Guidelines for the Load and Resistance Factor Design and Rating of Riveted and Bolted Gusset-Plate Connections for Steel Bridges (2013)

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120 CHAPTER 4. RESISTANCE FACTOR CALIBRATION For the suggested limit-state equations to be properly integrated into the BDS and MBE, they must be calibrated. Since the goal is not to define new load factors, this only requires calibrating new resistance factors. This section will demonstrate how the calibration task was completed, and the results obtained by it. CALIBRATION METHOD For this research a partial reliability analysis was performed, meaning only resistance factors were determined through the reliability analysis. This was done so as not to add more load combinations to the existing suite within the AASHTO design specifications, and use the existing load combinations to only derive new resistance factors. Resistance factors were derived via Monte Carlo analysis for each limit-state according to the philosophy outlined in NCHRP 20- 07(186).(54) In this analysis method, statistical parameters of bias and coefficient of variation (COV) are determined for the dead and live loads, as well as the limit-state resistance variables. A spreadsheet is used to create a list of randomly generated numbers that seed a statistical generation of loads and resistance values. Each line of the spreadsheet then contains a randomly generated set of dead and live loads, and associated resistance. A limit-state check is performed for each line of the spreadsheet where it is determined if the combination of loads exceeds the available resistance and if so, it is considered a failure. The goal of the calibrations is to derive -factors that can attain a desired reliability index, . The meaning of the reliability index is best illustrated with the assistance of Figure 67. The left- handed plot demonstrates the variability in applied load (Q) and available resistance (R) for a generic structural element. The right-handed plot of Figure 67 shows the probability density function for the function, R-Q. Any time the function R-Q is negative represents when the applied load is greater than the resistance and failure occurs. The overall distribution has a mean that is located away from the failure axis. The reliability index, , describes how many standard deviations the mean of the R-Q distribution is away from the failure axis. When reading in reliability textbooks and when calculating reliability with spreadsheet functions, it is important to note that the reliability index is reported as a negative number because the probability is related to a standard normal distribution with a mean of 0.0 and standard deviation of 1.0, and the area of interest is the left-hand tail probability which is on the negative side of the standard normal distribution. In LRFD and LRFR discussions, reliability is presented as a positive number as a matter of convenience. The notional probability of failure associated with reliability indices of 3.5 and 4.5 are 0.02326% and 0.00034%, respectively. All calibrations were performed at two reliability indices of 3.5 and 4.5. The reliability index of 3.5 is the current target level AASHTO uses for member-level reliability. (7, 54, 55) The reliability index of 4.5 is an alternative target for connection limit-states using the philosophy of the American Institute of Steel Construction (AISC), which will be described in a later section.

121 The Monte Carlo simulations were performed in a spreadsheet that conducted 600,000 random limit-state checks. The random analysis was conducted five times for a total of 3,000,000 simulations. The number of simulations came from the recommendation of the NCHRP 20- 07(186) report that at least 10 failures should be observed at the target reliability index. To produce 10 failures with a probability of failure 0.00034% at a reliability index of 4.5 requires 2,943,191 simulations. In addition, all simulations were performed using assumed dead-to-live load ratios with the AASHTO Strength I and IV load combinations. Prior AASHTO calibrations have used only the Strength I load combination, but the results of the Strength IV combination are presented because it may be a controlling combination for many long-span trusses. As will be described in subsequent sections, the -factors derived from the Strength I and IV load combinations converge around a dead-to-live load ratio of 6.0 and perhaps this should be observed considering that trusses are typically dominated by dead load. Additionally, the reliability indices of the derived -factors were analyzed to see how far they deviate from the assumed 2.5 when used with the MBE Operating load combination. Figure 67. Illustrative explanation for the reliability index, . Assumed Calibration Statistics The Monte Carlo analysis requires statistical data in the form of bias and coefficient of variation based on real data to provide an accurate estimation of the real variations expected in the limit- state variables. The bias is the ratio between the mean and the specified value, and the COV is the ratio of the standard deviation divided by the mean. In the previous chapter, much of the statistical data was reported as the professional factor, which is the same as the bias because the nominal mean value is 1.00. Statistical analysis of live and dead loads on a suite of trusses was not performed as part of this calibration process. Rather these load statistics were assumed based on judgment and from the data presented in the NCHRP 20-07(186) report.(54) The dead load was assumed to have a bias of 1.05 and a COV of 0.10 which comes from the data reported for cast-in-place elements because 0.000 0.050 0.100 0.150 0.200 50 100 150 200 250 300 P ro ba bl ity o f O cc ur an ce Load or Resistance (kips) Load (Q) Resistance (R) 0.000 0.025 0.050 0.075 -80 -40 0 40 80 120 160 200 P ro ba bl ity o f O cc ur an ce Resistance - Load (kips) R-Q P ro ba bi lit y of O cc ur re nc e P ro ba bi lit y of O cc ur re nc e

