Environmental Engineering Reference
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β
1
=2.7948
β
7
β
2
= 2.8933
β
3
β
6
β
8
Clay1
β
4
β
5
Clay2
n
1
/β
1
n
3
/β
3
n
2
/β
2
n
5
/β
5
n
4
/β
4
n
6
/β
6
n
7
/β
7
n
8
/β
8
-1.0000
-1.0000
-0.4535
-0.4539-0.4561-1.0000
-1.0000
-0.5314
0.0000
0.0000
-0.8912
-0.8910
-0.8899
0.0000
0.0000
-0.8471
ρ
of eight failure modes:
ρ
=
A
T
R
-1
A
1.0000
1.0000
0.4535
0.4539
0.4561
1.0000
1.0000
0.5314
β
1.0000
1.0000
0.4535
0.4539
0.4561
1.0000
1.0000
0.5314
0.4535
0.4535
1.0000
1.0000
1.0000
0.4535
0.4535
0.9960
0.4539
0.4539
1.0000
1.0000
1.0000
0.4539
0.4539
0.9960
0.4561
0.4561
1.0000
1.0000
1.0000
0.4561
0.4561
0.9963
1.0000
1.0000
0.4535
0.4539
0.4561
1.0000
1.0000
0.5314
1.0000
1.0000
0.4535
0.4539
0.4561
1.0000
1.0000
0.5314
0.5314
0.5314
0.9960
0.9960
0.9963
0.5314
0.5314
1.0000
2.7948 β
1
2.8367 β
3
2.8933 β
2
2.9024 β
5
2.9428 β
4
3.0467 β
6
3.1118 β
7
3.5388 β
8
System
P
f
bounds
Lower
0.416%
Upper
0.441%
Figure 9.11
System reliability analysis considering eight failure modes, including the two reliability-based
critical modes of
Figure 9.10
.
9.7 MultICrIterIa rbD oF a laterallY loaDeD PIle In
S PatIa llY autoCo r r e l ate D C l aY
Low et al. (2001) studied the deflection and bending moment of laterally loaded single piles
in which soil-pile interaction was based on the nonlinear and strain-softening Matlock
(1970)
p
-
y
curves
(
Figure 9.12
)
. The soil-pile interaction problem was solved using a rigor-
ous numerical procedure based on constrained optimization in the spreadsheet platform.
The numerical procedure was then extended to reliability analysis in which the Hasofer and
Lind index was computed. The soil resistance was modeled stochastically to reflect spatial
variation. Multicriteria RBD of a laterally loaded pile was also illustrated. The work was
further extended in Chan and Low (2012b) to a probabilistic analysis of laterally loaded
piles using response surface and neural network approaches. A case of deterministic analysis
followed by multicriteria RBD from Low et al. (2001) is presented in this section.
This deterministic problem in
Figure 9.12a
is described in Tomlinson (1994, Example
8.2). A steel tubular pile having an outside diameter
d
of 1.3 m and a wall thickness of
0.03 m forms part of a pile group in a breasting dolphin. The flexural rigidity
E
p
I
p
of the
pile is 4,829,082 kNm
2
. The pile is embedded 23 m in the stiff overconsolidated clay with
undrained shear strength
c
u
= 150 kN/m
2
and protrudes 26 m above the seabed. For the case
where a cyclic force of 421 kN is applied at 26 m above the seabed, only the embedded por-
tion of the pile requires soil-structure interaction analysis. The deflection of the pile head
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