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35 most formulas in codes of practice and therefore, not suit- ciency levels that were strengthened with FRP shear rein- able for use in practice. forcement (i.e., a set of 3 × 2 × 3 = 18 bridges). One set used 2. Availability and Reliability of Model Parameter Data. anchored FRP reinforcement where FRP rupture is the domi- The models proposed by most researchers require knowl- nating mode of failure, and another set used non-anchored FRP edge of the measurable features of the test beam and FRP reinforcement where debonding is to be expected. Table 2.11 application, such as the development length of the FRP lists the nominal properties of these bridges, illustrating the reinforcement. While such requirements improve accu- wide range in the ratio of FRP contribution (Vf) to the nominal racy, design provisions must include items that are avail- strength of the existing structure. The live-to-dead load shear able or can be appropriately assumed. The parameters demand ratio for these bridges ranges from 0.83 to 2.32. proposed in a Vf relationship for use in the LRFD specifi- The nominal values of the bridge properties were used to cations should recognize that FRP shear reinforcement is obtain random variables for each of the main parameters in most likely to be used for strengthening of an existing old the design equation. This step required knowledge of the sta- structure for which full design details may not be available. tistical characteristics of the random variables. Some of the 3. Calibration with Full Shear Strength Ratio Considering required information was obtained from the literature, and Specific Vc and Vs Relationships. A comparative evaluation other information specific to this study had to be determined of Vf models, can be conducted using the strength ratios of by estimating the random variables' bias (ratio of random Vf,test/Vf. However, to include in a code of practice, the rela- variable's mean value to its nominal value) and COV (ratio of tionship for Vf must be calibrated based on the bias (strength random variable's standard deviation to its mean value). For ratio) and COV of the Vn,test/Vn ratio (Vn = Vc + Vs + Vf). example, the statistical characteristics of the analysis model uncertainty, P, were obtained by comparing the predicted shear strength using the proposed equations to the experi- 2.9 Reliability Assessment mentally, measured values as reported in the database. The A study was conducted to assess the reliability of the pro- effect of material and fabrication tolerances on the resistance posed design equations using procedures similar to those used model was determined using Monte Carlo simulations and in the calibration of AASHTO-LRFD specifications (AASHTO, some recently published data (Nowak and Szerszen, 2003). 2008). The First Order Reliability Method (FORM) was used The statistical characteristics of load-related random vari- for calculating the reliability index (r) of 36 bridges that were ables were taken from NCHRP Report 368 (Nowak, 1999). designed for such purposes. These bridges covered three span Table 2.12 shows the bias and coefficient of variation for the lengths, interior and exterior girders, and three shear defi- main parameters used in the limit state function (Z) that Table 2.11. Shear resistance components. Bridge t f (in) Vf Av dv s w f s f Vc Vs V f Span Case Girder (in ) (in) (in) Rupture Debond (kips) (kips) (kips) Vc + Vs 2 Length L45F1I 0.847 12.0 0.00774 0.01863 50.82 33.88 0.272 Interior L45F2I 0.730 10.0 0.01142 0.02749 43.79 40.91 0.349 L45F3I 0.613 8.0 0.01477 0.03556 36.76 47.94 0.435 45 ft L45F1E 0.697 12.0 0.01461 0.03519 73.57 41.79 41.79 0.362 Exterior L45F2E 0.532 10.0 0.02315 0.05575 31.92 51.66 0.490 L45F3E 0.368 8.0 0.03146 0.07576 22.05 61.53 0.643 L60F1I 0.696 14.0 0.00644 0.01023 41.77 41.77 0.273 Interior L60F2I 0.580 12.0 0.00644 0.01402 34.78 48.76 0.334 L60F3I 0.463 10.0 0.00728 0.01753 27.79 55.76 0.401 60 ft L60F1E 0.565 4.0 14.0 0.01461 0.01855 111.24 33.89 50.84 0.350 Exterior L60F2E 0.468 12.0 0.02315 0.02210 28.06 56.67 0.407 L60F3E 0.371 10.0 0.03146 0.02477 22.23 62.49 0.468 L75F1I 0.501 16.0 0.00596 0.00596 30.06 45.09 0.233 Interior L75F2I 0.319 14.0 0.00648 0.00776 19.12 56.03 0.306 L75F3I 0.136 12.0 0.00664 0.01142 8.18 66.97 0.389 75 ft L75F1E 0.394 16.0 0.00730 0.00849 163.81 23.67 55.23 0.295 Exterior L75F2E 0.214 14.0 0.00764 0.01277 12.86 66.04 0.374 L75F3E 0.034 12.0 0.00762 0.01734 2.05 76.85 0.463
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36 Table 2.12. Summary of bias and COV values used in calibration study. Random Variable Bias COV Material and Fabrication Tolerances, MF Varies (see Appendix) FRP Rupture 1.680 0.330 Analysis Model, P Other 1.410 0.269 Wearing Surface DL, WS 1.000 0.250 Component DL, DC 1.050 0.100 L=45 ft 1.041 0.180 Highway LL (including impact), LL+IM L=60 ft 1.046 0.180 L=75 ft 1.071 0.180 Interior 1.134 0.157 LRFD Girder Distribution Factor, GDF Exterior 1.307 0.239 accounts for variabilities and uncertainties in material and were validated using Monte Carlo simulations for one bridge. fabrication tolerances (MF), analysis model accuracy (p), After several trials, the proposed design expression was adjusted and girder distribution factors (GDF) as well as the different to achieve r values close to the level targeted by most AASHTO loading types (wearing surface, dead load, and live load). calibration studies (i.e., r,target = 3.50). The simplified approximate expressions used in previous The performance of the proposed expression is illustrated AASHTO calibration studies were not used because of the in Figure 2.17 for a range of girder span lengths and spacings. high COV for the analysis model, which exceed the limits of The figure shows that the reliability index is nearly the same the applicability of the simplified approximated expressions. for all girder spacings and is about 3.50 for shorter span A detailed iterative FORM analysis was performed; the results lengths but decreases for longer span lengths. 3.75 Girder Spacing, S = 5 ft Girder Spacing, S = 7 ft 3.50 Girder Spacing, S = 9 ft Reliability Index, r 3.25 3.00 2.75 20 30 40 50 60 70 80 90 Girder Span Length, L (ft) Figure 2.17. Reliability index (r) versus span lengths for different girder spacings.