122 much of the weight in the truss is from a cast-in-place deck. The live load bias was assumed to be 1.15 with a COV of 0.12. The chosen bias was a gross average of all the reported live load biases in NCHRP 20-07(186). The COV of 0.12 is reported not to include the effects of impact, therefore the impact factor is not included in the Monte Carlo analysis, also as reported in NCHRP 20-07(186). For the resistance side of the limit-state, the overall resistance is the nominal resistance multiplied by professional (P), material (M), and fabrication (F) factors as shown in Equation 14. R = Rn * P * M * F (Eq. 14) The professional factor accounts for variations in real resistance versus calculated resistance from an equation. The material factor accounts for the variation in material properties such as yield and tensile strength. The fabrication factor takes into account the variations in the accuracy of elements from which a structure is created; this would include such things as variation in plate thickness and fabrication tolerances. The statistical data for professional factors was presented in Chapter 3 for all the various limit- states. As for the M-factor, the only variable considered in the many gusset plate limit-states is either yield or ultimate stress. The authors of the NCHRP 20-07(186) report referenced back to the NBS 577 (21) document for the values used in that calibration. NBS 577 reported various values of bias and COV values of hot-rolled steel products, but did mention that material from webs of hot-rolled product and structural plates had bias of 1.10 and COV of 0.11 for both yield and tensile strength. These values have been revised in more recent studies to account for changes in modern plate processing, but it was felt that since the majority of trusses being rated predate the publishing of the NBS 577 report, those values would be most appropriate. In addition, NBS 577 also recommended a bias of 1.11 and COV of 0.10 for shear yielding; however, this represents little difference over tensile yield properties and since the shear limit- state equations use tensile yield properties, tensile yield statistics were used. Finally, as also reported in NBS 577, the F-factor was assumed to have values of 1.00 for bias and 0.05 for COV. No additional work was performed as part of this research to quantify the dimensional variations of gusset plates and there is no better published data to query to refine the F-factor statistics. Table 32 outlines all the assumed bias and COVs for various parameters used in this research’s resistance factor calibration.

123 Table 32 Assumed Calibration Statistics Bias Factor () COV Dead Load 1.05 0.10 Live + Impact 1.15 0.12 Yield or Tensile Strength (Fy or Fu) 1.10 0.11 Fabrication Factor (F) 1.00 0.05 System Factors The Monte Carlo simulations did not account for any additional reductions due to system or condition factors that are currently considered optional in the Manual of Bridge Evaluation. Therefore, it is assumed that both of these factors have values of 1.00. Dead-to-Live Load Ratios for Trusses To make the calibration process easier without conducting a series of live load analyses through a suite of various trusses, it is often easier to assume a dead-to-live load ratio. The question then becomes what is the spectrum of dead-to-live load ratios for truss bridges. As described in Chapter 1, the research team received construction plan sets (and sometimes shop drawings) for 20 different trusses, from eight different states. The trusses utilized various configurations and were all built between the years of 1929 and 1990, the majority being built in the 1950’s, 60’s, and 70’s. More information on these trusses can be found in Appendix A. The dead and live loads reported within the construction plans were assumed to be correct and recorded in a spreadsheet for further statistical analysis. It is important to note that many different live load models were assumed in the design of this suite of trusses including H15, H20, and HS-20. This is important to note because the calibration task is being conducted using the load factors derived around the HL-93 live load model, which is more stringent than the three mentioned legacy live load models. Shown in Figure 68 is a histogram of all the dead-to-live load ratios (DL/LL) for all the data broken out into the three member types of chords, diagonals, and verticals. Chord members are fairly uniformly distributed between ratios of 0.5 to 6.0. Diagonals demonstrated a wide-banded Gaussian distribution between ratios of 0.5 and 6.0 peaking with an average around 3.0. Vertical members show a narrow-banded Gaussian distribution between ratios of 0.5 and 3.0 with a peak at 2.0. The high frequencies of members with a ratio of 10.0 are an aberration; these are meant to represent members that did not have any reported live load (i.e., the divide-by-zero anomaly was

124 represented by the dead-to-live ratio of 10.0). The majority (94%) of members with no live load were vertical members. Figure 69 shows a histogram of all the data that is independent of the member type which shows a skewed, wide-banded Gaussian distribution with an average of 2.3, and a standard deviation of 1.4 (neglecting the members with no live load). The histograms of dead-to-live load ratios are based on the older live load models (H15, H20, and HS-20); therefore, the spectrum of ratios would be expected to shift to the left given an HL- 93 live load model. Given this data, it was decided to investigate the calibration using dead-to- live load ratios of 0.5, 1.0, 2.0, 3.0, 4.0, 6.0, and 8.0. The original LRFD calibration of AASHTO used actual bridge designs; however, NCHRP 20-07(186) reported the DL/LL ratios of the girders varied from 0.3 to 4.5. Therefore, the DL/LL of trusses is slightly higher than girders, as expected. Figure 68. Dead-to-live load ratios for three member types. 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Fr eq ue nc y DL/LL Chords Verticals Diagonals

