1298
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
functions are known from literature, from which the functions
that adopt an exponential shape are commonly used. In this case
the following expression is used (Breysse, 2004 and DeGroot,
1993):
c
Ld
z
z
e
/
)2 ln( );1 ln(
(8)
In which:
d
= horizontal distance between two springs [m]
L
c
= autocorrelation length of ln(z) [m]
The covariance can be determined according to (CUR190,
1997):
)
))
ln( ),
(ln(
2 ln(
)1 ln(
)2 ln( );1 ln(
2
1
z
z
z
z
z
z
Cov
(9)
From this the covariance matrix
C
can be constructed.
4.3.3
Autocorrelation length
The autocorrelation length L
c
can be interpreted as the
distance over which a certain parameter is significantly
correlated. In literature several indicative values for the
horizontal and vertical correlation length for soil parameters are
given. In this case especially the horizontal correlation length is
relevant. Typical values for the horizontal autocorrelation length
for soil properties are in the range L
c
≈ 20 to 100 m (DeGroot,
1993; TAW, 2001 and Gruijters, 2009).
Before determining the autocorrelation length the influence
of this parameter is checked. If L
c
→ 0, the logarithm of the
settlement at two locations is independent. Because of the
averaging effect of the stiff foundation, the rotation is expected
to approach the value as found in a deterministic approach, e.g.
zero rotation if a homogeneous soil is modelled. If L
c
→ ∞, the
logarithm of the settlement at two locations is fully correlated.
In this case the rotation is also expected to approach the value as
found in an deterministic approach, e.g. zero rotation if a
homogeneous soil is modelled. The maximum rotation is found
for an intermediate value of L
c
, typically half the foundation
size.
For the deformation analysis especially the spatial variation
of the compressibility of the different soil layers and the
thickness of the clay layer are important. From the soil
investigation it turned out that the horizontal correlation length
with respect to the thickness of the clay layer is typically in the
order of 10 to 20 m. However, in general the horizontal
correlation length with respect to soil properties is typically in
the order of 50 to 100 m. Therefore the most critical value for
the horizontal correlation length within the range between 10 to
100 m was selected. In this case a correlation length of 20 m has
been used.
4.4
Step 4 - Realisations of spring values
For the probabilistic analysis a Monte Carlo (MC) procedure is
used (CUR190, 1997 and Haugh, 2004). To generate correlated
values for the spring values an algorithm in a spreadsheet
program was set up. The following procedure is applied for
each realisation 1 to
n
:
1)
Generate a vector with realization of the standard normal
distribution
X
N(0,
I
)
. In which
I
is the identity matrix
and the size of the vector is equal to the number of springs
s = I.J
2)
Decompose the covariance matrix
C
ln(z)
=
A
.
A
T
(Cholesky
decomposition (Haugh, 2004))
3)
Determine the correlated vector
Z
’ =
A
.
X
+
μ
ln(z)
N(μ
ln(z)
,C
ln(z)
)
4)
Determine the vector with correlated settlement values
Z
i;j
=
exp(Z
i;j
’)
5)
Determine the vector with the correlated spring values from
eq. (1)
4.5
Step 5 – Determining settlement and rotation foundation
Since an infinitely stiff foundation is assumed in step 1, the
rotation of the foundation can be determined from the vertical
force equilibrium and the moment equilibrium in 2 directions:
I
i
J
j
ji
z
qLW R
F
1
1 .
(10)
ay
I
i
J
j
ji ji
y
M qLW
xR
M
_
2
1
1
.
.
21
(11)
ax
I
i
J
j
ji ji
x
MWqL
yR
M
_
1
1
2
.
.
21
(12)
In which:
R
i;j
= force in spring
S
i;j
[kN]
x
i;j
= x coordinate of spring
S
i;j
[m]
y
i;j
= y coordinate of spring
S
i;j
[m]
M
y_a
= acting bending moment around the y axis [kNm]
M
x_a
= acting bending moment around the x axis [kNm]
The force in every spring can be determined according to:
ji ji
ji
uk R
.
;
;
(13)
In which:
u
i;j
= deformation in spring
S
i;j
[m]
The deformation in every spring can be expressed as:
ji y
ji x
ji
y
x
u u
;
;
0;0
;
(14)
In which:
u
0;0
= deformation in the point
x
= 0,
y
= 0 [m]
θ
x
= rotation around the y axis, long axis [-]
θ
y
= rotation around the x axis, short axis [-]
These are exactly the variables of interest, which can be
filled in into the equilibrium equations. This leads to a system
of linear equations, which can be presented in matrix notation:
8
5
4
9 7 3
7 6 2
3 2 1
0;0
A
A
A u
A A A
A A A
A A A
y
x
(15)
Wherein the parameters A1 to A9 can be derived from eq.
(10), (11) , (12), (13) and (14).
The matrix equation can be solved by Cramer’s rule (Lay,
2003), which states that:
D D U
u
/
0;0
0;0
(16)
DD
x
x
/
(17)
DD
y
y
/
(18)
In which:
D
=
determinant of the coefficient matrix
D
u0;0
= determinant of the matrix formed by replacing the
u
0;0
column of the coefficient matrix by the answer
matrix
D
θx
= determinant of the matrix formed by replacing the
θ
x
column of the coefficient matrix by the answer
matrix
D
θy
= determinant of the matrix formed by replacing the
θ
y
column of the coefficient matrix by the answer
matrix
