Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 357

exponentials. The five random variables are the shear strength parameters c and ϕ of the
failure surface (joint plane), tension crack depth z , height of water z w in the tension crack,
and the coefficient of horizontal earthquake acceleration α. The deterministic formulations
are in Hoek (2007). The FORM analysis invokes Excel Solver to automatically change the u
vector (initially zeros) so as to obtain the reliability index β and the design point x * via the n
vector and the u vector. The SORM analysis uses the Chan and Low (2012a) Excel spread-
sheet approach. Five Monte Carlo simulations (MCS), each with 500,000 trials yielded P (if
values within the range 2.24-2.28%. The FORM P (if is 2.96%, higher than the Monte Carlo
average P (if of about 2.25%. In contrast, the average SORM P (if on the basis of FORM β and
four estimated components of curvature at the FORM design point is 2.18% (as shown) if
the eight discrete points selected for curvature estimation correspond to k = 2 (coarser grid),
and 2.05% if k = 1. The Breitung result of 2.28% is the closest to the Monte Carlo P (if for this
case where the value of β is nearer to the practical higher design range of β.
There is no unique SORM P (if value. It depends on the method used for estimating the
curvatures at the design point and on the formula used to compute P (if based on FORM β
and the curvatures at the FORM design point. Nevertheless, in the practical high reliability
range ( P (if < 2%), the seven SORM formulas give consistent P (if values for the case in Figure 9.5
and are more accurate than FORM P (if . One can as a rule extend the FORM analysis into the
SORM analysis. Should the curvatures of the LSS turn out to be negligible, all the SORM
formulas will approach Equation 9.5 , with the result that the computed SORM probability
of failure will be the same as FORM probability of failure.
An alternative solution of the slope of Figure 9.5 was given in Li et al. (2011), using 378
collocation sampling points, obtaining a P (if of 2.30%.
It may be noted that n = Lu by Equation 9.4b , and hence, n ≠ u as shown in the top two
right-most columns of Figure 9.5 when some of the random variables are correlated.
The Low and Tang (2004) x space approach and the Low and Tang (2007) n space
approach obtain the same reliability index β and the design point x *. Hence, all three meth-
ods of Figure 9.2 are efficient for the case in hand.
9.3.2 Positive reliability index only if the mean-value point is in
the safe domain
In reliability analysis and RBD, one needs to distinguish negative from positive reliability
index. The computed β index can be regarded as positive only if the performance function
value is positive at the mean-value point. Although the discussions in the next paragraph
assume normally distributed random variables, they are equally valid for the equivalent
normals of non-normal random variables in FORM.
The five random variables of the rock slope in Figure 9.5 include the shear strength
parameters c and ϕ of the discontinuity plane that is inclined at ψ p . In the two-dimensional
schematic illustration of Figure 9.6 , the LSS is defined by performance function g ( x ) = 0.
The safe domain is where g ( x ) > 0, and the unsafe domain is where g ( x ) < 0. The mean-
value point of Case I is in the safe domain and at the center of a one-standard-deviation
dispersion ellipse, or ellipsoid in higher dimensions. As the dispersion ellipsoid expands, the
probability density on its surface diminishes. The first point of contact with the LSS is the
most probable failure point, also called the design point. The reliability index β of Case I is
therefore positive and represents the distance (in units of directional standard deviations)
from the safe mean-value point to the unsafe boundary (the LSS) in the space of the random
variables. The corresponding probability of failure Φ(−β) is <0.5. If the mean-value point
sits right on the LSS, the probability of failure is 0.5 (if the LSS is planar) because β = 0
when the β-ellipsoid reduces to a point on the LSS. In contrast, Case II's mean-value point