125 Figure 69. Dead-to-live load ratios for all members. EXISTING FHWA GUIDE LEVEL OF RELIABILITY The experimental and analytical failures described in Chapter 3 were analyzed using the FHWA Guide document to determine the existing level of reliability of gusseted connections. This exercise helps put prospective on the target level of reliability for the proposed new approach and its associated calibration of -factors. The existing levels of reliability were only evaluated for the limit-states of buckling, full plane shear yielding, and block shear. Since only three analytical connections failed in a tension member mode, there were not enough data to determine reliability statistics for tensile failure modes. The following section will describe the results for each failure mode. Buckling Failure There were 124 total observed pure buckling failures, neglecting the buckling failures of models with simulated corrosion, multi-layered plates, and edge stiffeners. To assess the reliability associated with the FHWA Guide, the models that failed must be compared to the predictions using the resistance equations from the FHWA Guide to define the professional factors for use in the Monte Carlo simulations. Examples of the professional factor variation using the FHWA Guide were shown in Figure 49, where K-factors of 1.2 and 0.50 are used, respectively. 0 20 40 60 80 100 120 140 160 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Fr eq ue nc y DL/LL

126 The data sets had to be parsed into smaller subsets of avg values to properly account for the linear trend in the data (more so when K=1.2). In each case, a running average was used to define the professional factor bias and COV at discrete values of avg. Then for each set of statistics attained at each value of avg a Monte Carlo simulation was run using the FHWA Guide limit-state equation and the existing =0.90. The result of all these simulations is shown in Figure 70 for the two K-factors (1.2 and 0.50) used in the evaluation. What the plots in Figure 70 show is that the existing FHWA Guide limit-state equation produces a wide array of reliability indices varying from 0.75 to 5.0 depending on the value of avg. The extreme linear variation shown in the reliability index is directly attributed to the widely ranging professional factors that were shown in Figure 49. With low values of avg the professional factors are typically less than 1.0 leading to a low reliability index. However, when avg >1.0 the professional factors are much greater than 1.0 leading to favorable and conservative reliability indices. Figure 70. Existing level of reliability variation using c=0.90 with respect to avg and DL/LL ratio. (Left) K=1.2. (Right) K=0.50. Shear Failure The data presented in Chapter 3 showed that full plane shear yielding failures had average professional factors of 1.017 and a COV of 0.069 when =1.00. When =0.74, the average increases to 1.374 with no change in the COV. These two professional factor statistics were used in the Monte Carlo simulations to determine the reliability associated with the existing FHWA Guide resistance equation for shear yielding (using =0.95). The results from those simulations are shown in Table 33. When =0.74, the reliability varies from 4.22 to 5 depending on the DL/LL ratio. Likewise, when =1.00, the reliability index varies from 2.27 to 3.19. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 R el ia bi lit y In de x =12(1.2Lavg/t)(Fy/E) DL/LL=0.5 DL/LL=2 DL/LL=6 0 1 2 3 4 5 0 0.5 1 1.5 R el ia bi lit y In de x =12(0.5Lavg/t)(Fy/E) DL/LL=0.5 DL/LL=2 DL/LL=6

127 Table 33 Shear Yielding Reliability Using Existing FHWA Guide Reliability Index,  DL/LL 0.5 1 2 3 4 6 8 =0.74 =0.95 =1.375, COV=0.064) >5.00 4.97 4.55 4.75 4.43 4.27 4.22 =1.00 =0.95 =1.017, COV=0.064) 3.19 3.01 2.75 2.58 2.47 2.34 2.27 Block Shear The FHWA Guide adopted the block shear equations out of the BDS 4th Edition.(57) However, the BDS 5th Edition (7) changed the block shear equations to be unified with the 13th Edition of the AISC Steel Construction Manual.(56) Therefore, the existing reliability will be calculated for the new block shear equations from the 5th Edition as this reflects the state-of-the-art. As described in Chapter 3, the professional factor average and coefficients of variation for these block shear equations are 1.180 and 0.060, respectively. These statistical values were used in the Monte Carlo simulations to show the existing level of reliability with =0.80. The variation of reliability index versus the DL/LL ratio is shown in Table 34. The existing level of reliability is between 4.37 and 5 depending on the DL/LL ratio. Table 34 Block Shear Reliability Using Existing FHWA Guide Reliability Index,  DL/LL 0.5 1 2 3 4 6 8 =0.80 =1.180, COV=0.060) >5.00 4.97 5 4.97 4.48 4.50 4.37 Recommended Target Reliability Index The existing level of reliability of the FHWA Guide for shear, buckling, and block shear is variable between 1.0 and 5.0 depending on the limit-state and the DL/LL ratio. Because of the wide range in the level of reliability, the FHWA Guide is not a precise indicator of the target reliability. Specifying reliability indices higher than 4.5 would be unnecessarily conservative. It was beyond the scope of this research to perform a rigorous analysis of the existing level of reliability for truss structures for all applicable limit-states. Additional research may be desired to better grasp the reliability of truss systems and how it would interact with target reliability indices for use in -factor calibrations.

