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Construction of virtual sites for reliability-based design
Construction de sites virtuels à des fins de conception fiabiliste
Phoon K.K.
National University of Singapore, Singapore
Ching J.
National Taiwan University, Chinese Taipei
ABSTRACT: This paper presents the construction of “virtual sites” using multivariate normal distributions calibrated from actual soil
property databases. By doing so, the actual magnitude of uncertainty reduction from conducting better/more soil tests can be
estimated realistically, rather than theoretically.
RÉSUMÉ: Cet article présente la construction de "sites virtuels" en utilisant des distributions normales à plusieurs variables calibrées
à partir de bases de données de propriétés de sols réels. Par cette méthode, la réduction réelle de l'incertitude que l'on peut obtenir en
augmentant le nombre et/ou la qualité des essais de sol peut être estimée de manière réaliste, et non plus seulement théorique.
KEYWORDS: virtual site; uncertainties; soil properties; correlation; site investigation; reliability-based design.
1 INTRODUCTION
This paper presents the concept of a “virtual site”; the purpose
is to emulate site investigation efforts as realistically as
possible. It is not possible to emulate every aspect of a real site
at present. In this paper, the scope is to reproduce the
information content arising from a typical mix of laboratory and
field tests conducted in a site for the purpose of estimating a
design undrained shear strength (s
u
) for clays and friction angle
(
) for sands. The critical feature here is the consistent and
realistic coupling of different test data, which is achieved using
multivariate normal distributions. Data from different tests will
be correlated, because they are measuring the same mass of soil,
although they could be measuring different aspects of soil
behavior under different boundary conditions and over different
volumes. The purpose of developing a virtual site is not to
replace actual site investigation. The purpose is to quantify the
uncertainty reduction in s
u
and
by incorporating the test
results from better and/or more tests.
The idea of simulating a “virtual site” is not new. For
example, Jaksa et al. (2005) and Goldsworthy et al. (2007) used
three dimensional random fields and Monte Carlo simulation to
simulate the spatially variable elastic modulus of a “virtual”
site. Each spatially variable realization constitutes a plausible
full information scenario. Site investigation is then carried out
numerically by sampling the continuous random field at discrete
locations. The site investigation data so obtained constitute the
typical partial information scenario commonly encountered in
practice. The goal of these studies was to quantify the
difference in the designs based on these full and partial
information scenarios. In this paper, the virtual site simulation
is based on multivariate normal distributions that couple soil
parameters such as s
u
, overconsolidation ratio, standard
penetration test N-value, cone tip resistance, and Atterberg
limits. The distinct features of this paper are: (a) a more
realistic bag of multivariate information containing both
laboratory and field data and (b) the probability model is
constructed from an actual database of clays and sands. These
features are critical to the objective of this paper, which is to
quantify the uncertainty reduction in s
u
and
by incorporating
the test results from better and/or more tests. This objective is
only achievable if the information contained in the virtual site is
comparable to that contained in a real site, not merely pertaining
to a single laboratory/field parameter, but to a group of
parameters that are correlated in a realistic way. By doing so, it
is possible to evaluate the
actual
merits of reliability-based
design approximately, rather than elaborate on the theoretical
merits widely discussed in previous studies. This paper
summarizes the current development of such virtual sites.
2 MULTIVARIATE GEOTECHNICAL DATA
Multivariate information is usually available in a typical site
investigation. For instance, when undisturbed samples are
extracted for oedometer and triaxial tests, SPT and/or piezocone
test (CPTU) may be conducted in close proximity. Moreover,
data sources such as the unit weight, plastic limit (PL), liquid
limit (LL), and liquidity index (LI) are commonly determined
from relatively simple laboratory tests on disturbed samples.
These data could be correlated, and these correlations can be
exploited to reduce the coefficient of variation of a design
parameter. The impact on RBD is obvious. This section
presents statistical characterization of multivariate geotechnical
data.
Most soil parameters are not normally distributed, because
they are positive valued. Let Y denote a non-normally
distributed soil parameter. One well known cumulative
distribution function (CDF) transform approach can be applied
to convert Y into a standard normal variable X: X =
-1
[F(Y)],
where
(.) is the CDF of a standard normal random variable,
and F(.) is the CDF of Y. A set of multivariate soil parameters
Y = (Y
1
, Y
2
,
Y
n
)
can be transformed into X = (X
1
, X
2
,
X
n
)
.
By definition, X
1
, X
2
, … X
n
are
individually
standard normal
random variables. It is crucial to note here that
collectively
(X
1
,
X
2
, …X
n
)’ does not necessarily follow a multivariate normal
distribution even if each component is normally distributed.
Even so, recent studies by Ching et al. (2010) and Ching and
Phoon (2012a) showed that the multivariate normal distribution
is an acceptable approximation for selected parameters of clays,
and Ching et al. (2012b) arrived at the same observation for
selected parameters of sands.
The multivariate normal probability density function for X =
(X
1
, X
2
, …X
n
)
can be defined uniquely by a correlation matrix:
