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OCR for page 34
34 In the case of lognormal distributions for load and resis- bution to either be normal or lognormal, do not necessarily tance bias: compare very well with the results from the more robust Monte Carlo simulations. Therefore, numerical analyses (e.g., 2 ln Q R (1 + COVQ 2 ) (1 + COVR ) Monte Carlo simulations) are necessary to properly model = R Q (22) the distribution of R. ln [(1 + COVQ )(1 + COVR )] 2 2 For a given load factor and known load and resistance sta- Plain Steel Reinforcements tistics, Equations (21) and (22) are satisfied for selected val- Resistance factor calibrations are performed considering ues of resistance factors, rendering related pairs of reliability the use of plain steel (i.e., not galvanized) reinforcements. indices and resistance factors. From the computed pairs of The purpose of these calibrations is to define design parame- versus R, resistance factors can be selected corresponding to ters that are appropriate for plain steel reinforcements and the targeted level of reliability. demonstrate the advantage of using galvanized reinforce- Table 20 presents selected results and compares resistance ments. Since the use of plain steel reinforcements has been factors computed via Monte Carlo simulations to those limited, most of the data from the performance of plain steel computed via the Sagues service life model and closed-form reinforcements are from coupons placed at sites where galva- solutions for . The load bias used in these analyses refers to nized in-service reinforcements are employed. The few exam- the coherent gravity method and is a lognormal distribution ples where plain steel reinforcements have been used in the (D'Appolonia, 2007). A Weibull or normal distribution is United States are for grid-type reinforcements, as plain steel used to describe the variation of R. strip-type reinforcements are not readily available. Therefore, Results are presented for two cases: (1) where the probability density function (pdf) for R used in the Monte Carlo simula- the calibration is performed for grid-type reinforcements. The tion is normal, and (2) where the pdf is described with a Weibull calibration is performed considering reinforced fill quality that function. Because the closed-form solutions only consider meets AASHTO criteria for electrochemical properties, and probability density functions to be normal or lognormal this both good and high quality fill are considered. comparison demonstrates the importance of properly captur- ing the distribution of the pdf in the analysis, and the need for Good Quality Fill numerical simulations (i.e., Monte Carlo simulations). Table 20 demonstrates that when R is normally distributed, the compar- Based on the summary of statistics from corrosion rate mea- isons between the Monte Carlo simulations and the closed- surements depicted in Figure 8, a mean corrosion rate and form solutions are very good. Note that since the distribution standard deviation of 25 m/yr and 14 m/yr, respectively, of load bias is lognormal, the closed-form solution, assuming a represent the statistics for plain steel grid-type reinforcements normal distribution for both R and Q, does not always give the within good quality fill, and the distribution can be approxi- best results, even when R is normally distributed. mated as lognormal. The resistance bias is computed for dif- Alternatively, when R is described with the Weibull func- ferent sizes of grid-type reinforcements (W7, W9, W11, and tion, the closed-form solutions, which consider the distri- W14). The AASHTO metal loss model only applies to galva- Table 20. Summary of comparison between closed-form solutions and Monte Carlo simulations.1 Closed-Form Reinforced Fill Quality Life Thickness Monte Carlo Normal Lognormal w/Details of Steel Loss (yrs) (mm) pdf Model R Good w/Zinc Residuals 75 4 Normal 0.60 0.55 0.60 Good w/Conservative 75 15 Normal 0.45 0.50 0.50 Steel High Quality Fill 75 5 Normal 0.65 0.55 0.65 Good w/Conservative 75 4 Weibull 0.35 NA2 0.25 Steel Good w/Conservative 75 6 Weibull 0.40 0.10 0.35 Steel High Quality Fill 100 4 Weibull 0.85 NA2 0.55 1 Coherent gravity method applied to galvanized strip type reinforcements with assumed initial zinc thickness of 86 m. T =2.3 was used to compute . 2 NA means a result is not available because > 2.3 could not be achieved using the closed-form solution.