Practical risk assessment for embankments, dams, and slopes - Risk and Reliability in Geotechnical Engineering - page 446

[FS, P f ∗ ] exist such that Equations 11.6 and 11.7 are equivalent. The functional relationship
between the pair [FS, Pf∗] f ∗ ] has the form expressed in Equation 11.8 .

g () θ

=

[ ] −

n

PgZ , θ < 0 P

∗

(11.8)

f

FS

According to its definition, the normalized limit state function, denoted by G(Z,θ), can be
expressed by Equation 11.9.

θ= [ Z θ
θ

g
g

,]
()

GZ (,) (11.9)

n

And Equation 11.7 can be rewritten as Equation 11.10 .

g () θ

=

1

=

n

Pg [,] Z θ − < 0 P G [,] Z θ < P

∗

(11.10)

f

FS FS

This means that 1/FS is the (1 − Pf∗) f ∗ ) percentile of the random variable G. The relationship
between FS and P f ∗ can be determined using Monte Carlo simulation, drawing N samples
from Z and θ, and calculating the N values for G(i) (i) = G(Z (i) ,θ (i) ). For a chosen FS value, the
corresponding Pf∗ f ∗ value can be estimated with Equation 11.11 .

1
N

1
FS

≡

∑

ˆ

P

*

≈ IG

(i) <

P

∗

(11.11)

f f

Further details and discussion on this method can be found in the referenced paper
(Ching, 2009). The premise that the distribution of G(Z,θ) is invariant over the entire allow-
able design region D can be achieved only if the correct nominal limit state function g n (θ) is
found, which is usually a difficult undertaking, although finding a nominal limit state func-
tion for which the premise holds approximately is relatively simple, since g n (θ) = g(E(Z),θ) is
usually an acceptable choice.
It is important to note that for certain geotechnical problems, the FS can be defined in dif-
ferent ways. For instance, in shear problems (slope stability, sliding of a dam along its foun-
dation contact), the FS can be defined as the ratio between strength and stress, or resistance,
R, and loading, L. This is the case if limit equilibrium models are used ( Equation 11.12 ) .

R
L

(11.12)

FS =

On the other hand, it is common that if finite-element models are used, the FS is defined
as the ratio between a value of a strength parameter, φ, and the ultimate value before failure,
φ failure ( Equation 11.13 ) .

ϕ

FS = (11.13)

ϕ

failure