Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 202

1B C ' E F G H I J K

2
3
4
5

Spreadsheet Template for Calibrating Generalized Linear Models

Parameters to be estimated

Q p Q s w L w NL

b 0 b 1 b 2

6 21.932 -0.123 6.182 0.456 0.758 0.601583 2.247934

7
8

Observed data

M 1 in Table 4.4

CSR q t1n,cs

Case No. I Y P Li w i ln P ( d i |θ) L (θ| D )

9 1 1 0.40 42.50 11.00 0.999983 -1.008E-05 41.11 1

1
1
1
1
1
1
1
1

10 2 1 0.33 62.98 7.33 0.999343 -0.0003954

11 3 1 0.35 80.77 5.50 0.995923 -0.0024576

12 4 1 0.38 82.09 5.83 0.997074 -0.0017629

Log-likelihood

13 5 1 0.24 72.22 4.29 0.986523 -0.0081624

14 6 1 0.43 59.82 9.31 0.999909 -5.45E-05

15 7 1 0.29 47.83 8.41 0.999778 -0.0001337

16 8 1 0.42 54.90 9.77 0.999943 -3.438E-05

17 9 1 0.37 66.94 7.53 0.999463 -0.0003234

18 10 1 0.42 43.42 11.18 0.999986 -8.409E-06 1

169 161 0 0.29 187.62 -8.77 0.000155 -0.0003485 0

170 162 0 0.30 169.13 -6.31 0.001808 -0.004067

0

171 163 0 0.45 182.04 -5.33 0.004824 -0.0108703

0

172 164 0 0.39 161.39 -3.73 0.023489 -0.0534314 0

173 165 0 0.44 156.04 -2.26 0.094122 -0.2222098

0

174
175
176
177
178
179

Notes:
(1) To facilitate the spreadsheet implementation, an intemideiate variable Y is defined as follows:

Ybbq b SR =+ + ln( )

0 1 tNcs 1, 2 7.5, σ

(2) Rows 19 through 168 are skipped to save space.
(3) The setting in Solver is "Maximize the value in Cell J9 by changing the values in Cells H6, I6, and
J6, respectively."

Figure 4.6 Spreadsheet template for calibrating the generalized linear models.

potential. In this section, we want to show how a probabilistic version of this popular deter-
ministic model can be created. In the following example, we will illustrate how to develop a
P L - F S mapping function that relates the probability of liquefaction ( P L ) to the factor of safety
( F S ) computed with the deterministic Robertson and Wride method. The equations that are
required for computing CSR and CRR , and thus F S , using the deterministic Robertson and
Wride method are listed in Appendix I.

4.4.1 Probabilistic model
The semiempirical nature of the Robertson and Wride method ensures an inherent uncer-
tainty within this model. The calculated factor of safety F S can be related to the actual factor
of safety F Sa as follows:

F
z

S

F = (4.24)

Sa