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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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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).
