3444
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
some specific deterministic value). Thus, the probability of
occurrence of limiting settlements becomes:
)
)
(3)
The integrals of equations (1, 2 and 3) are commonly solved
using analytical approximations (or reliability methods). In the
following sections three methodologies using FOSM, SOSM
and MCS methods with Schmertmann’s (1970) equation are
shortly presented and discussed as a simple and practical way to
characterize the settlement solicitation curve for a case of a
single footing in a sandy soil.
2 ANALYZED METHODOLOGIES
2.1
Main concepts adopted
The main concepts adopted on the analyzed methodologies are:
The total predicted settlement (ρ) is given by
Schmertmann’s (1970), calculated through the sum of
the settlement increments (ρ
i
) of each sublayer:
(4)
where: i=1, N and N is the number of adopted sublayers.
If the increments (ρ
i
) are statistically independents and
V[ρ
i
] are the variance increments of the sublayers then,
the total variance also can be calculated as the sum of
V[ρ
i
]:
(5)
The proposed simplifications consider that the predicted
settlement is function of only one random variable (E
Si
, in each
sublayer), and is completely described by its first two moments
(mean and variance). The evaluation must have done
considering the soil stratification, first through the evaluation of
each sublayer, individually, and then accounting for the entire
stratum of the subsoil (sum of the increments).
where: E[ρ
1
] and E[ρ
2
] are mean predicted settlements of the
footings and V[ρ
1
] and V[ρ
2
] are its variances.
2.2
The FOSM and SOSM methods
Consider the given form of the performance function of the
random variables x
1
, x
2
, x
3
... x
i
, independent, such as:
G[X]=G(x
1
, x
2
, x
3
... x
i
). Developing the function G[X] about its
mean and the mean of the random variables x
i
, using the
Taylor’s expansion series, gives (Baecher and Christian 2003):
(6)
The mean (E[ρ]) and variance (V[ρ]) of the predicted
settlement can be obtained from equation (6), considering the
Schmertmann’s (1970) method as the performance function and
assuming the parameter E
S
as the unique random variable. For
the FOSM method, gives:
(7)
(8)
For the SOSM method, settlement mean and variance are:
N
i
Si
Si
Zi
Zi
N
i
Si
Zi
Zi
EV
E
I
CC
E
I
CC E
1
3
*
2 1
1
*
2 1
] [ .
] [
lim
lim
). (
]
[
dx x
p
E
(9)
(10)
2
2
1
3
*
2 1
2
1
2
*
2 1
] [ .
2
] [ .
] [
Si
N
i
Si
Zi
Zi
Si
N
i
Si
Zi
Zi
EV
E
I
CC
EV
E
I
CC V
The first terms at
the right side of equations (9 and 10) correspond exactly to the
mean and variance calculated by the FOSM method, while the
second terms represent the additional terms considered in the
Taylor’s series. This simple observation shows that the use of
the FOSM method underestimates the results of mean and
variance as increasing the importance of the second term of the
considered performance function.
With the calculated values of settlement mean, variance and
standard deviation (root square of variance) in hands, the
probabilistic analysis can be made by setting the lognormal
distribution to represent the predict settlement and specifying
deterministic values to limiting settlement.
N
i
i
1
The lognormal distribution was used here for being a strictly
positive distribution, while having a simple relationship with the
normal distribution (Bredja et al. 2000, Fenton and Griffiths
2002, Goldsworthy 2006).
] [
] [
1
N
i
i
V
V
The methodologies assume the analysis of an isolated
footing. Nevertheless, if two non-correlated footings are being
evaluated, differential settlement can be obtained by:
(11
] [ ] [ ] [
2
1
(12
2.3
The MCS method
The Monte Carlo Simulation method consists basically in the
simulation of all random variables and the resolution of the
performance function for all those generated values. Here again,
deformability modulus is the only random variable.
As a simplification, is proposed a number of 1.000
simulations of modulus for each sublayer, using lognormal
distribution. The simulation can be done by using random
number generator algorithms for Microsoft Excel. The main
steps are summarized below:
Analysis of mean and variance of q
ci
results, for each
sublayer.
Estimation of mean and variance of E .
Si
Simulation of E
Si
(using mean, variance and lognormal
distribution).
Calculus of settlement mean and variance increment for
each sublayer.
Calculus of total settlement mean and variance.
Probabilistic settlement analysis using lognormal
distribution and an adopted limiting settlement value(s).
2.4
Evaluation of deformability modulus in the sublayers
In reliability analysis, independent random variables are
influenced by uncertainties and it must be appropriate
quantified. In the proposed methodologies, only one random
variable is adopted (E
Si
) for each sublayer.
...
)³
( ³
!3
1 )²
( ²
!2
1
)
(
!1
1 ] [
] [
3
2
XX
x
G
XX
x
G
XX
x
G XG XG
E E E
] [ ] [ ] [
2
1
V V V
N
i
is
zi
zi
E
I
CC
E
1
2 1
*
] [
] [ .
] [
2
1
2
*
2 1
is
N
i
is
iz iz
EV
E
I
CC V
