewed No Approximately plumb webs are obtained. Skewed &/or Curved Cross-frame forces and girder flange lateral bending stresses Straight-Skewed Conditionally No As long as: (1) The first intermediate cross-frames are offset based on Equation 15, and (2) The cross-frames are symmetrical about their mid- length (e.g., no Z-type cross-frames), separate single-size intermediate and bearing-line cross- frames can be designed conservatively and used throughout the bridge based on the maximum member forces obtained from an accurate 2D-grid or 3D FE analysis neglecting lack-of-fit effects (top chord members designed for the maximum tension and the maximum compression determined in the top chord at the cross- frames throughout the bridge, bottom chord members designed similarly, and diagonal members designed similarly). One cross-frame type can be designed for all the intermediate cross-frames, and another for the bearing line cross-frames. In addition, the girder flange lateral bending stresses tend to be predicted conservatively from an accurate 2D-grid or 3D FE analysis neglecting lack-of-fit effects given above caveat Number 1 (e.g., see Figure 3-61). Unfortunately, for bridges with larger skew indices, the conservatism of designing single-size cross-frames in the above fashion can be prohibitive. Since the distribution of the internal cross-frame forces based on NLF detailing (see Figure 3-57) can be very different from that obtained based on SDLF or TDLF detailing (see Figure 3-58), the only alternative if the cross-frames are detailed for SDLF or TDLF is to account for the corresponding locked-in force effects in the analysis. In addition, note that generally, the total forces in the SDL condition (SDL + locked-in, e.g., see Figure 3-60) need to be considered. Skewed &/or Curved Yes The cross-frame forces and girder flange lateral bending stresses generated by the cross-frame detailing effects tend to be additive with the dead load effects. Table 3-11. Summary recommendations and rationale for when the initial lack of fit from SDLF or TDLF detailing should be considered in an accurate 2D-grid or 3D FE analysis (based on the assumption that the cambers are determined from an accurate 2D-grid or 3D FE analysis).

Findings and Applications 119 of the same straight-skewed bridge matches exactly with the above line-girder analysis results, but only if initial lack-of-fit effects are considered in the analysis (and only if the lack-of-fit effects are calculated based on the cambers from the line-girder analysis). Although there is no lack of fit between the cross-frames and the girders in the above SDL condition, there is a lack of fit between the cross-frames and the girders in the initially fabricated (cambered and plumb) girder geometry. Therefore, locked-in forces are generated by the SDLF detailing. These locked-in cross-frame forces are exactly equal and opposite to the cross-frame forces from the SDL, and the corresponding locked-in girder flange lateral bending stresses are exactly equal and opposite to the SDL girder flange lateral bending stresses. Assuming that the structural system remains elastic, and neglecting aspects such as friction at the supports and non-zero connection tolerances, the bridge response is unique. That is, regardless of the sequence in which the structure is erected, if the cross-frames are detailed for SDLF based on the cambers from the above line-girder analysis, the total internal cross-frame forces and the girder flange lateral bending stresses are theoretically zero in the final SDL configuration. In summary, an accurate 2D-grid or 3D FE analysis has to include the initial lack-of-fit effects (i.e., the corresponding locked-in forces need to be calculated in the analysis) to properly capture the bridge behavior corresponding to this “optimum” SDL camber-SDLF detailing combination. The key attributes of this “optimum” combination are summarized again in Table 3-14. It is important to note that the total dead load (TDL) line-girder analysis responses for a bridge fabricated with the above “optimum” SDL camber-SDLF detailing combination typically will not be accurate. The only way to obtain accurate TDL results in general for the above type of bridge is to conduct an accurate 2D-grid or 3D FE analysis including the initial lack-of-fit effects. •• Total dead load (TDL) response of straight-skewed bridges with the cross-frames fabricated for TDLF based on line-girder analysis cambers. It is very interesting to note that, in many situations, if the TDL cambers are obtained from a line-girder analysis, the total TDL cross- frame forces and girder flange lateral bending stresses tend to be minimized (relative to the results with other cross-frame detailing options). However, these forces and stresses generally are not zero. This is because of (1) the interaction between the girders and cross-frames in the 3D structural system once the cross-frames are engaged, (2) the influence of non-equal loading effects on the fascia girders and the interior girders, (3) the influence of eccentric loads applied to the fascia girders from overhang brackets, and (4) the interaction between the girders and a composite concrete deck, for any construction stages where the deck has gained significant early stiffness and strength. However, in cases with relatively equal load effects on the fascia and interior girders, and if the torsion from eccentric overhang loads on the fascia girders is estimated from a separate analysis, the TDL line-girder analysis results are reasonably accurate for the above case. This fact can be understood by considering the behavior for the “optimum” SDL camber-SDLF detailing combination, and then realizing that the comparable TDL camber-TDLF detailing combination works approximately the same. Similar to the previous summary, an accurate 2D-grid or 3D FE analysis has to include the initial lack-of-fit effects to properly capture the bridge behavior for this TDLF camber-TDLF detailing combination. The reader is referred to the Task 8 report (Appendix C of the contractors’ final report), Section 7.5.1, for an example illustrating these results. Ozgur (2011) provides additional detailed examples and results. Lastly, it is important to note that the steel dead load (SDL) line-girder analysis responses for a bridge fabricated with the above TDL camber-TDLF detailing combination typically will not be accurate. The only way to obtain accurate SDL results in general for the above type of bridge is to conduct an accurate 2D-grid or 3D FE analysis including the initial lack-of-fit effects. •• Use of cambers obtained from a V-load analysis on curved or curved and skewed I-girder bridges. As shown previously in Table 3-1 of Section 3.1.2, the vertical displacements obtained from a 1D line-girder (V-load) analysis can be in error by as much as 20 percent for curved radially supported bridges with IC ≤ 1 (C grade in Table 3-1), and by as much as 30 percent for

120 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges curved radially supported bridges with IC > 1 (D grade in Table 3-1). For curved and skewed I-girder bridges, Table 3-1 shows an F grade for the vertical displacements. Therefore, gener- ally the use of a V-load analysis to set the camber profiles in curved or curved and skewed bridges should be discouraged. For curved, or curved and skewed I-girder bridges, the dis- placements used to set the girder cambers and to establish the cross-frame drops for SDLF or TDLF detailing of the cross-frames generally should be based on an accurate 2D-grid or 3D FE analysis. Regarding whether the initial lack-of-fit effects should be included in the structural design analysis, Table 3-11 then applies. Section 7.5.2 of the Task 8 report (Appendix C of the contractors’ final report) discusses an interesting fact that the V-load analysis results for simple-span radially supported I-girder bridges approximate the physical responses obtained using TDLF detailing better than the responses for SDLF or NLF detailing. This appears to be due to the fact that the girder webs are held in an approximately plumb position at the cross-frame locations when TDLF detailing is used. Ozgur (2011) provides some additional discussion of this behavior. 3.3.5 Estimation of Steel Erection Fit-Up Forces Including SDLF or TDLF Effects The identification of potential fit-up difficulty during steel erection and the development of erection plans that avoid or alleviate this difficulty is a key task for the erection engineer. This task is often handled based on experience and using relatively simple analysis tools. However, in some situations with longer spans, tighter curves, and sharper skews, a relatively rigorous estimate of the forces required to fit-up the steel may be desirable at certain intermediate stages. This section outlines one recommended process for determining these estimates. The basic concepts are relatively simple and can be listed as follows: 1. Select a given erection stage where fit-up of the steel is a concern. Numerous factors enter into the decision about which erection stages may need to be evaluated. In very broad terms, fit-up difficulty is typically due to some combination of structural component or unit weights, deflections of the steel components under their self-weight, component resistances to being deformed by come-alongs, jacks, cranes, etc., such that their connections can be made, and site conditions or restrictions that limit the erector’s ability to provide temporary supports and/or to apply forces to make a given connection. In many situations, the greatest fit-up difficulty occurs when the connections are made for one of the last girders to be erected in a given span. This is because the incompatibility in the displacements may be significant between the portion of the structure that is already erected, particularly if the structure has significant deflections under its self-weight, and a relatively lightly loaded girder that is being assembled. In addition, the locked-in forces due to SDLF or TDLF detailing of the cross-frames tend to build up as more and more components are connected. It is important to note that there are as many steel dead load configurations as there are erection stages. SDLF detailing commonly is based on the final erected configuration. Therefore, if SDLF detailing is used, the girder webs generally are not close to plumb under steel dead load until all the steel is erected. Particularly when TDLF detailing is used, the locked-in forces can be a significant fraction of the internal forces in the steel during the steel erection. 2. Analyze the structure in the specific configuration attained immediately after the targeted connection is made. The calculated force in the targeted connection at this stage is a direct indicator of the forces that need to be applied to make the connection. This is because, just prior to making the connection, the connection force is, of course, zero. As noted previously, the locked-in forces from any initial SDLF or TDLF lack of fit between the cross-frames and the girders in their initial fabricated (cambered and plumb) geometry can be significant in some cases. These lack-of-fit effects can be included in the analysis with relative ease using