128 The BDS has been calibrated around a member-level reliability index of 3.5 using the Strength I load combination. This is also the same as the MBE Inventory Level, but the Operating Level has an implied reliability of 2.5. It would be wise to consider using a higher reliability index for connection-level limit-states as is done in AISC since AASHTO already provides a higher level of reliability for block shear and fastener limit-states (as they are adopted from AISC). AISC uses a member-level reliability of 2.8 for members and 4.0 for connection elements. Therefore, since AASHTO uses a member reliability of 3.5, bridge connection limit-states could have a reliability of 4.5. However, the higher level of reliability was mainly imposed on the connectors (bolts and welds), not for the other connection-specific failure modes (except for block shear). Until further research can justify a different target reliability index, it is recommended for design that a target reliability index of 4.5 be selected. This decision is fairly inconsequential for design because it is easy to add plate thickness before a bridge is fabricated. However, in load rating, the plate thickness is predetermined and expensive retrofitting or load posting must be performed if the rating factors are less than favorable. The NCHRP panel conducted some spot checks using the design criteria for rating and determined that much of the existing inventory would not rate. While it may appear that a higher reliability index for rating is justified, the economics of posting every truss are not. Economics was not the only consideration in the decision; accepting a lower reliability index also meant acknowledging the long performance history of gusset plates with no problems (except I-35W) and the nebulous nature of various design specifications that have higher levels of reliability for “connections,” though only enforced through fastener and weld limit-states. Therefore, a lower level of reliability for rating was justified. It is recommended that gusset plate limit-states in the MBE be calibrated at a reliability index of 3.5. This is also predicated on the notion that AASHTO mandate the use of the system factors for at least gusset plates, as currently system factors are considered optional. SAMPLE -FACTOR EXPLANATION Throughout the remainder of this chapter, plots such as those shown in Figure 71 will show the variation of the “calibrated” -factor versus the DL/LL ratio. The calibrations are performed using the AASHTO Strength I and IV load combinations. For the Strength IV load combination, there is a general increase in the required -factor as the DL/LL increases. It is difficult to describe exactly why this trend occurs because it is interrelated with the load factors and various bias and COV values. However, it must be recognized that in the Monte Carlo simulation the trial “design” (whether it is cross-sectional area, moment of inertia, etc.) is calculated based on the factored load combination using nominal values; however, the nominal loads are randomized based on their statistical properties to obtain real values. In particular for the Strength IV load combination, it may seem that there should be no variation in the -factor with the DL/LL ratio as that load combination only used dead load. But the factored load combination is only used to define a trial “design” value, yet the dead and live loads are still randomized in the simulation, hence why there is a variation in  with DL/LL. Likewise, using the Strength I load combination, the calibrated -factor must decrease as the DL/LL ratio increases. This is a well-recognized trend that was noted in the first LRFD calibration of AASHTO. It has to do with the dead load

129 factor being smaller than the live load factor, forcing the trial design to become “lighter” as the live load diminishes and the simulation is affected more by the statistical distribution of the dead load rather than the resistance. It is essential that users of this report are careful not to misinterpret why the -factors are being reported for both the Strength I and IV load combinations. The plots to be shown throughout this chapter are not meant to represent a different set of -factors to use with each load combination (i.e., very small -factors should not be used with the Strength IV load combination, especially at small DL/LL ratios). The reason both curves are shown is that for all the limit-states the two curves tend to intersect at DL/LL ratio between 6.0 and 8.0. This notion can be used to justify reporting -factors at a DL/LL ratio of 6.0 because it represents the minimized -factor that works for both load combinations. The controlling -factor is illustrated as the black line in Figure 72 which in this instance is controlled by Strength I up to DL/LL=6 and controlled by Strength IV at higher DL/LL ratios. Figure 71. Sample -factor variation with DL/LL ratio. SHEAR CALIBRATION The professional factor for all the specimens and models that failed in shear had an average of 1.017 with a COV of 0.0649 just considering 0.58FyAg as the resistance equation. The statistics for plates thinner than 3/8 of an inch were the same. The result of the Monte Carlo calibration is 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -f ac to r DL/LL Ratio Strength I Strength IV Design Curve

