Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 372

For the multidimensional LSS of the roof wedge stability model above and the rock-
bolt-reinforced tunnel below, one may extend the FORM analysis into the SORM analysis.
Should the curvatures of the LSS turn out to be negligible, the κ i values in Figure 9.16a and
b will be practically zeros, and all the SORM formulas will reduce to Equation 9.5 , with
the result that the computed SORM probability of failure will be the same as FORM prob-
ability of failure. As a practical alternative, one may also note that RBD typically requires a
target β index of 2.5 or 3.0, corresponding to FORM P f of 0.62 and 0.14%, respectively, and
hence, some inaccuracy in the FORM P f (from Equation 9.5 , P f ≈ 1 - Φ(β)) is of no practical
concern because the correct P f will still be below 1%. For the case in Figure 9.16a , the P f of
0.36%, estimated based on FORM β of 2.68, is lower than the more accurate 0.39%, but
this discrepancy may not justify much practical concern.
Low and Einstein (2013) also investigated FORM design of the length and spacings S z S θ
of rockbolts for a reliability index β of 2.5 against bolt tension exceeding the bolt rup-
ture strength. The implicit mechanical model for the mobilized bolt force as formulated in
Bobet and Einstein (2011) was used. Four lognormally distributed random variables were
modeled, namely the three-dimensional (3D) effect parameter β σ , and the shear strength
parameters ϕ and c and modulus E of the rock mass, with a negative correlation coefficient
ρ of −0.5 between ϕ and c. Figure 9.17 shows the FORM probability of failure for different

σ o

Rockbolts

Rockbolt head

S o
q

ρ
r o

σ o

σ i

Grout length

Probability of failure

SORM P f

FORM P f
= Φ (- β )

Six simulations with
importance sampling
each 3000 random sets

Six Monte Carlo
simulations each
3000 random sets

Rock-bolt
length
( ρ - r 0 )

Required S z S θ
for β = 2.5

0.43%, 0.47%, 0.45%,
0.46%, 0.45%, 0.41%

0.50%, 0.33%, 0.40%,
0.40%, 0.57%, 0.47%

2 m

0.59 m 2

0.62% 0.44%

0.46%, 0.48%, 0.44%,
0.48%, 0.47%, 0.45%

0.57%, 0.53%, 0.60%,
0.40%, 0.43%, 0.50%

3 m

0.70 m 2

0.62% 0.45%

0.47%, 0.47%, 0.48%,
0.47%, 0.46%, 0.46%

0.37%, 0.60%, 0.73%,
0.53%, 0.30%, 0.47%

4 m

0.81 m 2 0.62% 0.47%

Figure 9.17 Required length and spacing of rockbolts for a target reliability index β of 2.5, and comparisons
of probability of failure based on FORM, SORM, importance sampling, and ordinary Monte
Carlo simulations.