Findings and Applications 121 the procedures discussed previously in Section 3.2.5. The actual forces that the erector must apply are, of course, generally different from the above connection force. However, given the connection force that needs to be developed, along with the selection of rigging and other equipment, reasonable estimates can be made of the actual forces the erector will need to apply. If the estimated fit-up forces are too large, manipulate the temporary supports, holding cranes, and other devices to limit the displacement incompatibility (the lack of fit in the deformed geometry) that needs to be resolved at the stage just prior to making the connection. 3.4 Pros and Cons of Different Cross-Frame Detailing Methods There is much variety in the industry across the United States regarding practices and preferences pertaining to the detailing of the cross-frames to affect the constructed geometry of curved and/or skewed bridges and to achieve successful erection of the structural steel. In many cases, this variety of practices does not mean that one method is “wrong” or another answer is “better.” Rather, there is often more than one right answer, and successful practices vary widely based on local preferences, local strengths, and specific characteristics of a given bridge. In recognition of the above, this section aims to synthesize the wide range of information about each of the main cross-frame detailing methods (NLF, SDLF, and TDLF) learned from the NCHRP Project 12-79 studies in the form of “pro facts,” “con facts,” and commentary about these facts and their implications and applications to the two basic types of I-girder bridges addressed in the previous sections: straight-skewed and horizontally curved radially supported. The intent is to provide a reasonably comprehensive accounting of the various factors that can influence the choice of a method, so that designers, detailers, fabricators, erectors, and owners have information readily at hand. Tables 3-12 through 3-17 provide a synthesis of the pro facts, con facts, and commentary for straight-skewed bridges while Tables 3-18 through 3-23 provide this information for horizontally curved radially supported bridges. These tables are followed by a short discussion of horizontally curved bridges with significant skew of their bearing lines and detailing for special cases such as widening projects, phased construction, and specific tub-girder bridge considerations. As would be expected, horizontally curved bridges with significant skew generally exhibit a combination of the characteristics detailed in the above tables for straight-skewed and curved radially supported bridges. If the skews increase the girder length on the outside of the curve, Pro Facts: Commentary: The steel fits together with zero applied force in the no-load condition. This facilitates erection in a shored configuration approximating the theoretical no- load condition. However, erection under other shored or unshored conditions is practically always achievable for straight-skewed bridges, particularly if SDLF detailing is used. Furthermore, NLF detailing leads to other undesirable consequences on straight-skewed bridges as discussed in Table 3-13. The locked-in forces are zero. As a result, the structural analysis is simpler. When the cross-frames are detailed for NLF, the cross-frame forces are theoretically as analyzed by the designer for SDL, TDL, and LL+I. No additional locked-in forces are present. Table 3-12. Pro facts and commentary, straight-skewed I-girder bridges with NLF detailing—no lack of fit between the cross-frames and the girders in the fabricated (cambered and plumb) no-load geometry of the girders (Configuration 1 of Figure 3-30); plumb girder webs in the no-load state after connecting the cross-frames (Configurations 2 and 3 of Figure 3-30).

122 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Con Facts: Commentary: Due to differential displacements and rotations between the girders, the steel does not fit together in an unshored SDL condition without applying forces. This is not a problem for smaller spans and/or smaller skew indices. However, for longer span lengths and larger skew indices, temporary shoring or hold cranes will likely be required. The erector may encounter fit-up difficulty if the girders are not supported by holding cranes or temporary supports such that their dead load deflections are limited. In some cases, the erector may have to affect the relative vertical elevation of the girders, in addition to twisting a girder, to install a cross-frame. Affecting the relative girder vertical elevations typically is much more difficult to accomplish. In straight- skewed bridges, this problem can be alleviated by (1) temporary shoring or hold cranes, if NLF detailing is used, (2) the use of SDLF detailing and allowing the steel to deflect to its unshored SDL profile (this may require temporary shoring or holding to that profile; also, see the subsequent discussion of the “optimum” girder SDL cambers for SDLF detailing in straight-skewed bridges), and (3) generally, offsetting the first intermediate cross-frames relative to the bearing lines based on Equation 15 of Section 3.3.1.6, a > max(1.5D, 0.4b). The girders twist (i.e., lay over) under any dead load conditions. At highly skewed end bearing lines, the TDL layover of the girders tends to be large. More expensive bearings may be required in some instances at heavily skewed bearing lines, unless the dead load rotations are accommodated by modifying the bevels of the sole plates (note that beveled sole plates are already common in many bridges to accommodate grade changes along the length of the bridge). The deck dead load lateral deflections due to girder layover must be addressed in the alignment of any deck joints. Substantial layover of the girders under the TDL (in the final constructed condition) may be visually objectionable. The NCHRP Project 12-79 research, as well as other prior research studies, has shown that the influence of girder layovers on the system strength is negligible as long as the checks for global stability, stability between the cross-frame locations, and bracing of the girders are satisfied. If desired, the layover of the girders at the completion of the erection can be estimated accurately, based on the Table 3-1 guidance. These layovers may be specified on the engineering drawings to indicate the expected geometry. The locked-in forces are zero, since the girders are not “reverse twisted” during the installation of the cross-frames. At the end of the construction (i.e., under TDL conditions), the internal cross-frame forces and girder flange lateral bending stresses tend to be larger (compared to the results with other cross-frame detailing methods) due to the effects of torsion, since these forces and stresses are not offset by any locked-in force effects that would have been introduced by initial lack of fit in the no-load condition if the SDLF or TDLF cross-frame detailing methods were used. Table 3-13. Con facts and commentary, straight-skewed I-girder bridges with NLF detailing. the skew effects tend to amplify the horizontal curvature effects (i.e., the bridge tends to exhibit a significant overall twist rotation of its cross-section within the span). If there is a sharp skew that increases the length of the girder on the inside of the curve, the bridge tends to behave more like a straight-skewed structure. In cases involving widening projects and/or phased construction where new cambered girders are placed next to an existing decked girder line under total dead load, detailing the cross-frames to fit between the steel dead load profile of the new girders and the total dead load profile of the decked girders at the time of the erection is one option. Another option is to not provide any cross-frame diagonals to transfer shear between the new and existing girders until after the deck is placed on the new girders. Because of the inherent torsional stiffness of the tub girders, and to maintain equal heights of the webs on both sides of the tubs, these girder types are commonly detailed with their cross-section rotated parallel to the bridge cross-slope in the initial no-load configuration. The bearing-line diaphragms are commonly detailed so that they can be subassembled, then fitted