130 shown in Figure 72 that is presented in terms of the required -factor for DL/LL ratios varying from 0.5 to 6.0. Based only on the Strength I load combination, the combined -factor would have to range from 0.70 to 0.90 for both design and rating depending on the reliability index and DL/LL ratio. Shear Rupture No shear rupture was observed in the experimental testing and the analytical models did not have the fidelity to capture such a limit-state. Therefore, no changes to the shear rupture resistance equation used in the FHWA Guide can be derived from this research. Figure 72. Plot of required -factor to attain target reliability at various DL/LL ratios. BUCKLING CALIBRATION In Chapter 3, data were presented for all 124 specimens and models that failed by buckling; this is summarized in Tables 14 through 18. This calibration is for the new method of evaluation that uses a fixed K-value of 0.5 and Lmid. Of these 124 failures, only 62 were controlled by the Whitmore buckling criterion; the remaining were controlled by partial plane shear yielding. In looking at the plot of professional factors in Figure 53, there is a general increase in the professional factor as the mid value also increases. The column curve is controlled by two functions demarcated at mid=2.25. The critical case is for mid<2.25 because those are the professional factors nearest to 1.0. The professional factors of buckling failures predicted by 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d  -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5)

131 Whitmore and mid<2.25 had an average of 1.24 and COV of 0.127. The derived -factor would be conservative for cases where mid>2.25 as the professional factors are much higher than 1.0. A plot of the required -factors for rating and design versus the DL/LL ratio is shown in Figure 73 and 74 for both load combinations and reliability indices. The required -factor was found to vary from 0.96 to 1.00 depending on the target reliability index and DL/LL ratio. Since the buckling resistance relies on both the Whitmore buckling and partial plane shear yielding criteria, the partial plane shear yield also has to be calibrated. A plot of the required - factors for rating and design versus the DL/LL ratio is shown in Figures 75 and 76 for both load combinations and reliability indices. The required -factor was found to vary between 0.60 and 0.95 depending on the target reliability index and DL/LL ratio. Since the partial plane shear requirement is a shear-initiated limit-state, this -factor should be unified with the full plane shear yielding  term calibrated from the full plane shear yielding presented in the prior section. The two calibrations are similar; perhaps a  should be carefully selected to satisfy both full and partial plane check such that special provisions do not have to be written for partial shear plane checks. Figure 73. Required -factor for Whitmore compression resistance considering all data, to be used in rating. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5)

132 Figure 74. Required -factor for Whitmore compression resistance neglecting plates thinner than 0.375 inches for use in design. Figure 75. Required -factor for partial shear plane yielding considering all data, to be used in rating. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d  -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5)

133 Figure 76. Required -factor for partial shear plane yielding neglecting plates thinner than 0.375 inches for use in design. TENSION CALIBRATION Unfortunately, throughout the NCHRP 12-84 project, no tension failures were observed in the experimental program. In addition, there were only three models in the parametric study that failed around a tension member. With so little data, no firm conclusions could be drawn to refine or refute the existing FHWA Guide and no changes are recommended for the Whitmore gross and net section resistance equations around tension members. However, the available literature does contain relevant data for a calibration of block shear and the professional factor data had an average of 1.18 and COV of 0.060. Figure 77 shows the required -factor versus the DL/LL ratio. The required resistance factor was found to vary from 0.80 to 1.00 depending on the target reliability index and the DL/LL ratio. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d  -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5)

134 Figure 77. Plot of required -factor for block shear. CHORD SPLICES The procedure for evaluating chord splices presented in Chapter 3 produced two sets of professional factor statistics considering all the data (rating), and neglecting data from plates thinner than 3/8 inch thick (design). These statistics were derived from 33 failures of both tension and compression chord splice failures noted in the analytical studies. Figures 78 and 79 present the results of the Monte Carlo simulations in terms of the required φ-factor for various DL/LL ratios, for both the rating and design specifications respectively. The required φ-factor ranges from 0.60 to 0.92 depending on the reliability index and DL/LL ratio. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d  bs DL/LL Ratio Strength I (β = -4.5) Strength I (β= -3.5) Strength IV (β = -4.5) Strength IV (β = -3.5)

135 Figure 78. Required -factor for chord splices considering all data, for use in rating. Figure 79. Required -factor for chord splices neglecting plates thinner than 0.375 inches, for use in design. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5)

136 RIVET SHEAR CALIBRATION In Chapter 3, Equation 11 defines a generalized fastener resistance equation. The assumed values of the shear-to-tensile ratio and rivet stock tensile strength were presented along with the average, bias, and COV from test data found in the available literature. The rivets were segregated into three different strength levels and a -factor calibration needed to be performed for each of them. It was also shown that the connection length reduction factor also has a role and it is assumed that the nominal reduction will be 0.90. Provided both of the Tide inequalities are satisfied, then the bias and COV of the connection length reduction factor is 1.035 and 0.077, respectively. However, the statistics for the connection length effect are different if both Tide criteria fail where the bias and COV are 1.002 and 0.103, respectively. The statistics assuming both Tide criteria fail were used in the rivet calibration because those are less favorable from a calibration standpoint and represent the lower bound. Therefore, resistance factors will be conservative for connections that meet one or both of the Tide criteria. Figures 80 through 82 outline the required -factor for the three rivet strength levels. As expected, there is not one all-encompassing -factor to describe all rivets without unnecessarily penalizing one of the strength levels. The required -factor ranged from 0.50 to 0.80 depending on grade, target reliability index, and DL/LL ratio. Figure 80. Required -factor for rivets of unknown origin. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -f ac to r DL/LL Ratio Strength I (β= 4.5) Strength IV (β= 4.5) Strength I (β= 3.5) Strength IV (β= 3.5)

