Evolution of geotechnical risk analysis in North American practice - Risk and Reliability in Geotechnical Engineering - page 465

1.40

1.20

1.00

0.80

0.60

0.40

Pf = 0.05

0.20

0.00
0.0 0.5 1.0

Factory of safety

1.5 2.0 2.5

Figure 12.4 Probability density function of factor of safety for slope reliability with mean E[FS] = 1.5 and
standard deviation S[FS] = 0.3.

A stochastic finite element (SFE) or finite difference methods permit the analyst to model
the spatial details of a physical process instead of relying on a few variables to represent the
complexities of nature. They are particularly useful for dealing with variables that are spa-
tially correlated. Fenton and Griffiths (2008) present a detailed treatment of SFEs applied
to geotechnical problems.
It is relatively easy to assign properties randomly to each element in a mesh as long as
the properties are not spatially correlated. When spatial correlation must be addressed, the
analytical procedure must assign to each element values of the random properties that both
satisfy the underlying probability distributions and retain the correlation structure. There
are several methods for doing this, including the moving average (MA), discrete Fourier
transform (DFT), covariance matrix decomposition, fast Fourier transform (FFT), turning
bands (TBM), and local average subdivision (LAS). Each of these approaches has advantages
and disadvantages. At this time, there is no consensus on the best method to use.

12.3.4 lumped versus distributed parameter models
This lumped-parameter analysis emphasized the centrality of derived distributions in mod-
eling geotechnical reliability. A derived distribution is that which results from propagating
uncertainty in input parameters through a mathematical model to obtain corresponding
uncertainty in output parameters. A variety of tools were developed to achieve these derived
distributions. Cornell's first-order second-moment methods suited this purpose (Cornell
1971); Rosenblueth introduced point-estimate methods (Rosenblueth 1975). Today, most
people use Monte Carlo simulation to achieve the same end, with a great deal more power
and less thinking. Harr (1977) was an early proponent of Monte Carlo methods, but most
researchers at the time looked upon stochastic simulation with suspicion. Today, the error is
the reverse, if it is an error at all.
A significant advancement during this period was the realization that soil and rock mass
properties are spatially variable and exhibit a spatial correlation structure (Lumb 1966).
This has several important implications. The first is that the variability among the aver-
age properties of large soil volumes is considerably less than that among the properties of
small volumes, such as boring samples or in situ tests. Matheron in the related context of