Findings and Applications 123 to the girder bottom flange and web assemblies in the shop (i.e., they are detailed for no-load fit [NLF]). The stiffeners are kept normal to the bottom flange (AASHTO/NSBA, 2006). At a skewed bearing line, if the diaphragm plate also is kept normal to the flanges, this means the major-axis bending camber rotation of the girders (at the bearing) must be accounted for in determining the fabricated profile of the diaphragm plate. Otherwise, the profile geometry of the diaphragm plate will not fit-up with the profile geometry of the tub girders without some forcing. Intermediate external cross-frames (or diaphragms) in tub-girder bridges can be installed no-load fit (NLF) or a special steel dead load fit (SDLF) to the tub girders in their unshored deflected position under the steel self-weight (special because both the vertical deflections and the torsional rotations of the tub girders are considered). This latter detailing of the intermediate external cross-frames allows them to be installed theoretically without having to apply any force to the girders, assuming that the girders are in an unshored deflected position at the time of the installation. These cross-frames are subsequently effective to help restrain relative torsional movement between the tub girders during the placement of the deck (although, they are not effective in restraining the relative movements between the tub girders under the steel self-weight). If the cross-frames are detailed for this special SDLF, they will then Table 3-14. Pro facts and commentary, straight-skewed I-girder bridges with SDLF detailing—cross-frames fabricated such that they do not fit-up with the girders in their fabricated (cambered and plumb) no-load geometry (Configuration 1 of Figure 3-30); the erector must generally “reverse twist” the girders during the installation of the cross-frames to achieve fit-up (Configuration 3 of Figure 3-30). Pro Facts: Commentary: Locked-in forces are generated by the initial lack of fit between the cross-frames and the girders in their fabricated (cambered and plumb) no-load geometry. The girder webs will be approximately plumb at the end of the steel erection. This results in a web plumb condition, which is easy to measure and inspect at a time when the erector is still on site and the deck has not yet been cast (thus allowing better opportunity to correct any misalignments). The girder vertical displacements are relatively insensitive to the lack-of-fit effects from SDLF detailing in straight-skewed I-girder bridges. The internal cross-frame forces and girder flange lateral bending stresses tend to be minimized under the SDL conditions (compared to the results from other methods of cross-frame detailing). The corresponding TDL forces and stresses (at the end of the construction) generally still are significant, but are reduced relative to the results for NLF detailing. The tendency for uplift at bearings (e.g., uplift at bearings located at the acute corners of a simply supported bridge plan) is minimized under the SDL conditions (compared to the other methods of cross- frame detailing). This is a direct result of the internal cross-frame forces being minimized. At highly skewed end bearing lines, the TDL layover of the girders is smaller than that for NLF detailing. Depending on the skew angle of the bearing line and the rotation capacity of the bearings, the layover of the girders at the bearing line may be acceptable. The lack of fit between the cross- frames and the girders, under any unshored SDL erection conditions, tends to be small compared to the results from other methods of cross-frame detailing (this applies only to straight-skewed bridges). For straight-skewed bridges, the girder unshored SDL deflections during the steel erection tend to largely offset the SDL cambers, even though the SDL cambers are based on the state at the completion of the steel erection. The discussion below of the “optimum” SDL cambers requires a thorough understanding of the behavior, but aids in understanding these general statements. (continued on next page) (text continues on page 133)

124 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Line-girder analysis provides the “optimum” SDL cambers for SDLF detailing of the cross-frames in straight-skewed bridges. This statement applies only to straight-skewed bridges and only to SDLF detailing. “Optimum” means that the total cross-frame forces and girder flange lateral bending stresses in the SDL condition, produced by the sum of the SDL forces and the locked-in forces (from the lack of fit in the NL geometry due to SDLF detailing), are minimized. If the girder SDL camber is obtained from a line-girder analysis, then theoretically, there is zero lack of fit between the girders and the cross-frames in an idealized unshored SDL configuration where, prior to engaging the cross-frames, all the girders are placed on the supports and allowed to deflect under the self-weight of the steel. The girder webs are plumb in this condition, since there is no interaction with the cross-frames. If the girders are detailed for SDLF based on the above cambers, zero force is required to fit-up the cross-frames with the girders in the above idealized unshored SDL configuration. Correspondingly, once all the cross-frames are fully connected in this configuration to complete the steel erection, the internal cross- frame forces and the girder flange lateral bending stresses will be zero. Assuming that the structural system remains elastic, and neglecting aspects such as friction at the supports and non-zero connection tolerances, the bridge response is unique. That is, regardless of the sequence in which the structure is erected, if the cross-frames are detailed for SDLF based on the cambers from a line-girder analysis, the internal cross-frame forces and the girder flange lateral bending stresses are theoretically zero in the final SDL configuration. Although there is no lack of fit in the above idealized SDL condition, there is a lack of fit between the cross-frames and the girders in the initially fabricated (cambered and plumb) girder geometry. Therefore, locked-in forces are generated by the SDLF detailing. If the SDL cambers are obtained based on an accurate 2D-grid or 3D FE analysis, rather than a line-girder analysis, the sum of the SDL and lack-of-fit forces will be non-zero even though the cross- frames are detailed for SDLF. This is because the interaction between the girders and cross-frames in the 3D structural system modifies the girder SDL displacements from the values discussed above. The torsional effects of the distributed vertical loads (the self- weight of the steel) cannot be perfectly offset by the locked-in concentrated forces at the cross-frame locations. Generally, the above sum of the SDL forces and the locked-in forces is not negligible, but tends to be relatively small as long as the first intermediate cross-frames are sufficiently offset from the bearing lines based on Equation 15 of Section 3.3.1.6, a > max(1.5D, 0.4b). Pro Facts: Commentary: Table 3-14. (Continued).

Findings and Applications 125 Table 3-15. Con facts and commentary, straight-skewed I-girder bridges with SDLF detailing. Con Facts: Commentary: Locked-in forces are generated. In general, an accurate 2D-grid or 3D FE analysis is required to accurately assess the bridge responses at the end of the construction (i.e., under TDL conditions), as well as for any conditions other than SDL. Generally, for IS > 0.30, accurate calculation of the responses by line-girder analysis is possible only for the SDL condition, and only if the SDL cambers are set based on the line-girder analysis. The locked-in forces are not sufficient to offset the TDL forces in the final constructed condition. At the TDL level, the cross-frame forces and girder flange lateral bending stresses can be significant, and need to be considered in the design, although they are smaller than when NLF detailing is used. At the end of the construction (i.e., under TDL conditions), the cross-frame forces and girder flange lateral bending stresses tend to be larger than for the case of TDLF detailing, although these forces and stresses are smaller than for NLF detailing. Line-girder analysis provides an accurate characterization of SDL vertical deflections and major-axis bending stresses in straight-skewed bridges, if the cross-frames are detailed for SDLF using the line-girder analysis cambers. For any other conditions and/or combinations with SDLF detailing, line-girder analysis can provide erroneous predictions of the girder vertical deflections and major-axis bending stresses. The magnitude of the errors is strongly correlated with the skew index IS. When the concrete deck is placed and other appurtenances are added to the bridge, the resulting cross-frame forces can be substantial in bridges with a large skew index, due to the torsional interactions within the system. In general, if SDLF detailing is used, accurate calculation of the cross-frame forces and flange lateral bending stresses in the TDL condition (from the sum of the TDL effects plus the locked-in force effects) requires the use of an accurate 2D-grid or 3D FE analysis including the modeling of the initial lack of fit. At highly skewed end bearing lines, the TDL layover of the girders still may be large. More expensive bearings may be required in some instances at heavily skewed bearing lines, unless the dead load rotations are accommodated by modifying the bevels of the sole plates (note that beveled sole plates are already common in many bridges to accommodate grade changes along the length of the bridge). The deck dead load lateral deflections due to girder layover must be addressed in the alignment of any deck joints. Substantial layover of the girders under the TDL (in the final constructed condition) may be visually objectionable. The NCHRP Project 12-79 research, as well as other prior research studies, has shown that the influence of girder layovers on the system strength is negligible as long as the checks for global stability, stability between the cross-frame locations, and bracing of the girders are satisfied. For longer spans, the difference between the girder SDL displacements obtained from a line-girder analysis versus an accurate 2D-grid or 3D FE analysis can be significant. This is due to the interaction between the girders and the cross-frames in the 3D system. However, as noted in the pros for SDLF detailing, setting the cambers and the SDLF detailing based on a line-girder analysis tends to minimize the lack of fit between the girders and the cross-frames in any unshored SDL erection conditions. If the SDL camber is based on line-girder analysis, an accurate 2D-grid or 3D FE analysis can reproduce the line-girder analysis (and the physical/actual) vertical deflections, but only if lack-of-fit effects are included in the analysis. An accurate 2D- grid or 3D FE analysis does not give the correct girder vertical displacements for a bridge where the SDL cambers are based on line girder analysis, unless the lack-of- fit effects are included in the analysis.

