Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 206

suggested by Zhang et al. (2002) by including a likelihood of soil liquefaction (Juang et al.
2013). Zhang et al. (2002) coupled the CPT-based method by Robertson and Wride (1998)
with the volumetric strain relationship defined by Ishihara and Yoshimine (1992) to provide
a design chart for estimating the volumetric strain ε v . Through curve fitting, the chart by
Zhang et al. (2002) is approximated with the following equation (Juang et al. 2013):

ε v (%) =

‡

‰
‰
‰

0 if F ≥ 2

S

+
−−+

aa q
Fa

ln() 1

0 1

min

aq bb q ln( )) ,

+ ln( ) + bbq ln()

2

if 2 − << F 2

ˆ

12 ( ) (

0 1 2

aa q + ln()

S

‰
‰
‰

S 2 3 2 3

1

bb qb q + ln() + ln()

2

if F ≤≤− + 2

0 1 2 S

aa q ln()

Š

2 3

(4.33)

where

a = 0 3773 . , a = − 0 0337 . , a = 1 5672 . , a =− 0 1833 . , 1

0 2 3

b = 28 45 ., b = − 93 .372 , b = 0 7975 . , 1

0 2

q tNcs =

in kg/cm

2

( ≈ 100 kPa ).

,

1 ,

where F S = factor of safety calculated using the Robertson and Wride method.
As it is difficult to predict with certainty whether the soil will liquefy, the liquefaction
indicator I i in Equation 4.32 is modeled as a binomial random variable. Let P Li denote the
probability of liquefaction of layer i . Based on the property of a binomial distribution (e.g.,
Ang and Tang 2007), the first two moments of I i can be calculated as follows: E[ I i ] = P Li and
Var[ I i ] = P Li (1 − P Li ). According to Equation 4.31 , P L can be calculated based on the factor
of safety F S computed with the Robertson and Wride method.
In Equation 4.33 , deterministic nominal values are adopted for q t1N,cs and F S . Hence, ε vi in
Equation 4.32 is also a deterministic value. Thus, I is the only random variable in Equation
4.32 . Based on Equation 4.32 , the mean of s p can be determined as follows:

N N N

∑ ∑ ∑

µ = E ε ⋅ ∆ z ⋅ I = ε ⋅ ∆ z ⋅ EI [] = ε ⋅ ∆ z ⋅ P (4.34)

p vi i i vi i i vi i i

i = 1 i = 1 i = 1

Further, if the indicator functions ( I i , i = 1, …, N ) are assumed independent from each
other, then the variance of the predicted settlement can be determined as follows:

N N N

∑ ∑ ∑ ⋅

2 2 2 2 2

σ = Var ε ⋅ ∆ zI ⋅ = ε ⋅ ∆ z ⋅ Var [] I = ε ∆ zP ⋅ ( 1 − P ) (4.35)

p vi i i vi i i vi i i Li

i = 1 i = 1 i = 1

As the simplified modeling assumptions are involved, it is reasonable to expect that
Equation 4.32 is not perfect. To consider the model error in settlement prediction, a model
bias factor α can be applied to Equation 4.32 as follows:

s =α s

(4.36)

a p