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Bearing Capacity Prediction of Spatially Random
Soils
Canadian Geotechnical Journal
, 40(1), 54–65, Feb 2003
Gordon A. Fenton
Dalhousie University, Halifax, NS B3J 2X4
Gordon.Fenton
@
dal.ca
D. V. Grifths
Colorado School of Mines, Golden, CO 80401
D.V.Grifths
@
mines.edu
Abstract
Soils with spatially varying shear strengths are modeled using random eld theory and elasto-plastic
nite element analysis to evaluate the extent to which spatial variability and cross-correlation in
soil properties (
and
are independent, are derived
using a geometric averaging model and then veried via Monte Carlo simulation. The standard
deviation prediction is found to be quite accurate, while the mean prediction is found to require
some additional semi-empirical adjustment to give accurate results for ‘worst case’ correlation
lengths. Combined, the theory can be used to estimate the probability of bearing capacity failure,
but also sheds light on the stochastic behaviour of foundation bearing failure.
and
1. Introduction
The design of a footing involves two limit states; a serviceability limit state, which generally
translates into a maximum settlement or differential settlement, and an ultimate limit state. The
latter is concerned with the maximum load which can be placed on the footing just prior to a
bearing capacity failure. This paper looks at the ultimate bearing capacity of a smooth strip footing
founded on a soil having spatially random properties.
Most modern bearing capacity predictions involve a relationship of the form (Terzaghi, 1943)
[1]
=
+ ¯
+
1
2
where
is the ultimate bearing stress,
is the cohesion, ¯
is the overburden stress,
is the unit soil
are the bearing capacity factors. To simplify
the analysis in this paper, and to concentrate on the stochastic behaviour of the most important
term (at least as far as spatial variation is concerned), the soil is assumed weightless. Under this
assumption, the bearing capacity equation simplies to
,
, and
[2]
=
factors, are often based on plasticity
theory (see, e.g., Prandtl, 1921, Terzaghi, 1943, and Sokolovski, 1965) of a rigid base punching
into a softer material. These theories assume a
homogeneous
soil underlying the footing – that is,
1
) affect bearing capacity. The analysis is two dimensional, corresponding to
a strip footing with innite correlation length in the out-of-plane direction, and the soil is assumed
to be weightless with footing placed on the soil surface. Theoretical predictions of the mean and
standard deviation of bearing capacity, for the case where
weight,
is the footing width, and
Bearing capacity predictions, involving specication of the
the soil is assumed to have properties which are spatially constant. Under this assumption, most
bearing capacity theories (e.g., Prandtl, 1921, and Meyerhof, 1951, 1963) assume that the failure
slip surface takes on a logarithmic spiral shape to give
[3]
=
tan
tan
2
4
+
2
1
tan
This relationship has been found to give reasonable agreement with test results (Bowles, 1996)
under ideal conditions. In practice, however, it is well known that the actual failure conditions
will be somewhat more complicated than a simple logarithmic spiral. Due to spatial variation in
soil properties the failure surface under the footing will follow the weakest path through the soil,
constrained by the stress eld. For example, Figure 1 illustrates the bearing failure of a realistic
soil with spatially varying properties. It can be seen that the failure surface only approximately
follows a log-spiral on the right side and is certainly not symmetric. In this plot lighter regions
represent stronger soil and darker regions indicate weaker soil. The weak (dark) region near the
ground surface to the right of the footing has triggered a non-symmetric failure mechanism that is
typically at a lower bearing load than predicted by traditional homogeneous and symmetric failure
analysis.
Figure 1. Typical deformed mesh at failure, where the darker regions indicate weaker soil.
The problem of nding the minimum strength failure slip surface through a soil mass is very similar
in nature to the slope stability problem, and one which currently lacks a closed form stochastic
solution, so far as the authors are aware. In this paper the traditional relationships shown above
will be used as a starting point to this problem.
For a realistic soil, both
and
[4]
=
=
where
is the mean cohesion and
is the stochastic equivalent of
, ie.,
=
. The
stochastic problem is now boiled down to nding the distribution of
. A theoretical model for
the rst two moments (mean and variance) of
, based on geometric averaging, are given in the
2
are random, so that both quantities in the right hand side of Eq. (2)
are random. This equation can be non-dimensionalized by dividing through by the cohesion mean,
. This is followed by an example
illustrating how the results can be used to compute the probability of a bearing capacity failure.
Finally, an overview of the results is given, including their limitations.
2. The Random Soil Model
In this study, the soil cohesion,
, is assumed to be lognormally distributed with mean
, standard
. The lognormal distribution is selected because it
is commonly used to represent non-negative soil properties and since it has a simple relationship
with the normal. A lognormally distributed random eld is obtained from a normally distributed
random eld,
, and spatial correlation length
ln
ln
(
), having zero mean, unit variance, and spatial correlation length
ln
through
the transformation
[5]
(
) = exp
ln
+
ln
ln
(
)
are obtained from
the specied cohesion mean and variance using the lognormal distribution transformations,
is the spatial position at which
is desired. The parameters
ln
and
ln
2
[6
]
2
ln
= ln
1 +
2
[6
]
ln
= ln
1
2
2
ln
The correlation coefcient between the log-cohesion at a point
1
and a second point
2
is specied
is the absolute distance between the two
points. In this paper, a simple exponentially decaying (Markovian) correlation function will be
assumed, having the form
(
), where
=
ln
1
2
2
[7]
ln
(
) = exp
ln
The spatial correlation length,
ln
, is loosely dened as the separation distance within which
two values of ln
are signicantly correlated. Mathematically,
is dened as the area under
ln
) (Vanmarcke, 1984). (Note that geostatisticians often dene the
correlation length as the area under the non-negative half of the correlation function so that there is
a factor of two difference between the two lengths – under their denition, the factor of 2 appearing
in Eq. (7) is absent. The more general denition is retained here since it can be used also in higher
dimensions where the correlation function is not necessarily symmetric in all directions about the
origin.)
