**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

*Proposed Modification to AASHTO Cross-Frame Analysis and Design*. Washington, DC: The National Academies Press. doi: 10.17226/26074.

**Suggested Citation:**"Chapter 3 - Findings and Applications." National Academies of Sciences, Engineering, and Medicine. 2021.

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60 Findings and Applications As noted in Chapters 1 and 2, the five major objectives of NCHRP Project 12-113 are addressed with both experimental and analytical studies. The experimental portions included the instrumentation and monitoring of three bridges, while the analytical studies included four independent studies (Fatigue Loading Study, R-Factor Study, Commercial Design Software Study, and Stability Study). While Chapter 2 outlined the general means and methods for each study, this chapter summarizes the key findings and results of those studies. Similar to the previous chapters, the data presented herein is not intended to provide a comprehensive overview of the experimental and analytical results. Rather, sample results are provided that highlight the key observations to contextualize the conclusions and proposed modifications to AASHTO LRFD presented in Chapter 4. For a more detailed overview of the results, refer to Appendices E and F. This chapter is divided into five major sections similar to the organization of Chapter 2. Section 3.1 outlines the major results from the field experimental program. The controlled live load test and the model validation studies are summarized, followed by the in-service stress data that supplements the computational results. Section 3.2 presents the key results from the Fatigue Loading Study with a particular emphasis on AASHTO fatigue loading criteria and WIM records. Section 3.3 summarizes the investigation of the eccentric connections for cross- frames and the proposed R-factors. More specifically, the reliability of the stiffness modification approach for 3D modeling of cross-frames is explored, as well as a proposed eccentric-beam approach. Section 3.4 reviews the data related to the Commercial Design Software Study, where the limitations of various 2D modeling methods are examined. Lastly, Section 3.5 provides a summary of the major findings related to the Stability Study. Note that the results of the industry survey were previously discussed in Section 2.1, so no additional commentary is pro- vided in this chapter. 3.1 Field Experimental Program and Model Validation As outlined in Section 2.2, two different types of field experiments were conducted in Phase II of the project for three different bridges (a straight bridge with normal supports, a straight bridge with skewed supports, and a horizontally curved bridge). The controlled live load test, for which trucks of known axle configuration and weight were statically positioned at different locations along the deck width and length, provided valuable data to aid in the validation of an FEA modeling approach. The in-service monitoring study, which measured stress cycle counts in instrumented cross-frame members and girder flanges subjected to realistic traffic conditions, provided context to the computational results obtained from the Fatigue Loading Study. C H A P T E R 3

Findings and Applications 61 Given the distinct differences in the two experimental tests, this section of the report is divided into two major subsections. Section 3.1.1 highlights key findings from the controlled live load tests, and Section 3.1.2 summarizes the measured rainflow-counting data considering one month of measured traffic data on the instrumented components of the bridges. 3.1.1 Controlled Live Load Test and Model Validation As noted in Section 2.2.2.1 the controlled live load tests performed on each bridge consisted of seven static load cases as well as a series of moving load cases. The static load cases were primarily conducted to obtain clean and concise data by which 3D FEA models could be validated. The moving-load cases were conducted at slow speeds to obtain general influence-line measure- ments for the instrumented components as well as load-position sensitivity (i.e., how truck placement impacts cross-frame force response). Before addressing the model validation results, sample measured influence lines will first be examined to introduce load-position effects. Figure 3-1 presents the axial-stress influence-line plot for select instrumented cross-frame members near the maximum positive dead load moment region in the straight bridge with normal supports that is referred to as Bridge 1. The geometry and cross-frame layout of Bridge 1 were previously summarized in Section 2.2.1. For additional reference, a plan and cross-section view of the instrumented span (i.e., 194-foot end span of three-span continuous unit) is pro- vided in Figure 3-2. Note that the identification system used for cross-frames in this figure differs from what is presented in Appendix E to simplify discussions in the report. The remainder of BS2 D2-1 TS2 D2-2 BS2 D2-1 TS2 D2-2 0 50 100 150 200 TS2 -0.80 0.00 0.80 1.60 2.40 -0.80 0.00 0.80 1.60 2.40 -0.80 0.00 0.80 1.60 2.40 ( ssertSlaix A ks i) C F Li ne 4 Br g. Br g. Distance from Start of Bridge (ft) TS2 BS2 D2-1D2-2 Figure 3-1. Influence-line plots for various instrumented cross-frame members.

62 Proposed Modification to AASHTO Cross-Frame Analysis and Design the figures presented in this section, both the static and moving-load cases, are based on instru- mented cross-frames in Bridge 1 unless noted otherwise. The full set of field data related to all three bridges can be found in Appendix E. Three distinct loading scenarios are presented sequentially in the figure. In the first load case (top plot), a three-axle dump truck (with an approximate gross vehicle weight of 50 kips) slowly traversed the full length of the three-span continuous unit along the inside edge of the left bridge barrier. Similar dump trucks subsequently traversed the full length of the bridge at different transverse lane positions. For each scenario, the axial-stress time-history in the cross- frame members of interest was recorded by the DAQ system. The time-history responses were then converted into influence-line plots by aligning the time component of the measured data with the longitudinal position of the truck relative to the start of the span. This is demonstrated in the horizontal axis of Figure 3-1. Additional bench- mark distances, such as cross-frame (CF) line 4 where the instrumented members are located, and bridge supports (abbreviated as âBrg.â in the figures for âbearingsâ) are included for reference. Note that only the instrumented 194-foot end span of the three-span continuous unit is plotted along the horizontal x-axis. The influence of applied load on the adjacent two spans with respect to cross-frame force effects was negligible. It is also important to note that the data obtained from the strain gages illustrated in this fig- ure simply provide the change in strain/stress during the applied loading. The data does not indicate the state of stress prior to the gage being installed. As such, the measured response does not provide information on the permanent state of stress due to dead loads or residual stresses. In review of the sample influence-line results provided in Figure 3-1, there are several key observations: â¢ Load-induced cross-frame response is highly sensitive to the transverse position of the truck. For instance, the cross-frame diagonal designated as âD2-2â experiences a tensile Cross-frame with gages D1-1D1-2 BS2 D2-1 TS2 D2-2 BS3 D3-1D3-2D4-1D4-2 Cross-frame gages Bottom flange gage Figure 3-2. Plan and cross-section views (line 4) illustrating the instrumentation locations and static load case that correspond to the sample results in this section.

Findings and Applications 63 stress cycle when the truck traverses along the left-hand side of the panel but experiences a compressive stress cycle when the truck transverses along the right-hand side. â¢ The influence of longitudinal load position is localized, as demonstrated by the fact that the measured axial stresses are nearly zero when the truck is positioned beyond 50 feet from the panel of interest, regardless of the lane position. â¢ The various cross-frame members in the instrumented panel also have highly varied responses to a given load condition. In general, top strut (TS) force effects are negligible, which is attributed to the composite nature of the superstructure. Diagonal and bottom strut (BS) forces are generally more substantial, but the stress magnitude and sign are dependent on truck position and the corresponding load-induced deformation pattern of the cross- frame panel. â¢ Although not explicitly shown in the figure, similar behavior was observed from the results for the straight bridge with skewed supports (Bridge 2) and the horizontally curved bridge with radial supports (Bridge 3). The overall load path and load-induced response of the cross- frames in those more complex framing systems, however, produces more interesting results. The effects of skewed supports and horizontal curvature are examined computationally in Section 3.2. In terms of the static load cases performed during the field experimental program, four different dump trucks were incrementally positioned on the bridge deck one at a time. Thus, for each individual load case, the result was a stepped time-history response of increasing stress. An example of this is presented in Figure 3-3, where measured bottom flange stresses in Girder 1 (measured at cross-frame line 4 in accordance with Figure 3-2) are graphed as a function of time for a specific static load case. There are six distinct steps that are apparent in the time-history plot, as follows: A. No trucks positioned on bridge; no live load-induced stress. B. Truck 1 (lead truck) positioned on the bridge; live load-induced stress increases. C. Truck 2 positioned on the bridge behind Truck 1; live load-induced stress increases. D. Truck 3 positioned on the bridge behind Trucks 1 and 2; live load-induced stress increases. E. Truck 4 positioned on the bridge behind Trucks 1, 2, and 3; live load-induced stress increases. F. All four trucks are removed from bridge simultaneously; live load-induced stress returns to zero. Due to the relatively static nature of this test, the stress response for each instrumented element essentially follows a step function. Small spikes were periodically recorded because the -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0 200 400 600 800 1000 1200 1400 Bo tto m F la ng e St re ss (k si) Time (sec) A C B D F E Figure 3-3. Measured bottom flange stress for a representative static load case.

64 Proposed Modification to AASHTO Cross-Frame Analysis and Design load was not purely static. For example, a small spike was recorded at the beginning of plateau B in Figure 3-3. This spike occurs as the truck passes over the maximum point on the influence line for the girder bottom flange, which happens to not coincide with the final static position of Truck 1 in this sample load case. Had the final position of Truck 1 coincided with the maximum point on the influence-line plot, the spike would not have been measured. There are also a few spikes as the trucks are moved off the bridge, which can be explained the same way as the spike in plateau B. Given that these load cases were performed at very low speeds, the research team does not believe these spikes are related to a dynamic impact effect of the truck entering the bridge. The magnitudes of the plateaus are the major focus of the model validation process. These values indicate the stress imposed on the instrumented elements under static loading condi- tions. In order to cancel out potential effects of electromechanical noise, an average stress value, which served as the metric by which the FEA models were validated, was obtained from each of these load-step plateaus. An example of static loading results is presented in Figure 3-4. The measured axial stresses in the instrumented cross-frame members along line 4 are graphically depicted for each intermediate stage of a load case. The positioning of the trucks relative to the cross-frame line of interest is depicted in Figure 3-2 for reference. From these results, there are a few notable observations with regard to the cross-frame response: â¢ Similar to Figure 3-3, the single instrumented top strut member developed very little axial force, even for the final stage of the load case that represented over 200 kips of GVW. 2.24 1.99 0.65 2.01 1.80 0.59 1.42 1.30 0.43 0.47 0.43 0.09 1 Truck 2 Trucks 3 Trucks 4 Trucks Figure 3-4. States of axial stress measured in instrumented cross-frame during a static load case (Units: ksi).

Findings and Applications 65 â¢ Interestingly, all but two instrumented cross-frame members experienced a net tensile stress under the applied load. This behavior seems to indicate that the simplified postprocessing methods commonly utilized in 2D analysis programs as outlined in Section 2.5, do not accu- rately represent realistic conditions. In other words, the diagonal members and strut members do not have equal-and-opposite force effects under a given applied load. Cross-frame stresses (e.g., Figure 3-4), girder flange stresses, and girder deflections were compiled from every static load test performed at all three instrumented bridges. This measured data served as the metric by which the FEA modeling approach utilized throughout the Phase III computational studies was validated. Before the experimental program was executed, preliminary 3D FEA models were developed based largely on the properties and dimensions specified in the design plans as well as con- servative design code approaches. In general, these models consistently overpredicted girder stresses, deflections, and cross-frame forces. Using the measured data as a benchmark, the research team was able to adjust several of the key modeling assumptions. Ultimately, it was shown that three modeling parameters, when adjusted, had the most impact in achieving good agreement with the measured results. These three parameters, which are briefly intro- duced herein, included boundary conditions, contribution of concrete barriers, and the elastic modulus of the concrete deck. For a more detailed review of these modeling parameters, refer to Appendix E. In short, the boundary conditions were modified from pure pinned and roller conditions to a more accurate linear spring representation of the bridge bearings. The stiffness of the elastomeric bridge bearings was estimated based on AASHTO LRFD Equation 14.6.3.1-2 and the dimensions and parameters specified on the design plans. The stiffness of the bridge bearings was most influential for instrumented Bridge 3 (the horizontally curved system), which is intuitive given the torsional response of the horizontally curved superstructure to vertically applied loads. Conventional analysis and design practices generally neglect the contributions of a bridge barrier, continuous or discontinuous. The preliminary analyses, much like the conventional approach, also neglected the barriers in the 3D model. After several model iterations, it was evident that including the discontinuous bridge barriers markedly improved the cross-frame force predictions in the FEA models, particularly for Bridge 3, which is a narrow system. By providing another load path for truck loads, barriers reduce the force demands in the cross- frame elements. It is acknowledged, however, that including the barriers in analysis models increases the computational efforts by both designers and programmers. Thus, the conventional approach of not modeling barriers typically results in a conservative estimate of cross-frame force effects. Lastly, the assumed modulus of elasticity of the concrete deck and rails was also shown to significantly affect girder and cross-frame response in 3D FEA models. In the preliminary analyses, the research team based the assumed elastic modulus on the minimum concrete strength specified by the design plans and the traditional American Concrete Institute (ACI) equation (i.e., â²fc where f â²c is the concrete strength in psi). It was recognized, however, that design codes inherently provide lower-bound estimates. As such, the research team explored different publications and experimental data to justify a higher elastic modulus for concrete (i.e., a value more representative of mean conditions as opposed to lower-bound conditions) (Tadros et al. 2003), which in turn improved comparisons between the analytical solutions produced by the models and the experimentally measured data. In design practice, making these assumptions would be challenging given the uncertainty in the material properties. Thus, similar to the bridge barrier discussion above, the conventional approach of using code-based material properties typically results in a conservative estimate of cross-frame force demands.

66 Proposed Modification to AASHTO Cross-Frame Analysis and Design It is also important to note that in general terms, a user can manipulate a model in many ways to achieve the target solution; however, those changes may not be a good representation of the actual structural system. The team was interested in achieving good agreement between measurements and FEA predictions but not at the expense of using unreasonable assumptions. Based on a literature review and examining many different model configurations, the research team was able to select a consistent set of parameters and assumptions that not only improved the accuracy of the results compared to the measured data, but also made sense based on reason- able engineering assumptions. Based on the commentary above, the modeling approach was fine-tuned to improve the agreement between the measured and finite element results. To demonstrate these improve- ments for instrumented Bridge 1, Table 3-1, Table 3-2, and Table 3-3 compare the preliminary and validated analytical results with the measured data for critical girder deflections, girder stress, and cross-frame force effects, respectively. Only the load case illustrated in Figure 3-2 is presented in these tables; however, the results are representative of all load cases performed at Bridge 1. For reference, the percent error (relative to the measured data) associated with the model is also presented. From these tables, it is apparent that increasing the stiffness of the concrete deck and including a discontinuous concrete rail stiffened the bridge overall and consistently improved the analytical results. Critical girder deflections, which were once uniformly overestimated, improved to errors typically within 10%. Critical girder stresses were also consistently improved to within 10% to 20% of the measured data. Given the complexity of the bridge model and the potential uncertainty in the field measurements, these validated discrepancies are deemed acceptable. In general, Table 3-3 also demonstrates the improved accuracy of the model with respect to cross-frame forces. It is important to note that the error associated with cross-frame forces Element ID Measured Preliminary Analysis Validated Analysis Deflection (in) Deflection (in) Percent Error Deflection (in) Percent Error Girder 1 0.68 0.75 10% 0.66 â4% Girder 2 0.67 0.81 21% 0.71 6% Girder 3 0.70 0.85 23% 0.75 8% Girder 4 0.79 0.85 7% 0.74 â6% Girder 5 0.70 0.82 18% 0.71 2% Table 3-1. Sample Bridge 1 results (girder deflections) demonstrating the comparisons between preliminary and validated analysis results with measured data. Element ID Measured Preliminary Analysis Validated Analysis Stress (ksi) Stress (ksi) Percent Error Stress (ksi) Percent Error Girder 1 2.27 2.77 22% 2.73 20% Girder 2 2.49 3.04 22% 2.91 17% Girder 3 2.77 3.25 17% 3.10 12% Girder 4 2.53 3.19 26% 3.02 20% Girder 5 2.48 3.03 22% 2.94 18% Table 3-2. Sample Bridge 1 results (girder flange longitudinal stresses) demonstrating the comparisons between preliminary and validated analysis results with measured data.

Findings and Applications 67 is noticeably higher than what is observed for girder stresses or deflections. Even for the most sophisticated full-shell, 3D FEA model and a relatively simple bridge geometry, the errors associated with the critical cross-frame forces still ranged from 0 to 60%. Larger discrepancies were tabulated, but those correspond to less critical cross-frame members. Load paths and flexural behavior of girders are generally more straightforward than that of cross-frames. Furthermore, it should be noted that many of the stress magnitudes were relatively small such that any slight variations can produce very large percent differences between measured and predicted stresses. In general, these tables highlight the difficulty with trying to improve all of the measurements (i.e., girder deflections, girder stresses, and cross-frame stresses). The system is highly indeterminant when the various components are consideredâ multiple girders, many cross-frames, and variations in the concrete deck thickness along the length and width. Given the complexity of the bridge model and the potential uncertainty in the field measurements, these validated discrepancies were deemed acceptable. As noted above, the data presented in Table 3-1 through Table 3-3 represent just one static load considered at Bridge 1. Similar results were synthesized for every live load case at every instrumented bridge, which is summarized in Appendix E. By achieving good agreement between the measured and finite element results in this validation study, it ensured that the parametric studies executed in Phase III produced reliable results that were consistent with real load-induced behavior of cross-frames. 3.1.2 In-Service Monitoring At each instrumented bridge, stress cycle spectra were obtained for various cross-frame members and girder flanges during a one-month monitoring period. As noted in Section 2.2.2.2, these spectra provide useful insight on the stress cycle magnitudes that a component typically experiences due to live loads. The relative difference in truck traffic volume between different bridges can also be inferred from the data. Thus, the fatigue damage accumulated on the instrumented bridge components caused by the truck trafficâa function of stress magnitudes and cycle countsâcan be deduced. The spectra, however, are limited in that they provide no indication of the load spectrum (i.e., the weight and axle configuration of the trucks causing those cycles) or the corresponding transverse lane positions, which is especially critical for cross-frames given their observed sensitivity to load position. Element ID Measured Preliminary Analysis Validated Analysis Stress (ksi) Stress (ksi) Percent Error Stress (ksi) Percent Error D1-1 â2.75 â3.37 22% â3.33 21% D1-2 5.03 5.92 18% 5.53 10% D2-1 1.18 â0.69 â159% â0.38 â132% D2-2 7.67 9.77 27% 8.77 14% D3-1 7.45 9.07 22% 8.14 9% D3-2 1.47 0.44 â70% 0.63 â57% D4-1 5.37 6.66 24% 6.19 15% D4-2 â2.78 â3.85 39% â3.77 35% TS2 1.85 9.67 423% 2.94 59% BS2 5.68 9.67 70% 8.99 58% BS3 6.41 10.24 60% 9.50 48% Table 3-3. Sample Bridge 1 results (cross-frame axial stresses) demonstrating the comparisons between preliminary and validated analysis results with measured data.

68 Proposed Modification to AASHTO Cross-Frame Analysis and Design The primary goal for obtaining these data was to establish effective and maximum stress range metrics by which the computational studies and current AASHTO fatigue criteria could be assessed. Effective stress ranges are related to finite-life behavior (Fatigue II limit state), and maximum stress ranges are related to infinite-life behavior (Fatigue I). As such, this section presents representative response spectra data and summarizes the measured effective and maximum stress range values obtained from the field studies. These metrics are subsequently used for comparison in later sections of the report. Measured response spectra are best illustrated as histogram plots, for which the number of stress cycle counts, determined by rainflow-counting algorithms, are compiled and sorted into different stress magnitude bins, Sr,j (i.e., j th stress range bin). The effective stress range then mathematically represents the response of the bridge component to the entire truck population by equating the fatigue damage caused by the variable-amplitude response spectrum to a constant amplitude spectrum of equal cycle count. The maximum stress range represents the upper tail of the spectra. A more detailed discussion on the calculation of effective and maxi- mum stress range of the spectrum, as well as the truncation process and bin size parameters, is provided in Appendix E. A sample histogram illustrating the variable-amplitude stress range spectrum for a cross- frame member (D1-2) in instrumented Bridge 1 is presented in Figure 3-5 (left graph). The right graph provides the mathematical representation of the effective stress. That is, 2,683 cycles at 0.91 ksi produces fatigue damage equivalent to the spectra of stress magnitudes and cycle counts shown on the left graph. The maximum stress range is also approximately 2.7 ksi based on the upper tail of the spectrum on the left. For clarity, the vertical axis is presented using a log scale. Aside from the calculation of effective and maximum stress range metrics, there are several additional observations from Figure 3-5: â¢ This instrumented cross-frame member in Bridge 1 experienced load-induced stress magni- tudes ranging from 0 to 2.7 ksi over the entire monitoring period. Note that the stress cycles below 0.65 ksi were truncated based on the discussion provided in Appendix E, which is consistent with the method outlined in Connor and Fisher (2006). â¢ The vast majority of the stress cycles are well below the CAFL for a Category Eâ² detail (i.e., 2.6 ksi). In fact, 90% of the recorded stress cycles corresponded to magnitude less than 1.1 ksi. 100 2, 68 3 cy cl es a t S re = 0. 91 k si 0 1 10 100 1,000 10,000 100,000 0 1 2 30 1 2 3 Stress Range, Sr, j (ksi) Total Cycles = 2,683 N um be r of C yc le s Stress Range, Sr, j (ksi) Stress Range Spectrum, Sr, j Figure 3-5. Graphical depiction of converting variable stress range spectrum to single effective stress range (cross-frame member in Bridge 1).

Findings and Applications 69 Similar histograms were produced for girder flanges as well. Figure 3-6 presents a side- by-side comparison of the stress range spectra measured for a cross-frame member (D1-2 in Figure 3-2) and girder flange (Girder 2 in Figure 3-2) for instrumented Bridge 1. Both instru- mented elements were subjected to the same load spectrum (i.e., same truck population and lane positions); however, the response of each is significantly different. Not only are the stress range magnitudes higher for the girder flanges, but the number of cycles above the correspond- ing truncation stress was significantly more. Given that the cross-frames were designed for a much more stringent fatigue detail (i.e., lower CAFL value), it is intuitive that the stress magni- tudes in girder flanges exceed those in cross-frames. The variation in stress cycle counts, on the other hand, can be attributed to the sensitivity of cross-frames to transverse load position. Depending on the precise transverse position of a passing truck, the stress range response in a particular cross-frame element could be significant or negligible (i.e., below the established truncation stress or even hidden by the electromagnetic noise of the strain gage). Bending stresses in girders, however, are less influenced by lane position, as evidenced by the sample data in Figure 3-6. Given that (i) the current AASHTO LRFD fatigue load factors were only calibrated for girder response to truck traffic and (ii) transverse load distribution effects were not explicitly considered in that study, this measured data gives an initial indication that load cases specific to cross-frame fatigue are perhaps warranted. This behavior is explored in greater depth computationally in Section 3-3. As noted above, the research team produced and processed a number of histogram plots similar to those presented in this report. In order to evaluate the measured in-service data with AASHTO fatigue design criteria in subsequent sections, only the effective and maximum stress ranges are examined. As such, Table 3-4 summarizes the range of effective and maximum stress ranges computed for the instrumented cross-frame members in Bridges 1, 2, and 3. In other words, these stress range metrics were computed for every instrumented cross-frame member, and the range presented in the table represents the maximum and minimum values of the data set. A similar table with respect to girder flange data is provided in Appendix E for reference. From Table 3-4, it is obvious that critical cross-frames in Bridge 2 (skewed) consistently experienced higher effective stress ranges in comparison to critical cross-frames in the other 0 1 10 100 1,000 10,000 100,000 0 1 2 3 4 5 N um be r of C yc le s 0 1 10 100 1,000 10,000 100,000 0 1 2 3 4 5 Cross-Frame Girder Flange N um be r of C yc le s Stress Range, Sr, j (ksi) Figure 3-6. Sample variable stress range spectrum of (left) cross-frame member and (right) girder flange (Bridge 1).

