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32 of 0.8 m/yr and 0.5 m/yr, respectively, for strip-type rein- For example, consider applying the resistance factors as forcements, and 0.5 m/yr and 0.2 m/yr, respectively for they stand in the current version of the AASHTO specifica- grid-type reinforcements. Corrosion rates observed from tions (AASHTO, 2009), where = 0.75 and 0.65 for strip- and plain steel coupons older than 16 years correspond to mean grid-type reinforcements, respectively. Thus, for designs with and standard deviation values of 11.5 m/yr and 9.4 m/yr, strip-type reinforcements the probability (pf) that stress in and these parameters are used to represent the loss of base steel excess of yield will occur before the end of the intended subsequent to depletion of the zinc coating for this case. Both design life is 0.005 and 0.015, respectively for construction of these distributions are modeled as lognormal. The mean of employing high quality and good quality fill. Similarly, for the corresponding resistance bias is computed as ranging from designs with grid-type reinforcements, pf would correspond 1.4 to 2.0 with COV approximately 10%. The bias distribution to 0.008 and 0.018. Thus, MSE walls designed in accordance is approximately normal considering a 75-year service life, but with current AASHTO specifications, and constructed with is better represented by a Weibull distribution considering a high quality fills, have a more favorable pf compared to the 100-year service life. target of 0.01. Based on the statistics of the current inventory Table 18 is a summary of the resistance factors calibrated described by AMSE (2006), this exceptional performance with metal loss measurements from sites with high quality applies to approximately 80% of MSE walls in the existing reinforced fill. These resistance factors are equal to or higher inventory. The remaining 20%, constructed with good qual- than those currently specified by AASHTO (see Table 7). ity fill, are associated with a lower level of performance, and The efficiency ratio for this case is approximately 0.5. a pf that is nearly twice the target valued of 0.01. Changing the initial zinc thickness from 86 m to 150 m per side did not have a dramatic effect on the computed resistance factors compared to the case with good quality Verification of Monte Carlo Analysis backfill and the conservative steel model. This is because the Results from the Monte Carlo simulations used to cali- resistance factors computed with zi = 86 m considering brate resistance factors are verified via comparison incorpo- high quality fill are closer to one. rating alternative formulations for computing resistance bias The resistance factors summarized in Tables 17 and 18 and closed-form solutions for reliability index. Although the correspond to the target reliability index (T = 2.3 corre- closed-form solutions are limited to particular distributions sponding to pf = 0.01). However, other values of corre- sponding to different levels of reliability are also of interest. of the bias variables, they render estimates for comparison Table 19 compares the relationship between resistance fac- and illustrate the effect of incorporating more realistic distri- tors and reliability in terms of and pf, for different scenar- butions via Monte Carlo simulations. In general, the verifica- ios involving good or high quality fill, and strip or grid-type tion study is performed as follows: reinforcements. Table 19 considers typical galvanized strip reinforcements with S = 4 mm and grids with W11 longitu- Step 1. Select a design life and compute the distribution of dinal wires. On the basis of data depicted in Table 19, alterna- metal loss using the service life model described by tive approaches may be contemplated for selecting resistance Sagues et al. (2000). factors for design rather than calibrating to achieve a target Step 2. Compute the bias of the remaining cross-sectional reliability index. area, Ac, as the ratio of remaining cross section based Table 19. Comparison of relationship between and for different fill quality. Strip Reinforcements (S = 4 mm) Grid Reinforcements (W11) High Quality Good Quality High Quality Good Quality Fill Fill Fill Fill pf pf pf pf 0.55 3.12 0.001 2.73 0.003 2.99 0.0014 2.49 0.006 0.60 2.97 0.0015 2.64 0.004 2.63 0.004 2.24 0.012 0.65 2.93 0.002 2.42 0.008 2.41 0.008 2.09 0.018 0.70 2.72 0.003 2.30 0.010 2.13 0.016 1.93 0.026 0.75 2.56 0.005 2.17 0.015 2.11 0.017 1.79 0.036 0.80 2.48 0.006 2.04 0.020 1.96 0.025 1.69 0.045 0.85 2.27 0.011 1.91 0.028 1.80 0.035 1.52 0.063 0.90 2.19 0.014 1.80 0.036 1.67 0.047 1.39 0.082 calibrated for target reliability index T = 2.3 corresponding to pf = 0.01. based on current AASHTO specifications (AASHTO, 2009).
