An Analytical Approach to Probabilistic Modeling of Liquefaction Based on Shear Wave Velocity

Abstract

Evaluation of liquefaction potential of soils is an important step in many geotechnical investigations in regions susceptible to earthquake. For this purpose, the use of site shear wave velocity (V_s) provides a promising approach. The safety factors in the deterministic analysis of liquefaction potential are often difficult to interpret because of uncertainties in the soil and earthquake parameters. To deal with the uncertainties, probabilistic approaches have been employed. In this research, the jointly distributed random variables (JDRV) method is used as an analytical method for probabilistic assessment of liquefaction potential based on measurement of site shear wave velocity. The selected stochastic parameters are stress-corrected shear wave velocity and stress reduction factor, which are modeled using a truncated normal probability density function and the peak horizontal earthquake acceleration ratio and earthquake magnitude, which are considered to have a truncated exponential probability density function. Comparison of the results with those of Monte Carlo simulation indicates very good performance of the proposed method in assessment of reliability. Comparison of the results of the proposed model and a standard penetration test (SPT)-based model developed using JDRV shows that shear wave velocity (V_s)-based model provides a more conservative prediction of liquefaction potential than the SPT-based model.

Appendix

1.1 Derivation of Mathematical Functions K ₁ to K ₇ and FS and Their Domains is Presented in “Appendix”

$$f_{{K_{1} }} (k_{1} ) = f_{{V_{\text{S1}} }} \left( {L^{ - 1} (k_{1} )} \right) \times \left| {\frac{{{\text{d}}L^{ - 1} (k_{1} )}}{{{\text{d}}k_{1} }}} \right| = f_{{V_{{{\text{S}}1}} }} (V_{\text{S1}} ) \times \left| {\frac{1}{{\frac{{{\text{d}}K_{1} }}{{{\text{d}}V_{{{\text{S}}1}} }}}}} \right|$$

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$$L\left( {V_{{{\text{S1}}_{\hbox{max} } }} } \right) < k_{1} < L\left( {V_{{{\text{S}}1_{\hbox{min} } }} } \right)$$

$$\left\{ \begin{aligned} k_{{1_{\hbox{min} } }} = L\left( {V_{{{\text{S}}1_{\hbox{max} } }} } \right) \hfill \\ k_{{1_{\hbox{max} } }} = L\left( {V_{{{\text{S}}1_{\hbox{min} } }} } \right) \hfill \\ \end{aligned} \right.\text{ }$$

where

$$\left\{ {\begin{array}{*{20}l} {V_{{{\text{S1}}_{\hbox{min} } }} = V_{{{\text{S1}}_{\text{mean}} }} - 4\sigma_{{V_{\text{S1}} }} > 0} \hfill \\ {V_{{{\text{S1}}_{ \hbox{max} } }} = V_{{{\text{S1}}_{\text{mean}} }} + 4\sigma_{{V_{\text{S1}} }} } \hfill \\ \end{array} } \right.$$

V_S1mean is average value of stress-corrected shear wave velocity. σV_S1 is standard deviation of stress-corrected shear wave velocity. V_S1min is minimum value of stress-corrected shear wave velocity. V_S1max is maximum value of stress-corrected shear wave velocity.

$$f_{{K_{2} }} (k_{2} ) = f_{{{\text{r}}_{\text{d}} }} \left( {\frac{1}{{k_{2} }}} \right)\left| {\frac{\text{d}}{{{\text{d}}k_{2} }}\left( {\frac{1}{{k_{2} }}} \right)} \right| = \frac{1}{{\sigma_{{r_{\text{d}} }} \sqrt {2\pi } \cdot k_{2}^{2} }}\exp \left( { - 0.5\left( {\frac{{1 - r_{{{\text{d}}_{\text{mean}} }} \cdot k_{2} }}{{\sigma_{{r_{\text{d}} }} \cdot k_{2} }}} \right)^{2} } \right)$$