126 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Pro Facts: Commentary: Locked-in forces are generated by the initial lack of fit between the cross-frames and the fabricated (cambered and plumb) no-load geometry of the girders. The girder webs will be approximately plumb at the end of the construction (i.e., at the TDL level). The girder vertical displacements are relatively insensitive to the lack-of- fit effects from TDLF detailing in straight-skewed I-girder bridges. The internal cross-frame forces and girder flange lateral bending stresses (due to the sum of the dead load effects plus the locked-in force effects) tend to be minimized under the TDL conditions (compared to the results from other methods of cross-frame detailing). The tendency for uplift at bearings (e.g., uplift at bearings located at the acute corners of a simply supported bridge plan) is minimized under the TDL conditions, compared to the other methods of cross-frame detailing. The final internal force and stress state and the girder deflections in the TDL condition can be approximated from a line-girder analysis, if the TDL cambers are set based on a line-girder analysis and the cross-frames are detailed for TDLF. However, the accuracy of the line-girder analysis degrades as a function of the bridge skew index. A line-girder analysis predicts zero cross-frame forces and zero flange lateral bending stresses. The sum of the TDL effects plus the locked-in force effects generally is not zero, although this sum tends to be minimized as noted above. Line-girder analysis provides an accurate characterization of the TDL vertical deflections and major-axis bending stresses in straight-skewed bridges, as long as the cross-frames are detailed for TLDF using the TDL line-girder analysis cambers. For any other loading conditions combined with TDLF detailing (e.g., the SDL vertical deflections of a bridge using TDLF detailing), line-girder analysis can provide erroneous predictions of the girder vertical deflections and major-axis bending stresses. The magnitude of these errors is strongly correlated with the skew index IS. The TDL layover of the girders is approximately zero, even at highly skewed end bearing lines. This minimizes the total, long-term (permanent) rotation demand on the bearings. Also, this allows the use of a target TDL geometry in which the girders are assumed to be plumb for the layout of deck joints, etc. On bridges with constant cross- slope, detailing for TDLF allows the cross-frames to be built identically, with one fabrication set-up and one piece-mark for multiple frames. This makes the fabrication of the cross-frames more efficient and economical and facilitates the handling of the cross-frames during the erection. Table 3-16. Pro facts and commentary, straight-skewed I-girder bridges with TDLF detailing—cross-frames fabricated such that they do not fit-up with the girders in their fabricated (cambered and plumb) no-load geometry (Configuration 1 of Figure 3-30); the erector must essentially “reverse-twist” the girders during the installation of the cross-frames to achieve fit-up (Configuration 3 of Figure 3-30).

Findings and Applications 127 Table 3-17. Con facts and commentary, straight-skewed I-girder bridges with TDLF detailing. Con Facts: Commentary: • Significant locked-in forces are generated. • In general, a more complex analysis is required for any conditions other than TDL. Even for evaluating the TDL condition, an accurate 2D-grid or 3D FE analysis, including the influence of the initial lack of fit between the cross-frames with the girders in their fabricated (cambered and plumb) no-load geometry, is necessary to obtain accurate cross-frame forces and girder flange lateral bending stresses for bridges with larger skew indices. • In some cases, the cross-frame forces and girder flange lateral bending stresses under the SDL (due to the sum of the steel dead weight plus the locked-in force effects) at a given stage of steel erection can be larger than the corresponding TDL values at the end of the construction. If the bridge girders are connected to the cross-frames and supported in a configuration between the SDL and NL conditions, these forces and stresses can be even larger. Therefore, during interim stages of the steel erection, the locked-in force effects can be significant and should be considered in the design. It should be noted that the locked-in forces tend to be smaller at the earlier stages of the steel erection. These forces build up as more and more components are assembled into the structural system. • The corresponding forces and stresses under the TDL (at the end of the construction), due to the sum of the dead load effects plus the locked-in force effects, are relatively small compared to the results from the other methods of cross-frame detailing. However, generally, they are not negligible unless the skew index is smaller than IS = 0.30. These non-zero cross- frame forces and girder flange lateral bending stresses are due to the interaction between the girders and the cross-frames (as well as the girders and the slab for construction stages after portions of the concrete deck start to act compositely). The torsional effects of the distributed dead loads cannot be perfectly offset by the locked-in concentrated forces at the cross-frame locations. • The girder webs will not be plumb under the NL or SDL conditions, once the girders are connected to the cross-frames. The girders will be laid over in the opposite direction from the direction in which they twist under the application of the dead loads. • The bearings at support lines with significant skew can be subjected to relatively large rotation demands under various NL and SDL conditions during erection and prior to placement of the concrete deck. However, these rotations are temporary and are not additive with the rotation demands due to live loads. If necessary, blocking may be used to protect the bearings at these locations. • If desired, the layover of the girders at the completion of the steel erection can be estimated from an accurate 2D-grid or 3D FE analysis. These layovers can be specified on the engineering drawings to indicate the expected geometry. • Caution: The girders can be plumb only under one TDL condition. • If there are significant DC2 loads (such as a substantial utility load, barrier rail load, or wall load), the designer must decide under which TDL the girders should be plumb. • Caution: Various incidental effects may have an influence on the bridge TDL responses. • If early stiffness gain on the concrete deck from prior deck casting stages, or from set-up of the concrete during a given stage, is expected to be a factor, these effects would need to be considered in the calculation of the TDL displacements, internal forces, and internal stresses. In addition, other incidental effects such as tipping restraint at the bearings, participation of the metal deck forms, temporary timber struts between girders, welding of rebar between shear studs on adjacent girders, etc., can influence the response and may need to evaluated when estimating the TDL displacements, internal forces, and internal stresses. • The lack of fit between the cross-frames and the girders is maximized under any shored or unshored SDL erection conditions, compared to the results from other methods of cross-frame detailing. • For longer spans and larger skew indices, the forces required to fit-up the cross-frames with the girders during the steel erection can be substantial. This is because the TDL major-axis bending deflections have not yet occurred (since only the steel self-weight load is on the structure). Furthermore, for longer spans, the girders tend to be deeper, the girders tend to be more closely spaced relative to their depth, the flanges tend to be larger, and overall, the girders tend to be harder to deform to resolve the incompatibility in displacements between the cross-frames and the girders under the SDL. • The girder SDL deflections typically can be used to reduce the lack of fit between the girders and the cross-frames in an unshored SDL erection condition. In many situations, this may be sufficient to limit the magnitude of the forces the erector will need to apply to get the steel to fit up.

128 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Pro Facts: Commentary: The steel fits together with zero applied force in the no-load condition. This facilitates erection in a shored configuration approximating the theoretical no-load condition. However, erection under other shored or unshored conditions should be feasible for smaller spans and/or smaller curvature. The locked-in forces are zero. As a result, the structural analysis is simpler. When the cross-frames are detailed for NLF, the cross-frame forces are theoretically as analyzed by the designer for SDL, TDL, and LL+I. No additional locked-in forces are present. NLF detailing tends to minimize the total cross-frame forces, as well as the girder flange “negative” lateral bending moments over the cross-frame locations. This is because the locked-in cross-frame forces due to SDLF or TDLF detailing of the cross-frames tend to be additive with the SDL and TDL cross-frame forces in curved radially supported bridges (see the subsequent “con” discussions for SDLF and TDLF cross-frame detailing). This statement is true both for situations where temporary shoring or hold cranes are used to support the girders in a near NL condition, as well as for unshored SDL or TDL conditions. The girder flange lateral bending moments tend to be the largest at the cross-frame locations. The girder flanges act in lateral bending effectively like continuous-span beams. The cross-frames act as the supports for these analogous continuous-span beams (see the subsequent “con” discussions for SDLF and TDLF cross-frame detailing). This statement applies only to curved radially supported bridges. NLF detailing tends to minimize the fit-up forces required during the steel erection, since the girders are not “reverse twisted” during the installation of the cross- frames. In curved radially supported bridges, since the cross-frame connection forces at any intermediate stage of the steel erection tend to be smallest when NLF detailing is used, the force required to install a given cross-frame at a given intermediate stage also tends to be minimized by NLF detailing. This is because the cross-frame connection force at the intermediate stage is equal to the force that has to be developed into the cross-frame if it were installed just prior to this stage. Before the cross-frame is installed, the connection force is zero, since the cross-frame is unconnected. This statement is true both for situations where temporary shoring or hold cranes are used to support the girders in a near NL condition, as well as for unshored SDL conditions This statement applies only to curved radially supported bridges. The differential vertical displacements between the girders are comparable for both NLF and SDLF in curved radially supported bridges. In some cases, the erector may have to affect the relative vertical elevation of the girders, in addition to twisting a girder, in order to install a given cross-frame. Curved girders tend to twist as well as deflect vertically under their self-weight. The girder twisting tends to increase the girder vertical deflections. This is beneficial in facilitating the fit-up to steel that has already been erected (if working from the inside of the curve to the outside of the curve in erecting the girders). The steel that has already been erected will be over-rotated relative to its final SDL configuration. If working from the outside to the inside of the curve, the girders can be interconnected first near the mid-span, and the self-weight of the added girder can be used to reduce the over- rotation of the partially erected bridge cross-section. These attributes work essentially the same in bridges with either NLF or SDLF detailing. See the subsequent discussion under the pros for SDLF detailing. This statement applies only to curved radially supported bridges. Table 3-18. Pro facts and commentary, curved radially supported I-girder bridges with NLF detailing—no lack of fit between the cross-frames and the girders in the fabricated (cambered and plumb) no-load geometry of the girders (Configuration 1 of Figure 3-30); girder webs plumb in the no-load state after connecting the cross-frames (Configurations 2 and 3 of Figure 3-30).

