Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 371

(a)

FORM SORM

x * μ σ n Formula Pf (SORM)

κ i

-0.0326

Tvedt 0.39% 28.012 25 2 1.5059

α
ϕ

28.34 35 3 2.2199 -0.0011 Breitung 0.38%

0.0000

H & R 0.39% 0.100 0.1 .025 -0.0001

k s / k n

-0.0020

K & N 0.38% p 0.9975 1 0.1 -0.0245

1.4744 1.5 .375 -0.0681 C & E 0.39%

K 0

Hong 0.39% β Pf(FORM) Pf(Monte Carlo)

2.6835 0.36% 0.39% ±

(b)

FORM SORM

x * μ σ n Formula Pf(SORM) κ i

0.2952 -0.1528 Tvedt 34.13% 25.59 25 2

α
ϕ

0.2940 -0.1252 Breitung 31.08%

34.118 35 3

-0.0048

H & R 34.64% 0.100 0.1 .025 -0.0168

k s / k n

p 0.495 0.5 .05-0.0995 -0.0253

K & N 33.25%

C & E 33.93%

0.4526 0.5 .125 -0.3789

K 0

Hong 34.10%

β Pf (FORM) Pf (Monte Carlo)

0.5721 28.4% 33.6% ±

Figure 9.16 FORM and SORM reliability analysis of tunnel roof wedges: (a) R = 2 m, high p and K 0 ; (b)
R = 6 m, low p and K 0 .

α are the most sensitive of the five random variables. The insensitivity of wedge stability to
the in situ stress parameters p and K 0 in this case is due to the large mean N/W value of 2.779
MN/0.0391MN = 71. Given the large gripping force arising from in situ stresses, and in the
presence of the assumed parametric uncertainties for the five random variables (α, ϕ, k s /k n ,
p, and K 0 ), the most probable failure point (such as Figure 9.15 , but in the 5D space) is with
increasing α and decreasing ϕ. In contrast, Figure 9.16b has mean values of p = 0.5 MPa and
K 0 = 0.5, instead of the mean values of p = 1.0 MPa and K 0 = 1.5 used in Figure 9.16a , but the
same coefficients of variations (σ/μ) as Figure 9.16a . Also, R = 6.0 m instead of 2.0 m (but
h/R remains at 0.85). This reduces the value of mean N/W to 1.131MN/0.352MN = 3.214
(vs. 71 of Figure 9.16a) . The mean factors of safety are FS 1 = 1.36 and FS 2 = 1.10. The reli-
ability index is 0.572 (vs. 2.684 of Figure 9.16a) , and K 0 , with the largest numerical value of
n, is the most sensitive random variable of the five. The ability of the FORM reliability index
to reflect different parametric sensitivities and uncertainties from case to case without relying
on rigid partial factors is an important advantage of reliability analysis and RBD.
For the case in Figure 9.16a , the probability of roof wedge failure based on Equation 9.5 is
P f = 0.36%, compared with an SORM P f of about 0.39% using the Chan and Low (2012a)
spreadsheet codes. The SORM P f agrees very well with the P f values (0.39, 0.39, and 0.40%)
from three MCS of the wedge stability problem, each with 100,000 realizations using the
software @Risk ( http:/ / www. palisade. com ).
For the case in Figure 9.16b , the probability of roof wedge failure based on Equation
9.5 i s P f = 28.4%, compared with an SORM P f of about 33.5%. The SORM P f agrees very
well with the P f values (33.84, 33.73, and 33.37%) from three MCS, each with 50,000
realizations.