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$$r_{{{\text{d}}_{ \hbox{min} } }} \le r_{\text{d}} \le r_{{{\text{d}}_{ \hbox{max} } }} \text{ }\quad \text{ } \to \quad \text{ }\frac{1}{{r_{{{\text{d}}_{ \hbox{max} } }} }} \le k_{2} \le \frac{1}{{r_{{{\text{d}}_{ \hbox{min} } }} }}$$

$$\left\{ \begin{aligned} k_{{2_{\hbox{min} } }} = \frac{1}{{r_{{{\text{d}}_{ \hbox{max} } }} }} \hfill \\ k_{{2_{\hbox{max} } }} = \frac{1}{{r_{{{\text{d}}_{\hbox{min} } }} }} \hfill \\ \end{aligned} \right.\text{ }$$

$$\left\{ {\begin{array}{*{20}l} {r_{{{\text{d}}_{ \hbox{min} } }} = r_{{{\text{d}}_{\text{mean}} }} - 4\sigma_{{r_{\text{d}} }} > 0} \hfill \\ {r_{{{\text{d}}_{ \hbox{max} } }} = r_{{{\text{d}}_{\text{mean}} }} + 4\sigma_{{r_{\text{d}} }} } \hfill \\ \end{array} } \right.$$

r_dmean is average value of stress reduction factor. σ_rd is standard deviation of stress reduction factor. r_dmin is minimum value of stress reduction factor. r_dmax is maximum value of stress reduction factor.

$$f_{{K_{3} }} (k_{3} ) = f_{{M_{\text{w}} }} \left( {k_{3}^{{\text{ } - \frac{25}{64}}} } \right) \times \left| {\frac{\text{d}}{{{\text{d}}k_{3} }}\left( {k_{3}^{{\left( { - \text{ }\frac{25}{64}} \right)}} } \right)} \right| = \frac{{25 \times \lambda_{{M_{\text{w}} }} \cdot \exp \left( { - \lambda_{{M_{w} }} k_{3}^{{ - \text{ }\frac{25}{64}}} } \right)}}{{64 \times k_{3}^{{\left( {\frac{89}{64}} \right)}} \left( {\exp ( - \lambda_{{M_{\text{w}} }} M_{{w_{\hbox{min} } }} ) - \exp ( - \lambda_{{M_{\text{w}} }} M_{{w_{\hbox{max} } }} )} \right)}}$$

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$$M_{{{\text{w}}_{ \hbox{min} } }} \le M_{\text{w}} \le M_{{{\text{w}}_{\hbox{max} } }} \text{ } \to \text{ }\left( {M_{{{\text{w}}_{\hbox{max} } }} } \right)^{ - 2.56} \le k_{3} \le \text{ }\left( {M_{{{\text{w}}_{ \hbox{min} } }} } \right)^{ - 2.56}$$

$$\left\{ \begin{aligned} k_{{3_{\hbox{min} } }} = \text{ }\left( {M_{{{\text{w}}_{ \hbox{max} } }} } \right)^{ - 2.56} \hfill \\ k_{{3_{\hbox{max} } }} = \left( {M_{{{\text{w}}_{ \hbox{min} } }} } \right)^{ - 2.56} \hfill \\ \end{aligned} \right.\text{ }$$

where M_Wmin is minimum value of moment magnitude. M_Wmax is maximum value of moment magnitude. λ_Mw is rate of change in moment magnitude (rate parameter) = 1/β_Mw.β_Mw is scale parameter of moment magnitude.

$$f_{{K_{4} }} (k_{4} ) = f_{\alpha } \left( {\frac{1}{{k_{4} }}} \right)\left| {\frac{\text{d}}{{{\text{d}}k_{4} }}\left( {\frac{1}{{k_{4} }}} \right)} \right| = \frac{{\lambda_{\alpha } \cdot \exp \left( {\frac{{ - \lambda_{\alpha } }}{{k_{4} }}} \right)}}{{k_{4}^{2} .\exp \left( { - \lambda_{\alpha } \cdot \alpha_{\hbox{min} } } \right) - \exp \left( { - \lambda_{\alpha } \cdot \alpha_{\hbox{max} } } \right)}}$$

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$$\frac{1}{{\alpha_{\hbox{max} } }} \le k_{4} \le \frac{1}{{\alpha_{\hbox{min} } }}$$

$$\left\{ \begin{aligned} k_{{4_{\hbox{min} } }} = \frac{1}{{\alpha_{\hbox{max} } }} \hfill \\ k_{{4_{\hbox{max} } }} = \frac{1}{{\alpha_{\hbox{min} } }} \hfill \\ \end{aligned} \right.$$

where α_min is minimum value of earthquake acceleration ratio. α_max is maximum value of earthquake acceleration ratio. λ_α is rate of change in earthquake acceleration ratio (rate parameter) = 1/β_α.β_α is scale parameter of earthquake acceleration ratio.