137 Figure 81. Required -factor for A141 or A502 Grade 1 rivets. Figure 82. Required -factor for A195 or A502 Grade 2 rivets. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -f ac to r DL/LL Ratio Strength I (β= 4.5) Strength IV (β= 4.5) Strength I (β= 3.5) Strength IV (β= 3.5) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -f ac to r DL/LL Ratio Strength I (β= 4.5) Strength IV (β= 4.5) Strength I (β= 3.5) Strength IV (β= 3.5)

138 ANALYSIS FACTOR CALIBRATION The various resistance equations have been calibrated based on experimental and analytical failure results from a variety of connection geometries. As such, “one size fits all” resistance equations were defined and calibrated and they may produce conservative results for some connection geometries. Therefore, AASHTO may wish to consider allowing for higher level analysis to determine the resistance of some connections in lieu of using the simplified resistance equations. This would require detailed finite element modeling of connections using the modeling philosophy in Chapter 2, and could be helpful in determining more accurate shear or buckling resistance of some connections which have unfavorable ratings using the simplified resistance equations. When performing an LRFD analysis of a refined connection model, the variability of the loads, material factors, and fabrication factors are not taken into account. Therefore, the failure loads attained from the model must be factored to account for these unknowns that cannot be explicitly accounted for in the model. A special resistance factor for use with simulation analysis must be derived and to do this a Monte Carlo analysis is run considering a professional factor average of 1.00 and COV of 0.00. This assumes the model failure prediction is perfect and the analysis factor only accounts for the uncertainty in the load, material, and fabrication models. The results of the Monte Carlo simulations are shown in Figure 83 for various DL/LL factors. The required analysis factor was found to vary from 0.70 to 0.95 depending on the target reliability and DL/LL ratio. Figure 83. Plot of required analysis factor. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 R eq ui re d -F ac to r DL/LL Ratio Strength I (β= -4.5) Strength I (β= -3.5) Strength IV (β= -4.5) Strength IV (β= -3.5)

139 SUMMARY OF RESISTANCE FACTORS This project was able to provide six resistance equations calibrated to a LRFD/LRFR philosophy. An additional calibrated analysis factor is also provided to demonstrate how the variability associated with refined analysis may be integrated into a LRFD/LRFR philosophy. It was beyond the scope of this project to determine the target reliability index associated with gusset plates, primarily for lack of data. As such, the calibrations were performed at two levels of reliability of 3.5 and 4.5, where it is optimal that design use a reliability index of 4.5 and rating use a reliability of 3.5. Table 35 outlines the associated -factors for each resistance equation, at both reliability indices and four different DL/LL ratios. The -factors presented in the table are exact to attain the desired reliability index, though for implementation in design and rating specifications they should be rounded to the nearest 0.05 as shown in Table 36. Since the -factor has a dependency on the DL/LL ratio, a decision had to be made about where to select the -factor. For the purposes of design, the -factors were selected at a DL/LL ratio of 6.0. This would provide -factors that are always conservative in all situations despite the span length of the truss, because in design it may be difficult to estimate the DL/LL ratio. However, for load rating the loads are known and a variable -factor may be used to account for the variation with the DL/LL ratio. For rating, the -factors were selected at DL/LL ratio of 1.0. The following section will describe the reasons behind this. Reliability at Manual for Bridge Evaluation (MBE) Operating Level The National Bridge Inspection Standards require bridge ratings be reported at the Strength I Inventory and Operating Levels. In LRFR the only differences between the two are the live load factors which should provide a reliability index of 3.5 at the Inventory level, and 2.5 at the Operating level. Since the -factor calibrations were performed using the BDS Strength I (and hence to MBE Inventory Level too), those -factors should be checked to ensure they provide a 1.0 decrease in the reliability index with the 1.35 MBE Operating level live load factor. The results of the simulations that performed this are reported in Table 37. These simulations used the exact -factors derived at a reliability index of 3.5 but assessed the reliability using the MBE Operating load combination. It would be expected that the reliability index would be approximately 2.5 to reflect the 1.0 decrease in reliability using the lower live load factor. In fact, this assumption of the MBE is extremely sensitive to the DL/LL ratio. It can be seen that there is little change in reliability between the Inventory and Operating levels at the high DL/LL ratio of 6.0. However, for many of the limit-states, a DL/LL ratio of 0.5 produces reliabilities less than expected. It would appear that to maintain consistency of the MBE assumption of Inventory and Operating level reliability, -factors should be selected at a DL/LL of approximately 1.0. However, to ensure long-span trusses are also properly rated, an additional reduction factor should be prescribed to account for the decrease in the -factor for trusses with DL/LL ratios