70 Proposed Modification to AASHTO Cross-Frame Analysis and Design two bridges. These results imply that the cross-frames in this skewed system, especially with a contiguous layout, generally experience slightly higher stress ranges than the normal and horizontally curved systems. Although not explicitly addressed in the table, it was also observed in several instances that cross-frame stress ranges were dependent on its location relative to the right lane (i.e., the typical drive lane for heavy truck traffic). The discussion related to load-position effects and bridge geometry is expanded in Section 3.2 with respect to the finite element studies. 3.2 Fatigue Loading Study This section summarizes the analytical results related to the Fatigue Loading Study that was previously introduced in Section 2.4. This study, which focuses on addressing Objectives (a) and (b) of NCHRP Project 12-113, generally explores the fatigue loading model used for cross- frame analysis and design. More specifically, the Fatigue Loading Study seeks quantitatively based answers to five major questions that were outlined in Section 1.2. These questions, along with the dedicated subsection herein, are as follows: â¢ What is the influence of composite bridge geometry (e.g., support skew, horizontal curvature) and cross-frame layout (e.g., cross-frame spacing, staggered layout) on the load-induced fatigue behavior of cross-frame systems (including governing force effects, critical lane load- ing, and critical members under AASHTO fatigue loading criteria)? [Section 3.2.1] â¢ Based on the findings of the item above, is it necessary to perform a refined analysis, either simplified 2D or 3D methods, for straight and non-skewed bridges? [Section 3.2.1] â¢ Is the current fatigue load model in terms of truck position (i.e., single design truck passages positioned in various transverse lane positions) appropriate for cross-frame analysis and design? Or do multiple presence effects need to be considered? [Section 3.2.3] â¢ Is the ânâ value (i.e., number of cycles per truck passage) currently assumed for the generic âtransverse memberâ designation applicable for cross-frames? [Section 3.2.2] â¢ Are the current AASHTO LRFD Fatigue I and II load factors that were calibrated for girder force effects and recent WIM data appropriate for cross-frame analysis and design? [Section 3.2.2] As noted in the list above, these questions are systematically addressed throughout this section of the report. In the context of AASHTO fatigue loading criteria, Section 3.2.1 inves- tigates the correlation between key bridge parameters (e.g., support skew) and the load-induced force effects in critical cross-frames. These results are further evaluated to determine if and under what circumstances live load effects can be ignored in the design of cross-frames (i.e., cases in which live load force magnitudes are insignificant compared to other load cases). Sections 3.2.2 through 3.2.3 focus on cross-frame response to measured WIM records. Section 3.2.2 examines the cross-frame response to the WIM records, namely the number of stress/force cycles induced by the various load spectra, and investigates the use of modified Bridge No. Measured Range {Min, Max} Effective Stress, (ksi) Maximum Stress, (ksi) 1 {0.70â0.98} {1.35â3.61} 2 {0.78â1.19} {0.99â3.77} 3 {0.77â0.95} {0.99â2.84} Table 3-4. Ranges of effective stress ranges and maximum stress ranges measured for cross-frames at each instrumented bridge.

Findings and Applications 71 fatigue load factors for cross-frame design. Section 3.2.3 reviews the WIM data and explores the frequency at which a âdouble truckâ case (i.e., a loading condition that maximizes the force reversal in a cross-frame member) occurs. Section 3.2.4 provides a summary of the key findings. 3.2.1 Influence of Bridge Geometry As noted above, this subsection addresses the influence of bridge geometry on the live load demands in the cross-frame members in the context of the AASHTO fatigue loading criteria. More specifically, four different aspects are summarized from the study. First, cross-frame force effects are compared on a member-by-member basis in Section 3.2.1.1. That is, the research team investigated which member (i.e., type and location on span) typically governs the load- induced fatigue design of cross-frames. Next, the transverse lane passage that maximizes load- induced cross-frame forces is examined in Section 3.2.1.2. Section 3.2.1.3 studies the impact of skewed and/or curved geometries on those force effects, and Section 3.2.1.4 evaluates the fatigue loading criteria with respect to the fatigue resistance criteria (i.e., the cross-frames are effectively designed based on 9th Edition AASHTO LRFD guidance). By effectively designing the cross- frames, the overall efficiency and inherent conservatism of the AASHTO Specifications in terms of load-induced cross-frame fatigue design can be assessed. 3.2.1.1 Governing Cross-Frame Member To examine which cross-frame member typically governs load-induced fatigue design, the research team evaluated the unfactored design force computed for each selected cross-frame member in the 4,104 unique bridges. From these data, the cross-frame members that most often govern the design (i.e., the cross-frame with the largest force range recorded) were identified based on the procedure outlined in Section 2.3.2. This process is summarized in Figure 3-7. The process compiles the unfactored design force ranges for all 68,352 cross-frames in the parametric study in the form of a box-and-whiskers plot that indicates the maximum and minimum in the range as well as the median and first and third quarter values. This is discussed in more detail below. Each box-and-whiskers component is organized by cross-frame location and then further by cross-frame member. Thus, each figure presents the results of 16 different types of cross-frame members. For the sake of clarity, cross- frame identification is abbreviated. As far as cross-frame location, âESâ represents the cross- frame panel near the end support (colored in red), âM-Iâ the interior bay near the maximum positive dead load moment region (blue), âM-Eâ the edge bay near the maximum positive dead load moment region (green), and âISâ near the intermediate support (purple). Refer to Section 2.3.1.5 for more information on these general locations. In terms of the cross-frame members, âTSâ represents top strut members, âD1â for diagonal 1 (diagonal framing into the top flange of the left girder where left is based on the coordinate system previously established), âD2â for diagonal 2, and âBSâ for bottom struts. Note, that for K-type cross-frames, there are two independent bottom strut members. For purposes of the results hereafter, only the governing member is presented for consistency. The plots are also organized by bridge type to illustrate the effects of support skew and bridge curvature on the cross-frame response. For instance, data labeled as âstraight, normalâ was generated from all bridges with no bridge curvature (i.e., infinite radius of curvature) and zero support skew. To fully understand and interpret the box-and-whiskers plot, there are additional items to consider: â¢ Each box-and-whisker segment represents the results of all bridges and cross-frames that satisfy the applied filters (i.e., a probability design function). For example, the box-and- whiskers labeled âTSâ under the âM-Eâ category in the âStraight, Normalâ graph represents

72 Proposed Modification to AASHTO Cross-Frame Analysis and Design the compiled design force ranges for all top strut members evaluated near the maximum positive dead load moment region (edge bay) of straight and normal bridges. In total, 312 data points (corresponding to the unfactored design force ranges for 312 unique, straight and normal bridges) are represented. â¢ For each box-and-whiskers plot, six key statistical parameters are displayed: (i) the minimum value in the data set, (ii) the first quartile or the 25th percentile, (iii) the second quartile or the median value, (iv) the third quartile or the 75th percentile, (v) the maximum value, and (vi) the mean value. The maximum and minimum values are represented by the whiskers, the quartile values are represented by the bounds of the box, and the âXâ represents the mean. A graphical representation of this procedure is presented in Appendix F for reference. â¢ The critical lane passage that corresponds to the design force ranges presented are different for each point in the data set. In this figure, only the magnitudes of the design force ranges are of interest. The critical lane positions are covered in a subsequent section. â¢ A reference sketch is also provided in the figure to clarify the relative position of the cross- frame considered; the results for the color-coded cross-frame locations are plotted. Note that the exact location of the cross-frame, the number of girders, the number of spans, the support skew angle, and the radius of curvature in the sketch are for visual reference only; the bridge geometries of the various data points may differ from the representative sketch. By presenting the box-and-whiskers side-by-side, the cross-frame members that generally experience the largest force ranges due to the passage of the AASHTO fatigue truck in its critical Figure 3-7. Unfactored design force ranges compiled for every cross-frame member evaluated.

Findings and Applications 73 lane can be evaluated. By examining the results independently for bridge types, the impacts of support skewness and bridge curvature can also be assessed. From Figure 3-7, the following observations can be made about governing cross-frame members: â¢ There is substantial variability observed in all the box-and-whiskers plots presented. For example, the design forces for bottom struts near end supports of horizontally curved and skewed bridges (curved, skewed) range from a minimum of about 0.5 kips to a maximum of nearly 18 kips. This scatter is attributed to the large number of bridge parameters inherently considered in the results. â¢ Top strut members, as anticipated, generally have the lowest force demands observed since the concrete slab tends to restrain out-of-plane deformations at the top of the girder. â¢ The force ranges observed in skewed and/or horizontally curved bridges are generally greater than equivalent straight and normal bridges, particularly in bottom strut and diagonal members near skewed supports. For example, the mean force range for bottom struts at end supports (BS; ES) increases from 3 kips for straight and normal bridges to over 4 kips for straight and skewed bridges. â¢ This same relationship is not as impactful for cross-frames not in the vicinity of supports (i.e., near the maximum positive dead load moment region), as cross-frames in these regions are less impacted by the effects of support skew. â¢ In general, bottom strut members near M-I regions (interior bay near maximum positive dead load moment region) have the largest design force ranges for non-skewed (straight or horizontally curved) bridges. The maximum, 75th percentile, and mean values are all largest for this condition. â¢ For skewed bridges, the bottom strut members near end supports tend to have the largest force demands, but there is still significant scatter. From all the data obtained from this study, it is apparent that bottom strut members tend to be the most critical in terms of fatigue force demands. In past studies that focused on stability bracing applications of noncomposite girders, it has been shown that the top and bottom struts generally behave as zero-force members with both diagonal members equally effective (i.e., adjacent girders rotating equally). However, for cross-frames in composite systems and sub- jected to live loads, bottom struts are often heavily engaged. In many cases, the cross-frames behave similarly to floorbeams in a stringer-floorbeam system to distribute loads from girder- to-girder, where adjacent girders rotate differentially in this case. The bottom strut is analo- gous to the bottom tension flange of a composite floorbeam, the concrete deck is the top compression flange, and the top strut is in close proximity to the pseudo-neutral axis. At skewed supports, the global, torsional response of the superstructure tends to engage the nearby bottom struts. Contiguous lines of cross-frame panels act like a stiff, closed section that resists the torsional moments on the bridge cross-section. Even with these discernible trends, there is still significant variability in cross-frame response. The governing cross-frame member is still a function of many bridge parameters, albeit support skew and curvature are the most important. Given a random bridge geometry, the critical cross-frame panel could likely be identified within reasonable limits before any analysis is performed. Still, it is recommended that the critical cross-frame is not âmissedâ by taking a shortcut. Rather, conducting an influence-surface analysis and performing a comprehensive design of all intermediate cross-frames in the bridge ensures a fully vetted design. Many commercial design software packages have the built-in functionality to analyze and design all cross-frames for design loads. If the program does not automatically address this,

74 Proposed Modification to AASHTO Cross-Frame Analysis and Design manually developing a spreadsheet to evaluate each cross-frame due to AASHTO fatigue loading criteria is possible. 3.2.1.2 Governing Lane Passage Where Section 3.2.1.1 evaluated which cross-frame members govern load-induced fatigue design, the section herein addresses the lane position corresponding to those critically loaded members. As currently documented in AASHTO LRFD Article 3.6.1.4.3a, bridge designers must consider all transverse and longitudinal truck positions when evaluating details and components for the fatigue limit state. Other than a 2-foot clear distance from the inside face of the bridge barrier, the AASHTO fatigue truck passages must be evaluated in all possible trans- verse lanes, forward and backwardâa provision that is included primarily due to the uncertainty in future lane striping and/or bridge widening. Therefore, without assurance that the traffic lanes are to remain unchanged throughout the service life a bridge, the designer must take the conservative approach. As documented previously, the AASHTO fatigue truck traversed over all 4,104 bridges in 1-foot increments of transverse lane positions. Given the number of lane passages and cross-frame members studied, drawing concise and meaningful conclusions on the data was challenging. In the evaluation of the data, however, the research team observed that many of the critical cross-frame members were governed by truck passages along the inside faces of the barriers (i.e., centerline of the wheel line within 1 foot of the closest barrier). The minimum distance of 1 foot from the inside faces of the barriers was conservatively used in this study as described previously. Thus, to simplify the discussions in this section, lane passages are grouped as either âoverhangâ loads or ânon-overhangâ loads. Overhang loads represent truck passages where one of the transverse wheel lines is applied outboard of the fascia girder centerline; these load cases include âleftâ and ârightâ overhang loads, as well as both the forward and backward directions. Non-overhang loads represent truck passages for which both wheel lines are within the centerlines of the fascia girders. Overhang loads, as noted above, occur on a much less frequent basis for bridges in service. Although these loading cases are often critical for strength limit states, truck drivers are less likely to frequently drive within a few feet of the barrier (i.e., along the overhanging portion of the deck). As such, overhang loads are less representative of the fatigue limit state, which focuses heavily on loading most likely to occur at a high frequency throughout the design life. Non-overhang loads, on the other hand, are much more representative of the fatigue limit state. With that said, the focus of this subsection is to identify the types of bridges most commonly governed by overhang load cases. Assuming the designer is uncertain about future lane striping and/or bridge widening, these special bridge types are the most affected by the relatively conser- vative language in Article 3.6.1.4.3a. Figure 3-8 compiles the lane position associated with the results presented in Figure 3-7. For each of the 4,104 unique bridges, the critical lane passage corresponding to the governing cross-frame member was recorded and was later categorized as an overhang or non-overhang load based on the definition above. The tallied results are presented in the form of a bar graph in Figure 3-8, where the y-axis represents the frequency of occurrence with respect to the number of bridges in each type (e.g., 312 straight and normal bridges). Each of the bars is then further delineated as overhang and non-overhang loading conditions. The figure independently evaluates bridge types, similar to Figure 3-7, but does not explicitly present data from individual member types (i.e., bottom struts, diagonals, top struts). Instead, any of the governing cross-frame members from Figure 3-7, regardless of member type, are grouped based on their general location in the framing plan (i.e., ES, M-I, M-E, and IS).

Findings and Applications 75 To clarify the intent of the figure, the straight, normal data set is considered as an example. For all 312 straight and normal bridges evaluated, the governing cross-frame was located in an interior bay near the maximum positive dead load moment region on 204 occasions (65%). Of those 204 occasions, the critical cross-frame member was always governed by a non-overhang load (i.e., 100% non-overhang and 0% overhang). This is reflected in the âM-Iâ bar in the straight, normal portion of Figure 3-8. A more interesting example is the intermediate support âISâ case of the straight and skewed data set. In total, 1,440 straight and skewed bridges were analyzed in the Fatigue Loading Study. Of those 1,440 bridges, 565 were governed by cross-frame members near the intermediate support (39%). Of those 565 cases, 514 were governed by overhang truck passages and only 51 by non-overhang loads. From Figure 3-8, the following observations can be made about bridge type and critical lane passages: â¢ Non-overhang loads, although not overly descriptive, generally correspond to âlocalizedâ load effects. In other words, the force effects in the critical cross-frame were maximized by a truck passing just to the left or right of the panel, similar to the trends observed in the influence-surface plots in Section 2.3. â¢ For non-skewed bridges, the critical cross-frame members are almost always governed by non-overhang loads. In fact, the forces in critical cross-frame members in straight, normal bridges were maximized 100% of the time by non-overhang loads. The exception to this rule is M-I cross-frames in horizontally curved bridges (interior bay near the maximum positive dead load moment region), which were often governed by truck passages along the outer radius of the curve. Thus, it can be concluded that cross-frame fatigue design in bridges without support skews is largely governed by truck passages in close transverse proximity to the cross-frame panel. â¢ For skewed bridges, the overhang loads represent a large percentage of the critical truck passages, particularly for âESâ and âISâ cross-frames. The torsional response of the super- structure and the corresponding girder rotations often heavily engage the bottoms struts 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Straight, Normal Straight, Skewed Curved, SkewedCurved, Normal ES M -I M -E IS ES M -I M -E IS ES M -I M -E IS ES M -I M -E IS Reference Sketch: M-I M-E ES IS Key: Non-Overhang Load Overhang Load 312 Models 1,440 Models 624 Models 1,728 Models Bridge Type Fr eq ue nc y of O cc ur re nc e Figure 3-8. Bar graph presenting the critical lane position associated with the governing cross-frame member.

76 Proposed Modification to AASHTO Cross-Frame Analysis and Design and diagonals in these areas. As such, loads that maximize the induced torque on the bridge cross-section are often critical. For straight bridges, the critical loads correspond to those applied along either deck edge where the moment arm about the bridge centroid is largest. For horizontally curved bridges, those loads correspond to either the inner or outer radius about the straight-line chord of the curved segment. It is evident that force demands in cross-frames of skewed bridges, especially those near the end or intermediate supports, are sensitive to overhang loads. Because these cases are rather infrequent, basing a fatigue limit state design on an overhang truck passage could potentially result in overly conservative design loads. With that in mind, overhang and non-overhang truck passages are evaluated in the context of measured WIM data in Section 3.2.2, where the appro- priate lane position for AASHTO implementation is examined in greater depth. 3.2.1.3 Impact of Bridge Parameters As discussed in the preceding subsections, there is significant variability in load-induced cross-frame response. This was particularly evident in Figure 3-7, which presented the governing force demand in every cross-frame member evaluated in the 4,104-model matrix. The variability in the response is attributed to many of the parameters that comprise the bridge structures. This section herein examines the impacts of these parameters on the force effects in critical cross-frame members. Before evaluating the effects of each parameter, Figure 3-9 presents one isolated example for reference. In this plot, the response of a diagonal cross-frame member near the maximum positive dead load moment region is illustrated for three different bridge structures. The influence line shown corresponds to a passing AASHTO fatigue truck, whose left wheel line is positioned 5 feet from the bridge centerline. Therefore, the truck passes just to the left of the cross-frame panel of interest, as depicted in the framing plan sketch in Figure 3-9. The three bridges share identical parameters except for curvature: straight, 1,500-ft radius, and 750-ft radius. Thus, any differences in the cross-frame response can be attributed directly to the introduction of a curved layout. Based on this sample data, it is evident that the force demand in the cross-frame member of interest increases as the bridge radius decreases (or its curvature increases). Relative to the D2 -2 0 2 4 6 8 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 )spik( ecroFlaix A Longitudinal Position of Truck (ft) 1500-ft Radius Straight 7.0 kips 750-ft Radius Radius of Curvature (ft) 750 1,500 Straight Force Range (kips) 7.00 6.45 5.82 Figure 3-9. Sample data demonstrating the impacts of bridge curvature on cross-frame force demands.

Findings and Applications 77 straight bridge, the resulting force range from the passing truck increases 10% when a 1,500-ft radius is introduced and an additional 9% when the radius is halved to 750 feet. The general response of the cross-frame is the same, except for the magnitudes. Thus, based on this limited data set, it could be stated that bridge curvature is positively correlated to cross-frame force demands (i.e., bridge radius is negatively correlated). Figure 3-9 represents only a select number of unique conditions. The response could also have been presented for different lane positions, different cross-frame members, and differ- ent bridges altogether. Ultimately, a primary interest of most designers is knowledge of how a bridge parameter impacts the overall fatigue design of cross-frames, not the localized effects on non-governing members that are deemed less important. Therefore, the results presented here- after are focused on the governing cross-frame members and lane positions (i.e., the unfactored design forces in accordance with current AASHTO fatigue loading criteria). To expand on the sample results presented in Figure 3-9, critical cross-frame force effects and lane passages were evaluated for every bridge in the analytical testing matrix. That is to say, box-and-whiskers plots that represent the spectrum of design forces were generated and sub- sequently categorized by bridge parameter (e.g., all cross-frame force effects in straight bridges, bridges with a 1,500-foot radius of curvature, and bridges with a 750-foot radius of curvature were evaluated and compared). Figure 3-10 presents the summarized results from this exercise highlighting several key parameters, including skew index (defined below), horizontal curvature, cross-frame layout, number of girders, and concrete deck thickness. These parameters were selected, as they signi- ficantly influence the load-induced force effects in the critical cross-frames. In these figures, box-and-whiskers plots and response spectra are represented as a series of three distinct line graphs. In the figure, the solid black line represents the mean of each data set, and the dashed lines represent the 95th and 5th percentile values. The respective percentile lines give an indication where 90% of the data set is populated. Despite only presenting five parameters in Figure 3-10, every independent variable that was previously identified in Table 2-2 was evaluated similarly. For additional information on the development of these plots and the results for all bridge parameters (including spot checks of bridge models without barriers), refer to Appendix F. Before introducing the results, it is prudent to define skew index, as well as connectivity index. Skew (Is) and connectivity (Ic) indexes are respective measures of bridge skewness and curvature, as defined in the AASHTO G13.1 Guidelines for Steel Bridge Analysis (2019). These indexes are used to categorize bridge geometries for purposes of recommended analysis practices. Bridges with larger skew and connectivity indexes require more advanced analysis procedures, either improved 2D or 3D techniques. Although not presented in Figure 3-10, the connectivity index is utilized in subsequent sections of the report. The skew and connectivity indexes are defined by the following expressions: tan 3.1I w L s g s = q 15,000 1 3.2I R n m c cf( ) = + where Is, Ic = skew index and connectivity index, respectively, wg = width of bridge, measured between the centerlines of the fascia girders,

78 Proposed Modification to AASHTO Cross-Frame Analysis and Design q = largest skew angle on bridge relative to the axis normal to the longitudinal girders, Ls = span length at the bridge centerline, R = minimum radius of the horizontal curvature, ncf = number of intermediate cross-frames in the span, and m = constant, taken as 1 for simple-span bridges and 2 for continuous-span bridges. As the equations show, the skew index increases for shorter, wider spans with larger skew angles. Similarly, the connectivity index increases for tighter curves (i.e., smaller radius of curvature) and fewer cross-frames connecting the girders together. From Figure 3-10, the following observations can be made about cross-frame force demands and the select bridge parameters outlined above: â¢ Skew index: It is evident from the results that skew index substantially impacts the response of cross-frames. A +0.52 correlation coefficient was obtained, indicating a strong, positive relationship between skew index and cross-frame force effects. In simpler terms, a +0.1 increase Cross-Frame Layout Cross-Frame Layout 0 3 6 9 12 15 18 0 5 10 15 Curvature (10â4/ft) Curvature 0 0.35 0.7 Skew Index Skew Index 0 3 6 9 12 15 18 3 5 7 M ax F or ce R an ge (k ip s) No. of Girders No. of Girders 0 3 6 9 12 15 18 0 3 6 9 12 15 18 0 3 6 9 12 15 18 8 10 Deck Thickness (in) Deck Thickness Figure 3-10. Unfactored design force spectra demonstrating the impacts of several key bridge parameters.

