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15 ( zi - 2 rz1 ) for calibrating load and resistance factors for the LRFD spec- C = 2 yrs + (13b) ifications (Allen et al., 2005; D'Appolonia, 2007). The simula- rz 2 tions are performed in terms of a given load factor, , load where bias, Q, and resistance bias, R. The Monte Carlo technique rs is the corrosion rate of steel after zinc has been consumed, utilizes a random number generator to extrapolate the limit tf is the intended service life, state function, g, for calibration of yield resistance. Random C is the time to initiation of steel loss, values of g are generated using the mean, standard deviation, zi is the zinc initial thickness per side, and the distribution (e.g., normal, lognormal, or Weibull) of rz1 is the initial corrosion rate for zinc, and the load bias and the resistance bias. The extrapolation of g rz2 is the corrosion rate for zinc after the first two years. makes estimating possible for a given combination of and . A value of is adopted that is compatible with the static For plain steel reinforcements: earth load calculations (AASHTO, 2009). A range of values S = 2 rs t f (14) is assumed and estimated values of (by iteration) are checked against a value of 2.3 as used in previous LRFD calibrations Variables for the resistance calculation include Fy, Ac, rs, rz1, (Allen et al., 2005; D'Appolonia, 2007). Monte Carlo simula- rz2, and zi. The spacings of the reinforcements (SH and SV) are tions were facilitated by the Lumenaut software package considered to be constants. Using the statistics and observed (Lumenaut, 2007), which performs Monte Carlo simulations distribution for measurements of corrosion rate, the bias of through a link with Microsoft Excel. the remaining strength is computed and used as input for the The vertical pressure due to the weight of the reinforced reliability-based calibration of resistance factor. The bias is soil zone (v in Equation 8) is assigned load type "EV" as computed as described by Berg et al. (2009). AASHTO (2009) specifies equal to 1.35 for EV at the strength limit state, therefore, = Fy Ac 1.35 is adopted for calibration of the resistance factor similar R = (15) Fy Ac to D'Appolonia Engineers (2007) and Berg et al. (2009). The load bias depends on use of the simplified or coherent grav- The denominator includes nominal values used in design; ity method and may depend on reinforcement type (strip or Ac is based on the metal loss model recommended by AASHTO grid) as described by Allen et al. (2001), Allen et al. (2005), for design of metallic MSE reinforcements, and Fy is the and D'Appolonia (2007). Results from these studies demon- nominal yield strength. The statistics of the observed corro- strate that the load bias has a lognormal distribution, mean, sion rates from the database described in Chapter 3 are used and standard deviation as summarized in Table 8. to describe the variable A c , and the statistics for F y are taken Resistance bias is computed using Equations (10) to (15), from Galambos and Ravindra (1978) and Bounopane et al. nominal values for yield strength and remaining cross section, (2003). Bounopane et al. consider yield strengths to be the yield strength variation, and the mean, standard deviation, normally distributed with mean 1.05 times the nominal and and distribution from observations of metal loss archived in COV = 0.1. the performance database. Thus, the resistance bias depends on the nominal strength, which depends on the metal loss model used in design, as well as the manner in which observations of Resistance Factor Calibration corrosion rate and metal loss are extrapolated to render the Monte Carlo simulations are employed to compute the estimated remaining strength at the end of service. relationship between the reliability index, , and resistance Monte Carlo simulations were performed using these factor, . The Monte Carlo simulation method is used because values for load bias and load factor, and several different the approach is more adaptable and rigorous compared to scenarios were considered. Different scenarios treat metal loss other techniques, and it has become the preferred approach as deterministic or variable, and contrast "as built" versus Table 8. Mean () and standard deviation () of lognormal load bias. Strip Grid Parameter Simplified Coherent Simplified Coherent Method1 Gravity Method1 Gravity Method2 Method2 Q 0.973 1.294 0.973 1.084 Q 0.449 0.499 0.449 0.737 1 Allen et al. (2005). 2 D'Appolonia (2007).

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16 Figure 5. Monte Carlo simulation example showing input and output reports.

