Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 205

AB C D E F G H I J K

2
3
4
5

Spreadsheet Template for Converting a Deterministic Model into Its
Probabil istic Versi on

Parameters to
be estimated

Q p Q s

w L w NL µ δ

6 0.601583 2.247934 0.914337 0.28116825 0.456 0.758

7
8

Observed data

Q 2 in Table 4.6

w i ln P ( d i |θ) L (θ| D )

Case No. I CSR 7.5 CRR F S P L

9 1 1 0.396 0.215 1 -1.013E-07 -47.3646

0.085 1

1
1
1
1
1
1
1
1

10 2 1 0.329 0.314 0.999906 -5.637E-05 0.103

11 3 1 0.348 0.371 0.99914 -0.0005174 0.129

Log-likelihood

12 4 1 0.377 0.349 0.999606 -0.000237 0.131

13 5 1 0.242 0.476 0.987077 -0.0078249

0.115

14 6 1 0.425 0.235 0.999999 -5.043E-07

0.100

15 7 1 0.290 0.310 0.999923 -4.618E-05 0.090

16 8 1 0.416 0.230 0.999999 -3.303E-07 0.095

17 9 1 0.367

0.108

0.294 0.999965 -2.088E-05

18 10 1 0.416

0.086

0.207 1 -4.786E-08

1

168 160 0 0.433

0.458

1.059 0.25156 -0.65137

0

169 161 0 0.288

0.694

2.407 0.000132 -0.0002974 0

170 162 0 0.297

0.530

1.782 0.005272 -0.0118826 0

171 163 0 0.451 1.423 0.040832 -0.0937148

0.641 0

172 164 0 0.387 1.215 0.121059 -0.290067 0.471

0

173 165 0 0.442 0.981 0.346507 -0.9563221 0.433

0

174
175
176
177

Notes:
(1) Rows 19 through 167 are skipped to save space.
(2) The setting in Solver is "Maximize the value in Cell J9 by changing the values in Cells H6 and I6,
respectively."

Figure 4.7 Spreadsheet template for calibrating the probabilistic model that is based on the Robertson and
Wride method.

and field case histories (e.g., Lee and Albaisa 1974; Tokimatsu and Seed 1984; Ishihara and
Yoshimine 1992; Zhang et al. 2002; Dashti et al. 2010). For a site with level ground, far
from any free water surface, it is reasonable to assume little or no lateral displacement
occurs after an earthquake. Thus, the volumetric strain is approximately equal to the verti-
cal strain. The liquefaction-induced settlement caused by a given earthquake can then be
determined as a summation of the product of the volumetric strain in each liquefied soil
layer and the corresponding depth, symbolically expressed as follows (Juang et al. 2013):

N

∑ ε
1

s = ⋅ ∆ z ⋅ I

(4.32)

p vi i i

i =

where s p = predicted settlement at the ground surface, N = total number of soil layers,
Δ z i = thickness of the i ith layer, ε vi = volumetric strain of the i ith layer, and I i = an indica-
tor of liquefaction occurrence in the i ith layer, which is equal to 0 if the i ith layer does not
liquefy and equal to 1 if the i ith layer liquefies. Equation 4.32 is an extension of the method