Findings and Applications 79 in skew index generally resulted in a 25% increase in the critical cross-frame force demand. In general, these observations are consistent with the field experimental data outlined in Section 3.1. â¢ Curvature: A positive correlation is observed between bridge curvature and cross-frame force effects (i.e., correlation coefficient of +0.32). On average, an increase in curvature of 10-4/ft results in an increase of 3% for the governing cross-frame force effect. Thus, the trend observed in Figure 3-9 for a small sample set is consistent with the full data set. â¢ Cross-frame layout: Staggering the cross-frame layout in skewed bridges significantly reduces the governing design force in cross-frames (35% on average). This trend is attributed to the stiffness reduction associated with a discontinuous line of cross-frames. These results are consistent with the guidance provided in AASHTO LRFD Article C6.7.4.2. â¢ Number of girders: A positive correlation coefficient of +0.60 was observed between number of girders (i.e., bridge width) and cross-frame force demands. A more variable response was also observed as the number of girders increased, as evidenced by the distance between the 95th and 5th percentile lines. On average, increasing the overall width of the bridge results in a 30% increase in the governing force effect. These trends are attributed to the overhang truck loads outlined in Section 3.2.1.2. Eccentric loading and torque on a straight or curved system are amplified for wider bridges, which in turn amplifies the governing cross-frame response. â¢ Deck thickness: Deck thickness and cross-frame force effects are negatively correlated (â0.22 correlation coefficient). On average, a 1-inch increase in deck thickness results in a 14% decrease in force demands. A reduction in deck thickness often indicates that the cross-frames will attract a higher proportion of the load distribution, which increases member force effects. Additionally, a thinner deck corresponds to a reduction in overall superstructure stiffness, which subsequently increases girder displacements and cross-frame deformations. In general terms, it is evident that skewed and/or curved composite bridge systems typically result in larger load-induced force demands in their cross-frames. Although this behavior has been widely recognized by designers throughout the last few decades, the results above validate these assertions quantitatively. To mitigate these force effects and potential load- induced fatigue cracking problems in new designs, designers can utilize these trends to avoid an iterative âchase-your-tailâ design solution (i.e., increase the size of the cross-frame member to accommodate the design forces, re-analyze, and redesign for larger forces in the second pass, etc.). For instance, using a discontinuous, staggered cross-frame layout is a practical and economical solution for skewed systems. Increasing the deck thickness or decreasing girder spacing can also lessen load-induced cross-frame forces in composite bridges, but adjusting those parameters has significant impacts on the rest of the design. 3.2.1.4 Design Parameters Based on the procedures outlined in Section 2.3.2, governing cross-frame force effects were subsequently factored and converted to axial stresses, and AASHTO fatigue resistances were evaluated. By effectively designing the cross-frames in accordance with current AASHTO LRFD Specifications, the overall efficiency and inherent conservatism of the specifications in terms of load-induced cross-frame design can be assessed. Figure 3-11 presents the factored Fatigue II stress ranges that govern the design of all 4,104 representative bridges in the form of a histogram. The stress ranges were categorized in 0.1-ksi bins, and the counts were summed. For reference, the factored design stresses for the three instrumented bridges (based on analytical procedures outlined previously) as well as the measured effective stress ranges (Section 3.1.2) are also included. Recall that the Fatigue II load effects derived from AASHTO LRFD are synonymous with the effective stress range of the truck population; as such, a direct comparison can be made between the two entities. By comparing

80 Proposed Modification to AASHTO Cross-Frame Analysis and Design analytical and experimental results of instrumented Bridges 1, 2, and 3, a few key observations are established. First, the governing design stress ranges for Bridges 1 and 2 are close to the mean response (2.35 ksi), and the stress range for Bridge 3 is in the lower tail of the spectrum. Second and more importantly, it is evident that the analytical results generally exceed the measured results by a considerable margin. This is a preliminary indication that the AASHTO fatigue model for the Fatigue II limit state may be conservative when compared to real loading conditions expe- rienced by the instrumented bridges in the context of cross-frame behavior. This assertion is verified for a wider range of bridges herein; however, note that the physical evidence from measured data, rather than just relying entirely on analytical data, strengthens this observation. Figure 3-12 then illustrates the governing demand-to-capacity ratio (i.e., DFn/Î³Df) with respect to the Fatigue II limit state. The factored stress ranges from Figure 3-11 are compared to the factored resistances computed based on the procedure outlined in Section 2.3.2. Demand- to-capacity (D/C) ratios below unity indicate designs in conformance with AASHTO LRFD, 0 40 80 120 160 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Governing Fatigue II Design Stress, Î³Îf (ksi) Bridge 3 Bridge 1 Bridge 2 Key: Fatigue II âAnalyticalâ Design Stress Measured Effective StressBridge 2 Bridge 3 Bridge 1 N um be r of O cc ur re nc e Figure 3-11. Governing cross-frame fatigue stress ranges for all bridges in Fatigue Loading Study. Pr ob ab ili ty o f O cc ur re nc e 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Demand-to-Capacity Ratio Llano, TX, Traffic Mean = 0.54 % Above 1.0 = 7.4% Houston, TX, Traffic Mean = 1.40 % Above 1.0 = 65% Figure 3-12. Fatigue II cross-frame demand-to-capacity ratios for two extreme traffic conditions.

Findings and Applications 81 whereas ratios exceeding unity represent designs in violation of AASHTO LRFD. As outlined previously, it is assumed that the design of all intermediate cross-frame members is governed by the critical, maximum case. Thus, any bridge exceeding a D/C of 1.0 means that one (i.e., the critical cross-frame member) or more members are inadequate in terms of Fatigue II design criteria. Figure 3-12 presents D/C histograms for the two extreme traffic conditions: rural Ranch Road 152 in Llano, Texas, and urban I-10 in Houston, Texas. As outlined in Section 2.3.2, in these examples, the Llano and Houston bridges are just hypothetical locations of a rural bridge with relatively light traffic versus a busy urban environment with relatively high traffic. Based upon TxDOT data (2020), the single-lane ADTTSL for the Llano location can be estimated as 182 compared to 3,000 for the Houston location. For each individual histogram, the factored demands are identical; however, the factored capacities differ based on the large disparities in projected truck traffic volumes. From Figure 3-12, it is apparent that the projected traffic conditions have a significant impact on the results. For the low-volume conditions, the results indicate that the majority of the 4,104 bridges are conservative in terms of cross-frame fatigue design. In fact, the average D/C ratio was 0.54, which implies substantial reserve capacity for the majority of bridges. Only 7.4% of the models exceeded a D/C ratio of unity. In contrast, the results indicate a more severe trend for the high-volume conditions under the Fatigue II limit state (i.e., the 75-year ADTTSL does not exceed 8,485 trucks per day for a Category Eâ² detail per AASHTO LRFD Table 6.6.1.2.3-2). On average, the D/C ratios exceeded 1.0 (mean of 1.40), and 65% of the bridges resulted in a design in violation of AASHTO LRFD. If these bridges were to be implemented for a real construction project, modifications to the cross-frame properties (e.g., an increase in cross-sectional area) or layout (e.g., use a dis- continuous cross-frame layout in heavily skewed bridges) would likely be required to bring the design in conformance with AASHTO LRFD. To expand further on Figure 3-12, Figure 3-13 highlights the differences between straight and curved bridges, as well as bridges with normal and skewed supports. The full histogram related to the Houston, Texas, traffic in Figure 3-12 is broken down into four bridge types: straight bridge with normal supports, straight bridge with skewed supports, curved bridge with normal supports, and curved bridge with skewed supports. Each subset histogram (shown in black) is overlaid on the full histogram (shown in light blue) to demonstrate how bridge curvature and support skewness affect the D/C ratios. The average D/C ratio increases when skewed supports and/or horizontal curvature are introduced. For instance, the mean D/C for straight, normal bridges is 0.88; that value increases to 1.23 for straight, skewed bridges and 1.15 for curved, normal bridges. A similar trend is observed for the percent unconservative metric. In broader terms, the results from Figure 3-12 and Figure 3-13 suggest the following: â¢ For roadways with low truck traffic, load-induced fatigue does not appear to be a major concern for the 4,104-model bridge sample (which represents the most common bridge conditions in the United States) according to AASHTO 9th Edition criteria. â¢ For bridges serving heavy truck traffic, current AASHTO criteria indicate a potential load- induced fatigue problem. Development of appropriate design criteria for cross-frames, like other structural elements, requires achieving acceptable structural safety but must also consider economy. As learned from the industry survey discussed in Section 2.1 and Appendix D, actual load-induced fatigue prob- lems in cross-frame members have seldom been documented by bridge owners. Thus, experience has shown that cross-frames are largely satisfying structural safety requirements in terms of load-induced fatigue. A question of interest then is whether further

82 Proposed Modification to AASHTO Cross-Frame Analysis and Design economies are possible in cross-frame design, while still providing adequate structural safety. The absence of observed load-induced fatigue failures does not necessarily imply the current AASHTO LRFD design criteria are overly conservative and wasteful from a cost perspective. Nonetheless, with a goal of developing improved design criteria for cross-frames, the question of whether current AASHTO LRFD design criteria for load-induced fatigue of cross-frames are too conservative is important to examine. The analysis presented earlier in this section provides at least some indication that AASHTO LRFD may be too conservative and merits closer scrutiny. When considering whether AASHTO LRFD may be too conservative for load-induced fatigue design of cross-frames, the following three factors may be potentially considered: 1. The AASHTO fatigue resistance model is potentially too conservative (DFn is too low). The resistance model primarily consists of detail Category Eâ² and its associated constants (Table 6.6.1.2.3-1). 2. 3D FEA models perhaps consistently produce overly conservative force predictions. Among many modeling assumptions, the primary focus is the stiffness modification factors (R-factors) for cross-frames in 3D models (Article C4.6.3.3.4). 3. The AASHTO fatigue loading model is potentially too conservative (Î³Df is too high). The load- ing model comprehensively consists of the following: fatigue truck (Article 3.6.1.4.1), truck positioning (Article 3.6.1.4.3a; Article C6.6.1.2.1), dynamic load allowance (Article 3.6.2), load factors (Article 3.4.1), and shear lag factor [Table 6.6.1.2.3-1 (Condition 7.2)]. Note that, by simply listing the above factors, it is not advocated that changes to the design specifications are necessary related to all these factors. The list is intended to be comprehensive in nature and identify all possible sources. For instance, this research does not suggest that welded cross-frame connections be reclassified as a higher fatigue category. Pr ob ab ili ty o f O cc ur re nc e 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 0 1 2 3 4 50 1 2 3 4 5 Straight, Normal Mean = 0.88 % Above 1.0 = 35% Straight, Skewed Mean = 1.23 % Above 1.0 = 54% Curved, Normal Mean = 1.15 % Above 1.0 = 57% Curved, Skewed Mean = 1.73 % Above 1.0 = 83% Demand-to-Capacity Ratio Figure 3-13. Fatigue II cross-frame demand-to-capacity ratios for various bridge types (Houston I-10 traffic).

Findings and Applications 83 As noted previously, fatigue resistance is beyond the scope of the project and is not discussed further. Instead, the research team narrowed its efforts to examining the modeling approach (stiffness modification and simplified analysis techniques) and the fatigue loading model herein. Section 3.3 studies the applicability of the current R-factor approach (0.65AE per AASHTO Article C4.6.3.3.4) for composite systems. Section 3.4 explores the limitations of simplified 2D modeling approaches. Sections 3.2.2 through 3.2.3 examine the loading model (primarily the fatigue truck, truck positioning, and load factors) with respect to high-resolution WIM data. 3.2.2 Cross-Frame-Specific Load Factors 3.2.2.1 Fatigue I Stress Ranges (Normalized to Fatigue Truck) As noted in Section 2.3.3, the governing Fatigue I stress range calculated for all WIM sites was determined based on 99.99th percentile criteria. This stress range corresponds to the lowest magnitude of the top 0.01% of all stress ranges recorded (for the governing cross-frame for each bridge given the defined lane position). Figure 3-14 illustrates the selection of the Fatigue I stress range from a CDF of stress ranges for a sample cross-frame member. In this figure, the stress range corresponding to the largest stress range recorded is approximately 7 ksi; however, the lowest stress range within the top 0.01 percent stress ranges is shown to be 5.15 ksi. Thus, 5.15 ksi is considered the maximum stress for which AASHTO Fatigue I load criteria is evaluated. Once all Fatigue I stress ranges were calculated for each WIM record and each bridge (i.e., one value per bridge per WIM record), the stress ranges were normalized to the largest stress range produced by applying the unfactored AASHTO design load (i.e., the fatigue truck) to the same bridge in all possible transverse positions within the appropriate clear distance of the barriers. Note that the positioning of the fatigue truck to create the largest stress range does not necessarily correspond to the location of the WIM traffic stream. This is consistent with the current design approach, where the bridge (and its cross-frames) is designed based on the maximum force effect produced by locating the fatigue truck in the critical position between the barriers (in accordance with AASHTO Article 3.6.1.4.3a). The Fatigue I stress ranges and biases were calculated for each of the 20 bridges using all of the high-resolution WIM site records, as well as the arithmetic mean of all stress ranges produced by the 18 WIM site records and the âmean plus 1.5 standard deviationsâ of all stress ranges produced by the 18 WIM site records (the standard deviation was calculated for the Stress Range (ksi) 0 1 2 3 4 5 6 7 0.99995 0.99990 0.99985 0.99980 1.0 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 C um ul at iv e D es ig n Fu nc tio n Figure 3-14. Illustration showing selection of 99.99th percentile stress range.

84 Proposed Modification to AASHTO Cross-Frame Analysis and Design distribution of all stress ranges produced by the 18 WIM site records). Load bias refers to the ratio between the actual load effects and the design load effects (i.e., the WIM load effects divided by the load effects caused by the fatigue design truck). As documented in the SHRP 2 R19B report (Modjeski and Masters 2015), the value associated with the âmean plus 1.5 standard deviationsâ was taken as both the bias of the data (SHRP 2 R19B states this was a conservative measure, since it was unknown how accurately the WIM records used in the project reflected truck traffic across the nation) as well as an appro- priate load factor. This resulted in a proposed load factor of 2.0 for Fatigue I; however, the load factor adopted in the 8th Edition of AASHTO LRFD was 1.75. Based on discussions with indi- viduals knowledgeable on the selection of this load factor, it appears that the value of 1.75 was ultimately selected by using the arithmetic mean of the normalized stress ranges produced by the WIM site records (i.e., the bias and load factor are both taken to be equal to the mean bias value). This was rationalized by acknowledging perceived conservatism inherent in the resistance data for Fatigue I and the fact that the WIM data reviewed was deemed to be sufficiently abundant to represent national traffic loads. Figure 3-15 summarizes the ratios of the 99.99th percentile WIM cross-frame stress ranges for each of the 20 bridges divided by the unfactored fatigue truck stress range for each bridge. Each bin value in this figure represents the average of all Fatigue I stress ranges for all WIM sites applied to the identified bridge (as explained in the preceding paragraph). The solid line in this figure represents the arithmetic mean of all Fatigue I stress ranges for all WIM sites applied to the identified bridge. The dashed line in this figure represents the mean value of all âmean plus 1.5 standard deviationsâ (i.e., the mean plus 1.5 standard deviation for all values represented by the individual bars). Consistent with the approach that led to the 8th Edition AASHTO LRFD updates to the Fatigue I load factor, the arithmetic mean value of all Fatigue I stress ranges for all WIM sites would correspond to an appropriate Fatigue I load factor for cross-frames. In other words, the stress ranges produced by the unfactored fatigue truck must be amplified by this factor to produce an equivalent 99.99th percentile stress range that represents the 99.99th percentile load effects caused by the WIM traffic streams. Without consideration of material fatigue resis- tances (and therefore the reliability index), the mean values associated with the solid lines in Figure 3-15 imply an appropriate load factor for the Fatigue I limit state is 1.01. The coefficient Key: Mean Mean + 1.5 Standard Deviations 1.75 0 1.50 1.25 1.00 0.75 0.50 0.25 Representative Bridge Model ID M ax im um N or m al iz ed S R 99 .9 9% (W IM /D es ig nT ru ck ) Figure 3-15. Summary of Fatigue I normalized 99.99% cross-frame stress ranges for all WIM records/sites.

Findings and Applications 85 of variation associated with this value is 0.23. This load factor is less than the current Fatigue I load factor of 1.75, which indicates a potential source of conservatism in the design load criteria for cross-frame fatigue. Recall from Section 2.3.3.1 that the GVWs for the WIM records are not normal; rather, many WIM sites appear to have multi-modal distributions that likely correspond to natural groupings of different vehicle types and payload. Appendix F discusses examining the distribution on a normal probability plot in order to assess the normality of the data and applying a best fit line to the CDF to estimate the mean and standard deviations. This procedure is performed for all distribution summaries to compare directly to the means and standard deviations computed using standard arithmetic formulas. Table 3-5 compares the standard deviations and means obtained by both methods. 3.2.2.2 Fatigue II Damage Ratios (Normalized to Fatigue Truck) As discussed in Section 2.3.3.3, the Fatigue II design criteria can be evaluated in terms of accumulated damage, as opposed to simply the effective stress range of the truck popula- tion spectra. The accumulated damage metric inherently considers both variable stress range magnitudes and number of cycles. Thus, the total damage accumulated by the various WIM streams on the critical, governing cross-frame members is compared directly to the damage accumulated by the AASHTO fatigue truck (based on AASHTO criteria for the relationship between number of trucks and number of cycles). Appendix F derives the expression used for describing the fatigue damage ratio, l.Values of l less than 1 indicate that the damage accumulated by the WIM data is less than that of the assumed values based on AASHTO design criteria (i.e., the design criteria is overly conservative); the opposite is true to values of l larger than 1. For the number of cycles used with the AASHTO fatigue truck in design, the current AASHTO LRFD (Table 6.6.1.2.5-2) specifies a value of n equal to 1 for transverse members spaced greater than 20 feet, and a value of n equal to 2 for transverse members spaced equal to or less than 20 feet. For this project, the number of design cycles per truck was taken as 1.0 regardless of the cross-frame spacing, which is consistent with the analytical and experimental results obtained as part of this study. Figure 3-16 summarizes the fatigue damage ratios of the WIM traffic streams for each repre- sentative bridge. Figure 3-16 is the same as Figure 3-15 with respect to the definition of the dashed and solid horizontal lines. Without consideration of material resistances (and therefore the reliability index), the value associated with the âmean plus 1.5 standard deviationsâ implies that an appropriate load factor for the Fatigue II limit state is 0.49. The coefficient of variation associated with this value is 0.16. This load factor is significantly less than the 0.80 load factor currently specified in AASHTO LRFD for Fatigue II. The use of the âmean plus 1.5 standard deviationsâ as an indicator of an appropriate load factor is consistent with research documented in SHRP 2 Project R19B (Modjeski and Masters 2015) and the subsequent development of the 0.8 load Mean Coefficient of Variation Mean Plus 1 Standard Deviation Mean Plus 1.5 Standard Deviations Value Obtained by Direct Calculation 1.01 0.23 1.24 1.35 Value Obtained from Normal Probability Plot 1.04 0.17 1.22 1.31 Table 3-5. Comparison of statistical parameters obtained for the Fatigue I cross-frame stress ranges by two methods.

86 Proposed Modification to AASHTO Cross-Frame Analysis and Design factor for Fatigue II. This suggests that the 0.8 load factor may be conservative for characterizing the response of cross-frames to the U.S. truck spectra. Similar to the comparison made in Section 3.2.2.1, the fatigue damage ratios for all WIM records and all bridges were plotted on a normal probability plot to assess the normality of the damage ratios (further discussed in Appendix F). Table 3-6 compares the standard deviations and means obtained by standard arithmetic formulas and the normal probability plot approach. 3.2.2.3 Stochastic Simulation via the Monte Carlo Technique A Monte Carlo simulation was performed using the statistical summaries for resistance and cross-frame force effects previously described. For the simulation, a total of 10,000 samples were randomly generated from the distributions of load and resistance described by the statis- tical parameters. The procedure used in the development of this simulation is discussed in Appendix F and was performed for the cross-frame statistics in Table 3-7 for each detail category. The resulting reliability indices are shown in Table 3-8. Based on the reliability analysis performed, the resulting reliability indices for the Fatigue II limit state for cross-frames using the reduced load factors introduced in Section 3.2.2.2 generally exceed the target reliability assumed inherent within the AASHTO Specifications (b = 1), as presented in the SHRP 2 R19B study (Modjeski and Masters 2015). In contrast, the result- ing reliability indices associated with the reduced Fatigue I load factors introduced in Sec- tion 3.2.2.1 vary, for which the indices of several detail categories are less than unity. 0.8 0 0.7 0.6 0.5 0.4 Key: Mean Mean + 1.5 Standard Deviations 0.3 0.2 0.1 Representative Bridge Model ID M ax im um F at ig ue D am ag eR at io s (W IM /D es ig nT ru ck ) Figure 3-16. Summary of Fatigue II cross-frame damage ratios for all WIM records/sites. Mean Coefficient of Variation Mean Plus 1 Standard Deviation Mean Plus 1.5 Standard Deviations Value Obtained by Direct Calculation 0.40 0.16 0.46 0.49 Value Obtained from Normal Probability Plot 0.40 0.21 0.48 0.53 Table 3-6. Comparison of statistical parameters obtained for the Fatigue II cross-frame damage ratios by two methods.

Findings and Applications 87 In order to obtain reliability indices close to the assumed target reliability of unity, adjust- ments are required to either the load or resistance factors. As an alternative to altering the resistance factors [or the associated changes in constant amplitude fatigue thresholds, (F)TH, for Fatigue I, or detail constants, A, for Fatigue II] it is also possible to adjust the load factor for fatigue such that the minimum target reliability of unity is achieved for each category. Additionally, as an alternative to introducing two new load factors for the fatigue limit state for cross-frames, it is possible to apply a single adjustment factor to the existing load factors, similar to the procedure used for fatigue design of orthotropic decks established in AASHTO LRFD (2020) Article 3.4.4. Using an adjustment factor of 0.65 applied to the Fatigue I load factor of 1.75 (i.e., a resul- tant load factor for cross-frames of 1.14 for Fatigue I) and the Fatigue II load factor of 0.8 (i.e., a resultant load factor for cross-frames of 0.52), the resulting reliability indices are calculated via Monte Carlo simulation and shown in Table 3-9. With this adjustment, each detail category satisfies the minimum assumed target reliability of unity. 3.2.2.4 Reevaluation of AASHTO Fatigue Design Criteria Based on the fact that reduced cross-frame-specific load factors have been proposed to eliminate a source of conservatism observed in the fatigue loading model, it is worthwhile revisiting Figure 3-13. Recall that Figure 3-13 evaluated the fatigue design performance of cross-frames in each of the 4,104 bridges studied. Based on those results, it was apparent that current AASHTO design criteria indicate a potential load-induced fatigue problem in common cross-frame configurations, despite the lack of physical evidence in constructed super- structures across the United States. As such, Figure 3-13 is replicated in Figure 3-17 with two notable exceptions. Limit State Coefficient of Variation of Load Bias for Load Data Load Factor Fatigue I 0.26 1.01 1.01 Fatigue II 0.21 0.40 0.53 Table 3-7. Statistical parameters describing the load effects in cross-frames for Fatigue I and II limit states. Detail Category Reliability Index Fatigue I Fatigue II A 1.01 1.77 B 0.88 1.74 Bâ² 0.98 2.47 C 0.93 1.82 Câ² 0.93 1.82 D 1.27 2.89 E 0.60 1.95 Eâ² 1.41 2.31 Table 3-8. Reliability of cross-frames for Fatigue I and II limit states using the load factors introduced in Sections 3.2.2.1 and 3.2.2.2.

88 Proposed Modification to AASHTO Cross-Frame Analysis and Design Detail Category Reliability Index Fatigue I Fatigue II A 1.32 1.74 B 1.22 1.70 Bâ² 1.49 2.39 C 1.29 1.79 Câ² 1.30 1.77 D 1.82 2.81 E 1.08 1.87 Eâ² 1.77 2.25 Table 3-9. Reliability of cross-frames for Fatigue I and II limit states using adjustment factor of 0.65 to existing load factors. 0 0.02 0.04 0.06 0.08 0 1 2 3 4 5 0 0.02 0.04 0.06 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.08 Demand-to-Capacity Ratio Straight, Normal IS = 0 IC = 0 Mean = 0.55 % Above 1.0 = 5% Slightly Curved IS = 0 IC â¤ 1 Mean = 0.62 % Above 1.0 = 9% Heavily Curved IS = 0 IC > 1 Mean = 0.78 % Above 1.0 = 19% Slightly Skewed IS â¤ 0.3 IC = 0 Mean = 0.66 % Above 1.0 = 16% Heavily Skewed IS > 0.3 IC = 0 Mean = 1.02 % Above 1.0 = 44% Skewed & Curved IS > 0.1 IC > 0.5 Mean = 1.19 % Above 1.0 = 56% Pr ob ab ili ty o f O cc ur re nc e Figure 3-17. Fatigue II cross-frame demand-to-capacity ratios for various bridge types (Houston I-10 traffic) using reduced load factors.