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17 design conditions. For each scenario different aspects of the the target reliability index. Appendix E includes details simi- design, including reinforcement size and service life, are con- lar to Figures 5 and 6 in support of all Monte Carlo simula- sidered and 10,000 iterations are performed to arrive at the tions described in this report. distribution of the limit state function. McVay et al. (2000) proposed representing design efficiency The Monte Carlo analysis for calibration of resistance factor as the ratio of resistance factor and mean bias (/R). This computes values for the limit state function, g = R - Q, consid- measure of efficiency accompanies computed resistance fac- ering the uncertainty of R and Q, and renders the probability tors to avoid the misconception that high resistance factors that g < 0. The variables R and Q can be related to nominal are correlated with the economy of a design method. Paikowski values as Q = Qn Q, and based on the LRFD equation (2004) demonstrates that this ratio is systematically higher R Q Qn for methods that predict more accurately, regardless of the [Equation (6)], R = R Rn = . The analysis pro- bias. Using different models for the nominal resistance in the ceeds by selecting a value for Qn; a value of unity is used for denominator of Equation (15) may lead to different values of convenience. The Monte Carlo analysis then computes a range the mean bias (R) and corresponding resistance factors (), of values for g, using randomly generated values for R and Q but if the COVs are the same, the resulting efficiency factors are based on their statistics, and a trial value of . The Lumenaut identical. Higher efficiency factors are only obtained with meth- software outputs the results of the iterations in the form of a ods that produce a lower COV with respect to the computed bias. This may be accomplished by improving the quality and histogram and corresponding intervals are summarized in a quantity of measured or estimated resistance, the models and table. Figure 5 is an example output from a typical Monte methods used to represent and extrapolate the data, or both. Carlo simulation for computing based on a trial selection If the metal loss is assumed to be deterministic, the only vari- of . The top graph is a histogram depicting the distribution able (i.e., not a constant) describing the tensile strength remain- of g from results of the Monte Carlo simulation. The proba- ing at the end of service condition corresponds to yield strength, bility that g < 0 may be computed by sorting and ranking Fy. Thus, the bias for remaining tensile strength is considered as these results and computing the cumulative probability at g = 0. normally distributed with mean equal to 1.05 and standard The table beneath the histogram summarizes the statistics deviation of 0.105 (Bounopane et al., 2003). These results are associated with g and the figures at the bottom of the page presented to serve as a baseline to assess the impact that varia- describe the statistics and associate distribution provided as tions in metal loss have on the computed values for . input (in this case for R and Q). The calibration is performed to identify the resistance fac- A worksheet, including the actual results from the 10,000 tor corresponding to of approximately 2.3 (pf 0.01). Resis- iterations, is also generated. The g values are then sorted and tance factors, summarized in Table 9, are rounded to the ranked in an ascending order and the cumulative probability nearest increment of 0.05. at each g is calculated. Figure 6 shows a typical result depict- These results may be compared to the current AASHTO ing the standardized normal variable (z) versus randomly gen- specifications where is specified as 0.75 for strip-type rein- erated g. The reliability index, is equal to (-z) at g = o. For forcements and 0.65 for grids that are attached to a rigid wall the example shown in Figure 6, is equal to 2.27 correspond- facing. These factors are the same for the simplified and ing to pf = 1.15E-02. The analysis is repeated with different coherent gravity methods in the current edition of the speci- trial values for until the probability that g < 0 corresponds to fications. It appears that the current AASHTO specifications implicitly consider some variation with respect to estimated 5 sacrificial steel requirements for galvanized reinforcements. 4 This is in spite of the fact that the resistance factors are not 3 determined from a reliability-based calibration, but are cali- 2 brated with respect to safety factors corresponding to the ear- 1 0 lier specifications for ASD. z -2 -1 0 2 4 -2 = 2.27 Table 9. Resistance factors -3 -4 considering deterministic -5 metal loss model. g (R,Q) Type Simplified/Coherent Figure 6. Typical Monte Carlo result from Strip 0.55/0.45 resistance factor calibration. Grid 0.50/0.35