140 greater than 1.0. By inspection of -factors in Table 35 it can be seen that for all the limit-states, there is approximately a 0.90 reduction of the resistance factor at DL/LL=1.0 to the resistance factor at DL/LL=6.0. Inspection of all the graphs shown in Figures 72 through 83 show that a linear approximation would be prudent in this region and it is recommended that for rating a variable -factor reduction be applied when the DL/LL is between 1.0 and 6.0. Again, it is difficult to explain why there is sensitivity to the DL/LL ratio, but the reason is the same as the explanation of why -factors vary with the DL/LL ratio. As the DL/LL ratio increases, the trial design becomes lighter due to the diminishing influence of the live load factor. Therefore, the variability in the loading decreases since the live load has a higher bias and COV, hence a lower change in the reliability index. -Factor for Shear Yielding Limit-State In an earlier discussion on shear yielding, the resistance factor was coupled with the -factor. One option was to use the use the existing y=1.00 that AASHTO uses for shear and use the - factor to provide the required reliability. However, this leads to the conundrum in the MBE that there would then be a variable definition of the -factor to account for the DL/LL variation, which could distort the meaning of . Therefore, a fixed value of the -factor should be selected and new gusset plate shear yielding -factors should be defined for both design and rating. On average  was 1.02 and it would be logical to select =1.00 as the factor, though specifying a factor to be 1.00 serves no purpose, whereas the point of specifying an -factor is to make a distinction that shear yielding in gusset plates is different than in beams. As such,  was selected to be 0.88 because this would lead to gusset shear yielding resistance factors in design and rating that would be nearly rounded to the nearest 0.05. For instance, in Table 36 the combined design and rating shear yielding resistance factors () were 0.70 and 0.85 respectively. Considering the fixed value of =0.88 leads to the shear yielding resistance factors of 0.80 and 1.00 respectively for design and rating. Rivets Since the current 2011 MBE Interim specifying rivet strength lists the factored shear strengths, a similar approach is recommended so each rivet type can be assigned its own -factor. Table 38 lists the factored shear resistance (Vn) for each of the rivet strengths at a reliability level of 4.5 and a DL/LL ratio of 1.0. It was described earlier in this chapter that other design codes specify higher reliability for connection limit-states, though they do this by imposing higher reliability on the connectors, such as the rivets. For this reason, the recommended factored shear strength of rivets is reported at the higher reliability of 4.5 and this aligns with the current calibration of high-strength bolts.(58) Two factored shear strength values are presented considering the two different connection length factors discussed in Chapter 3. Also shown in Table 38 are the factored rivet shear values from the FHWA Guide, and those from the 2011 MBE Interims. Both use a different approach to account for the connection length

141 effect than is done with the Tide approach presented in the previous chapter. Therefore, the comparison of all the factored shear strengths in this table must be made carefully. For instance, the new values passing the Tide criteria are best compared to those in the 2011 MBE Interims with no length reduction. Likewise, the new value failing both Tide criteria is best compared with the 2011 MBE Interims with a full length reduction. What the results in the table show are that the current approach used in the 2011 MBE Interims is adequate. The new calibrated values are lower for unknown rivets, about the same for the A141/A502 Gr. 1 rivets, and much stronger for the A195/A502 Gr.2 rivets. However, this statement must be qualified. The data used to define the strength statistics for unknown rivets may be biased as that came from research reports from the same institution over a short period of time and it was unclear how many lots of rivets they encompassed. Likewise, most of the data for A195/A502 Gr. 2 rivets were from one source and do not give an indication of the true variation possible. The strength data for the A141/A502 Gr. 1 rivets were taken from a broad variety of sources and encompassed a large period of time; therefore, that is probably the most reliable data to compare to. Since the new approach and that published in the 2011 MBE Interims is nearly the same for the A141/A502 Gr.1 rivets, and the change to the Tide criteria would be unprecedented, no change of rivets shear strengths could be recommended. If further strength data could be uncovered for unknown and A195/A502 Gr. 2 rivets to buttress the statistical variations, then this topic could be revisited.

