Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 168

Table 3.14 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

Sliding in clay 1.95 Monte Carlo 2.2

Taylor Series, normal dist. of F 2.8

Bearing capacity 1.97 Monte Carlo 1.8

Taylor Series, normal dist. of F 1.5

of the footing, the difference is quite significant—more than a factor of two in the value of
probability of failure. These results show that the assumptions involved in the Taylor Series
method can result in considerable differences in results from the Monte Carlo method,
which is considered the better method.

3.10 taYlor SerIeS MethoD WIth a lognorMal
DIStrIbutIon oF the FaCtor oF SaFetY

If a lognormal distribution for the factor of safety is assumed, the same procedure is followed
until step 4. Then, equations are used to calculate the reliability index for a lognormal distri-
bution of the factor of safety. The assumption of a lognormal distribution for factor of safety
does not imply that the values of the individual variables (γ eq , tan δ, and γ bf ) must be distrib-
uted lognormally. The probability of failure using the Taylor Series method with a lognormal
distribution for the factor of safety is calculated using the following steps:

1. Same as for a normal distribution for the factor of safety

2. Same as for a normal distribution for the factor of safety

3. Same as for a normal distribution for the factor of safety

4. With both F M LV and V F known, the probability of failure can be determined as follows.
First, the reliability index is calculated using Equation 3.35.

F 1 400

10179

.
(.

MLV
COV
COV

ln ln

(3.35)

1
1

+
+

( )

2

+ )

2

F

β log = = = 181 .

normal

ln( )

2

ln( 110179 + . )

2

F

The probability of failure can be calculated using the NORMSDIST function in Excel
based on the reliability index using Equation 3.36.

P f (log =− 1 NORMSDIST ( β ) =− 1 NORMSDIST (. ) 1813 = . 53% (3.36)

normal ) log normal

The Taylor Series analysis was also used to compute the probability of failure by sliding
on the clay, and by bearing capacity failure in the clay. The standard deviation of the und-
rained shear strength of the clay was estimated to be 24 kN/m 2 . The calculated probability
of failure for sliding in the clay was found to be 0.56%, and the calculated probability of
failure for bearing capacity failure was found to be 0.17% when the factor of safety was
assumed to be lognormally distributed.