Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 165

3.9 taYlor SerIeS MethoD WIth aSSuMeD norMal
DIStrIbutIon oF the FaCtor oF SaFetY

The Taylor Series method has been described by Duncan (2000), Baecher and Christian
(2003), Ang and Tang (2007), and Harr (1987). The method is simple, easy to understand,
and easy to apply.
The Taylor Series method is a “first-order second moment” (FOSM) analysis. Only the
first two “moments” (the mean and the standard deviation) are considered in the analysis.
The application of the Taylor Series method in geotechnical engineering has been described
by Wolff (1994), U.S. Army Corps of Engineers (1997), Duncan (2000), and by Baecher and
Christian (2003). The method requires an assumption on the distribution of the factor of
safety. When using the Taylor Series method, 2N + 1 calculations of the factor of safety are
required, where N is the number of variables.
The method consists of two main parts:

1. Use the Taylor Series method to compute the COV of the computed result (e.g., factor
of safety or settlement), and

2. Assume a normal or lognormal distribution of the computed result, to determine the
probability of failure.

The method is illustrated here by application to the cantilever retaining wall shown in
Figure 3.18 . This example was previously analyzed using the Monte Carlo and Hasofer Lind
methods. Here, the probability of failure will be computed assuming first a normal distribu-
tion for the factor of safety and second a lognormal distribution of factor of safety. For the
conditions shown in Figure 3.18 , the value of the safety factor is 1.40, as shown on the top
of Table 3.13. In the realm of probabilistic analyses, this is called the “most likely value” of
factor of safety, F M LV .
To calculate the probability of failure using the Taylor Series method, the following steps
are used:

1. Estimate the standard deviations of the quantities involved in Equation 3.20. Simple
methods for estimating standard deviations have been discussed previously in this
chapter. Using those methods, the following values of standard deviation of the param-
eters involved in this example have been estimated:

σγ eq = standard deviation of the equivalent fluid pressure = 1.06 kN/m 3 ,
σtanδ = 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 Taylor Series technique (Wolff, 1994; U.S. Army Corps of Engineers 1997;
Duncan 2000; and by Baecher and Christian, 2003) to estimate the standard deviation
and the COV of the factor of safety using these formulas:

2 2 2

∆ F ∆ F ∆ F

+ +

(3.31)

1 2 3

σ F =

2 2 2

σ

F

V = (3.32)

F

F

MLV

in which ∆ FFFF
1 1 1 1 is the factor of safety calculated with the value of the

first parameter (in this case, the equivalent fluid pressure) increased by one standard

=− (

+ −

).

+