Findings and Applications 89 First, the Fatigue II load factors inherently built into the force demands (i.e., the numerator of the D/C ratio) are reduced from 0.80 to 0.52 based on the findings outlined above. Note that the resistance model (i.e., the denominator in the D/C ratio) remains unchanged. The second major exception is in how skewed and curved bridges are organized and presented. Rather than simply grouping the bridges based on the presence of support skew and/or horizontal curvature, the indexes developed in NCHRP Research Report 725 (White et al. 2012) and previously defined in Eqs. 3.1 and 3.2 are utilized. Thus, each bridge in the 4,104-model data set is grouped based on the skew and connectivity index bounds established in NCHRP Research Report 725. For instance, heavily curved bridges are differentiated from moderately curved bridges by Ic > 1 and Ic â¤ 1, respectively. By orga- nizing the figure in this manner, the criticality of a particular bridge type in terms of load- induced fatigue forces in cross-frames can be clearly delineated. It is also important to note that Figure 3-17 only investigates fatigue performance in the 4,104 bridges with respect to the high-volume truck traffic in Houston, Texas. The results for low-volume Llano traffic would be much less severe. With consideration of the proposed cross-frame-specific load factors and I-10 Houston traffic, the following observations can be made from Figure 3-17: â¢ In straight bridges with normal supports, the average D/C ratio was 0.55, where only 5% of the 312 qualifying bridges exceeded unity. This indicates that load-induced fatigue would not govern the design of cross-frames in these bridge types even for heavy truck traffic volumes. This implies that conducting a 2D or 3D refined analysis to obtain live load force effects in cross-frames is likely not warranted. This observation is consistent with the current AASHTO LRFD design approach. â¢ For slightly curved bridges (Ic â¤ 1) the mean D/C ratio (0.62) increased slightly compared to straight bridges, and only 9% of the qualifying data points exceeded unity. This indicates that design of cross-frames in moderately curved bridges with normal supports is also not likely to be governed by load-induced fatigue. In contrast, heavily curved bridges (Ic > 1) showed an increase in mean D/C ratio to 0.78 and an increase in percent exceeding unity to 19%. â¢ It is evident that support skew affects cross-frame force demands more significantly than horizontal curvature. Slightly skewed bridges (Is â¤ 0.3) and heavily skewed bridges (Is > 0.3) produced mean D/C ratios of 0.66 and 1.02, respectively. Additional statistical analysis showed that obtaining fatigue loads and designing for load-induced fatigue should be considered for cross-frames in bridges with a skew index exceeding 0.15. â¢ For bridges with both support skew and horizontal curvature (i.e., Ic > 0.5 and Is > 0.1), the mean D/C reported was 1.19, which indicates a significant load-induced fatigue issue even with the reduced load factors. Consequently, these bridges would likely need to be redesigned or the cross-frames reconfigured to mitigate the fatigue force demands. A refined 2D and 3D analysis would be necessary to accurately obtain those design forces. 3.2.3 Multiple Presence Study The assumptions of multiple presence in the original LRFD calibration studies (Nowak 1999, Kulicki et al. 2007) were initially based on engineering judgement and visual observations of truck traffic with unknown weights. These initial assumptions were that a âside-by-sideâ scenario (i.e., adjacent lane loaded with a passing truck in general alignment with the drive lane truck) occurred once every 15 load events. More recent studies have shown this assump- tion to be excessively conservative (Sivakumar, Ghosn, and Moses 2011). While the research conducted in SHRP 2 R19B (Modjeski and Masters 2015) confirmed this is an unlikely event, prior to this study the effect of multiple presence has not been reviewed in the context of cross-frame force effects.

90 Proposed Modification to AASHTO Cross-Frame Analysis and Design As discussed in Section 2.3.3.5, a cluster analysis was performed on the multi-lane WIM data to consider if a bridge may be loaded with other truck traffic during a primary drive lane load event (i.e., passage of one or more axles of a vehicle). Typical results of the adjacent lane scenarios are summarized in Figure 3-18 through Figure 3-20, which provide a sample histo- gram illustrating various aspects of multiple presence studied. For various load-position parameters (e.g., clear distance between drive lane and passing truck), the number of occur- rences for each multi-lane WIM site is compiled and plotted. Specifically, Figure 3-18 illustrates the clear distances between passing lane trucks and drive lane trucks, Figure 3-19 illustrates the spectrum of GVW for passing lane trucks, and Figure 3-20 illustrates the ratio of passing lane truck GVW to drive lane truck GVW. Appendix F provides additional histograms from these studies. Note that the occurrences on the primary vertical axis for these histograms are the number of total occurrences for the 12-month data set. This value should be taken into context with the total annual truck traffic (ATT) for the specific site. Table 3-10 summarizes Clear Distance Between Trucks (ft) Frequency Cumulative % Key: Indiana US 31 SB450 0 400 350 300 250 200 150 100 50 120% 0% 100% 80% 60% 40% 20% Fr eq ue nc y of O cc ur re nc e - - - - - - - - - - Figure 3-18. Histogram from two-lane WIM records showing the clear distance between a passing truck and the drive lane truck (Indiana US-31 SB). Frequency Cumulative % Key: Passing Lane GVW (kips) 120% 0% 100% 80% 60% 40% 20% 0 2500 2000 1500 1000 500 Indiana US 31 SB Fr eq ue nc y of O cc ur re nc e Figure 3-19. Histogram from two-lane WIM records showing GVW of a passing truck (Indiana US-31 SB).

Findings and Applications 91 how often (for the year of data considered) any truck was within a certain window of the drive lane truck, relative to the total volume of traffic in the drive lane for each specific site. Based on these results, it is evident that a passing truck in close proximity to a drive lane truck is a rare occurrence. The frequency of occurrence is less than the assumptions used in the original calibration studies (Nowak 1999). The largest frequency of occurrence is demonstrated by the Tennessee IH-40 westbound (WB) data, where approximately 30% of traffic is accom- panied by another vehicle located with a headway distance of less than 1,000 feet, and 1.8% of traffic is accompanied by another vehicle located with a headway distance less than 20 feet. For the most critical passing lane position for cross-frames (one directly behind another or zero clear distance), the largest frequency of occurrence is demonstrated to be 0.03%. Recalling that cross-frame forces generally reduce rapidly as a truck moves away from the cross-frame in the Frequency Cumulative % Key: Ratio of Passing Lane GVW to Drive Lane GVW 0 120% 0% 100% 80% 60% 40% 20% 1600 1400 1200 1000 800 600 400 200 Indiana US 31 SB Fr eq ue nc y of O cc ur re nc e Figure 3-20. Histogram from two-lane WIM records showing the ratio of the passing lane GVW to the drive lane GVW (Indiana US-31 SB). WIM Site Frequency of Occurrence within Clear Distance Window, Relative to Drive Lane Annual Truck Traffic (ATT)a +/- 1,000 ft +/- 280 ft +/- 50 ft +/- 20 ft +/- 0 ft Indiana US-31 NB 4.0% 2.1% 0.5% 0.3% 0.01% Indiana US-31 SB 4.2% 2.2% 0.5% 0.3% 0.00% Tennessee IH-40 EB 16.0% 7.4% 1.7% 1.0% 0.02% Tennessee IH-40 WB 29.9% 13.2% 3.0% 1.8% 0.03% Virginia US-29 3.9% 2.0% 0.5% 0.3% 0.01% Reference Figure 2-11 for illustration of positive and negative clear distances.a Table 3-10. Summary of multiple presence statistics for adjacent lane loaded.

92 Proposed Modification to AASHTO Cross-Frame Analysis and Design longitudinal direction, it is apparent that, for cross-frames, the largest frequency of occurrence for passing vehicles occurring simultaneously is appreciably low (i.e., larger headway distances generally do not result in superimposed cross-frame forces). Another observation is that the distribution of the passing lane truck GVW appears to be bi-modal for several sites, with the heavier truck mode being equal to or heavier than the drive-lane truck GVW. Sensitivity studies were conducted by White (2020) that explicitly accounted for the cross-frame force effects caused by passing lane vehicles, and the effects on the accumulated fatigue damage were found to be negligible. The WIM study performed in this research confirmed that the dual truck event initially considered in the 7th Edition AASHTO LRFD is a rare occurrence. As such, the current load criteria (i.e., a single design truck positioned in all longitudinal and transverse positions) is more appropriate. This multiple presence study indicates that even when considered, the effects of passing lane traffic on both the Fatigue I and Fatigue II limit state parameters are negligible. A comprehensive multiple lane analysis using all available two-lane traffic records demonstrates that the probability of a single truck record (regardless of GVW) being located in a critical position for magnifying cross-frame force effects is exceptionally lowâless than 0.02% for the most critical location (i.e., the steering axle of one truck is located just behind the rear axles of another truck in an adjacent lane) and less than 3% when the truck is located within 20 to 50 feet from the rear axle of another truck in an adjacent lane. 3.2.4 Major Outcomes Based on the results of the Fatigue Loading Study presented in the preceding subsections, there are several major conclusions that can be drawn with respect to Objectives (a) and (b) and the five questions posed in the introduction of Section 3.2: â¢ Cross-frames are sensitive to transverse truck placement. For cross-frames near the maxi- mum positive dead load moment region, the longitudinal load influence tends to be localized. For cross-frames in straight or curved girder systems near skewed end or intermediate sup- ports, the load influence response is significantly broadened. In general terms, the influence of load position is highly variable and difficult to predict unless an influence-surface analysis is conducted. â¢ Similarly, critical lane position also depends on a variety of parameters. In general, truck passages along the outer edges of the deck (i.e., overhang loads) tend to maximize cross- frame forces in skewed and curved bridges (i.e., load applies net torque on superstructure which engages the cross-frames) but not in straight, normal bridges. Additionally, cross- frame response is generally linearly dependent on the location of the applied load relative to the centerline of the fascia girder; that is, wheel loads that are 3 feet outboard of the fascia girder produce larger force demands on critical cross-frame members than loads that are 1 foot outboard. â¢ The WIM study confirmed that the dual truck event (i.e., multiple presence) initially considered in the 2016 Interims to the 7th Edition AASHTO LRFD Specifications is a rare occurrence. In terms of implementation of AASHTO LRFD, the results also indicate that the current specification language in Article 3.6.1.4.3a (âa single design truck shall be positioned transversely and longitudinally to maximize stress range at the detail under considerationâ) is the best method to ensure the critical load position for the governing cross-frame is considered by the designer. â¢ The governing cross-frame in a given bridge depends on many variables, including girder spacing, support skew, and lane positions. Bottom struts tend to be the governing cross- frame member in most bridges, as cross-frames tend to act as composite âfloorbeamsâ when distributing applied live loads transversely to adjacent girders. Still, the response is likely too

Findings and Applications 93 variable to pinpoint which cross-frames are critical without some refined, influence-surface analysis (i.e., to examine all possible truck positions and cross-frame force effects). â¢ The correlation between specific variables and cross-frame fatigue design forces was explored. In general, increased skew and tighter bridge curvature tends to increase cross- frame force demands in composite systems. These general rules-of-thumb can potentially be utilized by engineers to mitigate cross-frame forces in the design phase of a project. In other words, the engineer can potentially avoid an iterative âchase-your-tailâ design solution by adjusting other key parameters. For instance, using discontinuous, staggered cross-frame layout is a practical and economical solution, whereas increasing the deck thickness or decreasing girder spacing has significant impacts on the rest of the design. â¢ From the fatigue truck study, it was concluded that additional economies in the AASHTO fatigue criteria are possible for cross-frame design. This is largely attributed to the sensi- tivity of cross-frame response to transverse load position. Current design criteria require all possible lane positions be considered in accordance with AASHTO Article 3.6.1.4.3a, regardless of design lanes or actual lane striping. A more realistic scenario is to consider only drive lanes (i.e., lanes within the limits of the fascia girders, for which most truck traffic traverses). Alternatively, reduction factors specific to cross-frames could be applied to the existing load factors. Based on feedback received from the review panel, cross-frame- specific load factors outlined in the subsequent bulleted item were explored as the preferred approach. â¢ To maintain the current guidance in terms of truck position (i.e., the AASHTO fatigue truck is located in every transverse position on the bridge deck between the vehicle barriers), cross-frame-specific fatigue load factors were investigated to eliminate the source of conser- vatism in the current load factors. The modified load factors developed in this study are based on a comprehensive WIM study utilizing calibrated bridge models representative of a variety of straight bridges with normal supports, straight bridges with skewed supports, and horizontally curved bridges. Assuming the WIM records represent typical truck weights throughout the country, the improved load factors reflect more realistic force effects experi- enced by cross-frames in these types of bridges. Using published statistical data for resistance, the reliability of using these improved load factors was investigated. The Monte Carlo simulation using the available resistance data and cross-frame statistics summarized in this report result in reliability indices as shown in Table 3-9 when using an adjusted load factor of 0.65 applied to both of the current Fatigue I and II limit state load factors (i.e., a respec- tive Fatigue I and II load factor for cross-frames of 1.14 and 0.52). Based on this analysis, the resulting reliability indices for cross-frames using these adjusted load factors meets or exceeds the assumed target reliability of unity (Modjeski and Masters 2015). â¢ For the analytical and experimental analyses conducted in this study, the number of design stress cycles per truck passage (n in AASHTO LRFD Table 6.6.1.2.5-2) was taken as 1.0. The proposed load factors above were subsequently calibrated using this n = 1.0 assumption. â¢ In implementing the reduced cross-frame-specific load factors proposed, it was observed that load-induced fatigue problems are likely only an issue in moderately to heavily skewed and/or curved bridges (i.e., Ic > 1, Is > 0.15, or Ic > 0.5 & Is > 0.1) in areas of significant truck traffic volume. For bridges that do not fall into one of these categories, a refined 2D or 3D analysis is not warranted. These limits are similar to those established in NCHRP Research Report 725 (White et al. 2012), which focused primarily on analysis of noncomposite systems, except that the skew index limit proposed in that study is lowered herein from 0.3 to 0.15. A discussion on the appropriate analysis method is outlined in the subsequent sections. These findings, which are further summarized in Chapter 4, served as the basis for many of the proposed modifications to AASHTO LRFD. Refer to Appendix A for the proposed language and commentary.

94 Proposed Modification to AASHTO Cross-Frame Analysis and Design 3.3 R-Factor Study (3D Analysis) This section summarizes the results related to the R-Factor Study, which focuses on addressing Objective (c) of NCHRP Project 12-113. More specifically, the limitations of the traditional truss-element modeling approach for cross-frames in 3D models are investigated in the context of composite, in-service bridges. The applicability of a proposed alternative approach (eccentric-beam model) is also assessed for a variety of bridge geometries. These goals were summarized by the following questions, which were initially posed Section 1.2: â¢ Is the current established R-factor (0.65AE), which was based on analytical and experimental studies of a noncomposite system, appropriate for cross-frames in the composite condition? â¢ Are there alternative 3D modeling approaches for cross-frames? These questions are systematically addressed throughout this section of the report. Section 3.3.1 summarizes key findings from a panel-level study, which analytically examined the response of noncomposite cross-frame panels to various load-induced deformation patterns anticipated in in-service bridges. Section 3.3.2 presents the results of a 3D FEA parametric study that inves- tigated the response of cross-frames in composite bridge systems. The accuracy of the stiffness modification factors and the eccentric-beam approach are assessed in terms of predicted cross- frame force effects for a variety of bridge geometries. Lastly, Section 3.3.3 summarizes the key findings from the study. 3.3.1 Panel-Level Studies (Noncomposite) As noted in Section 2.4.4.1, a parametric study was performed to evaluate the stiffness response of different X-type cross-frame configurations and connection details due to various deforma- tion patterns in bridge girders, including equal rotation, differential vertical displacement, and differential rotation. In general, stiffness modification (R) factors were derived by comparing the computed torsional stiffness response of a benchmark shell-element model and the response of the corresponding truss-element model. This process was repeated for a variety of cross-frame panels and connections (approximately 2,000 unique cases). Figure 3-21 summarizes the computational results. The computed R-factor is plotted separately for the given loading condition. Each data point is grouped by angle leg thickness (1â4-inch up to 5â8-inch) to highlight the relative impact of the parameters on the results. Within each grouping, various angle leg lengths (3 inches and 4 inches) and girder spacing- to-panel height ratios are grouped and presented. Figure 3-21 provides specific callouts to demonstrate how the various parameters are presented. Additionally, the results presented correspond to a constant gusset plate thickness of 0.5 inches only. However, note that an increase in gusset plate thickness has been shown to generally produce higher modification factors (i.e., the increase in out-of-plane flexural stiffness influences the overall stiffness more than the additional eccentricity introduced). A key emphasis of the panel-level analytical studies, as previously introduced in Section 2.4, was related to the orientation of the cross-frame diagonals relative to the gusset and connection plates. As such, the differential-rotation deformation pattern was considered with two distinct cases similar to those schematically shown in Figure 2-16. These cases are identified as âsame sideâ connections and âopposite sideâ connections. The same side case corresponds to the loading condition in which only the diagonal that frames into the same face of the gusset as the strut is engaged; the opposite side case corresponds to the loading condition in which the diagonal that frames into the opposite face of the gusset is engaged. The other loading con- ditions, equal rotation and differential vertical displacement, engage both diagonal members equally such that the resulting effects are not pronounced in the results.

Findings and Applications 95 From Figure 3-21, the following observations can be made with respect to the panel-level study: â¢ In general, the parameters of the single-angle section (thickness, leg width) and the aspect ratio of the panel (girder spacing, panel height) affect the computed modification factor, R. The R-factor equation presented in Battistini et al. (2016) shows that increasing the S/hb ratio (girder spacing-to-panel height), yâ (distance between the face of the connected leg and the neutral axis of the angle section), and t (angle leg thickness) results in a decreased R-factor for the equal rotation condition. Similar trends for composite systems were generally observed from the results of the studies described in the next section. â¢ Significant scatter in the results is observed for the differential-rotation case depending on which cross-frame diagonal is engaged. The same side connections generally result in a more flexible response due to the additive effect of the end connection eccentricities, whereas the opposite side connections are much stiffer. â¢ Even for parameters dictated by designers (e.g., angle thickness, angle leg width, girder spacing, panel height, gusset thickness), there can be substantial variability in the R-factor response. Additional variability is also introduced by parameters not controlled by designers (e.g., deformation pattern and the force distribution due to truck position). Additional spot checks were performed to investigate the effects of different connection plate dimensions (i.e., slenderness ratios) and K-type cross-frames. In brief, it was evident that the connection plate slenderness had minimal effect on the computed R-factor for all loading conditions evaluated. In terms of the K-type cross-frames, the âsame and opposite sideâ effects were not observed given that diagonals in K-frames typically connect to the same face of the corresponding gussets. As such, it can be concluded that K-frames are not as sensitive to the deformation pattern given the symmetry of the end connections. For a more thorough review of these spot check analyses, refer to Appendix F. Given the variability and uncertainty in the response due to different geometries and load- ing conditions, developing a closed-form, general-use solution similar to that suggested in Differential Rotation â Same Side Differential Vertical Displacement Angle Thickness (in) 1/4 3/8 1/2 5/8 1/4 3/8 1/2 5/8 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Differential Rotation â Opposite Side Equal Rotation St iff ne ss M od ifi ca tio n Fa ct or , R Figure 3-21. Computed stiffness modification factors for various loading conditions and angle thicknesses.

96 Proposed Modification to AASHTO Cross-Frame Analysis and Design Battistini et al. (2016) would be challenging and impractical for design. From a simplicity stand- point, assigning a single modification factor for each cross-frame member, similar to the current approach in AASHTO LRFD, is desirable. This is particularly true since the orientation of the cross-frame connections (i.e., increasing or decreasing eccentricities) may not be known during the design process. With that in mind, the system-level studies in the next subsection evaluate the most appropriate modification factor to assign for the majority of bridge superstructures in service. An independent parametric study that quantifies the impact of an assumed R-factor on the cross-frame force prediction in composite bridge systems is outlined. 3.3.2 System-Level Studies (Composite) Two independent parametric studies were performed as part of this system-level study as outlined in Section 2.4.4. One study, consisting of only straight and normal bridges, was performed to emphasize the influence of connection and gusset plate dimensions. The other study was conducted to investigate the effects of support skew and horizontal curvature on the R-factor and eccentric-beam approaches. Given that cross-frame force effects are more critical in skewed and/or curved bridge systems, the results presented herein focus on the second study only. For an extended overview of the entire data set, refer to Appendix F. In general, multiple iterations of the same 3D bridge model were developed, and the cross- frame force results were subsequently compared. A rigorous shell-element model served as the benchmark model, and various truss-element and eccentric-beam models were produced to assess the accuracy and potential limitations of these simplified methods. Before discussing the implications of employing the proposed eccentric-beam approach to modeling cross-frames, Figure 3-22 summarizes a sample set of results for the truss-element, R-factor approach. In this figure, the design force range for each model iteration of a sample bridge is reported (based on the AASHTO fatigue truck traversing in the critical lane). Along the x-axis, the results for the various truss-element models are considered as a function of the applied modification factor, R = {0.5, 0.6, 0.7, 0.8, 0.9, and 1.0}. Along the y-axis, the Fsimplified/Fshell ratio is plotted. A value of unity indicates that the assigned R-factor resulted in perfect agreement with the shell-element model. Values above unity imply that the truss-element model is too stiff (i.e., the truss member attracts more force than the shell-element model) and that the R-factor should be reduced. This case represents a conservative design assumption with respect to the fatigue limit state (i.e., the analysis model produces excessive design forces) but is unconser- vative from the perspective of stability bracing or torsional behavior in a curved girder. The opposite is true for values below unity. In the callouts on Figure 3-22, note that the âexactâ R-factor, where the line intersects Fsimplified/Fshell = 1.0, is included for each member presented. For example, an R of 0.73 yields identical behavior between the shell and truss models for the bottom strut in the 1/2-inch gusset plate case. Figure 3-22 presents this data for three select cross-frame members in the sample bridge model. The select cross-frames are as follows: (i) a bottom strut near the maximum positive dead load moment region, (ii) a top strut from the same cross-frame panel, and (iii) a diagonal near the intermediate skewed support. The figure also differentiates the models by gusset plate thickness (1â2-inch or 1-inch thick). Note that the gusset plate thickness only affects the shell- element model directly, as the truss-element models do not explicitly represent gusset plates. Thus, only the denominator in the Fsimplified/Fshell ratio is impacted between the top ( 1â2-inch thick) and the bottom (1-inch thick) plots in Figure 3-22. For reference, a sketch of the framing plan and typical cross-section are provided (not to scale); the skewed diaphragms at the end and intermediate supports are included in the model

Findings and Applications 97 but neglected in the figures for clarity. For more information regarding the pertinent parameters of the superstructure, refer to Appendix F. The following observations can be made from Figure 3-22: â¢ The cross-frame force effect is not 1:1 dependent on the stiffness modification due to the high degree of indeterminacy in the bridge system. In other words, a 10% reduction in the R-factor does not result in a 10% reduction in the estimated cross-frame force. â¢ Rather, a 10% reduction in the R-factor generally results in approximately a 5% reduction on average in the estimated cross-frame force. That relationship typically holds true for modification factors above 0.6. Below R = 0.6, the rate at which the Fsimplified/Fshell approaches zero increases. Ultimately, a modification factor of zero results in Fsimplified/Fshell = 0, despite not being shown in the figure. This assumed 2:1 relationship (x% reduction in R â x/2% reduction in force) has design implications. For example, if an acceptable level of error in the predicted cross-frame force is established as 10%, the R-factor likely needs to be within 20% of the exact value (recall that it is conservative from a fatigue perspective to assign higher R-factors). â¢ Even for cross-frame members of similar geometries and eccentricities, the âexactâ R-factor for these critical loading conditions can vary. For example, the exact R-factor for the bottom strut is 0.73 compared to 0.63 for the top strut in the same cross-frame panel. This variability highlights the impact of load position and deformation patterns on the assumed stiffness. â¢ The gusset plate thickness also impacts the response of the cross-frame in the shell-element model. As the gusset plate thickness increases, the stiffness of the cross-frame tends to also increase. As noted earlier, although the eccentricity is larger with a thicker gusset plate, the Key: 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 F s im pl ifi ed / F s he ll ( C ri tic al L oa di ng ) Assigned R-Factor (Truss Model) 1 2 3 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.2 1.4 1.6 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Top Strut (R = 0.63) Bottom Strut (R = 0.73) Diagonal 1 (R = 0.63) Diagonal 1 (R = 0.70) Bottom Strut (R = 0.95) Top Strut (R = 0.56) 1/2-inch Gusset 1-inch Gusset Figure 3-22. Example data from two-span, continuous skewed bridge demonstrating the predicted cross-frame design force range from the truss-element approach (using various R-factors) and the shell-element approach.

