Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 360

types of undrained shear strength data. However, obtaining probability of failure from
MCS for critical noncircular slip surfaces (for comparison with FORM P f ) would be very
time-consuming because a search for critical noncircular slip surface is required for each
set of random numbers generated in MCS. In contrast, it is much simpler to obtain Monte
Carlo P f values with search for the critical circular slip surface , and these (Table 5 in Low
and Duncan 2013) are practically identical to the FORM P f values of 9.7, 19.4, and 45.6%
mentioned above.
9.5 relIabIlItY analYSIS oF a norWegIan SloPe aCCountIng
Fo r S PatIa l autoCo r r e l atI o n
Spatial autocorrelation (also termed spatial variability) arises in the geological material by
virtue of its formation by natural processes acting over unimaginably long time (millions
of years). This endows the geomaterial with some unique statistical features (e.g., spatial
autocorrelation) not commonly found in the structural material manufactured under strict
quality control. For example, by the nature of the slow precipitation (over many seasons) of
fine-grained soil particles in water in nearly horizontal layers, two points in close horizontal
or vertical proximity to one another are likely to be more positively correlated (e.g., likely to
have similar undrained shear strength c u values) than two points further apart in the verti-
cal direction.
A clay slope in southern Norway was analyzed deterministically and probabilistically by
Low et al. (2007) using the Low (2003a) spreadsheet-based reformulations of the Spencer
method and the intuitive FORM of Low and Tang (2004). The reformulation allows switch-
ing among the Spencer, Bishop simplified, and wedge methods on the same template, by
specifying different side-force inclination options and different constraints of optimization.
Search for the critical circular or noncircular slip surface is possible. The deterministic pro-
cedure was extended probabilistically by implementing the FORM via constrained optimi-
zation of the equivalent dispersion ellipsoid in the original space of the random variables.
The procedure was illustrated for an embankment on soft ground and for a clay slope in
southern Norway, both involving spatially correlated soil properties. The effects of autocor-
relation distance on the results of reliability analysis were studied. Shear strength anisotropy
was modeled via user-created simple function codes in the programming environment of the
spreadsheet.
Figure 9.8 shows the results of reliability analysis involving 24 spatially correlated c u
values and 24 spatially correlated unit weight values. The size of the correlation matrix is
48 × 48. The design point obtained by Excel Solver represents the most probable combina-
tion of the 24 values of c u and the 24 values of γ that would cause failure. As expected for
resistance parameters, the 24 values of undrained shear strength c u at the design point are
all lower than their respective mean values. On the other hand, when the autocorrelation
distance δ is 10 m or lower, as shown in Figure 9.8 , the design point index of γ—defined as
(γ i * − μ γ )/σ γ , where γ i * is the design point value of unit weight γ for i from 1 to 24—shows
most values of unit weight γ* above their mean value μ γ , as expected for loading parameters,
but, somewhat paradoxically, there are some design point values of γ near the toe that are
below their mean values. The implication is that the slope is less safe when the unit weights
near the toes are lower. This implication can be verified by deterministic runs using higher
γ values near the toe, resulting in higher factors of safety. It would be difficult for the design
code committee to recommend partial factors such that the design values of γ are above the
mean along some portions of the slip surface and below their mean along other portions. In
contrast, the design point is automatically located in FORM analysis by the spreadsheet's