Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 373

combinations of bolt length and spacing. For comparison, probabilities of failure are also
obtained from SORM, IS, and ordinary MCS.
The results of SORM and the averages of IS simulations in Figure 9.17 are in very good
agreement and indicate a probability of failure of about 0.4-0.5% depending on rock-bolt
length, compared with a FORM P f of about 0.6% based on β = 2.5. Engineers have to decide
whether for practical purposes (i) it is adequate (e.g., using a target FORM β of 2.5) that
the implied small probability of failure (0.6% by Equation 9.5 ) is approximate but similarly
small as the more accurate probability of failure (about 0.45% for the cases in Figure 9.17 ) ,
or (ii) engage in more precise RBD via SORM or IS for a target probability of failure. If the
latter is the desired action, a practical approach is as follows. In the light of the comparison
of FORM P f of 0.62% versus actual P f of 0.45% in Figure 9.17 , suppose a probability of
failure of 0.2% is the target P f for the case in hand, one may do a FORM design of the rock-
bolt length and spacing for a reliability index corresponding to a probability of failure (by
Equation 9.5 ) of (0.62%/0.45%)*0.2% = 0.28%, that is, β = Φ −1 (1 − 0.0028) = Excel func-
tion NormSInv(0.9972) = 2.77. For the case with rock-bolt length of 3 m, the required S z S θ
is found (via Excel Solver runs) to be about 0.67 m 2 for a target β = 2.77. This solution of
0.67 m 2 takes about a minute. To verify, with the input S z S θ = 0.67 m 2 and rock-bolt length
of 3 m, three MCS incorporating IS each of 10,000 random sets yielded (in 25 min) prob-
abilities of failure of 0.206, 0.199, and 0.203%, respectively; the average is practically the
target P f of 0.2%.
The above RBD procedure may be summarized as follows:

i. Conduct RBD based on the initial target reliability index β i of 2.5 or 3.0. The implied
FORM P f is Φ(-β i ), equal to 0.62 and 0.14%, respectively.

ii. For the design solution of (i), obtain more accurate P f using SORM or IS on the foun-
dation of FORM design point. This P f is denoted as P f , SORM .

iii. Decide on a target P f (<1%), and repeat the RBD based on an updated target reliability
index (β rev ) as follows:

Φ ( − β )

− 1 i

β = Φ 1 − × target P f (9.13)

rev

P , SORM

The final design solution is obtained via repeated FORM analyses using different designs
until the reliability index is equal to β rev . Linear interpolation between two designs and their
respective reliability indices can also be done. The final design based on β rev will have failure
probability close to the target P f .
If the design S z S θ is to be obtained by direct trial and error of different designs, each
checked against MCS (with or without IS), the computation time will be much longer than
the FORM-and-SORM approach involving Equation 9.13 .

9.10 ProbabIlIStIC SettleMent analYSIS oF a hong kong
trIal eMbankMent on SoFt ClaY

The Chek Lap Kok test reclamation fill on marine clay was part of a larger reclamation
project for the construction of a Hong Kong replacement airport (which started operation
in 1997). The objective of the test fill was to investigate the feasibility of reclamation over
soft marine clay and the effectiveness of vertical drains in accelerating consolidation. The
main test area, located about 200 m offshore and 100 m square in plan, was divided into