98 Proposed Modification to AASHTO Cross-Frame Analysis and Design bending stiffness of the plate increases. The reason for this is that the eccentricity is a linear function while the bending stiffness changes as a cubic function. As such, the exact R-factor for the bridge with 1-inch thick gusset plates tends to exceed the exact R-factor for the bridge with 1â2-inch thick gussets. For example, the exact R-factor for the diagonal member evaluated was 0.63 for 1â2-inch gussets and 0.70 for 1-inch gussets. â¢ In general, the sample results presented in Figure 3-22 are representative of the full data set in terms of the observed variability and trends. Figure 3-22 represents the results of just one bridge in the sample set. Rather than evaluate the effectiveness of the R-factor approach for every cross-frame member and every transverse lane passage, it is more important to just focus on the governing cross-frame member in every bridge (i.e., the cross-frame with the maximum force range). In this manner, more generalized observations about the stiffness modification factors can be made. Note that a similar approach was taken when compiling the data obtained from the Fatigue Loading Study. Thus, for each iteration of a given bridge model, Fsimplified/Fshell ratios were computed and compiled in the form of box-and-whiskers plots. The results of the eccentric-beam approach (ârefined modelâ only; refer to Figure 2-15) were also included to examine its potential benefits. Note that, while the results herein focus on the refined eccentric-beam model only, an examina- tion of the simplified versions is presented later in this section. For Figure 3-23, the box-and-whiskers components are organized by the assigned R-factor in the truss-element model, as well as the eccentric-beam model (which is independent of an assigned R-factor). Therefore, each box-and-whiskers represents a data set of Fsimplified/Fshell ratios corresponding to the truss-element models with R = {0.5, 0.6, 0.7, 0.8, 0.9, and 1.0} and the eccentric-beam model. By presenting the box-and-whiskers of the results for various R-factors side-by-side, the research team can evaluate which fixed factor generally produces the most accurate representations of the âtrueâ cross-frames stiffness (i.e., the shell-element model). Similarly, the variability of the R-factor approach can be compared to the eccentric-beam approach, which more accurately represents the stiffness of the cross-frame and its connections. Statistical parameters such as the mean, minimum, and maximum values of the data set are also provided. For the R = 0.5 example, the respective mean and maximum Fsimplified/Fshell ratios were 0.82 and 0.98, which indicates that assigned R = 0.5 always underpredicted the design force when compared to the benchmark shell-element model. Assigned R-Factor (Truss Model) 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0 Ecc. Beam 1/2-inch Gusset 25th Percentile F s im pl ifi ed / F s he ll ( G ov er ni ng M em be r) 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 Figure 3-23. Box-and-whiskers plot indicating the overall level of accuracy for the eccentric-beam modeling approach compared to the truss-element approach (1/2-inch gusset plate).

Findings and Applications 99 The figure only presents the results related to the bridges analyzed with 1â2-inch-thick gusset plates. However, bridges with 1-inch-thick gusset plates were evaluated, and the results are subsequently provided in Appendix F. In addition to the box-and-whiskers, a line graph representing the 25th percentile of each data set is provided, whose meaning is outlined below. Similar to the sample bridge in Figure 3-22, the following observations can be made about the stiffness modification (R) factor approach with respect to the full data set presented in Figure 3-23: â¢ As the assigned R-factor increases, the design force in the truss-element model increases, thereby increasing the Fsimplified/Fshell ratio. In other words, the R-factor and Fsimplified/Fshell ratio are positively correlated. â¢ Despite the positive correlation, there is still significant variability in the observed response. The accuracy of the truss-element model in terms of predicted cross-frame forces is not only a function of the assigned modification factor but also the bridge geometry and loading conditions. For example, the respective maximum and minimum Fsimplified/Fshell ratios for the R = 0.5 case are 0.98 and 0.73. â¢ Although not explicitly shown in this report, the Fsimplified/Fshell ratio decreases as the gusset plate thickness increases. This is attributed to the fact that an increased gusset thickness results in a stiffer cross-frame, which then attracts more force to the shell-element model. From the observations above, it is apparent that developing an expression that precisely predicts the exact R-factor for any general bridge model (i.e., the R-factor for which Fsimplified/Fshell always achieves unity) is not practical or feasible. The exact R-factor is a function of many parameters, including bridge geometry, loading conditions, cross-frame details (e.g., gusset plate thickness), and member type (e.g., bottom strut versus diagonal). Thus, it is more appro- priate to assign a uniform R-factor to all cross-frame members that statistically represents the majority of bridge conditions, similar to the R = 0.65 approach currently adopted in AASHTO LRFD that was developed for the construction condition of the bridge. Based on these results, there are three potential approaches for handling the variable response of the R-factor approach in terms of AASHTO LRFD. These are listed below from most sophisticated to most simplistic as well as potential pros and cons to each approach: â¢ Option 1: Develop a member-specific and bridge-specific expression that provides a precise R-factor. Given the variability in the response and the uncertain nature of live loads on bridge structures, this approach is neither feasible nor practical for implementation in AASHTO. The expression would inevitably be extremely complex and would still likely produce scattered results. â¢ Option 2: Develop an R-factor that is a direct function of the gusset plate thickness. Rather than develop an expression that explicitly considers all pertinent bridge and cross-frame parameters (e.g., girder spacing, angle thickness), this approach simply focuses on gusset plate thickness, which has been shown to substantially impact the exact R-factor. This procedure could be easily implemented as an equation or a table. However, given that the gusset plate thickness may be variable during the design process, this approach potentially lends itself to an iterative analysis procedure. Even with this extra level of refinement, the results could still be variable. â¢ Option 3: Develop a single R-factor that generally represents the behavior of all cross-frame members in a conservative manner (similar to the current 0.65 approach in the 9th Edition AASHTO LRFD that was developed for the construction condition). As demonstrated with all previous results, pinpointing an exact R-factor is very difficult given its inherent dependence on the loading conditions and the associated deformation patterns. Regardless of how the cross- frames are modeled and R-factors are assigned, some error in cross-frame force predictions

100 Proposed Modification to AASHTO Cross-Frame Analysis and Design is to be expected when compared to the shell-element modeling approach. With that said, assigning a single R-factor for all cross-frame members, regardless of connection details, bridge geometry, and loading conditions, is a viable option. After weighing the options, it was elected to recommend the use of a single R-factor (Option 3) similar to the current approach taken in AASHTO Article C4.6.3.3.4. With that in mind, it is also not appropriate to ensure every bridge model results in conservative estimates of cross-frame fatigue forces (Fsimplified/Fshell â¥ 1). Instead, the research team investigated different statistical parameters but ultimately elected to base the proposed modification factor on the 25th percentile value of the data sets. In other words, the intersection of the 25th percentile line and Fsimplified/Fshell = 1.0 represents the assigned R-factor that provides a conservative estimate of cross-frame design forces for 75% of the bridges. For the 25% of the bridges that result in unconservative estimates, it has been shown that the truss-element model is still within 10% of the rigorous shell-element model, which was deemed reasonable by the research team. Using that metric, the most appropriate R-factor to assign for the bridges with 1â2-inch gussets is 0.75. As such, in addition to the stiffness modification factor of 0.65 that was derived for a noncomposite system during construction, a second factor of 0.75 is proposed for structural analysis of the composite condition. Table 3-11 tabulates the results outlined above, as well as additional data corresponding to different gusset plate thickness values and statistical parameters. For instance, if the in-service R-factor is based on the 50th percentile (i.e., mean) of the data set with 1â2-inch-thick gusset plate, then R = 0.66 is more appropriate, which is approximately the same as the established modification factor in AASHTO LRFD (R = 0.65). Although the R-factor approach provides a method of correcting truss-element models for the reduction in stiffness due to eccentric connections, some designers or software producers may opt to make use of a model that directly considers the connection eccentricity. In terms of the proposed eccentric-beam results in Figure 3-23, it was observed that the refined eccentric- beam models predict the cross-frame fatigue design forces with reasonable levels of accuracy. For the 1â2-inch gusset case, the Fsimplified/Fshell ratio is 0.96 on average with a maximum of 1.03 and a minimum of 0.86. In other words, the eccentric-beam produces results within 4% of the shell-element model on average. As outlined above, the results presented in Figure 3-23 are based on the ârefinedâ eccentric- beam model only. To illustrate the impacts of the more simplified eccentric-beam models, Figure 3-24 presents sample influence-line results from the same two-span, continuous skewed bridge from Figure 3-22. In this figure, two different responses are examined: (i) the axial force in the bottom strut member near the skewed intermediate support when the AASHTO fatigue truck traverses the bridge along the barrier and (ii) the axial force in the bottom strut member (interior bay) near the maximum positive dead load moment region when the AASHTO fatigue truck traverses the bridge along its centerline. Gusset thickness [in] Appropriate R-factor Mean 25th Percentile 1/4 0.60 0.65 1/2 0.66 0.75 3/4 0.70 0.81 Table 3-11. Summary of R-factor results from system-level parametric study.

Findings and Applications 101 For each cross-frame response, several different results are presented, including the shell- element model (which serves as the control), the truss-element model range (i.e., R = 0.5 and R = 1.0), the refined eccentric-beam model, the simplified eccentric-beam model, and the truss-only eccentric-beam models (both fixed-end and pinned-end). The various responses are superimposed on the same graph. Additionally, the total force ranges and the percent error relative to the control model are tabulated for reference. Note the scale along the vertical axis is different for the top and bottom plots in Figure 3-24. Refer to Appendix F for a detailed description of each eccentric-beam modeling approach. The following observations can be made from the sample data set presented in Figure 3-24: â¢ Similar to the results presented above, the truss-element model approach has significant variability depending on which R-factor is assigned. Note that a modification factor of approximately 0.75 would produce the most accurate results in both cases shown in Figure 3-24, which is consistent with the conclusions of the R-Factor Study presented in the previous section. â¢ In both loading scenarios, the refined eccentric-beam model produces relatively accurate results when compared to the shell-element model (6% conservative). This is a similar observation to the results of the parametric study presented above. For instance, these two 0 210 420 R = 0.5 R = 1.0 -9 -6 -3 0 3 6 9 -3 0 3 6 0 210 420 BS BS Shell-Elements (Control) Truss-Elements; R={0.5, 1.0} Ecc. Beam; Simplified Ecc. Beam; Angle (Fixed Ends) Ecc. Beam; Refined Ecc. Beam; Angle (Pinned Ends) 11.0 {8.7, 13.3} 10.7 12.8 10.4 6.8 -- {-21%, +21%} -3% +16% -6% -38% % ErrorFRa (kips)CF Model aFR = Force Range R = 0.5 R = 1.0 Shell-Elements (Control) Truss-Elements; R={0.5, 1.0} Ecc. Beam; Simplified Ecc. Beam; Angle (Fixed Ends) Ecc. Beam; Refined Ecc. Beam; Angle (Pinned Ends) 4.8 {3.9, 5.6} 4.1 4.8 4.6 3.2 -- {-21%, +14%} -16% -2% -6% -35% % ErrorFRa (kips)CF Model Case 2 Case 1 Case 1 Case 2 A xi al F or ce (k ip s) Figure 3-24. Example data demonstrating the predicted cross-frame force range for the various eccentric-beam models.

102 Proposed Modification to AASHTO Cross-Frame Analysis and Design data points are close to the mean of the eccentric-beam data set (box-and-whiskers) shown in Figure 3-23 (i.e., a Fsimplified/Fshell ratio of 0.94 compared to 0.96 in Figure 3-23). â¢ The simplified eccentric-beam model, which neglects the individual eccentric offsets caused by the connection and gusset plates, has more variability but generally produces reasonably accurate and conservative results. This approach is much simpler to employ in practice and requires much less information about the cross-frame connection details. â¢ The angle-only models are largely dependent on the assumed end restraints. When the beam ends are fixed rotationally to the girder web, the stiffness of the cross-frame panel tends to be overestimated, which results in slightly overconservative force effects (but still less conservative than the truss-element model assuming R = 1.0). Despite the eccentric offset being explicitly modeled, the out-of-plane flexural rigidity is greatly overestimated at the end connections by using the angle section properties for the full length of the modeled element. â¢ When the beam ends are pinned (to account for the rotational flexibility of the connection plates), the overall cross-frame stiffness tends to be vastly underestimated (even more under- estimated than the truss-element model with R = 0.5). Thus, the cross-frame force effects are significantly unconservative with respect to the shell-element model. It can be observed from this sample data that an eccentric-beam approach, for which rotation is released at its ends, is not an appropriate 3D modeling technique for cross-frames. â¢ In review of the angle-only models, the ârealâ rotational restraint at the ends of the beam elements is somewhere between the idealized fixed and pinned scenario. Although not studied here, a rotational spring with finite stiffness could be explored as an alternative as well. The most significant observation from the sample results in Figure 3-24 is that accurately representing the out-of-plane flexibility associated with the connection and gusset plates is vital for eccentric-beam models. The angle-only models generally do a poor job of representing that end flexibility (assuming idealized fixed- or pinned-end conditions), whereas the simplified and refined models showed much more promise. In general, the eccentric-beam approaches eliminate several sources of uncertainty associated with the R-factor approach (e.g., eccentric distances, relative stiffness of gusset and connection plates, effect of load position on the deformed shape). However, modeling a cross-frame based on the refined methodology requires knowledge of the gusset and connection plate thicknesses prior to designing those elements. This potentially lends itself to an iterative process where the engineer starts with an assumed geometry (likely based on standard DOT details) and updates the model based on project-specific design decisions. This iterative process, though, affects the shell-element and refined eccentric-beam approaches equally. In contrast, the simplified methodology, although less precise and accurate, reasonably approximates the flexibility of the connection plate without having full detailed dimensions of the cross-frame connections. Based on these perceived advantages, this modeling technique has potential to serve as an alternative for bridge designers and commercial design software packages. As a reference, the current commentary language in AASHTO LRFD (C4.6.3.3.4) that discusses the use of the R-factor is provided herein: âIn addition, the axial rigidity of single-angle members and flange-connected tee-section cross-frame members is reduced due to end connection eccentricities (Wang et al., 2012). In lieu of a more accurate analysis, (AE)eq of equal leg single angles, unequal leg single angles connected to the long leg, and flange- connected tee-section members may be taken as 0.65AE.â As underlined in the provision above, the eccentric-beam approaches (namely the refined and simplified approaches) qualify as a âmore accurate analysisâ but do not require bridge designers to assemble labor-intensive, 3D FEA models with cross-frames modeled as shell elements. The overall conclusions with regards to this proposed analysis procedure are provided in Section 3.3.3 below.

Findings and Applications 103 3.3.3 Major Outcomes Based on the results of the R-Factor Study presented in the preceding subsections, there are three major conclusions that can be drawn with respect to simplified 3D analysis methods for cross-frame design: â¢ The appropriate R-factor to be assigned in truss-element models is largely a function of bridge geometry, cross-frame details, and uncertain loading conditions. Consequently, considerable scatter was observed in all phases of the analytical studies. Still, the R-factor approach is a simple solution to a complex problem that produces reasonably accurate approxi- mations of the actual cross-frame stiffness. Considering that many designers often prefer simple alternatives over sophisticated refined analyses, it serves an important role in AASHTO LRFD guidance moving forward. â¢ In terms of implementation into AASHTO LRFD, a reasonable approach considering the numerous uncertainties is the use of a single R-factor, similar to the current approach adopted into the specifications. Rather than use the current AASHTO recommendation of R = 0.65 that was developed based upon the construction condition, R = 0.75 has been shown to pro- duce more accurate results in composite systems when compared to benchmark solutions. As such, it is recommended to propose two separate factors, Rcon for construction stages (0.65) and Rser for in-service conditions (0.75). However, it should be noted that additional statistical parameters have been investigated that result in different potential modification factors. â¢ The proposed eccentric-beam model represents an approach that is slightly more refined than the conventional use of pin-ended truss elements, but less complex than modeling cross-frames with shell elements. As long as the out-of-plane connection plate flexibility is properly considered, the results demonstrated that the proposed method improves repeat- ability and reliability of 3D models by eliminating several sources of uncertainty associated with the R-factor approach. Although the R-factor approach serves a vital purpose in practice due to its ease of use and familiarity, the proposed eccentric-beam method offers another approach to engineers seeking a more refined solution. These findings, which are further summarized in Chapter 4, served as the basis for many of the proposed modifications to AASHTO LRFD. Refer to Appendix A for the proposed language and commentary. 3.4 Commercial Design Software Study (2D Analysis) This section summarizes the major findings of the Commercial Design Software Study that was previously introduced and outlined in Section 2.5. The study, which specifically addresses Objective (d) of NCHRP Project 12-113, investigates the common 3D and 2D analysis tech- niques employed by commercial design software programs with respect to estimating cross-frame force effects. This objective is otherwise summarized by the following questions that were initially posed in Section 1.2: â¢ What are the limitations of simplified 2D analysis techniques commonly used by popular commercial design software programs in terms of predicting cross-frame force effects in composite systems? â¢ Are there methods available to improve these simplified techniques? By comparing the performance of 3D models, 2D PEB models, and 2D grillage models, these questions are systematically addressed throughout this section of the report. Similar to the Fatigue Loading Study, an abundance of cross-frame data was obtained from the Commercial Design Software Study. To simplify the results and discussion for the reader, a similar outline with respect to data presentation is followed for the discussion of all models. Representative

104 Proposed Modification to AASHTO Cross-Frame Analysis and Design data are first shown in Sections 3.4.1 and 3.4.2. These sections address different aspects of the results. Section 3.4.1 presents sample influence-line results to introduce the general limita- tions of 2D analysis methods, and Section 3.4.2 investigates the influence of composite action on the accuracy of these 2D models. Section 3.4.3 compiles and summarizes the results for the full data set. Section 3.4.4 proposes an alternative postprocessing procedure for 2D PEB analyses, and Section 3.4.5 summarizes the major outcomes. 3.4.1 Sample Influence-Line Results As a starting point, Figure 3-25 compares the predicted cross-frame response for various analysis methods under a specified moving load. This sample highlights key trends and obser- vations that are further validated in subsequent sections. Figure 3-25 represents a straight bridge with normal supports. Before investigating more complex framing systems, it is prudent to assess these simplified 2D analysis methods on a simple structure. Simplified 2D analyses have been generally perceived to produce erroneous cross-frame results in heavily curved bridges and near skewed supports, because they are unable 1 2 3 Longitudinal Position of Truck (ft) -2 0 2 4 6 8 0 240 -8 -6 -4 -2 0 2 0 240 -6 -4 -2 0 2 4 0 240 C-2 D-2 D-3C-3 C-1 Case 1 Case 3 Case 2 Key: [Approach; Equiv. Beam] 2-D PEB; ShearC-1 2-D PEB; Flexural C-2 2-D PEB; Timoshenko C-3 2-D Grid; Flexural D-1 2-D Grid; Timoshenko D-2 3-DB Control (Abaqus)A Note: Framing plan sketch above not to scale D-3 C-1 C-2 D-1 C-2/C-3 C-1 A/B A/B C-3 D-1/D-2/D-3 A/B 2-D Grid; Composite D-3 D-2 D-1 A xi al F or ce (k ip s) Figure 3-25. Comparison of analysis methods for three select cross-frame members in a straight bridge with normal supports.

Findings and Applications 105 to capture the torsional response of the bridge and the complex load paths through the structure in each case. These geometric effects are addressed in subsequent sections. The axial-force response of the three select cross-frame locations is shown: (i) a bottom strut near midspan referred to as Case 1, (ii) a top strut from the same cross-frame panel referred to as Case 2, and (iii) a diagonal near the support referred to as Case 3. This is demonstrated schematically in the framing plan sketch in Figure 3-25. The AASHTO fatigue truck was moved longitudinally along the centerline of the bridge as is depicted. Figure 3-25 plots the axial-force response of the respective cross-frame members as a function of the truck position along the bridge length, measured from the front axle to the start of the bridge. These graphs are similar to the influence-line plots presented in Figure 3-24. This process was repeated for several variations of the same bridge model including the 3D validated approach in Abaqus (control) and the following methods in the commercial design software program Software A (recall that Software A has the ability to create either a 3D model or a variety of 2D models including PEB and grillage models): â¢ 3D model, â¢ 2D PEB model with equivalent cross-frame beams based on the shear-analogy approach, â¢ 2D PEB model with equivalent cross-frame beams based on the flexural-analogy approach, â¢ Improved 2D PEB model with equivalent cross-frame beams based on the Timoshenko approach, â¢ 2D grid/grillage model with equivalent cross-frame beams based on the flexural-analogy approach, â¢ 2D grid/grillage model with equivalent cross-frame beams based on the Timoshenko approach, and â¢ Improved 2D grid/grillage model with equivalent cross-frame beams considering the transverse stiffness of the concrete deck and the modified postprocessing procedure per Figure 2-17 (abbreviated as 2D Grid Composite in the Figure 3-25). The specifics of these 2D modeling approaches (e.g., equivalent cross-frame beam analogies) were outlined in Section 2.5. Note that, for the PEB and grillage models, equal distribution of cross-frame shear was assumed between top and bottom nodes when postprocessing the respective analysis results. The implications of using a different distribution assumption are addressed in subsequent sections. For more information regarding the overall geometry and cross-frame layout of this sample bridge as well as additional influence-line results, refer to Appendix F. From Figure 3-25, the following observations can be made when comparing the results of a specific cross-frame location across each bridge model: Case 1 (bottom strut near the maximum positive dead load moment region): â¢ The 3D commercial design software model produces excellent results when compared to the validated 3D FEA model in Abaqus (i.e., estimated force range within a few percent). Given that the commercial design software modeling approach is nearly identical to that of the control model, this trend was anticipated. This also indicates that the connection point of the cross-frame members (i.e., into a shared node on the web-flange junction versus at an offset distance along the web) is less impactful for 3D model results. â¢ The 2D PEB results were not significantly influenced by the equivalent beam method employed for these composite conditions. In general, these models produced accurate estimates of bottom strut forces when compared to the refined 3D models. â¢ The 2D grid models generally did a poor job at predicting the cross-frame forces, except when the Timoshenko beam approach was implemented.