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33 on metal loss measurements to the remaining cross Fs is the cumulative density function (cdf ) representing section based on nominal metal loss used in design. steel corrosion rates. Step 3. Compute the bias of the remaining tensile strength, Equation (18) was programmed into an Excel spreadsheet R, as the product of the random variables including and the integration performed numerically. The numerical the bias of remaining cross section determined in integration was performed in increments between 0.1 and Step 2, and the bias for yield strength. 0.01 times r0. In most cases convergence to within E-06 was Step 4. Compute as a function of resistance factor using achieved within 100 increments. The numerical integration the bias of remaining tensile strength determined in was performed for a range of X and the corresponding prob- Step 3 and available closed-form solutions. abilities of exceedance computed for a given service life. For each value of X the bias of the remaining cross section (strip- Sagues Formulation type reinforcement) is computed as Sagues et al. (2000) formulated a probabilistic deterioration (S - 2 × X ) Ac = (19) model for service life forecasting of galvanized soil reinforce- [S - 2 × 12 × (t f - C )] ments. This formulation is based on the following assump- tions, which are the same as those employed in the Monte wherein the AASHTO metal loss model (Eq. 5) is used in the Carlo simulations: denominator to compute nominal remaining cross section. A mean and standard deviation were determined from the dis- · The distribution of corrosion loss over all elements in the tribution of the computed bias to describe the variation of structure mirrors the overall distribution of corrosion Ac. The bias of the remaining tensile strength was then com- measured in the field; puted as: · During the early life of the structure, the corrosion rate dis- tribution reflects that of the galvanized elements; R = Ac × Fy (20) · Loss of base steel is initiated after the zinc coating is where Fy is the bias of the yield stress. Assuming that Fy and consumed; Ac are uncorrelated and their statistics are known, the mean · The highest rate of metal loss takes place in the region of and standard deviation of R could be computed using well maximum reinforcement stress, and the service life of a known relationships between functions of random variables given element is over when the sacrificial steel in the high- as described by Baecher and Christian (2003). est stressed region is consumed; and · Corrosion rates are constant with time. Closed-Form Solutions for Reliability Index The resulting formula to compute the probability that For a specific limit state and a single load source, the reliabil- metal loss, X, exceeds a given threshold, X, is given by ity index () and the resistance factor () can be related using the following formula (Allen et al., 2005), which assumes that P [ X > X t f , z i , rz , z , rs , s ] the load and resistance bias both have normal distributions: = f z ( rz )(1 - Fs (( X ) ( t f - z i rz ))) drz (18) Q R - Q R r0 = (21) 2 where COVR Q R + ( COVQ Q ) 2 P is probability; R X is loss of steel defined by tf, zi, rz, z, rs, s; X is a given amount of steel loss; where tf is service life; = reliability index (dimensionless), zi is the initial zinc thickness; Q = load factor (dimensionless), rz is the mean zinc corrosion rate; R = resistance factor (dimensionless), z is the standard deviation of zinc corrosion rate; Q = mean of load bias (dimensionless), rs is the mean steel corrosion rate; R = mean of resistance bias (dimensionless), s is the standard deviation of steel corrosion rate; COVQ = coefficient of variation of load bias (dimension- r0 = zi/tf and is the lowest rate of zinc corrosion for which less), and base steel will be consumed within tf; COVR = coefficient of variation of resistance bias (dimen- fz(rz) is the pdf representing zinc corrosion rates, rz; and sionless).