Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 306

Table 7.2 Summary of Bayesian analysis for the polynomial function example
X 2 value

1.14 1.16 1.18

X 2 bin j (1.13, 1.15] (1.15, 1.17] (1.17, 1.19]

Failure sample number n j within the jth bin (i.e., x 2 ∈ bin j ) 166 1574 1562

0.02999 0.02224 0.01585

P ( x 2 ∈ bin j ) using Equation 7.16

166/100,000 1574/100,000 1562/100,000

P ( x 2 ∈ bin j | F ) P ( F ) using Equation 7.17

P ( F | x 2 ∈ bin j ) using Equation 7.15 0.055 0.708 0.985

Note : Total sample number n t = 100,000.

corresponding X 2 values. The minimum and maximum X 2 values are found to be 1.13
and 1.46, respectively. Consider estimating the P ( F ) value when the X 2 value is taken as a
deterministic value (e.g., 1.14, 1.16, or 1.18 in Table 7.2 ) and X 1 remains random. A rela-
tively small bin size (e.g., 0.02) is first selected, such as (1.13, 1.15] for 1.14 in the second
column of Table 7.2 . The failure samples that fall within the corresponding bins are then
identified from the 6213 failure samples. The number of failure samples within each bin
is subsequently counted as 166, 1574, and 1562 for the X 2 value of 1.14, 1.16, or 1.18,
respectively (see the third row in Table 7.2 ) . Equations 7.16 and 7.17 are used to calculate
Px

(

2 ∈

bin j ) and Px (

2 ∈

binFPF | )()( = nn / ) in the fourth and fifth rows, respectively.

j j t

Note that Px (

2 ∈

bin j ) is calculated analytically without any information from MCS. For

instance, Px ∈ = − where CDF in this example is a

Normal distribution CDF with a mean and standard deviation of 1 and 0.1, respectively.
Finally, Equation 7.15 is used to estimate the P ( F ) value at a given deterministic X 2 value,
such as PF

( ( 113115 . , . ]) CDF (. ) 115 CDF ( . , 1 13

2

= ≈ ∈ = in the sixth row of Table 7.2 .
Figure 7.2 shows a X 2 histogram from the 6213 failure samples, together with the nomi-
nal (unconditional) probability distribution of X 2 (i.e., a normal probability distribution
with a mean and standard deviation of 1 and 0.1, respectively). The majority of the 6213
failure samples has X 2 values larger than 1.13, leading to a histogram peaking at the upper
tail of the nominal distribution. The failure sample distribution of X 2 and its nominal prob-
ability distribution are quite different, indicating that the effect of X 2 on failure probability
is significant. This is consistent with the ranking obtained from the hypothesis tests (see the
last row in Table 7.1 ) . In contrast, Figure 7.3 shows a X 1 histogram from the 6213 failure

(| x 114 .) P Fx (| (. ,. ]) 1 13 1150055 .

2 2

1800 10%

Failure sample frequency
Nominal distribution %

1500

8%

1200

6%

900

4%

600

2%

300

0 0%

0.8 0.9 1 1.1 1.2 1.3

X 2 bin

Figure 7.2 Histogram of the X 2 failure samples from direct MCS.