Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 261

Polynomial chaos expansions and
stochastic finite-element methods
Bruno Sudret
6.1 IntroDuCtIon
Soil and rock masses naturally present heterogeneity at various scales of description. This
heterogeneity may be of two types. On a large scale, soil properties may be considered
piece-wise homogeneous once regions (e.g., layers) have been identified. On a lower scale,
the local spatial variability of the properties shall be accounted for. In any case, the use
of deterministic values for representing the soil characteristics is poor, since it ignores the
natural randomness of the medium. Alternatively, this randomness may be properly mod-
eled using probabilistic models.
In the first of the two cases identified above, the material properties (e.g., Young's modu-
lus, cohesion, friction angle, etc.) may be modeled in each region as random variables whose
distributions (and possibly mutual correlation) have to be specified. In the second case, the
introduction of random fields is necessary. In this respect, probabilistic soil modeling is a
long-term story, see, for example, Vanmarcke (1977); DeGroot and Baecher (1993); Fenton
(1999a,b); Rackwitz (2000); and Popescu et al. (2005).
Usually, soil characteristics are investigated to feed models of geotechnical structures in
the context of the engineering design. Examples of such structures are dams, embankments,
pile or raft foundations, tunnels, and so on. The design then consists of choosing charac-
teristics of the structure (dimensions, material properties) so that the latter fulfills some
requirements (e.g., retain water, support a building, etc.) under a given set of environmental
actions that we will call “loading.” The design is practically carried out by satisfying some
design criteria that usually apply onto model response quantities (e.g., global equilibrium
equation, settlements, bearing capacity, etc.). The conservatism of the design according to
codes of practice is ensured first by introducing safety coefficients, and second by using
penalized values of the model parameters. In this approach, the natural spatial variability of
the soil is completely hidden.
From another point of view, when the uncertainties and variability of the soil properties
have been identified, methods that allow propagating these uncertainties throughout the
model have to be used. Perturbation methods used in the 1980s and 1990s (Baecher and
Ingra, 1981; Phoon et al., 1990) allow estimating the mean value and standard deviation
of the system response. First-/second-order reliability methods (FORM/SORMs) are used
for assessing the probability of failure of the system with respect to performance criteria
(Ditlevsen and Madsen, 1996). Numerous applications of the latter can be found, for exam-
ple, in Phoon (2003); Low (2005); Low and Tang (2007); and Li and Low (2010) among
others.
In the early 1990s, a new approach called stochastic finite-element method (SFEM) has
emerged, which allows one to solve boundary value problems with uncertain coefficients
and is especially suited to spatially variable inputs (Ghanem and Spanos, 1991). The key
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