Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 170

σ
MLV

F

V = (3.32)

F

F

4. Use the value of F M LV from step 1 and the value of V F from step 3 to determine the
value of P f , by means of Table 3.4 or by determining the reliability index and the
'NORMSDIST' function in Excel.

F − 1

mean

β = (3.37)

NORMAL

σ

FS

F

MLV
COV
COV

ln

(3.11)

1
1

+
+

( )

2

F

β Lognormal =

ln( ( ))

2

F

P f =− 1 NORMSDIST( β

(3.8)

3.11 PeM WIth a norMal DIStrIbutIon
For the FaCtor oF SaFetY

PEM is also a FOSM reliability method. The method was introduced by Rosenblueth (1975)
and its use in geotechnical engineering is discussed by Baecher and Christian (2003), Harr
(1987), and Wolff (1996). This method requires 2 N calculations of the factor of safety where
N is the number of variables. The factor of safety is calculated using combinations of vari-
ables where each is a standard deviation above or below the mean. These combinations of
variables are used to calculate the factor of safety. A table after Harr (1987) is shown below
that aids in the factor of safety calculations. On the table the (−) and (+) signs indicate the
mean value minus one standard deviation and the mean value plus one standard deviation.
For example, if only one variable has uncertainty in the calculation for the factor of safety
the number of cases to consider is 2 1 = 2. The first case would be to calculate the factor of
safety with the mean value of the variable minus the standard deviation. The second case
would be to calculate the factor of safety with the mean value of the variable plus the stan-
dard deviation.
The same cantilever retaining wall example for the factor of safety against sliding used for
the Taylor Series method will be presented here for the PEM assuming a normal distribution
for the factor of safety.
The PEM follows these steps:

1. Estimate the standard deviations of the quantities involved in Equation 3.20 . This is
the same first step as the Taylor Series Method. The values of the standard deviations
that will be used are

σγ eq = standard deviation of the equivalent fluid pressure = 1.06 kN/m 3 ,

σ tan δ = standard deviation of tan δ = 0.05, and

σγ bf = standard deviation of the unit weight of backfill = 0.565 kN/m 3 .

2. Use the point estimate technique (Rosenblueth 1975) to estimate the standard devia-
tion of the factor of safety. This is done by using Table 3.16 after Harr (1987) to
generate variables that are the mean plus or minus the standard deviation. With three