Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 169

Table 3.15 Comparison of the Taylor Series method assuming normal and lognormal distribution
of the factor of safety with the Monte Carlo methods for the retaining wall example

Deterministic factor
of safety

Probability of
failure (%) Failure mode Probability method

Sliding on sand 1.40 Monte Carlo 2.4

Taylor Series, normal dist. of F 5.5

Taylor Series, lognormal F 3.5

Sliding in clay 1.95 Monte Carlo 2.2

Taylor Series, normal dist. of F 2.8

Taylor Series, lognormal F 0.6

Bearing capacity 1.97 Monte Carlo 1.8

Taylor Series, normal dist. of F 1.5

Taylor Series, lognormal F 0.2

The results from the Taylor Series method assuming normal and lognormal distributions
for the factor of safety are compared to the results of the Monte Carlo method in Table 3.15.
None of the Taylor Series results agree very well with the Monte Carlo results, and in the case
of the bearing capacity failure mode, they differ by nearly an order of magnitude.
Thus, while the Taylor Series method offers a relatively easy way to compute prob-
abilities of failure, the results can differ significantly from the results of the Monte Carlo
method, which is considered to be a better method. In addition, the Taylor Series method
requires that the form of the safety factor distribution be assumed, and there is no logical
way to determine the best form of this distribution. It seems reasonable to conclude that
the strongest point in favor of using the Taylor Series method is its simplicity, but prob-
abilities of failure computed this way should be viewed as rough estimates that may be
higher or lower than values computed by better methods—the Monte Carlo method and
the Hasofer Lind method.

3.10.1 Summary of the taylor Series method
The steps involved in using the Taylor Series method are as follows:

1. Determine the most likely values of the parameters involved, and compute the factor
of safety by the normal (deterministic) method. This is F M LV .

2. Estimate the standard deviations of the parameters that involve uncertainty.

3. Compute the factor of safety with each parameter increased by one standard devia-
tion and then decreased by one standard deviation from its most likely value, with
the values of the other parameters equal to their most likely values. This involves
2N calculations, where N is the number of parameters whose values are being var-
ied. These calculations result in N values of F + and N values of F − . Using these
values of F + and F − , compute the values of Δ F for each parameter, and compute the
standard deviation and COV of the factor of safety (σ F and V F ) using the following
equations:

2 2 2 2

∆ F ∆ F ∆ F ∆ F

+ + +

(3.31)

σ F =

1 2 3 4

2 2 2 2