106 Proposed Modification to AASHTO Cross-Frame Analysis and Design Case 2 (top strut near the maximum positive dead load moment region): â¢ Similar to Case 1, the 3D commercial design software model generally produces excellent results when compared to the validated 3D FEA model in Abaqus. â¢ For the 2D analyses that do not consider the contributions of the deck when postprocessing the analysis results, the top strut results are equal in magnitude and opposite in sign to those results of Case 1. This is a function of the simplified postprocessing used when converting the equivalent beam back into a truss. As illustrated in this example, this simplification generally results in erroneous results for top strut forces. â¢ This error is corrected in the 2D grillage model using cross-frame beams with equivalent section properties to account for the contributions of the deck. Albeit the correct sign was obtained, the magnitudes are still generally overestimated. Case 3 (diagonal near the intermediate/end support): â¢ In general, the observations for Case 3 are similar to those for Case 1 except that the 2D approaches consistently underpredicted the cross-frame force effects. In reviewing data beyond these sample results (specifically for bridges with X-type cross- frames), it was evident that the predicted response of bottom strut member forces in 2D models was much more accurate than the predicted response of diagonal or top strut members. This behavior was apparent in both simple and complex bridge geometries. In other words, the variability observed in the response was not only a function of the bridge geometry (i.e., straight versus curved, normal supports versus skewed supports) but also the cross-frame member type (i.e., bottom strut versus diagonals). The variability associated with bridge geometry is attributed more to analysis error (i.e., how the analysis determines end shears and moments on the equivalent cross-frame beam), whereas variability associated with member type is attributed more to postprocessing error (i.e., how those end shears and moments are converted into cross-frame member forces). With that in mind, Section 3.4.2 examines this behavior in the context of noncomposite and composite sys- tems to determine the root cause of these discrepancies. 3.4.2 Postprocessing Error With general trends laid out in Section 3.4.1, this section explores the error associated with the postprocessing assumptions inherent to 2D modeling approaches. Analyses were performed for noncomposite and composite systems to better understand how a composite deck affects the distribution of forces in cross-frame systems. Namely, the research team sought answers to the following questions: are the assumptions inherent to noncomposite systems (e.g., equal shear distribution to top and bottom nodes) valid for composite systems; and, if not, how does an engineer make adjustments to correct those assumptions? Before examining more complex bridge systems, Figure 3-26 presents the results for the same single-span, straight bridge with normal supports presented in Figure 3-25. By starting with a simple bridge geometry, the postprocessing procedures can be isolated from any complexities introduced with support skews and horizontal curvature. The response of a midspan cross-frame panel is evaluated for a static loading condition (i.e., applying a 100-kip load just to the left of the cross-frame panel of interest). Note that an additional static load case was considered in this study, and those results are summarized in Appendix F. In total, four different analysis methods are compared: â¢ 3D Software A model of a composite system (top right quadrant of Figure 3-26), â¢ 3D Software A model of a noncomposite system (bottom right quadrant),

Findings and Applications 107 â¢ 2D PEB model of a composite system utilizing the Timoshenko beam approach for equiva- lent cross-frame beam properties and the equivalent torsion constant (Jeq) for girder beam properties (top left quadrant), â¢ 2D PEB model of a noncomposite system utilizing the Timoshenko beam approach for equivalent cross-frame beam properties and the equivalent torsion constant (Jeq) for girder beam properties (bottom left quadrant). Given that the preliminary results presented in Figure 3-25 demonstrated the benefits of PEB models over grillage models, the results in this section are focused on just PEB methods. For the 3D analyses (right side of figure), the axial-force output from the cross-frame members is presented, followed by the nodal forces required to equilibrate the panel. For the 2D PEB analyses (left side of figure), the shear and moments produced from the equivalent beam analysis are presented, followed by the cross-frame force effects computed based on the common postprocessing assumptions previously outlined. 100 k 2090 k-in 14.7 k 14.7 k 323 k-in 34.8k 34.8 k 7.4 k 7.4 k 20.1 k 20.1 k 5.4 k 5.4 k 7.4 k 7.4 k 20.8 k 21.1 k 35.6 k 35.5 k 7.4 k 7.2 k 20.8 k 21.1 k 6.3 k 6.6 k 7.2 k 7.4 k 756 k-in 4.2 k 4.2 k 258 k-in 60â 120â 12.6 k 12.6 k 2.1 k 2.1 k 8.5 k 8.5 k 4.3 k 4.3 k 2.1 k 2.1 k 8.4 k 2.6 k 60â 120â 0.5 k 17.7 k 4.7 k 1.0 k 8.4 k 2.6 k 12.3 k 6.3 k 1.0 k 4.7 k 2D PEB (Composite System) 3D (Composite System) 2D PEB (Noncomposite System) 3D (Noncomposite System) Figure 3-26. Sample 2D PEB and 3D cross-frame results for straight bridge with normal supports (noncomposite and composite) under static load (k = kips, k-in = kip inches).

108 Proposed Modification to AASHTO Cross-Frame Analysis and Design For each figure, there are three metrics by which the results can be compared, as listed below: 1. The total shear force acting on the cross-frame panel relative to the shear force in the concrete deck: This value provides an indication of how accurate the equivalent beam properties in the 2D model are. For instance, if the total shear force determined from the 3D model exceeds that of the 2D PEB model, then the stiffness of the equivalent cross-frame is likely underestimated. The opposite would be true if the total shear force of the 2D PEB model exceeds the 3D model. 2. The distribution of that total shear force to the top and bottom nodes: Girder displacements in noncomposite systems, particularly rotations, are different than girder displacements in composite systems. To demonstrate this behavior in simple terms, an idealized deformation pattern is provided in Figure 3-27. Figure 3-27 schematically depicts a common displaced shape of an X-type cross-frame panel under applied vertical loads; a noncomposite system is compared to a composite system. Note that the displacements are greatly exaggerated and that rigid-body motion has been removed for clarity. In comparing the noncomposite and composite systems, it is evident that noncomposite girders generally rotate out of plane about their centroid under applied vertical loads. Consequently, the top and bottom struts tend to deform equal magnitudes but in oppo- site directions, which results in equal-and-opposite axial-force effects in those members. In contrast, composite girders tend to rotate out-of-plane about an axis higher on the cross-section due to composite action with the concrete deck. As a result, the deformation demand on the top strut is small, as it is located near the point of girder rotation. In relative terms, the bottom strut deforms significantly more, which is illustrated in Figure 3-27 with the original position of the system lightly shadowing the deformed position. Under this idealized displaced shape, the top strut sees little to no force effects whereas the bottom strut sees substantial axial tension. To maintain compatibility with the girder displacements, both diagonal members in the X-frame must also deform under axial tension. Thus, when this idealized differential-rotation deformation pattern is paired with differen- tial vertical displacement (which generates substantial diagonal forces approximately equal- and-opposite in magnitude), the net effect is unbalanced diagonal forces. Cases in which the diagonals are both in tension or both in compression are, therefore, not uncommon. This generally explains why the equal-and-opposite force distribution in X-type cross-frames is observed in noncomposite systems but not in composite systems where realistic deformation patterns are highly complex. For K-type cross-frames, this discussion is less influential as equal-and-opposite force distribution in diagonal members is anticipated given the layout of the members. Figure 3-27. Idealized deformation pattern of noncomposite (left) and composite (right) systems.

Findings and Applications 109 Assuming equal shear force distribution in the postprocessing phase of 2D PEB models generally leads to poor results for cross-frame diagonals. With that in mind, evaluating the âtrueâ shear force distribution from the 3D models provides an indication of how significant the effects illustrated in Figure 3-27 are. 3. The relative difference between top strut forces and bottom strut forces: Similar to the discussion in Item #2 above, it is anticipated that bottom strut force effects will greatly exceed top strut force effects in composite systems. By evaluating the relative difference between these two magnitudes, the effects of composite action can be quantified. The sign of the top strut force can also provide an indication as to where the point of girder rotation exists along the depth of the cross-section (e.g., if the top and bottom struts are both in tension, then the point of rotation is likely above the top strut). In the context of the three comparative metrics established above, the following observations can be made from Figure 3-26: â¢ Total shear force: For the noncomposite system, the total cross-frame shear force estimated from the 2D PEB model is nearly identical to that of the 3D model (i.e., 14.8 kips versus 14.6 kips). This indicates that the Timoshenko beam analogy accurately represented the effective cross-frame stiffness by considering both shear and flexural deformations. For reference, the shear-analogy approach produced a total cross-frame shear of 10.6 kips for this same example, despite not being shown in the figure. This indicates that the shear-analogy underestimates the cross-frame stiffness. For the composite system, the total cross-frame shear force is underestimated in the 2D PEB models (e.g., 4.2 kips versus 5.7 kips). This relationship is explored further in this section. â¢ Shear force distribution: For the noncomposite system, the 3D model demonstrates that shear force is equally distributed to the top and bottom struts (subsequently resulting in equal-and-opposite diagonal force effects). As a result, the â50-50â distribution assumption commonly utilized in 2D postprocessing is valid for noncomposite conditions. Thus, the 2D models accurately predict the force effects in the diagonal members for the specified loading condition. For the composite system, the 3D analysis model demonstrates that load-induced forces in diagonal members are not equal and opposite. Figure 3-26 shows 2.3 kips of compression in one diagonal and 10.4 kips of tension in the other, resulting in an â80â20â distribution of the total shear force. Thus, when comparing cross-frame forces results, it is evident that the 2D PEB postprocessing procedure produces inaccurate results. â¢ Moment force couple distribution: For the noncomposite system, the 3D analysis results indicate that the load-induced forces in the top and bottom struts are nearly equal and opposite, as expected (i.e., 21.1 kips of compression in the top strut and 20.8 kips of tension in the bottom strut). Therefore, the 2D PEB model accurately predicts top and bottom strut forces when compared to the 3D counterpart model. For the composite system, this equal-and-opposite relationship does not hold true. The 3D model indicates that the top strut force effects share the same sign as the bottom strut force effects, but the magnitude is significantly less. Based on the postprocessing assumption, the 2D PEB model produces equal-and-opposite forces in the top and bottom struts. Thus, there is substantial error associated with the force prediction in the top strut member in the composite system for the 2D PEB model. However, the error in the force prediction in the bottom strut member is minimal (i.e., within 5% conservative in both cases). Despite the inconsistencies and oversimplifications in the 2D analysis and postprocessing, this same general trend related to bottom strut members is typically observed assuming the equivalent beam properties are accurate (i.e., the Timoshenko beam approach is utilized). The bottom strut results in Figure 3-25 are additional examples of this trend.

110 Proposed Modification to AASHTO Cross-Frame Analysis and Design Given that 2D PEB models generally produce reasonably accurate estimates for bottom strut force effects, especially for simple bridge geometries, it is important to note when these limitations and shortcomings become critical. If a bottom strut cross-frame member maximizes load-induced force effects and governs fatigue design, then this simplified analysis and post- processing procedure likely produces reasonable estimates of the governing design forces. However, if diagonal members (or much less likely, top struts) govern the fatigue design, then 2D PEB approaches may grossly underestimate the governing force effects. To expand on the findings from Figure 3-26, which focuses on a simple straight bridge with normal supports, the research team repeated these same studies to examine more complex bridge geometries and loading scenarios. Two additional bridges were examined similar to the sample bridge presented above: a straight bridge with skewed supports (X-type cross-frames) and a curved bridge with normal supports (X-type cross-frames). Additional studies were performed on different bridge geometries and cross-frame configurations (i.e., K-type versus X-type); those results are presented in Appendix F. For the straight and skewed bridge, an interior-bay cross-frame panel near the intermediate skewed support is examined. For the curved bridge, an interior-bay cross-frame panel near the maximum positive dead load moment region is examined. The results outlined above investigated only two different extreme conditions: noncomposite and full in-service composite (i.e., concrete modulus, Ec, taken as 5,000 ksi). Rather than evaluate just the two extremes, the research team expanded the study by investigating different variations of Ec to better quantify how composite action affects cross-frame behavior. The concrete modulus, Ec, was varied as opposed to the deck thickness as to maintain a consistent centroid location for the concrete deck and to simplify discussions. The Software A analysis procedures previously described in this subsection are otherwise unchanged. As such, a total of 272 unique models were developed as part of this study, including different bridges, various 3D and 2D modeling approaches, and multiple iterations of Ec. Note that the Ec = 0 ksi case represents noncomposite conditions. Instead of completely removing the deck from the model, the deck was simulated with virtually no stiffness to expedite the repetitive calculations. The response of the specified cross-frame members in each bridge was evaluated for two different static loading cases (100 kips in magnitude) that are arbitrarily identified as Load Case 1 and 2. For reference, the location of load application relative to the framing plan is sketched in Figure 3-28 and Figure 3-29. The results presented herein follow the same orga- nization as the three comparative metrics outlined above: total cross-frame shear, shear force distribution, and moment force couple distribution. Figure 3-28 and Figure 3-29 show how the total shear force developed in the cross-frame panel of interest varies as a function of the concrete deck modulus (or the degree of composite action, in more general terms). Figure 3-28 focuses on the straight bridge with skewed supports and X-type cross-frames, and Figure 3-29 focuses on the curved bridge. In these figures, only the results of 2D PEB models associated with the flexural-analogy and the Timoshenko beam approach are presented, given the inherent inaccuracies with the shear-analogy approach outlined previously. Figure 3-30 demonstrates how the total shear force is distributed to the top and bottom nodes per the 3D analysis model. Since the 2D models assume a â50-50â distribution, results related to 2D models are excluded as they do not add value to the graph. In Figure 3-31, the ratio between the top strut force and the bottom strut force is plotted as a function of the assumed concrete deck modulus. The results of the two sample bridges are compiled and provided. Note that these results correspond to the 3D models only, as the 2D models assume an equal-and- opposite force couple in the postprocessing phase. Thus, the 2D analysis results do not add significant value to the graph.

Findings and Applications 111 -1 0 1 2 3 4 5 3D (Control) 2D PEB; Timoshenko Beam 2D PEB; Flexural-Analogy Model; Equivalent Beam Case 1 Case 2 Case 2 (100 k) Case 1 (100 k) VCF VCF Concrete Modulus (ksi; log scale) C ro ss -F ra m e Sh ea r, V CF (k ip s) 1 10 100 1000 100001 10 100 1000 10000 00 Figure 3-28. Total cross-frame shear force as a function of concrete deck modulus (straight bridge with skewed supports and X-type cross-frames). 0 5 10 15 20 25 30 Concrete Modulus (ksi; log scale) C ro ss -F ra m e Sh ea r, V CF (k ip s) 3D (Control) 2D PEB; Timoshenko Beam 2D PEB; Flexural-Analogy Model; Equivalent Beam Case 1 Case 2 VCF VCF 1 10 100 1000 100001 10 100 1000 10000 00 Figure 3-29. Total cross-frame shear force as a function of concrete deck modulus (horizontally curved bridge). The following additional notes provide important insight to the development and construc- tion of Figure 3-28 through Figure 3-31: â¢ A cross-frame diagram is provided in each plot to illustrate the nodal force or internal axial force examined in the graph. For instance, Figure 3-28 and Figure 3-29 present the definition of the total cross-frame shear force, VCF. â¢ As stated above, a modulus value of 0 ksi represents a noncomposite system, whereas values between 3,000 and 5,000 ksi are representative of most in-service, composite bridges in the United States. The horizontal axis is presented on a log scale to clarify the noticeable change in behavior between 100 ksi and 1,000 ksi.

112 Proposed Modification to AASHTO Cross-Frame Analysis and Design â¢ In Figure 3-30, the shear force acting on the top node (Vtop) and the bottom node (Vbot) are compared independently with the total cross-frame shear (VCF). Thus, the sum of all Vtop/VCF and Vbot /VCF ratios equals one. Note that Vtop and Vbot are arbitrarily taken relative to the left side of the cross-frame panel as shown in the sketch. Had the shear forces been taken from the opposite side, the values of Vtop and Vbot would be flipped. The general behavior, however, would remain the same given that the shear force is constant across the width of the cross-frame panel. â¢ In Figure 3-31, the ratio of the top strut force (PTS) and the bottom strut force (PBS) is compared to a similar ratio determined by elastic beam theory. In other words, the concrete deck, top strut, and bottom strut are treated as three force couples that develop the moment strength of a pseudo-composite transverse beam, similar to a reinforced concrete beam. The elastic neutral axis of this pseudo-composite beam is derived after transforming the Concrete Modulus (ksi; log scale) Curved, Normal -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Straight, Skewed Load Case 1; Vtop / VCF Load Case 1; Vbot / VCF Load Case 2; Vbot / VCF Load Case 2; Vtop / VCF Key: VCF VCF Vtop Vbot C ro ss -F ra m e Sh ea r D ist ri bu tio n, V i / V CF 1 10 100 1000 1000001 10 100 1000 100000 Figure 3-30. Shear force distribution as a function of concrete deck modulus for two sample bridges with X-type cross-frames. Concrete Modulus (ksi; log scale) Straight, Normal Elastic Beam Theory Load Case 1 Results Load Case 2 Results Key: Curved, Normal PTS PBS -1.5 -1.0 -0.5 0.0 0.5 1.0 Skewed 1 10 100 1000 1000001 10 100 1000 100000 C ro ss -F ra m e Fo rc e C ou pl e D ist ri bu tio n, P TS / P B S Figure 3-31. Force couple distribution as a function of concrete deck modulus for the two sample bridges.

Findings and Applications 113 effective concrete area with the corresponding modular ratio. Assuming an elastic stress distribution, the PTS/PBS ratio is related to beam curvature and is simplified as yTS/yBS, where yTS and yBS are the distances measured from the centroid of the respective strut to the computed neutral axis. Therefore, by comparing the analysis results to this idealized beam theory approach, the impact that composite action has on the distribution of bending moments through the deck and cross-frame panel can be evaluated. The following observations can be made with regard to the total shear force estimated by the various analysis methods (Figure 3-28 and Figure 3-29): â¢ As the concrete modulus increases beyond zero (noncomposite), it is intuitive that the cross-frame forces generally decrease. Not only does the relative stiffness of the concrete deck increase and thus attracts more force in the highly indeterminate system, but the girder displacements diminish as the global stiffness increases. However, as the concrete modulus approaches values exceeding 3,000 ksi or more, the relative change is less significant. â¢ In general, the Timoshenko beam approach is accurate for noncomposite systems when compared to the 3D analysis, but the variability increases slightly for composite systems. Still, it is consistently the most accurate 2D equivalent beam method in terms of cross-frame force predictions, as is presented later in this subsection. â¢ The observed behavior for Load Case 2 in Figure 3-28 is less predictable. Note that the vertical axis scale is much smaller than in Figure 3-29 due to the nature of the loading. In Figure 3-29, the applied load is in close proximity to the cross-frame panel of interest, such that the shear demands are more significant. In Figure 3-28, though, the applied loads induce a torsional response to the superstructure, such that the net shear on the cross-frame of interest is generally small. Consequently, the discrepancy in the total cross-frame shear is less influential in terms of predicted cross-frame force effects, as is presented later. The following observations can be made regarding the shear force distribution determined by 3D analyses (Figure 3-30): â¢ For noncomposite systems (i.e., Ec = 0 ksi), especially for the non-skewed bridge, the shear force is equally distributed top and bottom (same result as Figure 3-26 before). The distribu- tion for the straight bridge with skewed supports is slightly less balanced given the nature of the cross-frame deformation pattern. â¢ As the concrete deck becomes stiffer and composite action becomes more substantial, it is evident that the shear force distribution gradually diverges from the â50-50â assumption. For instance, the top shear reaction in the straight, skewed bridge for Load Case 2 is approximately 1.6 VCF, whereas the bottom shear reaction is larger than 0.6 VCF in the opposite sign. In this particular instance, the observed behavior indicates that both diagonals were in significant compression. The following observations can be made regarding the force couple distribution determined by 3D analyses (Figure 3-31): â¢ For the noncomposite systems, the PTS/PBS ratio consistently converges to -1, which represents an equal-and-opposite force distribution in the struts. As the deck stiffens and composite action become more significant, the ratio tends to a small positive value. This indicates that the top strut shares the same sign as the bottom strut, but the magnitude is substantially smaller (i.e., a clear indication of composite behavior). In fact, the transition between a nega- tive ratio and a positive ratio as Ec increases represents the condition in which the center of out-of-plane girder rotation moves above the top strut. At that point, the top and bottom struts would be both compressive or both tensile forces. â¢ In general, the analysis results follow the same trends as the âelastic beam theoryâ bench- mark solution, but not exactly. Thus, the flexural action assumed in the development of this

114 Proposed Modification to AASHTO Cross-Frame Analysis and Design idealized curve is apparent in the realistic data, but the deviation from that curve is due to the fact that the âtrueâ deformation patterns are not identical to the idealized one. With general observations established about the validity of common analysis and postpro- cessing procedures, the next step is to compare the predicted force effects between 3D and 2D models. Utilizing those common assumptions (i.e., equivalent beam moment resolved as a force couple and equivalent beam shear equally distributed to top and bottom nodes of X-frames), 2D PEB results are compared to the 3D solutions. The results of the two sample bridges are compiled and presented in Figure 3-32. In total, four different graphs are presented. The two columns represent the two distinct load cases examined for each bridge. The two rows represent the different bridges evaluated as part of this study. The horizontal axis represents the elastic modulus of the concrete deck and is presented on log scale. The vertical axis represents the relative error of the 2D PEB model relative to the corresponding 3D model. Note that only the results using the Timoshenko beam analogy are shown. Relative error values of zero indicate perfect agreement between the 3D model and the simplified 2D model. Relative error values exceeding 1.0 indicate the 2D models over- estimate the force effects in excess of 100%; the opposite is true for relative errors below â1.0. For clarity, data points beyond -1.0 and 1.0 are eliminated from view. The relative errors associated with the diagonal members and the bottom strut member are also graphed separately. The lines are color-coordinated based on the cross-frame sketch provided in the corners of each graph. The results for top struts are excluded given that they are generally lightly loaded members. The following observations can be made regarding the relative error associated with the simplified 2D modeling procedures (Figure 3-32): â¢ For noncomposite bridges, relative error approaches zero for all member types and all bridge types. Although not explicitly shown in the report, this is especially true for straight and Case 2Case 1 0.1 1 100 1000 10000 -1.0 -0.5 0.0 0.5 1.0 0.1 1 10 100 1000 Straight, Skew ed -1.0 -0.5 0.0 0.5 1.0 C urved, N orm al Key: Key: Key: Key: Concrete Modulus (ksi; log scale) R el at iv e Er ro r (C om pa re d to 3 D M od el ) 1 10 100 1000 100001 10 100 1000 100000 0 Figure 3-32. Analysis error of the 2D PEB model as a function of concrete deck modulus for the two sample bridges.

Findings and Applications 115 normal bridges. This is largely attributed to two ideas: (i) the Timoshenko beam approach accurately represents the cross-frame stiffness in the analysis, and (ii) the common post- processing procedures are valid for noncomposite systems. The error increases slightly with more complex bridge systems (i.e., skewed or curved bridges) but is still relatively accurate. For example, the relative error associated with the âredâ X-frame diagonal in the skewed bridge for Ec = 0 ksi is approximately 20% conservative under Load Case 1. Under Load Case 2, the relative error is not shown largely due to the fact the absolute magnitude of the force is small. In this case, any slight discrepancies typically result in large error values. â¢ For the composite systems evaluated (i.e., Ec exceeding 3,000 ksi), the error associated with bottom strut members is typically small (within 5%â10%). Despite its limitations and oversimplifications, 2D methods produce accurate force estimates for bottom strut members. This is consistent with the results presented previously in this section. â¢ For diagonal members in X-type cross-frames, the errors generally become excessive as Ec increases. This trend was also observed for straight bridges with normal supports. One diagonal tends to a large positive value, and the other tends to a large negative value. For example, the relative error associated with the diagonal members of the curved bridge under Load Case 2 approach +100% and -100% error, respectively, for Ec = 5,000 ksi. This is attributed mainly to the equal shear force distribution assumed with the postprocessing methods. â¢ Although not explicitly shown in the report, the errors for diagonals in K-type cross-frames are much more manageable than those with X-type cross-frames. For these cross-frame types, recall that the distribution of shear force distribution is not a meaningful discussion. â¢ Although not explicitly shown, relative error magnitudes associated with 2D PEB models using the flexural-analogy approach for equivalent beams are more substantial than those shown in Figure 3-32 for the Timoshenko beam approach. Thus, it can be concluded that the Timoshenko beam approach is more appropriate for noncomposite and composite systems. From Figure 3-32, it is apparent that 2D analysis methods produce poor cross-frame force estimates in two areas: (i) in top strut members and (ii) in X-frame diagonal members regardless of bridge type. Erroneous, overly conservative results in top struts are not generally impactful in terms of cross-frame fatigue design. These members typically do not govern fatigue design such that those results, accurate or not, can be disregarded. Erroneous diagonal member forces, on the other hand, can have significant impact on the cross-frame fatigue design of a bridge. As such, it is prudent to reevaluate how designers typically handle the postprocessing of 2D analysis models to ensure more reliable results. The major challenge in developing a simple postprocessing modification is the degree in which the shear force distribution varies. In noncomposite systems, diagonal forces typically are nearly equal-and-opposite. In the composite systems evaluated as part of Figure 3-30, shear force distribution was shown to vary from nearly equal to 150% VCF at one node and â50% at the other. The distribution is a function of bridge type, loading, and subsequently the induced deformation pattern, which is difficult to quantify without developing a 3D model. An alter- native method is explored in Section 3.4.4. 3.4.3 Parametric Study To further assess the performance of 2D analysis methods, a 20-bridge parametric study (detailed in Section 2.5.4) was conducted. For each cross-frame member evaluated, the maxi- mum force range was computed based on the critical lane passage of the AASHTO fatigue truck. Thus, the results in this section differ slightly than the previous results in Section 3.4.2, which were based on static loads. It should be noted that, in some cases, the governing lane conditions between the 3D and simplified 2D models differed. For example, a 3D model indicated that a specific cross-frame

116 Proposed Modification to AASHTO Cross-Frame Analysis and Design member under a specific transverse lane passage maximized force effects, whereas the cor- responding 2D model indicated that the force in the same member was maximized by a different lane passage. To maintain consistency, the governing lane position was established based on the control model (i.e., the 3D validated model in Abaqus); the results of the simplified analysis models were then obtained from those established parameters. Figure 3-33 presents this data in the form of a box-and-whiskers graph, similar to those pre- viously shown in the report. Each box-and-whiskers plot represents a set of unfactored design force effects obtained from the various analysis models. Along the vertical y-axis, the governing force effects derived for each iteration of the 20 representative bridges are compared to the control model. The percent error with respect to the results of the control model is reported. Zero percent error indicates perfect agreement between the simplified commercial design software models and the control model. Positive error indicates that the simplified analysis conservatively overestimates the predicted force effects, and negative error indicates the opposite. In terms of accuracy, it is desirable that the simplified methods provide solutions within about 10% of the validated solution with marginal variance in the box-and-whisker. In terms of a conservative design approach, it is preferred to have positive error over negative error. In Figure 3-33, the results corresponding to bottom strut members are differentiated from diagonal members based on the discussion in Section 3.4.2. In both plots, the results are catego- rized by the modeling technique along the horizontal x-axis (3D; 2D grillage with flexural- analogy equivalent beams; and 2D PEB with shear-analogy, flexural-analogy, and Timoshenko equivalent beams). From Figure 3-33, the following generalized observations can be made when comparing the analysis error reported across the full data set: â¢ The 3D commercial design software analysis generally provides cross-frame results (struts and diagonals) that agree well with the control model. On average, the predicted design force range was 2% conservative. The observed scatter in the results was also small. â¢ For the 2D analysis methods, significant variability is observed. This behavior is attributed to: (i) skewed and/or curved geometry and (ii) the postprocessing errors inherent with the 2D analysis methods outlined in Section 3.4.2. DiagonalsBottom Struts100% 80% 60% 40% 20% â20% â40% â60% â80% â100% 0% Pe rc en t E rr or Figure 3-33. Box-and-whiskers plot demonstrating the range of analysis error obtained from the Commercial Design Software Study (bottom struts versus diagonals).

