Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 368

denote the design point on the LSS and represent the most likely combination of parametric
values that will cause failure. The expected embedment depth is d = 12.15- 6.4 -μ z = 3.35 m.
At the failure combination of parametric values, the design value of z is z * = 2.9632, and
d * = 12.15- 6.4 - z * = 2.79 m. This corresponds to an “overdig” allowance of 0.56 m. Unlike
EC7, this “overdig” is determined automatically and reflects uncertainties and sensitivities
from case to case in a way that specified “overdig” cannot.
The nx column indicates that, for the given mean values and uncertainties, rotational
stability is, not surprisingly, most sensitive to ϕ′ and the dredge level (which affects z and d
and hence the passive resistance). It is interesting to note that at the design point where the
six-dimensional dispersion ellipsoid touches the LSS, both unit weights γ and γ sat (16.20 and
18.44) are lower than their corresponding mean values, contrary to the expectation that
higher unit weights will increase active pressure and hence greater instability. This apparent
paradox is resolved if one notes that smaller γ sat will (via smaller γ′) reduce passive resis-
tance, smaller ϕ′ will cause greater active pressure and smaller passive pressure, and that γ,
γ sat , and ϕ′ are logically positively correlated.
In an RBD (such as the case in Figure 9.13 ) , one does not prescribe the ratios mean / x *—
such ratios, or ratios of ( characteristic values )/ x *, are prescribed in EC7 limit state design—
but leave it to the expanding dispersion ellipsoid to seek the most probable failure point on
the LSS, a process that automatically reflects the sensitivities of the parameters. Besides, one
can associate a probability of failure for each target reliability index value. The ability to
seek the most probable design point without presuming any partial factors and to automati-
cally reflect sensitivities from case to case is a desirable feature of the RBD approach.

9.9 relIabIlItY analYSIS oF rooF WeDgeS
anD roCkbolt ForCeS In tunnelS

A symmetric roof wedge of central height h and apical angle 2α in a circular tunnel of radius
R is shown in Figure 9.14 . An analytical approach for assessing the stability can be based
on Bray's 1977 two-stage relaxation procedure, as described, for example, in Sofianos et al.
(1999) and Brady and Brown (2006). The first stage computes the confining lateral force
H 0 on the wedge from the stress field and the geometries of the wedge and tunnel, for an
assumed homogeneous, isotropic, linearly elastic, and weightless medium. The second stage
then assumes deformable joints and a rigid rock mass, to arrive at the normal force N acting
on each joint surface.

p

N N

h

αα

H 0 H 0

S

S

W

K 0 p

R

θ

x

O

Figure 9.14 Notation for symmetric roof wedge in a circular tunnel.