Findings and Applications 129 Con Facts: Commentary: The girders will be twisted (laid over) under any dead load conditions. Layover of the girders is restrained essentially to zero by the bearing-line cross-frames at radial bearing lines. Layover of the girders within the span is more difficult to detect and therefore tends not to be visually objectionable. The NCHRP Project 12-79 research, as well as other prior research studies, has shown that the influence of girder layovers on the system strength is negligible as long as the checks for global stability, stability between the cross-frame locations, and bracing of the girders are satisfied. Because there are no locked-in forces, the girders see larger “positive” flange lateral bending moments between the cross-frames. The “negative” flange lateral bending moments at the cross-frame locations are typically the largest moments. Therefore, NLF detailing of the cross-frames tends to give smaller maximum flange lateral bending moments. Table 3-19. Con facts and commentary, curved radially supported I-girder bridges with NLF detailing. Pro Facts: Commentary: Locked-in forces are generated by the initial lack of fit between the cross-frames and the girders in their fabricated (cambered and plumb) no-load geometry. The girder webs will be approximately plumb at the end of the steel erection. This results in a web plumb condition that is easy to measure and inspect at a time when the erector is still on site and the deck has not yet been cast (thus allowing better opportunity to correct any misalignments). The layover of the girders within the span will be smaller than that for NLF detailing. In curved radially supported bridges, the “reverse twisting” of the girders required to install the cross-frames induces internal forces that twist the girders in the opposite direction from that which they tend to roll under the dead load. This occurs at all of the cross-frames along a given span. As a result, the overall “global” twisting of the girders, and the corresponding lateral bending of the girder flanges, is reduced along the full span lengths. The differential vertical displacements between the girders are comparable for both NLF and SDLF in curved radially supported bridges. In some cases, the erector also may have to affect the relative vertical elevation of the girders in order to install a given cross-frame. Curved girders tend to twist as well as deflect vertically under their self-weight. The girder twisting tends to increase the girder vertical deflections. This is beneficial in facilitating the fit-up to steel that has already been erected (if working from the inside of the curve to the outside of the curve in erecting the girders). The steel that has already been erected will be over-rotated relative to its final SDL configuration. If working from the outside to the inside of the curve, the girders can be interconnected first near the mid-span, and the self-weight of the added girder can be used to reduce the over-rotation of the partially erected bridge cross-section. These attributes work essentially the same in bridges with either SDLF or NLF detailing. Some fabricators and erectors believe that bridges with cross- frames detailed for SDLF generally are easier to fit-up under unshored SDL erection conditions. The analytical evidence suggests that this is the case for straight- skewed bridges. However, the analytical evidence suggests that the forces required to fit up the steel under unshored SDL erection conditions are somewhat larger when SDLF detailing is used in curved radially supported bridges. The fit-up forces often are very comparable for curved radially supported bridges with either SDLF or NLF detailing of the cross- frames. Table 3-20. Pro facts and commentary, curved radially supported I-girder bridges with SDLF detailing—cross-frames fabricated such that they do not fit-up with the girders in their fabricated (cambered and plumb) no-load geometry (Configuration 1 of Figure 3-30); the erector must essentially “reverse twist” the girders during the installation of the cross-frames to achieve fit-up (Configuration 3 of Figure 3-30).

130 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges Con Facts: Commentary: Locked-in forces are developed within the structural system. An accurate 2D-grid or 3D FE analysis, including the influence of the initial lack of fit between the cross-frames with the girders in their fabricated (cambered and plumb) no-load geometry, is necessary to account accurately for these effects. On average, the locked-in cross-frame forces due to SDLF or TDLF detailing are additive with the SDL and TDL cross-frame forces in curved radially supported bridges. For SDLF or TDLF detailing, the girder flanges in curved radially supported bridges work effectively like continuous-span beams over the cross-frames in the lateral direction. The SDLF or TDLF detailing effects are akin to pre- stressing these effective continuous-span beams by displacing their supports (the cross-frames) in the direction opposite to that which these supports displace under the SDL or TDL. This “pre-stressing” increases the continuous-span beam reactions (i.e., the cross-frame forces) and increases the beam negative moments over the supports (i.e., the flange lateral bending moments over the cross-frame locations). The predominant SDLF or TDLF detailing effect is in the cross-frame diagonals, and is associated with a racking of the cross-frames that accomplishes the compensating deflections necessary for the girder webs to be plumb in the targeted dead load condition. Assuming that the bridge remains elastic, and neglecting aspects such as friction at the supports and non-zero connection tolerances, these responses are independent of the sequence of erection. Generally, the locked-in forces need to be calculated in the analysis to obtain accurate cross-frame forces. The girder flange maximum lateral bending stresses tend to be increased by the effects of SDLF or TDLF detailing. The girders still will be laid over within the spans under the TDL, although the layover will be smaller than for NLF detailing of the cross-frames. Layover of the girders is restrained essentially to zero by the bearing-line cross-frames at radial bearing lines. Layover of the girders within the span is difficult to detect and therefore tends not to be visually objectionable. The NCHRP Project 12-79 research, as well as other prior research studies, has shown that the influence of girder layovers on the system strength is negligible as long as the checks for global stability, stability between the cross-frame locations, and bracing of the girders are satisfied. In curved bridges, SDLF detailing tends to reduce the vertical deflections. The required cambers will tend to be over-predicted by an analysis that neglects lack- of-fit force effects. The change in the vertical deflections due to SDLF detailing is usually relatively small. However, in extreme cases such as the Ford City Bridge example (see Figure 3-84), this change can be several inches. This can reduce the concrete deck haunch to a thickness that is not desirable or can lead to problems in achieving the desired deck elevations along the spans. Generally, the locked-in forces need to be calculated in the analysis to obtain accurate girder cambers. The differential vertical displacements between the girders are comparable for both SDLF and NLF in curved radially supported bridges. SDLF detailing has approximately the same effect on the vertical displacements for all the girders at any given cross-section of the bridge. The overall rotation of the bridge cross-section tends to not be significantly affected. Therefore, the influence of SDLF detailing on the differential vertical displacements between the girders is small (i.e., there is no significant benefit of SDLF versus NLF, or vice versa, in resolving vertical displacement incompatibilities during erection). In curved, radially supported bridges, SDLF detailing tends to increase the fit-up forces required during the steel erection somewhat relative to NLF detailing. In curved radially supported bridges, since the cross-frame connection forces at any intermediate stage of the steel erection tend to be increased due to the locked-in forces from SDLF detailing, the force required to install a given cross-frame into the system at a given intermediate stage tends to be increased. This is because the cross-frame connection force at a given intermediate stage is equal to the force that has to be developed into the cross-frame if it were installed just prior to this stage. Before the cross-frame is installed, the connection force is zero, since the cross-frame is unconnected. Table 3-21. Con facts and commentary, curved radially supported I-girder bridges with SDLF detailing.