Findings and Applications 117 â¢ The 2D PEB models (shear-analogy) consistently underpredict the governing design force range (-80% error on average). The 2D PEB models (flexural-analogy) improve upon the shear-analogy approach but still underpredict the design force ranges. â¢ The 2D PEB models (Timoshenko beams) provide the most accurate solutions with respect to 2D models but still result in significant errors. These models are generally conservative (between 0% and 20% conservative) for bottom struts but are unconservative for diagonals (between 40% and 60%). This is attributed to the equal shear assumption used in the post- processing procedure. â¢ 2D grillage models using the flexural-analogy for the development of equivalent cross- frame beams show considerable scatter in the predicted cross-frame response. In some cases, the model predicted design forces above 100% conservative and below 60% unconservative. The upper tail of the bottom strut box-and-whiskers results is beyond the scale of the vertical axis, given the large discrepancies observed. Figure 3-34 expands on Figure 3-33 by breaking up the bottom strut results based on bridge geometry. For the 2D PEB results, only models utilizing the Timoshenko beam approach are included for clarity. Note that the skew index and connectivity index established in NCHRP Research Report 725 (White et al. 2012) and defined in Eqs. 3.1 and 3.2 are used to differentiate straight and curved bridges with or without support skews. To explore the effects of bridge geometry, Figure 3-34 compares the results in terms of skew and connectivity indices. Recall that only bottom strut results are presented in this figure for clarity. The following generalized observations can be made from that figure with an emphasis on PEB methods: â¢ 3D models generally perform well for all cases. Thus, 3D models offer consistently reliable results in terms of cross-frame forces primarily because the superstructure depth is accurately represented. â¢ 2D PEB models are generally conservative in terms of their bottom strut force estimates. However, scatter in the results is more evident as the bridges become more skewed and/or more curved. For instance, the percent errors range from â20% unconservative to 20% conservative for straight and normal bridges, whereas the percent error ranges from â60% conservative to 80% conservative for curved and skewed bridges. â¢ Although not presented, the results for diagonal members are more variable than what is shown in Figure 3-34 for bottom struts. 3D 2D PEB 2D Grid IS â¤ 0.1 IC â¤ 0.5 Straight & Normal IS > 0.1 IC = 0 Straight & Skewed IS = 0 IC > 0.5 Curved & Normal IS > 0.1 IC > 0.5 Curved & Skewed 3D 2D PEB2D Grid 3D 2D PEB 2D Grid 3D 2D PEB 2D Grid 100% 80% 60% 40% 20% -20% -40% -60% -80% -100% 0% Pe rc en t E rr or Figure 3-34. Box-and-whiskers plot demonstrating the range of analysis error obtained for different bridge geometries (bottom struts only).

118 Proposed Modification to AASHTO Cross-Frame Analysis and Design It is evident from the results that 3D models offer increased repeatability and reliability in terms of cross-frame force effects. However, the biggest setback to simplified 2D analyses (particularly PEB models) is related to diagonal members and the postprocessing assumptions. This statement is independent of bridge type, although analysis error becomes more apparent for complex systems. Consequently, Section 3.4.4 explores a simplified method to enhance the postprocessing procedures for 2D analysis methods. 3.4.4 Alternative 2D Postprocessing Procedure The research team investigated potential methods of manipulating 2D grillage and PEB analysis results to produce more reasonable estimates of cross-frame design forces. As discussed previously, equivalent beam moments from grillage models (excluding those that use notional beams to represent the transverse stiffness of the deck) can be resolved based on the equivalent truss/deck model illustrated in the left side of Figure 2-17. Equivalent beam models from PEB models, despite providing equal-and-opposite strut force effects, have been shown to produce reasonable estimates of the more critical bottom strut members. However, the distribution of the shear force in both grillage and PEB models directly impacts the diagonal member force effects, which are highly variable. The research team explored procedures to resolve this issue. A simple alternative to the â50-50â distribution is to conservatively design each diagonal member for 100% of the total cross-frame shear. That is to say, the top and bottom struts are handled in the same manner as presented above (i.e., equal-and-opposite force distribution, but neglect the top struts in design), but the diagonals are handled differently. This procedure is schematically shown in Figure 3-35, which is a modified adaptation of Figure 3-26. First, note that the 2D analysis output (i.e., end shear and moments on the equivalent cross- frame beam) is identical to that of Figure 3-26. Second, the top and bottom strut forces are computed (i) assuming the end moment of 756 kip-in is applied as a force couple to the top and bottom nodes and (ii) assuming the end shear of 4.2 kips is equally distributed top and bottom. This produces a top strut design force equal to 8.5 kips of compression and a bottom strut design force equal to 8.5 kips in tension, which is the same as Figure 3-26. In this instance, the top strut force can be disregarded, and the bottom strut force from the 2D analysis (8.5 kips) is in good agreement with the 3D analysis (8.4 kips). Lastly, the diagonal forces are calculated independently assuming 100% of VCF is applied to the node of interest. In Figure 3-35, the full 4.2 kips of shear are applied to the bottom node to maximize force in the tension diagonal (9.4 kips), which shows much better agreement to the 3D analysis (10.4 kips) than was presented in Figure 3-26 (4.6 kips). A similar procedure for the top node was performed, although not explicitly shown in the figure, which results in 9.4 kips of compression in the opposite diagonal. This simple method, despite producing equal-and-opposite diagonal forces, produces design magnitudes generally closer to the 3D analysis (at least for the critical diagonal member). It is acknowledged that there are obvious limitations to this simplified method. Given that the true governing member of this cross-frame panel would be the tension diagonal, designing for 9.4 kips of compression is overly conservative. It must also be recognized that this procedure could potentially be overly conservative in some cases, where the true shear distribution is closer to â50-50â (e.g., Load Case 1 in curved bridge or noncomposite systems in general). Similarly, this procedure could still be unconservative for cases in which the shear at one node exceeds 100% VCF (e.g., Load Case 2 in a straight, skewed bridge). As stated above, this simple procedure is one of many possible solutions to this problem. Yet, the only way to ensure reliably accurate results, regardless of bridge type, is to develop a 3D model.

Findings and Applications 119 3.4.5 Major Outcomes Based on the results of the Commercial Design Software Study presented, there are several conclusions that can be drawn with respect to simplified analysis methods for cross-frame design: â¢ In general terms, designers must be cognizant of the potential trade-offs between sophisti- cated and simplified analysis (in terms of accuracy and ease of use). A central theme explored throughout the report is related to the balance between increased computational complexity and improved reliability versus simplified modeling and improved ease of use. The results presented in this section clearly highlight this trade-off. The designer must make informed decisions in terms of the analysis performed. The conclusions below are intended to provide practicing engineers the technical and quantitative background to help inform those decisions, while not necessarily advocating one or the other for a specific project need. â¢ In all cases, 3D refined analysis produces improved accuracy for predicting cross-frame forces compared to 2D analytical approaches. Although 3D models are more time-consuming to 100 k 756 k-in 4.2 k 4.2 k 258 k-in 8.4 k 2.6 k 2D PEB (Composite System) 3D (Composite System) Analysis Output: 60â 120â 12.6 k 12.6 k 2.1 k 2.1 k 8.5 k 8.5 k 4.3 k 4.3 k 2.1 k 2.1 k Postprocessing Strut Forces: 4.2 k 4.2 k Postprocessing Diagonal Forces: 8.5 k Combining Results: 8.5 k Figure 3-35. Alternative postprocessing procedure demonstrated through Load Case 1 on sample straight bridge with normal supports (work with Figure 3-36).

120 Proposed Modification to AASHTO Cross-Frame Analysis and Design develop than the 2D counterparts, these models offer solutions with improved accuracy, reliability, and repeatability. The traditional approach of connecting cross-frame members into a shared node along the web-to-flange juncture is also acceptable. â¢ 2D PEB models generally produce more accurate results than 2D grillage models because the concrete deck is explicitly considered. Although there are methods to consider the effective transverse stiffness of the deck, it is more accurate to represent the deck as a thin shell element that is rigidly connected to the girder to simulate composite action. â¢ Common postprocessing practices for 2D methods tend to produce erroneous results for top strut members and inaccurate results for diagonal members in X-frames, regardless of bridge type. A simple, conservative approach has been presented, but alternative methods can and should be explored by designers or programmers. â¢ If a designer elects to use a 2D modeling approach for these complex cases, it is recommended to incorporate simple modifications to the model (e.g., utilizing the Timoshenko approach for equivalent cross-frame beams and considering the transverse stiffness of the concrete for grillage-type models) and postprocessing (e.g., conservatively applying the full cross-frame shear force in the design of X-frame diagonals and considering the concrete deck in post- processing of grillage models similar to Figure 2-17). Still, even after implementing these improvements to 2D analyses, substantial error is likely in cross-frame force predictions, especially in bridges with significant horizontal curvature and/or support skews. â¢ Based on the conclusions above, it is recommended that any bridge satisfying the geometric limits established in Section 3.2.4 (i.e., bridge geometries for which live load force effects in cross-frames should be considered) should be evaluated using 3D modeling techniques. In general terms, this implies that heavily skewed and/or curved systems are best suited to be analyzed with 3D models. These findings, which are further summarized in Chapter 4, served as the basis for many of the proposed modifications to AASHTO LRFD. Refer to Appendix A for the proposed language and commentary. 3.5 Stability Study This section summarizes the results related to the Stability Study, which focused on Objective (e) of NCHRP Project 12-113. More specifically, the appropriate stability bracing strength and stiffness requirements for implementation into AASHTO LRFD are examined computationally. These goals were summarized by the following questions that were initially posed in Section 1.2: â¢ Can the AISC design guidelines for stability bracing be incorporated into AASHTO LRFD? â¢ Are special requirements needed for negative moment regions of continuous systems? â¢ How are these stability bracing requirements combined with other load conditions such as wind? These questions are systematically addressed throughout this section of the report. Before pre- senting the results of the bracing studies, Section 3.5.1 documents a computational study on the buckling behavior of girders in the composite condition. Section 3.5.2 presents the key results of the bracing strength study, and Section 3.5.3 outlines the key results of the bracing stiffness study as well as modifications to the AISC Specification expressions based on the major find- ings. Section 3.5.4 provides an overview of how to implement these cross-frame stability bracing forces in the context of a design. Lastly, Section 3.5.5 summarizes the outcomes of these studies. 3.5.1 Buckling in the Composite Condition As noted in Section 2.6.1, an elastic eigenvalue buckling analysis was carried out to examine the effects of continuous top flange restraint on the buckling behavior of bridge girders. In this

Findings and Applications 121 section, several sample results are presented that demonstrate the benefits of a composite con- crete deck on the buckling capacity of the steel girder section, even in the region around interior supports when shear studs may not be provided and the moments cause compression in the bottom flange. Figure 3-36 presents sample eigenvalue buckling results of a girder with an unbraced-length- to-depth ratio (Lb/d) of 5 and a well-stiffened web with various top flange restraints and moment gradients. For the three different top flange restraints considered (i.e., no additional restraint, continuous lateral restraint, and continuous lateral and torsional restraint), the critical buckling moment, Mcr, was obtained from the FEA analysis and is plotted. To demonstrate the magni- tude of these buckling loads compared to a limit state governed by cross-sectional yielding, Mcr is normalized by the moment at first yield of the section, My, where the yield strength is taken as 50 ksi. Along the horizontal axis, the results are organized by the corresponding straight-line moment gradient, which is represented as the ratio between applied end moments, MR/ML (applied moment at the right end of the unbraced beam segment/applied moment at the left end). For example, MR/ML = 1.0 represents a uniform-moment case in which the bottom flange is subjected to uniform compressive stresses. Sketches are provided above the figure to graphically illustrate these moment gradients. From Figure 3-36, the following observations can be made with regards to the effects of top flange restraint on girder buckling behavior: â¢ In general, the critical buckling moment increases as MR/ML decreases due to the benefits of moment gradient along an unbraced segment in terms of buckling capacity. The intent of the moment gradient factor, Cb, adopted by many design specifications is to account for these effects in design. â¢ In comparing the case with no additional top flange restraint and the case with continuous lateral restraint, it is evident that adding continuous lateral support increases the buckling capacity, particularly for cases with reverse-curvature bending (i.e., MR/ML = -1). These results are consistent with the modified Cb factor expressions in the AISC Commentary Section F1 (2016) that account for continuous lateral restraint. â¢ When adding continuous lateral and torsional restraint, the critical buckling moments consistently exceeded the yield moment of the section. For these cases, the buckled shapes of the beams were consistent with the web distortion mode presented in Figure 2-18. Even for the uniform-moment condition in which the restrained top flange is purely and uniformly 0.0 0.5 1.0 1.5 2.0 2.5 3.0 110â1 ML MR ML ML MR Key: Top Flange Restraint Continuous Lateral & Torsional Continuous Lateral None N or m al iz ed B uc kl in g M om en t, M cr / M y Applied End Moment Ratio, MR/ML Figure 3-36. Normalized critical buckling moment as a function of moment gradient and top flange restraint (Lb/d = 5; stiffened web).

122 Proposed Modification to AASHTO Cross-Frame Analysis and Design in tension, Mcr exceeded My by approximately 50%. Thus, these unbraced segments would yield prior to reaching an instability, which in turn mitigates the bracing demands on the cross-frames in these regions. To expand on Figure 3-36, Figure 3-37 evaluates two different conditions independently: an unbraced-length-to-depth ratio of 10 and an unstiffened web. The intent of this figure is to examine the effects of increased unbraced length and web slenderness on the LTB and web distortion buckling modes introduced schematically in Figure 2-18. Recall that the cases for which no top flange restraints are provided (blue line in Figure 3-37) and continuous lateral restraints are provided (red line) generally correspond to more conventional lateral-torsional buckled shapes. In contrast, the case simulating composite systems (i.e., continuous lateral and torsional restraint to the top flange) produced buckled shapes more consistent with web distortion modes. From Figure 3-37, the following observations can be made with regards to the effects of unbraced length and transverse stiffeners on LTB and web distortion buckling: â¢ As demonstrated by the left figure, increasing the unbraced length impacts the LTB modes (i.e., girders with no top flange restraint and girders with continuous lateral top flange restraint) much more significantly than the web distortion modes (i.e., girders with con- tinuous lateral and torsional top flange restraint). For instance, a 70% reduction in buckling capacity was consistently observed when the Lb/d ratio was doubled. These results verify the findings in Helwig and Yura (2015) that state that web distortion buckling is not as sensitive to the unbraced length. â¢ As demonstrated by the right figure, eliminating the web stiffeners drastically reduces the buckling capacity of the web distortion buckling modes. A nearly 70% decrease in Mcr on average was observed when compared to the finite element solutions for the girders with well-stiffened webs. For the LTB-related cases, this trend is less pronounced, particularly for the uniform-moment case where no shear force is present. For the cases with moment gradient, however, a slight reduction in Mcr is observed due to the effects of web distortion on the LTB response, which were intrinsically neglected in the development of the tradi- tional LTB solutions developed by Timoshenko and Gere (1961). Note that the solution derived by Timoshenko assumes a perfectly straight web (i.e., absent of any distortion) in its buckled shape. Studies have shown that the interaction between web distortion and LTB is most pronounced with smaller unbraced lengths, which are often controlled by yielding over buckling. -80% -60% -40% -20% 0% 20% Stiffened Web Unstiffened Web Key: Top Flange Restraint Continuous Lateral & Torsional Continuous Lateral None Lb /d = 10 Lb /d = 5 10-1 10-1 Applied End Moment Ratio, MR/ML R ed uc tio n in M cr Figure 3-37. Reduction in the critical buckling moment (compared to the results in Figure 3-36) for (left) a longer unbraced length and (right) an unstiffened web.

Findings and Applications 123 â¢ Despite the nearly 70% reduction demonstrated in the right figure for the continuous lateral and torsional restraint condition, it is important to note that these results are presented in relative terms. In absolute terms, a 70% reduction in Mcr for this restraint condition still produces buckling capacities that exceed the lateral-only or no top flange restraint conditions. With these factors in mind, it is evident that instability related to conventional LTB behavior is generally not critical once the composite deck has cured, and it is able to provide continuous lateral and torsional restraint to the girder top flange. In these instances, the critical unbraced segment in negative flexure is generally controlled by yielding of the cross-section, for which stability bracing demands in the cross-frames are greatly diminished. While including minimal shear studs in the negative moment region improves the twist restraint of the girders, the deck still provides considerable restraint even without shear con- nectors. The restraint comes in the form of âtipping restraint,â in which twist of the girder shifts the contact point between the deck and the girder out to the edge of the flange. Tipping restraint is very significant in most practical problems such that conventional LTB in the composite girder is not generally a critical mode in the finished bridge. As such, the subsequent sections focus primarily on bracing requirements for the noncomposite system, where stability is most critical. 3.5.2 Bracing Strength Study As outlined in Section 2.6.2, a finite element parametric study was developed and conducted to assess two different variations of the AISC cross-frame bracing strength requirements (i.e., Eq. 2.3 versus Eq. 2.4). A companion study was carried out on the stability brace strength requirements as documented in Liu and Helwig (2020). The results of these computational, second-order studies are briefly summarized herein. The relationship between internal girder moment and brace moment is specifically examined for a variety of bracing configurations, girder cross-sections, and loading conditions. For clarity, only select results are presented herein to highlight key trends that support the final suggestions and conclusions outlined in Chapter 4. For a full overview of the results, refer to Appendix F. In general, moment demands in the critical bracing elements were monitored as a function of increasing girder moments. That is to say, the out-of-plane girder deformations and cor- responding brace moments resulting from second-order effects on the initial imperfection were recorded during these incremental analyses. Thus, the figures presented in this section graph the brace moment, Mb, as a function of the maximum girder moment, M. As an example, Figure 3-38 illustrates this relationship for two different uniformly distributed loading cases. Uniform-moment conditions were also evaluated as part of this study. However, it was observed that uniformly distributed loads that introduce shear on the cross-section produced larger moment demands on the critical braces. As such, uniform-moment results are included in Appendix F for reference but are not included in the report. The graphs on the left represent top flange loading, whereas the graphs on the right represent mid-height loading. The figure also normalizes girder moments M to Mcr, which represents the theoretical Timoshenko and Gere (1961) solution for critical elastic buckling moment between brace points. Additionally, the brace moments, Mb, are normalized to the girder moments, M, such that a direct comparison to Eq. 2.3 and Eq. 2.4 can be made. In other words, the cor- responding Mb/M ratio as the applied moment, M, approaches Mcr (i.e., M/Mcr â 1.0) is the primary focus of the FEA results.

124 Proposed Modification to AASHTO Cross-Frame Analysis and Design Figure 3-38 also represents the results related to different girder cross-sections and bracing schemes. Girder Cross-sections 1, 2, and 3 are evaluated independently as denoted on the line graphs. In terms of bracing, note that only an odd number of intermediate braces (i.e., 1, 3, and 5 intermediate braces) between the end supports is presented, as these cases align a brace point at the location of maximum girder moment (i.e., at midspan). Cases with an even number of braces were also analyzed, but those conditions resulted in less severe brace forces. As noted in Section 2.6.2, it was observed that elastic materials (which were controlled by stability limit states and not yielding) always resulted in larger buckling-related deformations and thus larger brace moments compared to brace moments that occurred at the limit state of yielding in the girder. As such, Figure 3-38 and subsequent figures presented in this section are based on elastic material properties. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Cross-section 1 Cross-section 2 Cross-section 3 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Cross-section 1 Cross-section 2 Cross-section 3 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Cross-section 1 Cross-section 2 Cross-section 3 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Cross-section 1 Cross-section 2 Cross-section 3 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Cross-section 1 Cross-section 2 Cross-section 3 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Cross-section 1 Cross-section 2 Cross-section 3 n = 3, mid-height loading n = 5, mid-height loadingn = 5, top flange loading n = 3, top flange loading Mb / M M / M cr n = 1, mid-height loadingn = 1, top flange loading Figure 3-38. Brace moments as a function of applied bending moment (under transverse loading).

