## 3 CPT-Based Simplified Liquefaction Procedure and Consolidation Settlement Model  
### CPT-Based Liquefaction Probability Assessment Framework  
A CPT-based liquefaction probability framework was proposed by [PERSON] and [PERSON] [26] and subsequently updated by [PERSON] [27]. The ratio of soil liquefaction resistance (expressed as the cyclic resistance ratio (\(CRR\))) to seismic demand (expressed as the cyclic stress ratio (\(CSR\))) was quantified using the factor of safety (\(FS\)) for soil liquefaction:  
\[FS=\frac{CRR}{CSR} \tag{1}\]  
When \(FS<1\), liquefaction occurs at the assessment site for a given seismic scenario, \(CRR\) provides the soil resistance, which is a function of the equivalent cone penetration resistance in clean sand (\(\left(q_{\mathrm{c1N}}\right)_{cs}\)) and can be expressed as:  
\[CRR=\left\{ \begin{array}{l}93\Big{[}\frac{\left(q_{\mathrm{c1N}}\right)_{cs }}{1000}\Big{]}^{3}+0.08,\qquad 50\leq\left(q_{\mathrm{c1N}}\right)_{cs}<160\\ 0.833\Big{[}\frac{\left(q_{\mathrm{c1N}}\right)_{cs}}{1000}\Big{]}+0.05,\quad \left(q_{\mathrm{c1N}}\right)_{cs}<50\end{array} \right. \tag{2}\]  
where \(\left(q_{\mathrm{c1N}}\right)_{cs}=K_{\mathrm{c}}q_{\mathrm{c1N}}\) is a function of soil particle properties. \(K_{\mathrm{c}}\) is a correction factor, which can be expressed as a soil behavior type index (\(I_{\mathrm{c}}\)). \[K_{\mathrm{c}}=\left\{ \begin{array}{l}1.0,\qquad\qquad\qquad\qquad I_{ \mathrm{c}}\leq 1.64\\ -0.403I_{\mathrm{c}}{}^{4}+5.581I_{\mathrm{c}}{}^{3}-21.63I_{\mathrm{c}}{}^{2}+ 33.75I_{\mathrm{c}}-17.88,\quad I_{\mathrm{c}}>1.64\end{array} \right. \tag{3}\]  
where \(I_{\mathrm{c}}=\left[\left(3.47-\log Q\right)^{2}+\left(1.22+\log F\right)^{2 }\right]^{0.5}\), \(Q=\left[\left(q_{c}-\sigma_{v_{0}}\right)/P_{a}\right]\)\(\left[\left(P_{a}/\sigma_{v_{0}}^{\prime}\right)^{n}\right]\), and \(F=\left[f_{s}/\left(q_{c}-\sigma_{v_{0}}\right)\right]\times 100\%\). \(\sigma_{v_{0}}\) is the total vertical stress, \(q_{\mathrm{c1N}}\) is the normalized cone penetration resistance \(q_{\mathrm{c1N}}=\left(P_{a}/\sigma_{v_{0}}^{\prime}\right)^{n}\left(q_{c}/P_{ a}\right)\), \(P_{a}\approx 0.1\) MPa, and \(\sigma_{v_{0}}^{\prime}\) is the effective vertical stress and the index \(n=0.381\left(I_{\mathrm{c}}\right)+0.05\left(P_{a}/\sigma_{v_{0}}^{\prime} \right)-0.15\). Combined with that suggested by [PERSON] et al. [28], the simplified equation for \(CSR\) is as follows:  
\[CSR=0.65\bigg{(}\frac{a_{max}}{g}\bigg{)}\bigg{(}\frac{\sigma_{v_{0}}}{\sigma_ {v_{0}}^{\prime}}\bigg{)}r_{d}\frac{1}{K_{\sigma}}\frac{1}{MSF} \tag{4}\]  
where \(a_{max}\) represents the peak horizontal acceleration at the surface, \(g\) represents the acceleration of gravity, and \(K_{\sigma}=\left(\sigma_{v_{0}}^{\prime}/P_{a}\right)^{\left(f-1\right)}\) represents the overburden correction factor, assuming a determination factor of \(f=0.7\), and the quantified equation for the magnitude scaling factor as \(MSF=\left(M_{w}/7.5\right)^{-2.56}\). The stress reduction coefficient (\(r_{d}\)) is a function of depth \(z\):  
\[r_{d}=\frac{\left(1.0-0.4113z^{0.5}+0.04052z+0.001753z^{1.5}\right)}{\left(1.0- 0.4177z^{0.5}+0.05729z-0.006205z^{1.5}+0.001210z^{2}\right)} \tag{5}\]  
The simplified procedure for \(CSR\) is subject to significant uncertainty, but the calculated \(CSR\) is used as a reference for developing the \(CRR\) through site case calibration, attenuating this uncertainty [29]. Therefore, to obtain a reliable spatial distribution of liquefaction potential over the whole cross-section, 2D BCS is used to model the \(CRR\) within the cross-section from sparse CPT measurement data and to estimate the spatial distribution of liquefaction potential. [PERSON] et al. [25] developed a probabilistic version of [PERSON]'s method, denoted as:  
\[P_{L}=1-\Phi\bigg{[}\frac{0.102+\ln(FS)}{0.276}\bigg{]}\approx\bigg{[}\frac{1 }{1+\left(FS/0.9\right)^{6}}\bigg{]} \tag{6}\]  
where \(P_{L}\) represents the probability of liquefaction and \(\Phi\) represents the standard normal cumulative distribution function.