Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 192

Table 4.3 Four commonly used generalized linear models
Model

Link function Equation for calculating P L

Logistic

1

P

L

P

hP () ln

=

=

L

{ }

L

1

−

P 1

+

exp(

− ++++

bbxbx b x )

L 0 1 1 2 2 rr

Probit

P Φ ( b bx bx bx )

hP ()

Φ 1

() P

= + +++

=

L 0 1 1 2 2 r r

L L

Log-log

[ ]

{

[ ]

}

hP () ln ln() P =− −

P exp xp ( b bx bx bx )

= − −++++

L L

L 0 1 1 2 2 r r

C-log-log

[ ] [ ]

hP () ln ln( 1 P ) P 1 exp xp( b bx bx bx )

=−− =− − + +++

L L L 0 1 1 2 2 r r

liquefaction, P L , under the assumption that ln[ P L /(1 − P L )] is a linear function of explanatory
variables (e.g., Liao et al. 1988). Extending this idea, let h ( P L ) denote a function of P L and
assume h ( P L ) is a linear function of the explanatorily variables:

hP ()=+ + bbxbx + + b x

(4.21)

L 0 1 1 2 2 r r

where b 0 , b 1 , …, b r are the regression coefficients to be calibrated. Based on Equation 4.21 ,
P L can be calculated as follows:

Phbbxbx =

−1

( + + + + b x ) (4.22)

L 0 1 1 2 2 r r

where h −1 () is the inverse function of h (). The expressions for calculating P L based on various
generalized linear models are also shown in Table 4.3.
As noted previously, whether a soil will liquefy or not is determined by the load on the
soil and the resistance of the soil against liquefaction. The load on the soil that can cause
liquefaction is represented by CSR . The resistance of the soil to liquefaction, expressed as
CRR , may be assessed with such in situ tests as the SPT, CPT, or shear wave velocity (V s )
measurements (e.g., Youd et al. 2001; Juang et al. 2002).
The explanatory variables in Equation 4.21 depend on the type of in situ test data used
to characterize the liquefaction resistance of a soil. As an illustration, the liquefaction prob-
ability is predicted herein based on CPT data. In the simplest form, CRR is a function of
the clean-sand equivalence of normalized cone tip resistance, denoted as q t1N,cs (Robertson
and Wride 1998; Robertson 2009). Note that historically, the normalized cone tip resistance
( q t1N ), defined in Appendix I, was used for assessing liquefaction resistance CRR of clean
sand (i.e., sands with less than 5% of fine-grained material). In other words, CRR model
(see Appendix I) was first created for clean sands. To expand the use of this CRR model to
soils with high fines content (FC), the normalized cone tip resistance of such soils is first
converted into its clean-sand equivalence, denoted as q t1N,cs . Thus, q t1N,cs and CSR can be
used as explanatory variables in the analysis of liquefaction case histories. The equations
used to calculate q t1N,cs and CSR that are recommended by Robertson and his colleagues are
summarized in the Appendix I.
Generalized linear models are most appropriate when the explanatory variables follow
the normal distribution (e.g., Hoffmann 2004). For an accurate evaluation of liquefaction
probability, transformations can be applied to q t1N,cs and CSR such that the transformed
variables are largely normal. Based on previous research (e.g., Liao et al. 1988; Toprak et al.
1999; Juang et al. 2002, 2003, 2006; Lai et al. 2006), four liquefaction models are investi-
gated here using q t1N,cs and ln( CSR ) as explanatory variables. Note that ln( CSR ), in lieu of