Findings and Applications 125 From Figure 3-38, the following observations can be made with regards to brace strength requirements and uniformly distributed loading: â¢ For these single-curvature bending cases, the brace moments ranged from 3% to 10% of the girder moment depending on the position of the load on the cross-section (load-height effects) and the specific cross-section. As such, the current 15th Edition AISC (2016) brace strength equation (Eq. 2.4) that recommends 2% of the design moment is unconservative (i.e., 3% versus 2% is a 100% error). â¢ Top flange loading is generally more critical than mid-height loading, and girders with fewer intermediate braces result in larger normalized brace moments. In other words, the relative magnitude of the brace forces at the location of the maximum girder moment decreases with an increased number of intermediate braces. â¢ For most analyses, slender sections (i.e., smaller bf/d ratios) resulted in larger brace forces; however, in some cases the stockiest section (Cross-section 3) resulted in the largest brace force as a percentage of the applied moment. The much wider flange for Cross-section 3 leads to a larger critical brace moment compared to Cross-sections 1 and 2, such that the applied moment M is relatively large. â¢ Some of the analyses did not reach convergence beyond 90%â98% of Mcr due to excessive girder deflections. However, the original study focused on the current AISC stiffness recom- mendation. As noted later, the recommended stiffness for the AASHTO LRFD provisions is higher, which will result in better control (and analysis convergence) of brace forces. As Figure 3-38 presented sample results from the parametric study, Table 3-12 summarizes the torsional bracing strength requirements for the entire data set. As noted above, different cross-sections, numbers of intermediate braces, and loading conditions were evaluated. The pre- vious figure demonstrated the curves tend to flatten as M/Mcr approaches 1.0, and convergence was not reached above 90%â98% of Mcr in some instances. In Table 3-12, the brace moments are presented for either M/Mcr = 1 or the largest load that convergence was achieved, which was typically capped at M/Mcr = 0.9. Although some of the brace moments tabulated are approximately 2% at M/Mcr = 0.9 (for analyses that did not converge), it should be emphasized that brace moments tend to increase dramatically as the applied moment increases from M/Mcr = 0.9 to M/Mcr = 1. The brace moment has been shown to grow between 50% and 100% beyond an applied moment of 90% of Mcr. As such, these cases, which may not appear critical, can be rather significant. Another point to note is that the results documented in this section were from a com- panion study to NCHRP Project 12-113 that is documented in Liu and Helwig (2020). A later companion study that is discussed in the next subsection modified the stiffness requirements. Loading Conditions n 1 3 5 bf / d 1/6 1/4 1/3 1/6 1/4 1/3 1/6 1/4 1/3 Uniform Moment Mb / M 6.9% 3.9% 3.7% 2.7% 2.2% 2.2% 2.0% 2.0% 3.0% M / Mcr 100% 100% 100% 90% 90% 90% 90% 90% 90% Top Flange Loading Mb / M 10.4% 7.3% 7.7% 2.5% 2.0% 2.0% 4.0% 3.4% 2.0% M / Mcr 100% 100% 90% 90% 90% 90% 100% 100% 90% Mid-Height Loading Mb / M 7.3% 4.0% 3.3% 2.5% 2.0% 2.0% 4.0% 3.4% 2.0% M / Mcr 100% 100% 100% 90% 90% 90% 100% 100% 90% Table 3-12. Summary of strength requirements of torsional bracing for different cross-sections and load conditions.

126 Proposed Modification to AASHTO Cross-Frame Analysis and Design The larger stiffness requirements produce much better agreement of the strength behavior with regards to Eq. 2.3. In general, the brace moments reported are consistently much larger than the 0.02M requirement given by the current AISC expression (Eq. 2.4). In many cases, the FEA results were more than 100% larger. Therefore, it is recommended that designers use the brace strength equation from the 14th Edition Specification with the modifications outlined in Section 3.5.3. Even with the potential implementation of Eq. 2.3 in AASHTO LRFD, it is evident from Table 3-12 that the required brace moments can still exceed the brace moment given by Eq. 2.3 (i.e., a minimum brace moment of 2.6% of Mcr). However, there are mitigating factors that lessen the demands reported in the table. As previously noted, stability brace moments are extremely sensitive to the shape and distribution of the initial imperfection in the girders. In most situations, though, the actual imperfection will not match the critical shape considered in this study, resulting in smaller brace forces. Also, in many situations, the brace sizes often utilized in design will exceed the stiffness required by Eq. 2.1, which will typically result in smaller stability-induced forces. Lastly, a companion study related to bracing stiffness require- ments that is documented in Section 3.5.3 proposes an increase of the stiffness requirements such that better agreement is achieved with Eq. 2.3. With that in mind, Eq. 2.3, as modified in the next subsection, is deemed suitable for implementation into AASHTO LRFD. Given that Table 3-12 focuses entirely on single-curvature bending cases, it is prudent to verify these conclusions for reverse-curvature bending. To demonstrate bracing demands in beams subjected to significant moment gradient, sample results are presented in Figure 3-39. 0.00 0.01 0.02 0.03 0.04 0.05 Imperfect BF Key: Imperfect TF Key: 0.00 0.20 0.40 0.60 0.80 1.00 1.20 M M Imperfect TF Key: 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 0.01 0.02 0.03 0.04 0.05 Imperfect TF Key: [Note: Mcr based on largest moment] Mb / M M / M cr Figure 3-39. Brace moments as a function of reverse-curvature bending (Cross-section 1; n = 5; mid-height loading) (TF = top flange; BF = bottom flange).

Findings and Applications 127 Beams with Cross-section 1 properties were subjected to the reverse-curvature moment gradient outlined in Section 2.6. Five intermediate braces were incorporated to maximize the negative moment magnitude at the first intermediate cross-frame line. For the sake of comparison, Figure 3-39 presents the results for four different loading and critical imperfection scenarios: â¢ Uniform moment with the critical imperfection along the top flange at midspan; â¢ Single-curvature, uniformly distributed mid-height loading with the critical imperfection along the top flange at midspan; â¢ Reverse-curvature, uniformly distributed mid-height loading with the critical imperfection along the top flange at midspan; and â¢ Reverse-curvature, uniformly distributed mid-height loading with the critical imperfection along the bottom flange at the first intermediate cross-frame line. Note that in the third and fourth scenarios listed, the loading conditions were identical. Only the location of the critical imperfection was varied to study its impact on the results. In Figure 3-38, only the midspan cross-frame results were provided, as bracing demands were maximized at this location for single-curvature bending (due to girder moment magni- tudes and girder displacements). For reverse-curvature bending, however, that behavior is less clear. As such, Figure 3-39 presents the bracing force demands on three different cross- frames in the five-brace scheme. It should be noted that the results are not symmetric (e.g., cross-frame lines 1 and 5 produce different results) due to the asymmetric nature of the initial imperfection. Thus, only the results related to the more critical half of the beam are presented. Figure 3-39 is otherwise constructed similarly to Figure 3-38 with one key exception. Along the horizontal and vertical axes, the maximum applied moment, M, is taken as the maximum negative moment at the support for consistency. Mcr is then based on the end segment, which was shown to buckle before the segments in the positive moment region by eigenvalue analysis. This is clearly denoted at the top of the figure. Sketches are also provided for each loading and imperfection condition, for reference. The following observations can be made from the results presented in Figure 3-39: â¢ In the uniform-moment case, it is evident that the midspan brace experiences the largest bracing demands. This validates the assumption used throughout the development of the single-curvature cases (e.g., Figure 3-38). Additionally, the midspan brace moment approaches 2% of the maximum applied moment at M/Mcr = 0.9 (i.e., analysis did not converge at 100% of Mcr). This result is consistent with Table 3-12. â¢ In the single-curvature uniformly distributed load case, it is also evident that the midspan brace experiences the largest bracing demands. Note that the critical values are also consistent with what is reported in Table 3-12. â¢ For the reverse-curvature bending cases, the critical cross-frame coincides with the location of the critical imperfection, as expected. In the bottom left plot (imperfection along the top flange at midspan), the bracing moment approaches 1% of the maximum applied moment (taken as the negative moment value at the support) as Mcr is reached. The other two braces that are presented pick up nearly no bracing forces, even at the elevated load magnitudes. In the bottom right plot (imperfection along the bottom flange at the first brace line), the critical bracing moment approaches 1.5% of the maximum applied moment. These cases still produce less force demands than the comparable single-curvature conditions. Note that the beams at the supports in these analyses were assumed to have zero initial twist. This is consistent with actual practice where the most precise control on geometry will generally be achieved at the support region.

128 Proposed Modification to AASHTO Cross-Frame Analysis and Design â¢ For these spot check conditions, it was generally observed that bracing demands are more critical for single-curvature bending conditions with less moment gradient (i.e., a smaller Cb factor) than reverse-curvature conditions with significant moment gradient. This is reflected in Eq. 2.3, in which the Cb factor of the unbraced segment of interest is in the denominator. Based on these results, it can be concluded that Eq. 2.3 is valid and generally conservative for cases with reverse-curvature bending in noncomposite systems. That is to say, the bracing strength requirements are not necessarily a function of positive or negative moment. For cases with variable unbraced segment lengths and moment gradient factors, the bracing moment is to be based on the brace point and unbraced segment that maximizes the Mr/CbLb component of the equation regardless of which flange is in compression. In doing so, this ensures the critical condition is covered. All cross-frame braces in the span would be subsequently designed for this worst-case condition. These three variables are segment-specific, whereas the other vari- ables and constants are consistent for the given span (e.g., the span length L is taken as the full span length under consideration). One note to the above discussion is the often overly conservative expression for Cb that is currently in AASHTO LRFD as of this writing. There is significant work currently underway related to the moment gradient factor in the AASHTO Specification that is likely to result in improved accuracy over the current Cb equation given in the specification. Note that Lb in Eq. 2.3 need not be taken as less than the maximum unbraced length permitted for the girder segment based upon the required flexural strength, Mr. This criterion accounts for scenarios in which a closer spacing of the braces is provided compared to the spacing required to develop the required strength. In essence, this provision lessens the demands on cross-frames for girders that would not buckle under the specified cross-frame layout (i.e., the girder would partially or fully yield prior to buckling). In these instances, out-of-plane deformations would be limited such that cross-frame bracing demands are relatively small. Hence, using the increased Lb term in the denominator reflects that behavior. As outlined in Section 3.5.1, bracing demands in the composite condition were deemed less critical than in the noncomposite condition, given the lateral and torsional restraint pro- vided by the concrete deck as well as the bridge bearings in the negative moment region. Thus, the procedure introduced above is only necessary for the construction condition, where gird- ers are far more susceptible to LTB. Note that, although cross-frame bracing requirements are not critical for the composite condition, web stiffeners must be adequately proportioned to prevent web distortion buckling effects in the negative moment regions per Yura (2001). 3.5.3 Bracing Stiffness Study Similar to the bracing strength study previously outlined, a finite element parametric study was developed and conducted to assess the applicability of the AISC bracing stiffness require- ment (Eq. 2.1) for implementation into AASHTO LRFD. Second-order analyses were performed on a variety of noncomposite bridge systems and bracing schemes to evaluate the torsional brace stiffness required to limit out-of-plane girder deformations at critical buckling loads. Thus, evaluating the relationship between internal girder moment and girder twist as a function of the brace stiffness is the primary focus of this section. For clarity, only select results are presented to highlight key trends that support the final suggestions and conclusions outlined in Chapter 4. For a full overview of the results, refer to Appendix F. Before presenting general results of the full data, a sample set is shown to illustrate key aspects of the analytical results. Figure 3-40 presents the twist of an imperfect girder cross-section at midspan (i.e., the critical location) as a function of the applied moment magnitude (also taken

Findings and Applications 129 at midspan) for the following select parameters: a single-span, twin-girder system (ng = 2) comprised of Cross-section 2, one intermediate brace (n = 1), and uniformly distributed load- ing on the top flange. As the applied load was increased, the out-of-plane girder deformations resulting from second-order effects on the initial imperfection were recorded. Along the x-axis, the cross-sectional twist of the girder at midspan, q, is normalized by the initial imperfection, qo, as defined in Eq. 2.3. The q /qo ratio equals 1.0 at no applied load given the assumed initial imperfection. Along the y-axis, the applied moment, M, is normalized by the critical buckling capacity between brace points, Mcr. The applied moment for a single girder is taken as the average internal moment in the multi-girder system. In some cases, the applied moment does not reach Mcr due to in-plane stiffness effects. Instead, it is limited to 97% to 99% Mcr. This response is plotted for twin-girder systems with various levels of cross-frame stiffness: 2, 2.5, 3, and 4 times the ideal stiffness, b i. Recall that the values for b i and Mcr are obtained from eigenvalue analyses. Many of these same plots were produced for different sets of parameters. The primary goal of the plots is to determine the cross-frame stiffness that limits the q /qo ratio to 2.0 at the critical buckling load (i.e., M/Mcr = 1.0). A ratio of 2.0 represents the case in which the induced deformations at the buckling load are equal to the initial imperfection, which was the basis for the AISC design procedures and Eq. 2.3. From Figure 3-40, it is apparent that providing twice the ideal stiffness results in q /qo = 2.7 for this particular example. The desired q /qo ratio of 2.0 is, however, achieved when providing three times the ideal stiffness with even tighter tolerances achieved for a stiffer brace (i.e., four times ideal stiffness). Additionally, a stiffer brace not only reduces the relative deformations in the girders but also reduces the force demands in the brace members themselves, which impacts the brace strength requirements discussed in Section 3.5.1. Similar observations were also made for different sets of bracing and girder parameters. Additional figures that investigate the effects of girder spacing, number of girders, load- height effects, and intermediate bracing schemes are presented and summarized in Appendix F. 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 1.5 2.0 2.5 3.0 Î¸ / Î¸o Î²b = Î²2i Î²b = 2.5 Î²i Î²b = 3 Î²i Î²b = 4 Î²i M / M cr Figure 3-40. Girder cross-sectional twist as a function of applied uniform moment for various brace stiffness values (n = 1, Cross-section 2, and top flange loading).

130 Proposed Modification to AASHTO Cross-Frame Analysis and Design For clarity, these results are not included herein. However, the following items provide a cursory overview of the major observations: â¢ The relative girder twist induced at critical buckling loads decreases as the brace stiffness increases. In all cases, assigning stiffer cross-frame elements mitigates girder deformations. â¢ In general, adding intermediate braces (regardless of the stiffness) decreases the relative cross-section twists. â¢ Increasing the girder spacing, particularly in twin-girder systems, not only improves the system buckling capacity but also decreases the stiffness demands for the cross-frame braces. Similarly, systems with increased girder redundancy generally resulted in an increase in the in-plane stiffness of the bridge system, which effectively led to smaller girder deformations at critical buckling loads. â¢ Top flange loading and the mid-height loading resulted in similar deformation behavior. The effect of the transverse loading locations on the stiffness requirement for torsional bracing was marginal. â¢ In general, assigning three times the ideal stiffness tends to result in girder cross-sectional twists closer to two times qo than assigning twice the ideal stiffness. Table 3-13 summarizes the torsional bracing stiffness requirements based on the results of this parametric study. The final girder twist (as a ratio of the initial imperfection) is tabulated for various cross-frame stiffness values, loading conditions, and girder cross-sections. Note that these results correspond to twin girders only, because this system has been shown to be most critical. For each q /qo ratio reported, the corresponding M/Mcr value is also shown. In most cases, the applied moment reached the critical buckling moment (i.e., M/Mcr = 100%). However, there were a few instances in which the critical buckling moment was not achieved; in those cases, the girder twist at the maximum girder moment recorded is presented instead. Table 3-13. Summary of stiffness requirements of torsional bracing for different cross-sections and load conditions. Loading Type Cross- frame Stiff., Î²b na 1 (Cb = 1.30) 3 (Cb = 1.06) 5 (Cb = 1.03) bf / d 1/6 1/4 1/3 1/6 1/4 1/3 1/6 1/4 1/3 Top Flange Loading 2Î²i Î¸ / Î¸o 3.15 2.72 2.89 3.59 2.74 3.15 2.7 2.55 3.29 M / Mcr 100% 100% 100% 98% 99% 99% 100% 100% 98% 2.5Î²i Î¸ / Î¸o 2.77 2.32 2.46 2.9 2.27 2.64 1.96 2.03 3.2 M / Mcr 100% 100% 100% 99% 100% 100% 100% 100% 99% 3Î²i Î¸ / Î¸o 2.56 2.11 2.22 2.39 1.94 2.21 1.71 1.86 3.14 M / Mcr 100% 100% 100% 100% 100% 100% 100% 100% 99% 4Î²i Î¸ / Î¸o 2.3 1.89 1.97 1.8 1.67 1.91 1.49 1.68 3.07 M / Mcr 100% 100% 100% 100% 100% 100% 100% 100% 99% Mid- Height Loading 2Î²i Î¸ / Î¸o 3.41 2.92 3.11 3.29 2.61 2.85 2.42 2.4 3.51 M / Mcr 100% 100% 100% 99% 100% 100% 100% 94% 99% 2.5Î²i Î¸ / Î¸o 2.98 2.47 2.63 2.41 2.03 2.35 1.9 2.05 3.4 M / Mcr 100% 100% 100% 100% 100% 100% 100% 100% 99% 3Î²i Î¸ / Î¸o 2.76 2.24 2.38 1.89 1.83 2.04 1.68 1.87 2.89 M / Mcr 100% 100% 100% 100% 100% 100% 100% 100% 100% 4Î²i Î¸ / Î¸o 2.46 2 2.13 1.59 1.59 1.84 1.49 1.6 2.61 M / Mcr 100% 100% 100% 100% 100% 100% 100% 100% 100% Notes: aGiven the uniformly distributed loading (wL2/8 type loading), the moment gradient factor, Cb, is a function of the bracing scheme.

Findings and Applications 131 In general, the table shows that providing twice the ideal stiffness (2b i) typically results in load-induced girder twists exceeding the assumed twist in Eq. 2.3 (i.e., q/qo > 2). This implies that the current assumption built into AISC Specifications (i.e., Eqs. 2.1 and 2.3) is invalid. However, the cross-sectional twist can be reduced to magnitudes more consistent with the assumed values by requiring braces to possess three times the ideal stiffness (3b i). Therefore, for implementation into AASHTO LRFD, it is advised that Eq. 2.1 be modified to inherently consider three times the ideal stiffness as a means to limit girder deformations. This is presented mathematically by the following expression: 3.6 3.3, 2 2 LM nEI C T req r yeff b =b f A cross-frame must then be designed and detailed such that its total stiffness, as computed by Eq. 2.2, exceeds the required stiffness from Eq. 3.3. Note that the constant in the equation has increased 50% (i.e., from 2.4 to 3.6) to account for these recommended changes. This modification not only impacts the brace stiffness requirements, but also the brace strength requirements. Recall from Section 2.6 that the brace strength equations are effectively the required brace stiffness multiplied by the assumed initial imperfection. With that in mind, it is also recommended to modify Eq. 2.3 to account for the 3b i assumption outlined above. This is reflected with the revised expression below: 0.036 3.4M M L nC L br T o r b b = =b q To determine the axial-force demands in the cross-frame members, the bracing moment computed by the expression above can be resolved as a force couple to the top and bottom nodes. This procedure is demonstrated in Appendices B and C. It is important to note that, as an alternative, designers are also permitted to perform a large-displacement analysis to estimate stability bracing force demands, provided that initial imperfections are considered (Helwig and Yura 2015). For curved systems though, the effect of initial imperfections has been shown to be less impactful given the curved geometry of the girders. 3.5.4 Consideration of Stability Forces in Design As introduced in Section 1.1.1, cross-frames serve many functions throughout the construc- tion and service life of a steel I-girder bridge. AASHTO LRFD Article 6.7.4.1 summarizes these functions in the context of minimum design and analysis requirements. Despite not currently providing any guidance on the topic, AASHTO LRFD includes stability bracing as one of the design considerations for cross-frames. While Sections 3.5.1 and 3.5.3 examined different strength and stiffness requirements to address this gap in design guidance, it is also important to understand how these stability design forces interact with other cross-frame functions and load cases. As such, this section provides an overview on how stability bracing strength requirements can be considered in conjunction with wind loads, construction loads, etc. As noted in the preceding sections, stability bracing requirements are most critical during construction when the system is noncomposite. The critical construction stage usually occurs during placement of the concrete deck since this results in the maximum moment that the noncomposite steel girder must resist. The concrete deck, once hardened and composite with the bridge girders, provides continuous restraint to the top flange and substantial bracing benefits to the bottom flange in negative moment regions. As such, the discussion herein is limited to force effects generated in cross-frame members during steel erection and deck

132 Proposed Modification to AASHTO Cross-Frame Analysis and Design construction. Given that cross-frames are generally not required as stability braces in the finished bridge, it is only necessary to evaluate stability-related force effects with other con- struction force effects, including dead loads (e.g., wet concrete, formwork, fit-up forces), construction loads (e.g., overhang construction, screed), and wind loading acting on the bridge fascia. Cross-frame force effects related to the finished structure, such as live loads and wearing surface dead loads, are then to be considered independently in their own set of load combinations as currently established in AASHTO LRFD. Because the stability-related force effects are critical during construction, at which point the bridge presumably exhibits fully elastic behavior, the principle of superposition is applicable for these specific load combinations. That is, the bracing strength forces computed from Eq. 3.4 or a large-displacement analysis can be directly added to the force effects from dead, construction, and wind loads derived from an elastic analysis. Before introducing each load case, it is important to note that construction sequencing must be considered when applying principles of superposition. Depending on the responsible party at the specific construction condition, either the engineer-of-record (EOR) or the erector or contractorâs engineer may need to evaluate stability bracing requirements at different stages of construction. The behavior is often dependent on the state of cross-frame installation and/or external bracing systems (i.e., guy cables, shoring, or holding cranes) at the specific erection or construction stage. Therefore, the additive nature of these force effects is contingent on consistent and concurrent bridge conditions. For example, stability bracing force effects related to an intermediate phase of girder erection should not be combined with dead load force effects related to wet concrete, as these two activities occur at different stages of construction. It is also imperative that the EOR or the erector or contractorâs engineer maintain the proper sign when summing the force effects of each load case (i.e., tension and compression). With that in mind, dead loads consider both (i) internal force effects due to gravity loads and (ii) âfit-upâ forces due to externally applied loads by the erector to assemble the structural steel during erection. The fit-up forces are largely dependent on which fit condition is selected by the engineer and constructed by the steel erector (i.e., no-load fit, steel dead load fit, or total dead load fit). Dead load force effects in cross-frames due to gravity loads on noncomposite systems are commonly estimated using 2D or 3D analysis models. Many commercial design software programs have also been designed and programmed to perform staged construction analysis to handle the various states of stress throughout the construction process. This topic is covered extensively in White et al. (2012). Guidance for estimating fit-up forces (when necessary), on the other hand, is provided in AASHTO LRFD Article 6.7.2, which is largely based on the work of White et al. (2015). Construction-related force effects in cross-frames, such as those induced by overhang deck construction, are often estimated with hand solutions as documented in Appendices B and C. Lastly, wind loads on a steel superstructure during construction can be estimated with refined analyses or the simplified procedures provided in AASHTO LRFD Article 4.6.2.7. With the individual load cases outlined, the final step is related to load factors. Because stability bracing force effects are a function of the factored girder moment (i.e., Mr in Eq. 3.4), no additional load factors are needed when combining these effects with the other construction- related force effects. If large-displacement analysis is used to estimate the bracing force demands, then the designer must give special consideration to how load factors are applied in analysis and design checks. The remainder of the load cases are subsequently factored based on the guidance and tables provided in AASHTO LRFD Articles 3.4.1 and 3.4.2. This entire process, as summarized above, is illustrated through two design examples in Appendices B and C.

Findings and Applications 133 3.5.5 Major Outcomes Based on the results of the Stability Study presented in the preceding subsections, there are several conclusions that can be drawn with respect to the bracing requirements for cross-frames and the LTB behavior of nonprismatic girders: â¢ Given that a composite deck provides continuous restraint to the top flange and substantial restraint to the bottom flange, these bracing provisions are only necessary to evaluate during the construction stages. â¢ In terms of implementation into AASHTO LRFD, it was determined that the general form of the torsional brace strength equations from the 14th Edition of AISC (2010) are more appropriate than the current 15th Edition (2016). The 15th Edition, which requires a design brace moment of 2% of the maximum girder moment, was shown to significantly under- predict the required brace moment. In contrast, the 14th Edition version of the strength requirements is a function of the bracing layout and moment gradient factor. The proposed expression, which is a modified version of the 14th Edition AISC equation, is provided as Eq. 3.4 above. â¢ For cases with reverse-curvature bending, it is recommended to conservatively base the required bracing moment on the unbraced segment that maximizes Mr/CbLb in Eq. 3.4. This ensures that for each span (with a corresponding length, L, and number of intermediate braces, n), the critical bracing moment is considered. â¢ Historically, design specifications have required engineers to provide torsional brace stiffness equivalent to twice the ideal stiffness. This rule of thumb was developed by Winter (1960) and was largely validated for columns rather than beams. Through FEA parametric studies, it was concluded that providing three times the ideal stiffness better limits girder deformations and cross-frame forces at critical buckling loads in beams. A proposed modification to the current AISC approach is therefore provided in Eq. 3.3. â¢ Bracing strength demands, as computed by Eq. 3.4 or large-displacement analyses, can be combined with other construction-related force effects via linear superposition. This procedure is demonstrated through two design examples in Appendices B and C. These findings, which are further summarized in Chapter 4, served as the basis for many of the proposed modifications to AASHTO LRFD. Refer to Appendix A for the proposed language and commentary.