Findings and Applications 131 Pro Facts: Commentary: Locked-in forces are generated by the initial lack of fit between the cross-frames and the girders in their fabricated (cambered and plumb) no-load geometry. The girder webs will be approximately plumb at the end of the construction (i.e., at the TDL level). However, it is most important that the girder webs be plumb at the bearing lines. The cross-frames at radial bearing lines enforce this, regardless of the type of cross-frame detailing. In curved radially supported bridges, the “reverse twisting” of the girders required to install the cross-frames induces internal forces that twist the girders in the opposite direction from that which they tend to roll under the dead load. This occurs at all the cross-frames along a given span. As a result, the overall “global” twisting of the girders, and the corresponding lateral bending of the girder flanges, is reduced along the full span lengths. For TDLF detailing of the cross-frames, the twisting of the girders is approximately zero at the cross-frame locations at the end of the construction (i.e., at the TDL level). For curved radially supported bridges, the final internal cross-frame forces and the girder flange lateral bending stresses from a V-load analysis tend to correlate well with the corresponding physical responses associated with TDLF detailing of the cross- frames. However, the accuracy of V-load analysis for predicting the girder vertical deflections degrades as a function of the horizontal curvature. Table 3-22. Pro facts and commentary, curved radially supported I-girder bridges with TDLF detailing—cross-frames fabricated such that they do not fit-up with the girders in their fabricated (cambered and plumb) no-load geometry (Configuration 1 of Figure 3-30); the erector must essentially “reverse twist” the girders during the installation of the cross-frames to achieve fit-up (Configuration 3 of Figure 3-30). Con Facts: Commentary: Locked-in forces are developed within the structural system. An accurate 2D-grid or 3D FE analysis, including the influence of the initial lack of fit between the cross-frames with the girders in their fabricated (cambered and plumb) no-load geometry, is necessary to account accurately for all of the effects of the locked-in forces. For curved radially supported bridges, the final internal cross-frame forces and the girder flange lateral bending stresses from a V-load analysis tend to correlate well with the corresponding physical responses associated with TDLF detailing of the cross-frames. However, the accuracy of V-load analysis for predicting the girder vertical deflections degrades as a function of the horizontal curvature. On average, the locked-in cross-frame forces due to TDLF or SDLF are additive with the TDL and SDL cross- frame forces in curved radially supported bridges. For TDLF or SDLF detailing, the girder flanges in curved radially supported bridges work effectively like continuous-span beams over the cross-frames in the lateral direction. The TDLF or SDLF detailing effects are akin to pre- stressing these effective continuous-span beams by displacing their supports (the cross-frames) in the direction opposite to that which these supports displace under the TDL or SDL. This “pre-stressing” increases the continuous-span beam reactions (i.e., the cross-frame forces) and increases the beam negative moments over the supports (i.e., the flange lateral bending moments over the cross-frame locations). The predominant TDLF or SDLF detailing effect is in the cross-frame diagonals and is associated with a racking of the cross-frames that accomplishes the compensating deflections necessary for the girder webs to be plumb in the targeted dead load condition. Assuming that the bridge remains elastic, and neglecting aspects such as friction at the supports and non-zero connection tolerances, these responses are independent of the sequence of erection. Generally, the locked-in forces need to be calculated in the analysis to obtain accurate cross-frame forces. The girder flange maximum lateral bending stresses tend to be increased by the effects of TDLF or SDLF detailing. Table 3-23. Con facts and commentary, curved radially supported I-girder bridges with TDLF detailing. (continued on next page)

132 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges The lack of fit between the cross-frames and the girders is maximized under any shored or unshored SDL erection conditions, compared to the results from other cross-frame detailing methods. For longer spans and larger skew indices, the forces required to fit-up the cross-frames with the girders during the steel erection can be substantial. This is because the TDL major-axis bending deflections have not yet occurred (since only the steel self-weight load is on the structure). Caution: The girders can be plumb only under one TDL condition. If there are significant DC2 loads (such as a substantial utility load, barrier rail load or wall load), the designer must decide under which TDL the girders should be plumb. Caution: Various incidental effects may have an influence on the bridge TDL responses. If early stiffness gain on the concrete deck from prior deck casting stages, or from set-up of the concrete during a given stage, is expected to be a factor, these effects would need to be considered in the calculation of the TDL displacements, internal forces, and internal stresses. In addition, other incidental effects such as tipping restraint at the bearings, participation of the metal deck forms, temporary timber struts between girders, welding of rebar between shear studs on adjacent girders, etc. can influence the response and may need to evaluated when estimating the TDL displacements, internal forces, and internal stresses. Under SDL, the girders will be laid over in the opposite direction from the direction in which they twist under the application of the dead loads. These rotations are temporary and are not additive with the rotations due to live load. In curved radially supported bridges, TDLF detailing tends to reduce the vertical deflections. The required cambers will tend to be over- predicted by an analysis that neglects lack-of-fit force effects. The change in the vertical deflections due to TDLF detailing can potentially be of significance. In the extreme Ford City Bridge example (see Figure 3- 84), this change was as much as approximately 5 in. This can reduce the concrete deck haunch to a thickness that is not desirable or can lead to problems in matching the desired deck elevations at a given location along the spans. The calculation of locked-in forces generally should be included in the analysis to predict accurate girder cambers. The differential vertical displacements between the girders are not significantly affected by TDLF detailing. In curved radially supported bridges, TDLF detailing influences the girder twists and the girder vertical deflections. However, the overall rotation of the bridge cross-section tends not to be significantly affected. Therefore, the influence of TDLF detailing on the differential vertical displacements between the girders is small in these types of structures (i.e., there is no significant benefit of TDLF versus NLF or SDLF detailing in resolving vertical displacement incompatibilities during erection). In curved radially supported bridges, TDLF detailing tends to increase the fit-up In curved radially supported bridges, since the cross-frame connection forces at any intermediate stage of the steel erection tend to be increased due to the locked-in forces from TDLF detailing, the force required to install forces required during the steel erection relative to the results for both SDLF and NLF detailing. a given cross-frame into the system at a given intermediate stage tends to be increased. This is because the cross-frame connection force at a given intermediate stage is equal to the force that has to be developed into the cross-frame if it were installed just prior to this stage. Before the cross- frame is installed, the connection force is zero, since the cross-frame is unconnected. Con Facts: Commentary: Table 3-23. Continued.

Findings and Applications 133 need to be installed with the girders in their deflected steel dead load positions. If the cross- frames are installed before temporary supports or holding cranes are removed, NLF detailing is necessary. Due to the torsional stiffness of tub girders, force-fitting of the cross-frames generally is not an option. In most cases with longer-span tub-girder bridges, there are multiple field sec- tions along the spans and shoring to the approximate no-load condition is preferred. 3.5 Selection of Cross-Frame Detailing Methods for I-Girder Bridges Based on the summaries in Section 3.4, it is apparent that different methods of cross detailing work well for different I-girder bridge geometries. Furthermore, in many cases, steel I-girder bridges can be built successfully using a wide range of methods. Generally, the appropriate selection of a cross-frame detailing method depends in large part on the priority that one assigns to the various objectives and tradeoffs. Therefore, in the view of the NCHRP Project 12-79 research team, it is important to allow flexibility in any recommendations for selecting cross-frame detailing methods. However, given the detailed pros and cons discussed in Section 3.4, a few basic trends become apparent. These trends are explained in this section. For straight-skewed bridges with IS ≤ 0.30, TDLF detailing is typically a good option: •• The girder webs will be approximately plumb under the targeted TDL. •• The TDL cross-frame forces and girder flange lateral bending stresses will be canceled out in large part by the TDLF locked-in forces. As such, the cross-frame forces and girder flange lateral bending stresses tend to be minimized under the targeted TDL. In addition, these forces and stresses tend to be negligible, given IS ≤ 0.30. •• Fit-up concerns during the steel erection should be minimal, given IS ≤ 0.30. •• Line girder analysis provides a reasonable estimate of the responses under TDL, given that the cross-frame forces and the girder flange lateral bending stresses are negligible. •• The twist rotation of the girders in the SDL condition can be estimated as fz = fx tan q at skewed bearing lines, where fx is the sum of the initial camber and the SDL girder major-axis bending rotations and q is the skew angle, equal to zero for zero skew. The girder SDL twist rotation at cross-frames normal to the girders within the spans may be estimated as fz = Dy /s, where Dy is the differential vertical displacement between the cross-frame ends due to sum of the initial TDL camber and the SDL displacements. These layovers can be specified on the engineering drawings to indicate the expected geometry at the completion of the steel erection (the direction of the layovers under SDL will be opposite to those due to the TDL). •• Potential “incidental” effects such as non-calculated early stiffness gains of the concrete, tipping restraint at the bearings, participation of metal deck forms, temporary timber struts between girders, welding of rebar between studs on adjacent girders, etc., potentially should be considered when setting the TDL cambers. The accounting for these effects requires engineering judgment regarding specific construction practices and characteristics and cannot be well quantified as of this writing. The engineer may consider reducing the TDL cambers (based on ideal conditions) to ensure that the girders are not “over-cambered” or specifying a cross-frame detailing method somewhere between TDLF and SDLF, but not both of these ad hoc compensating measures. In ordinary practice, these types of effects are often neglected without any apparent detrimental influence. •• The first intermediate cross-frames generally should be positioned at an offset distance a ≥•max(1.5D, 0.4b), where D is the girder depth and b is the second unbraced length within the span adjacent to the offset from the bearing line. This is intended to alleviate local spikes in the cross-frame forces and corresponding potential fit-up difficulty due to nuisance stiffness effects. (In basic terms, the first intermediate cross-frame needs to be far enough away from

