Environmental Engineering Reference
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head and the undrained shear strength c u at 31 nodal points along the embedded pile length
below the seabed. The P H was assumed to be normally distributed, with a mean value of
421 kN and a coefficient of variation of 25%. Since the mean undrained shear strength c u
typically exhibits an increasing trend with depth, it was assumed that μ cu = 150 + 2z kPa.
The standard deviations of the 31 c u random variables were equal to 30% of their respective
mean values. The following established negative exponential model was adopted to model
the spatial variation of the c u values:
(
Depthi
()
Depthj
() /
)
δ
ρ
=
e
(9.8)
ij
The β index obtained was 1.514 with respect to yielding at the outer edge of the annular
steel cross-section.
At the most probable failure point (where the ellipsoid touches the LSS), the value of the
lateral load at pile head is P H = 580.3 kN, that is, at 1.513σ PH from the mean value of P H ,
while the 31 autocorrelated c u values deviate only very slightly from their mean values. This
is not surprising given the e = 26 m cantilever length above the seabed; in fact, the maximum
bending moment occurs at a depth of only 1.36 m below the seabed, or 27.36 m from the
pile head. Hence, for the case in hand, pile yielding caused by bending moment is sensitive to
the applied load at pile head and not sensitive to the uncertainty of the shear strength below
the seabed. However, separate reliability analysis for cases where the lateral load acts on the
pile head near the ground surface (with zero cantilever length) indicates that the response is
sensitive to both the lateral load at pile head and the soil-shearing resistance within the first
few meters of the ground surface. The different sensitivities from case to case are automati-
cally reflected in reliability analysis aiming at a target index value (presented next), but will
be difficult to consider in codes based on partial factors.
9.7.1 Illustrative example of multicriteria
rbD of a laterally loaded pile
The steel tubular pile that forms part of a pile group in a breasting dolphin ( Figure 9.12 )
has been examined probabilistically above based on the single performance function of the
bending failure mode. The reliability index is only 1.514. In design, a reliability index of
2.5 or 3.0 is often stipulated. Further, deflection criterion also needs to be considered. In
the following illustrative design example, it is assumed that 1.4 m is the maximum tolerable
pile head deflection, which means a secant-tilt angle of about 1.4 in 26, or about 3°, with
respect to the seabed.
Analysis using mean parametric values (including μ cu = 150 + 2z kPa) results in a pile
head deflection of 0.986 m (slightly smaller than the 0.994 m of the above-mentioned deter-
ministic case for which a c u value of 150 kPa was used as in Tomlinson (1994)). This aver-
age deflection of 0.986 m does not provide information on the reliability of not exceeding
the 1.4 m tolerable limit, because the uncertainties of the random variables have not been
reflected in estimates based on mean values.
To illustrate multicriteria RBD, suppose it is desired to select the external diameter d and
steel wall thickness t of the tubular pile so as to achieve a reliability index of 3.0 with respect
to both the pile head deflection limit state and pile-bending moment limit state. The mean
and covariance structure of P H and c u profile are as in the previous section. Pile embedment
length is 23 m. Note that the external diameter d and wall thickness t affect both ulti-
mate limit state and serviceability limit state functions, by affecting the moment of inertia
I , the Matlock p-y curves ( Figure 9.12b , in which B = d = 1.3 m), and the yield moment
M y = 2σ y I /d, where σ y is the yield stress (417 MPa) of high-tensile alloy steel.
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