It should also be noted that the correlation function selected above acts between values of ln
(
ln
.
is normally distributed and a normally distributed random eld is simply
dened by its mean and covariance structure. In practice, the correlation length
can be
estimated by evaluating spatial statistics of the log-cohesion data directly (see, e.g., Fenton, 1999).
Unfortunately, such studies are scarce so that little is currently known about the spatial correlation
ln
3
next section. Monte Carlo simulations are then performed to assess the quality of the predictions
and determine the approximate form of the distribution of
deviation
where
by a correlation function,
the correlation function,
This is because ln
structure of natural soils. For the problem considered here, it turns out that a worst case correlation
length exists which should be assumed in the absence of improved information.
The random eld is also assumed here to be statistically isotropic (the same correlation length in
any direction through the soil). Although the horizontal correlation length is often greater than
the vertical, due to soil layering, taking this into account was deemed to be a renement beyond
the scope of this study. The main aspects of the stochastic behaviour of bearing capacity needs to
be understood for the simplest case rst and more complex variations on the theme, such as site
specic anisotropy, left for later work.
The friction angle,
(
), according to
[8]
(
) =
+
2
(
)
1 + tanh
(
)
2
is a scale
factor which governs the friction angle variability between its two bounds. Figure 2 shows how
the distribution of
and
are the minimum and maximum friction angles, respectively, and
(normalized to the interval [0
1]) changes as
changes, going from an almost
uniform distribution at
= 5 to a very normal looking distribution for smaller
. In all cases,
the distribution is symmetric so that the midpoint between
and
is the mean. Values
greater than about 5 lead to a U-shaped distribution (higher at the boundaries), which is not
deemed realistic. Thus, varying
between about 0.1 and 5.0 leads to a wide range in the stochastic
behaviour of
.
s = 0.1
s = 0.2
s = 1.0
s = 2.0
s = 5.0
0
0.2
0.4
0.6
0.8
1
f
(standardized)
Figure 2. Bounded distribution of friction angle normalized to the interval [0
1].
The random eld,
(
), has zero mean and unit variance, as does
ln
(
). Conceivably,
(
)
could also have its own correlation length
distinct from
. However, it seems reasonable to
ln
4
, is assumed to be bounded both above and below, so that neither normal
nor lognormal distributions are appropriate. A beta distribution is often used for bounded random
variables. Unfortunately, a beta distributed random eld has a very complex joint distribution and
simulation is cumbersome and numerically difcult. To keep things simple, a bounded distribution
is selected which resembles a beta distribution but which arises as a simple transformation of a
standard normal random eld,
1
where
of
assume that if the spatial correlation structure is caused by changes in the constitutive nature of
the soil over space, then both cohesion and friction angle would have similar correlation lengths.
Thus,
is taken to be equal to
ln
in this study. Both lengths will be referred to generically
, remembering that this length reects correlation between points in the
underlying normally distributed random elds,
), and not directly between points
in the cohesion and friction elds. As mentioned above, both lengths can be estimated from data
sets obtained over some spatial domain by statistically analyzing the suitably transformed data
(inverses of Eq’s 5 and 8). After transforming to the
ln
(
) and
(
elds, the transformed correlation
lengths will no longer be the same, but since both transformations are monotonic (ie. larger values
of
and
= C.O.V. = 1
0,
the difference is less than 15% from each other and from the original correlation length). In that
all engineering soil properties are derived through various transformations of the physical soil
behaviour (eg. cohesion is a complex function of electrostatic forces between soil particles), the
nal correlation lengths between engineering properties cannot be expected to be identical, only
similar. For the purposes of a generic non-site specic study, the above assumptions are believed
reasonable.
The question as to whether the two parameters
give larger values of
, etc.), the correlation lengths will be similar (for
are correlated is still not clearly decided in
the literature, and no doubt depends very much on the soil being studied. Cherubini (2000) quotes
values of
and
0
70, as does Wolff (1985) (see also Yuceman et al., 1973,
Lumb, 1970, and Cherubini, 1997). As Wolff says (private correspondence, 2000),
The practical meaning of this [negative correlation] is that we are more certain of the undrained
strength at a certain conning pressure than the values of the two parameters we use to dene
it.
This observation arises from the fact that the variance of the shear strength is reduced if there is a
negative correlation between
ranging from
0
24 to
and
.
is not certain, this paper investigates the correlation extremes
to determine if cross-correlation makes a signicant difference. As will be seen, under the given
assumptions regarding the distributions of
and
(lognormal) and
(bounded), varying the cross-
1 to +1 was found to have only a minor inuence on the stochastic behaviour
of the bearing capacity.
from
3. Bearing Capacity Mean and Variance
The determination of the rst two moments of the bearing capacity (mean and variance) requires
rst a failure model. Equations 2 and 3 assume that the soil properties are spatially uniform. When
the soil properties are spatially varying, the slip surface no longer follows a smooth log-spiral and
the failure becomes unsymmetric. The problem of nding the constrained path having the lowest
total shear strength through the soil is mathematically difcult, especially since the constraints
are supplied by the stress eld. A simpler approximate model will be considered here wherein
geometric averages of
and
5
from now on simply as
ln
In that the correlation between
correlation
, over some region under the footing, are used in Equations 2 and
3. The geometric average is proposed because it is dominated more by low strengths than is the
arithmetic average. This is deemed reasonable since the failure slip surface preferentially travels
through lower strength areas.
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