134 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges the bearing line so that one end of the girder offset length can be pulled over relative to the other without requiring excessive force.) For straight-skewed bridges with small-to-moderate span lengths and IS > 0.30, TDLF detail- ing is typically a good option: •• The girder webs will be approximately plumb under the targeted TDL. •• The TDL internal forces and stresses due to the system torsional effects will be offset in large part by the TDLF locked-in forces. As such, the cross-frame forces and girder flange lateral bending stresses will tend to be minimized under the targeted TDL. •• Fit-up during the steel erection should be feasible, given the small-to-moderate span length. •• Generally, significant cross-frame forces and girder flange lateral bending stresses will exist in the TDL condition and in other loading conditions. Accurate calculation of these values requires an accurate 2D-grid or 3D FE analysis, including the calculation of locked-in forces due to the initial lack-of-fit effects. Since the locked-in forces are comparable in magnitude to the internal forces due to the TDL effects, the internal forces from an accurate 2D-grid or 3D FE analysis neglecting the initial lack-of-fit effects will be substantially in error (e.g., compare the forces in Figure 3-57 versus those in Figure 3-58). •• The twist rotation of the girders in the SDL condition can be estimated as fz = fx tan q at skewed bearing lines, where fx is the sum of the initial camber and the SDL girder major-axis bending rotations and q is the skew angle, equal to zero for zero skew. The girder SDL twist rotation at cross-frames normal to the girders within the spans may be estimated as fz = Dy/s, where Dy is the differential vertical displacement between the cross-frame ends due to the sum of the initial TDL camber and the SDL displacements. These layovers can be specified on the en- gineering drawings to indicate the expected geometry at the completion of the steel erection. •• Potential “incidental” effects such as non-calculated early stiffness gains of the concrete, tipping restraint at the bearings, participation of metal deck forms, temporary timber struts between girders, welding of rebar between studs on adjacent girders, etc., potentially should be considered when setting the TDL cambers. The accounting for these effects requires engineering judgment regarding specific construction practices and characteristics and cannot be well quantified as of this writing. The engineer may consider reducing the TDL cambers (based on ideal conditions) to ensure that the girders are not “over-cambered” or specifying a cross-frame detailing method somewhere between TDLF and SDLF, but not both of these ad hoc compensating measures. In ordinary practice, these types of effects are often neglected without any apparent detrimental influence. •• The first intermediate cross-frames generally should be positioned at an offset distance a ≥•max(1.5D, 0.4b), where D is the girder depth and b is the second unbraced length within the span adjacent to the offset from the bearing line. This is intended to alleviate local spikes in the cross-frame forces and corresponding potential fit-up difficulty due to nuisance stiffness effects. (In basic terms, the first intermediate cross-frame needs to be far enough away from the bearing line so that one end of the girder offset length can be pulled over relative to the other without requiring excessive force.) For straight-skewed bridges with large span lengths and IS > 0.30, SDLF detailing, or detailing between SDLF and TDLF, typically are good options: •• In these cases, a potential consideration is the alleviation of fit-up difficulty during the steel erection. SDLF detailing tends to minimize the fit-up difficulty for straight-skewed bridges, but may result in significant layover of the girders at highly skewed bearing lines under the TDL. In the experience of some erectors, long-span straight-skewed bridges with TDLF detailing do not present any major problems with respect to fit-up. Ozgur (2011) discusses a 267-ft. span skewed bridge erection procedure in which TDLF detailing was used and the bridge was erected quite successfully by using the steel dead load deflections to alleviate fit-up problems.

Findings and Applications 135 •• The tendency for excessive layover at highly skewed bearing lines can be addressed by a combination of the cross-frame detailing, the use of beveled sole plates, and/or by using bearings with a larger rotation capacity. If TDLF detailing is used, the layover is addressed entirely by the cross-frame detailing. •• For SDLF detailing, the girder webs will be approximately plumb under the SDL at the completion of the steel erection. •• For other than SDLF detailing of the cross-frames, the twist rotation of the girders in the SDL condition can be estimated as fz = fx tan q at skewed bearing lines, where fx is the sum of the initial camber and the SDL girder major-axis bending rotations and q is the skew angle, equal to zero for zero skew. The girder twist rotation at cross-frames normal to the girders within the spans may be estimated as fz = Dy /s, where Dy is the differential vertical displacement between the ends of the cross-frame due to the sum of the initial TDL camber and the SDL displace- ments. These layovers can be specified on the engineering drawings to indicate the expected geometry at the completion of the steel erection. •• Generally, significant cross-frame forces and girder flange lateral bending stresses will exist in the TDL condition and in other loading conditions. Accurate calculation of these values requires an accurate 2D-grid or 3D FE analysis, including the calculation of locked-in forces due to the initial lack-of-fit effects. Since the locked-in forces are comparable in magnitude to the internal forces due to the corresponding dead load effects (e.g., SDL for SDLF and TDL for TDLF), the internal forces from an accurate 2D-grid or 3D FE analysis neglecting the initial lack-of-fit effects will be substantially in error (e.g., compare the forces in Figure 3-57 versus those in Figure 3-58). •• The first intermediate cross-frames generally should be positioned at an offset distance a ≥•max(1.5D, 0.4b), where D is the girder depth and b is the second unbraced length within the span adjacent to the offset from the bearing line. This is intended to alleviate local spikes in the cross-frame forces and corresponding potential fit-up difficulty due to nuisance stiffness effects. (In basic terms, the first intermediate cross-frame needs to be far enough away from the bearing line so that one end of the girder offset length can be pulled over relative to the other without requiring excessive force.) For curved bridges with radial supports, NLF detailing, or detailing between NLF and SDLF, typically are good options: •• NLF detailing tends to minimize the cross-frame forces as well as the “negative” girder flange lateral bending moments over the cross-frame locations, since there are no additive locked-in force effects due to initial lack of fit. •• Because the cross-frame forces tend to be minimized, the analytical evidence shows that the fit-up forces required to erect the steel tend to be minimized. However, the experience of some fabricators and erectors is that curved radially supported bridges are easier to fit-up under unshored SDL erection conditions if SDLF detailing is used. The use of SDLF detailing on curved radially supported I-girder bridges is a common practice in the industry, although bridges of this type have been detailed and constructed without difficulty using NLF detailing. It is recommended that the expanded use of NLF detailing should be explored and monitored on selected projects to further validate the NCHRP Project 12-79 findings. •• Layover of the girder webs occurs within the spans, but this layover is more difficult to detect visually and is not of any significance with respect to the bridge structural resistance as long as the checks for global stability, stability between the cross-frame locations, and bracing of the girders are satisfied. If the girder layovers within the span are judged to be excessive, the engineer may wish to employ a flange level lateral bracing system to stiffen the structure, particularly for longer span bridges. •• For NLF detailing, the structural analysis is simplified, since there are no initial lack-of-fit effects.