142 Table 35 Summary of Exact Resistance Factors Reliability Index = 4.5 = 3.5 DL/LL ratio 0.5 1.0 2.0 6.0 0.5 1.0 2.0 6.0 Full Plane Shear Yielding () 0.76 0.76 0.73 0.68 0.89 0.87 0.84 0.79 Bu ck lin g Whitmore Buckling (all data considered) 0.80 0.78 0.76 0.70 0.98 0.95 0.91 0.86 Whitmore Buckling (neglecting plates thinner than 0.375 inch) 0.84 0.82 0.79 0.75 1.02 0.99 0.95 0.89 Partial Plane Shear (all data considered) 0.70 0.68 0.65 0.61 0.86 0.83 0.80 0.75 Partial Plane Shear (neglecting plates thinner than 0.375 inch) 0.79 0.76 0.74 0.70 0.95 0.93 0.89 0.83 Block Shear 0.92 0.89 0.86 0.80 1.02 0.98 0.93 Chord Splice (all data considered) 0.71 0.68 0.66 0.63 0.87 0.85 0.82 0.77 Chord Splice (neglecting plates thinner than 0.375 inch) 0.74 0.73 0.71 0.65 0.92 0.90 0.86 0.80 R iv et S he ar Unknown Origin 0.56 0.56 0.54 0.49 0.70 0.68 0.65 0.61 A141/A502 Gr. 1 0.70 0.67 0.65 0.61 0.81 0.79 0.76 0.71 A195/A502 Gr. 2 0.79 0.78 0.76 0.70 0.92 0.90 0.87 0.81 Simulation Analysis 0.81 0.80 0.77 0.71 0.93 0.91 0.87 0.82

143 Table 36 Summary of Resistance Factors Rounded to Nearest 0.05. Reliability Index = 4.5 = 3.5 DL/LL ratio 0.5 1.0 2.0 6.0 0.5 1.0 2.0 6.0 Full Plane Shear Yielding () 0.75 0.75 0.75 0.70 0.90 0.85 0.85 0.80 Bu ck lin g Whitmore Buckling (all data considered) 1.00 0.95 0.90 0.85 Whitmore Buckling (neglecting plates thinner than 0.375 inch) 0.85 0.80 0.80 0.75 Partial Plane Shear (all data considered) 0.85 0.85 0.80 0.75 Partial Plane Shear (neglecting plates thinner than 0.375 inch) 0.80 0.75 0.75 0.70 Block Shear 0.90 0.90 0.85 0.80 1.00 1.00 0.95 Chord Splice (all data considered) 0.85 0.85 0.80 0.75 Chord Splice (neglecting plates thinner than 0.375 inch) 0.75 0.75 0.70 0.65 R iv et S he ar Unknown Origin 0.55 0.55 0.55 0.50 0.70 0.70 0.65 0.60 A141/A502 Gr. 1 0.70 0.65 0.65 0.60 0.80 0.80 0.75 0.70 A195/A502 Gr. 2 0.80 0.80 0.75 0.70 0.90 0.90 0.85 0.80 Simulation Analysis 0.80 0.80 0.75 0.70 0.95 0.90 0.85 0.80

144 Table 37 Associated Reliability Indices for MBE Operating Level Rating with -Factors Derived at a Reliability Index of 3.5 DL/LL ratio 0.5 2.0 6.0 Full Plane Shear Yielding 2.39 2.86 3.19 Whitmore Buckling 3.19 3.65 3.90 Partial Plane Shear Yielding 2.58 3.01 3.28 Block Shear 2.83 3.18 Chord Splice 2.67 3.01 3.27 Unknown Rivet Shear 2.60 3.03 3.30 A141/A502 Gr. 1 Rivet Shear 2.44 2.91 3.27 A195/A502 Gr. 2 Rivet Shear 2.31 2.79 3.21 Analysis Factor 2.14 2.67 3.00 Expected Value 2.5

145 Table 38 Recommended LRFR Rivet Shear Strengths Reliability Index Rivet Type Unknown Origin (Fu=50 ksi) A141 or A502 Gr. 1 (Fu=60 ksi) A195 or A502 Gr. 2 (Fu=80 ksi) = 4.5 (DL/LL=1.0) -factor a 0.56 0.67 0.78 (fails both criteria)Vn (ksi) b 16 23 37 (otherwise) Vn (ksi) c 21 30 47 FHWA Guide (ksi) 18 27 32 2011 MBE Interims (ksi) (with full length reduction) 20 24 32 2011 MBE Interims (ksi) (no length reduction) 27 32 43 a – Should be reduced for DL/LL ratios greater than 1.0. The additional reduction decreases linearly from 1.00 to 0.90 as the DL/LL ratio changes from 1.0 to 6.0. The additional reduction shall not be less than 0.90. b – This aggregate term assumes 0.85Fu multiplied by an implied 0.70 connection length reduction factor considering both Tide criteria fail. It also does not consider additional reduction from the presence of filler plates. c -This aggregate term assumes 0.85Fu multiplied by an implied 0.90 connection length reduction factor considering one or both Tide criteria are met. It also does not consider additional reduction from the presence of filler plates.