LRFD calibration of simple limit state functions in geotechnical soil-structure design - Risk and Reliability in Geotechnical Engineering - page 339

8.5 eXaMPle
8.5.1 general
LRFD concepts and procedures for the case of a simple linear limit state function with one
load term are demonstrated in this section using the example of one internal limit state
for one class of mechanically stabilized earth (MSE) wall structures. MSE walls are par-
ticularly well suited for reliability-theory-based calibration because there are many instru-
mented walls reported in the literature. These data can be used to compare measured loads
to nominal (predicted) loads in the soil-reinforcing layers under operational conditions and
thus generate load bias data. Similarly, there are substantial collections of pullout (resis-
tance) data available from published and unpublished sources for different soil-reinforcing
types (e.g., steel strips, steel grids, steel anchors, and polymeric geogrids). Predicted pullout
capacities at failure can be generated using any number of pullout models available in the
literature. Predicted values can then be used with measured data from laboratory or in situ
pullout tests to generate resistance bias values.
In the example to follow, the focus is on the treatment of bias data and LRFD calibration
rather than the details of the actual load and resistance models. Load and resistance bias
values used in this example are given in rank order in Table 8.1 .
8.5.2 load data
Measured reinforcement loads from steel strip reinforced soil walls have been collected
by Miyata and Bathurst (2012a), Allen et al. (2001, 2004), and Bathurst et al. (2008b,
2009). Miyata and Bathurst (2012a) fitted an exponential equation to the measured data to
improve the accuracy of the model compared to existing bi-linear load models. Fitting was
carried out by selecting empirical coefficients such that the mean of the load bias values was
1 and the COV of load bias values was as small as possible. This explains why the mean of
the load bias μ Q = 1 using all the load bias data in this example. In most LRFD calibration
exercises, the mean of load bias values is unlikely to be unity and is most often less than 1.
This is because most load models were developed for ASD with factors of safety and thus,
they cautiously overestimate expected actual load values.
Load bias data are presented as normal and lognormal CDF plots in Figure 8.3a and 8.3b ,
respectively. Approximations are superimposed on the plots using the computed values of
the mean and COV for all the load bias values (μ Q and COV Q ). Visually, the normal dis-
tribution does well except at the top end of the data. The approximation to the data using
a lognormal CDF plot does better at the top end of the measured data, but poorly at the
bottom. Four data points are identified as possible outliers. Careful attention must be paid
to any potential outliers at the end of CDF plots since it is the tails of these plots that can
have a large influence on LRFD calibration outcomes. If these data are not representative of
the sample population used to plot the entire data set, then they should be discarded. As an
example, the outliers are removed in Figure 8.3c and approximations are made to the entire
filtered data set using the mean and COV of the filtered data set and just the upper tail using
μ Q = 1.20 and COV Q = 0.108. The tail statistics were selected by manual adjustment until
the approximation through the upper tail of the filtered data set was visually judged to be
good. Load bias statistics are summarized in Table 8.1 .
Filtered load bias data are plotted with cumulative and exceedance fraction axes in Figure
8.3d. Trial load factor values (γ Q ) can be selected for different exceedance values as illus-
trated in the figure. In the calculations to follow, a range of load factors from 1.00 to 1.50
is considered.