136 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges •• For other than NLF detailing, the locked-in force effects in the cross-frames and in the “negative” girder flange lateral bending moments at the cross-frame positions tend to be additive with the dead load effects (compare Figure 3-73 to Figure 3-72, 3-74 to 3-72, 3-76 to 3-75, and 3-77 to 3-75, see Figures 3-78 and 3-87, compare Figure 3-88b to 3-88a, and see Figures 3-90 and 3-91). Accurate calculation of these values requires an accurate 2D-grid or 3D FE analysis, including the calculation of locked-in forces due to the initial lack-of-fit effects. Since the locked-in forces tend to be additive with the internal forces due to the dead load effects, the internal forces from an accurate 2D-grid or 3D FE analysis neglecting the initial lack-of-fit effects tend to underestimate the true forces. For curved bridges with sharply skewed supports, minor horizontal curvature and small span lengths, TDLF detailing is typically a good option: •• In these cases, limiting the girder layover at the skewed bearing lines is the overriding consideration. •• The tendency for the cross-frame forces and the girder “negative” flange lateral bending moments (due to horizontal curvature effects) to be increased by the TDLF detailing can be accounted for by conducting an accurate 2D-grid or 3D FE analysis, including the calculation of locked-in forces due to the initial lack-of-fit effects. Since the locked-in forces are comparable in magnitude to the internal forces due to the dead load effects, the internal forces from an accurate 2D-grid or 3D FE analysis neglecting the initial lack-of-fit effects will be substantially in error. •• The first intermediate cross-frames generally should be positioned at an offset distance a ≥•max(1.5D, 0.4b), where D is the girder depth and b is the second unbraced length within the span adjacent to the offset from the bearing line. This is intended to alleviate local spikes in the cross-frame forces and corresponding potential fit-up difficulty due to nuisance stiffness effects. (In basic terms, the first intermediate cross-frame needs to be far enough away from the bearing line so that one end of the girder offset length can be pulled over relative to the other without requiring excessive force.) For curved bridges with moderately skewed supports, and small to moderate span lengths, detailing of the cross-frames anywhere between NLF and TDLF can be a good option: •• In this case, the engineer should select the cross-frame detailing method to balance between (1) limiting the dead load twist rotations at the skewed bearing lines, (2) alleviating the larger additive locked-in forces associated with TDLF detailing on a curved bridge, and (3) facilitating fit-up during the steel erection. •• Often SDLF detailing is a good “middle of the road” option for these bridge types. •• For other than NLF detailing, the locked-in force effects due to the horizontal curvature in the cross-frames and in the “negative” girder flange lateral bending moments at the cross-frame positions tend to be additive with the dead load effects. Accurate calculation of these values requires an accurate 2D-grid or 3D FE analysis, including the calculation of locked-in forces due to the initial lack-of-fit effects. Since the locked-in forces tend to be additive with the internal forces due to the dead load effects, the internal forces from an accurate 2D-grid or 3D FE analysis neglecting the initial lack-of-fit effects tend to underestimate the true forces. For curved bridges with skewed supports and large span length, SDLF detailing, or detailing between SDLF and NLF, is typically a good option. •• In these cases, the overriding consideration is the alleviation of fit-up difficulty during the steel erection. SDLF detailing tends to minimize the fit-up difficulty in the vicinity of highly skewed bearing lines and is often preferred by fabricators and erectors for these types of bridges. However, SDLF detailing may result in significant layover of the girders at highly skewed bearing lines under the TDL. NLF detailing tends to minimize the fit-up difficulty

Findings and Applications 137 with respect to horizontal curvature effects (based on analytical evidence), but provides no compensation for the layover of the girders at highly skewed bearing lines. •• The tendency for excessive layover at highly skewed bearing lines can be addressed by a combination of the cross-frame detailing, the use of beveled sole plates, and/or by using bearings with a larger rotation capacity. •• For SDLF detailing, the girder webs will be approximately plumb under the SDL at the completion of the steel erection. •• For other than NLF detailing, the locked-in force effects in the cross-frames and in the “negative” girder flange lateral bending moments at the cross-frame positions tend to be additive with the dead load effects due to the horizontal curvature. Accurate calculation of these values requires an accurate 2D-grid or 3D FE analysis, including the calculation of locked-in forces due to initial lack-of-fit effects. Since the locked-in forces associated with the combined skew and horizontal curvature can be comparable in magnitude to the internal forces due to the dead load effects, the internal forces from an accurate 2D-grid or 3D FE analysis neglect- ing the initial lack-of-fit effects can be substantially in error. •• The first intermediate cross-frames generally should be positioned at an offset distance a ≥ max(1.5D, 0.4b), where D is the girder depth and b is the second unbraced length within the span adjacent to the offset from the bearing line. This is intended to alleviate local spikes in the cross-frame forces and corresponding potential fit-up difficulty due to nuisance stiffness effects. (In basic terms, the first intermediate cross-frame needs to be far enough away from the bearing line so that one end of the girder offset length can be pulled over relative to the other without requiring excessive force.) 3.6 Construction Engineering Recommendations The main focus of this research was the improvement of analysis methods for erection analysis and prediction of constructed geometry of steel girder bridges. However, the construction engi- neering recommendations represent perhaps the most important results of this work, and they are likely to be the most easily implemented to provide direct benefit to the industry. The recommendations in regard to construction engineering are organized into four categories, represented by the subsections of this report presented below. The specific recommendations, however, are best presented in their full and complete form, which appears in the NCHRP Project 12-79 Task 9 report, “Recommendations for Construction Plan Details and Level of Construction Analysis.” This document is included as Appendix B of this report. This appendix provides specific guidelines and commentary on recommendations for construction plan details and recommendations for methods of structural analysis and calculations. These guidelines are comprehensive and address all aspects of erection engineering plans and calculations. An owner-agency could adopt the guidelines as a complete specification, could reference the guidelines in their erection specifications, or could adopt all or portions of the guidelines in their specifications. To further facilitate immediate implementation of these recommendations, this appendix has been deliberately written in a format and with language that can be directly adopted by AASHTO via a revision to the AASHTO/NSBA Steel Bridge Collaboration Guide Specification S10.1 – 2007, Steel Bridge Erection Guide Specification. 3.6.1 Recommendations for Construction Plan Details The reader is referred to Section 2 of Appendix B, which provides detailed and comprehensive recommendations with commentary, organized in a format that would easily lend itself to the

138 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges development of approval checklists. Figure 3-103 provides the summary checklist developed in this portion of NCHRP Project 12-79’s Task 9. 3.6.2 Recommendations for Methods of Structural Analysis and Other Calculations The reader is referred to Sections 3.1 through 3.6 of Appendix B for summary recommendations on methods of structural analysis and other calculations. Section 3.1 provides an introduction to this topic. Section 3.2 specifically provides quantitative guidance on the accuracy of various analysis methods, organized according to key parameters related to bridge geometry and framing. These analysis accuracy tables are supplemented with examples illustrating their use. Section 3.3 specifically provides guidelines on calculations for structural adequacy and espe- cially stability of the steel framing during construction, as well as guidance on myriad associated issues such as cantilever girders, uplift, temporary hold cranes and support loads, bearing, cross-frames, and bracing. Section 3.5 specifically addresses miscellaneous calculations and recommendations for crane pick locations, alignment of field splice and cross frame connections, and support conditions. Section 3.6 provides a useful calculation checklist. This checklist is shown as Figure 3-104 for ease of reference. 3.6.3 Design and Construction Considerations for Ease of Analysis via Improved Behavior There are a number of ways to improve analysis accuracy while simultaneously improv- ing the behavior and constructability of steel girder bridges by means of wisely establishing • Plan of work area o Permanent and temporary structures shown o All roads, railroad tracks, waterways, clearances, utilities, potential conflicts shown o Material (steel) storage areas shown • Erection sequence o Step-by-step procedure—figures and narrative dictating work o Delivery location of components shown o Crane locations shown o Temporary support, hold cranes, blocking, tie-downs shown o Load restrictions for certain stages (i.e., wind) • Crane information o Crane type, pick radii, boom length shown o Approximate crane pick points shown o Crane pick weights shown o Hold crane loads • Details of lifting devices and special procedures • Bolting requirements • Bearing blocking and tie-down details • Temporary supports o Details of structure shown o Load capacities • Jacking devices and procedures Figure 3-103. Erection plan and procedures checklist.

Findings and Applications 139 the framing plan for the structure and by avoiding problematic details. Problematic details are in fact a significant enough topic to warrant separate discussion in Section 3.6.4 of this report. A wisely established framing plan is one that provides clean, direct load paths and specifically avoids use of secondary bracing members (such as cross-frames) in locations where they would be anticipated to carry significantly high loads as a function of displacement compatibility. Examples include the use of lean-on bracing or omitting selected cross-frames near supports in severely skewed bridges, as cited in Krupicka and Poellot (1993). 3.6.4 Problematic Physical Characteristics and Details to Avoid The reader is referred to Section 3.7 of Appendix B for a discussion of problematic charac- teristics and details such as oversize or slotted holes, narrow bridges or bridge units, V-type cross- frames without top chords, bent-plate connections in I-girder bridges, long span I-girder bridges without top flange lateral bracing systems, partial-depth end diaphragms in tub-girder bridges, non-collinear external intermediate diaphragms in tub-girder bridges, and two-girder bearing systems at tub-girder supports. • Complete analysis of erection sequence 1. Proper level of analysis used 2. Support conditions modeled appropriately at all stages • Correct design criteria employed • Correct loads investigated • Complete checks of structural adequacy of bridge components • Complete checks of stability of girder and bridge system • Second-order amplification effects addressed as needed • Girder reactions checked for uplift • Temporary hold crane loads computed • Temporary support loads computed • Bearing capacity and rotation checks • Cross frame and bracing placement • Checks of structural adequacy of temporary supports and devices 1. Falsework towers 2. Girder tie-downs 3. Lifting beams 4. Jacking devices • Crane pick location calculations • Checks of displacements at field splices • Checks of displacements for cross frame placement Figure 3-104. Calculation checklist.