$$f_{{K_{5} }} (k_{5} ) = f_{{K_{1} \times K_{2} }} (k_{5} ) = \int\limits_{\alpha }^{\beta } {\left| {k_{1} } \right|f_{{K_{1} }} (k_{1} )f_{{K_{2} }} \left( {\frac{{k_{5} }}{{k_{1} }}} \right){\text{d}}k_{1} }$$

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$$k_{{1_{\hbox{min} } }} k_{{2_{\hbox{min} } }} \le k_{5} \le k_{{1_{\hbox{max} } }} k_{{2_{\hbox{max} } }}$$

$$\left\{ \begin{aligned} k_{{5_{\hbox{min} } }} = k_{{1_{\hbox{min} } }} k_{{2_{\hbox{min} } }} \hfill \\ k_{{5_{\hbox{max} } }} = k_{{1_{\hbox{max} } }} k_{{2_{\hbox{max} } }} \hfill \\ \end{aligned} \right.\quad {\text{and}}\quad \left\{ \begin{aligned} \alpha = \hbox{max} \text{ }\left[ {k_{{1_{\hbox{min} } }} \& \frac{{k_{5} }}{{k_{{2_{\hbox{max} } }} }}} \right] \hfill \\ \beta = \hbox{min} \text{ }\left[ {k_{{1_{\hbox{max} } }} \& \frac{{k_{5} }}{{k_{{2_{\hbox{min} } }} }}} \right] \hfill \\ \end{aligned} \right.$$

$$f_{{K_{6} }} (k_{6} ) = f_{{K_{5} \times K_{3} }} (k_{6} ) = \int\limits_{\alpha }^{\beta } {\left| {k_{3} } \right|f_{{K_{3} }} (k_{3} )f_{{K_{5} }} \left( {\frac{{k_{6} }}{{k_{3} }}} \right){\text{d}}k_{3} }$$

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$$k_{{5_{\hbox{min} } }} k_{{3_{\hbox{min} } }} \le k_{6} \le k_{{5_{\hbox{max} } }} k_{{3_{\hbox{max} } }}$$

$$\left\{ \begin{aligned} k_{{6_{\hbox{min} } }} = k_{{5_{\hbox{min} } }} k_{{3_{\hbox{min} } }} \hfill \\ k_{{6_{\hbox{max} } }} = k_{{5_{\hbox{max} } }} k_{{3_{\hbox{max} } }} \hfill \\ \end{aligned} \right.\quad {\text{and}}\quad \left\{ \begin{aligned} \alpha = \hbox{max} \text{ }\left[ {k_{{3_{\hbox{min} } }} \& \frac{{k_{6} }}{{k_{{5_{\hbox{max} } }} }}} \right] \hfill \\ \beta = \hbox{min} \text{ }\left[ {k_{{3_{\hbox{max} } }} \& \frac{{k_{6} }}{{k_{{5_{\hbox{min} } }} }}} \right] \hfill \\ \end{aligned} \right.$$

$$\text{ }f_{{K_{7} }} (k_{7} ) = f_{{K_{6} \times K_{4} }} (k_{7} ) = \int\limits_{\alpha }^{\beta } {\left| {k_{4} } \right|f_{{K_{4} }} (k_{4} )f_{{K_{6} }} \left( {\frac{{k_{7} }}{{k_{4} }}} \right){\text{d}}k_{4} }$$

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$$k_{{6_{\hbox{min} } }} k_{{4_{\hbox{min} } }} \le k_{7} \le k_{{6_{\hbox{max} } }} k_{{4_{\hbox{max} } }}$$

$$\left\{ \begin{aligned} k_{{7_{\hbox{min} } }} = k_{{6_{\hbox{min} } }} k_{{4_{\hbox{min} } }} \hfill \\ k_{{7_{\hbox{max} } }} = k_{{6_{\hbox{max} } }} k_{{4_{\hbox{max} } }} \hfill \\ \end{aligned} \right.\quad {\text{and}}\quad \left\{ \begin{aligned} \alpha = \hbox{max} \text{ }\left[ {k_{{4_{\hbox{min} } }} \& \frac{{k_{7} }}{{k_{{6_{\hbox{max} } }} }}} \right] \hfill \\ \beta = \hbox{min} \text{ }\left[ {k_{{4_{\hbox{max} } }} \& \frac{{k_{7} }}{{k_{{6_{\hbox{min} } }} }}} \right] \hfill \\ \end{aligned} \right.$$

And the cumulative distribution function of K₇ can be determined as below:

$$F_{{K_{7} }} (k_{7} ) = P\left\{ {K_{7} \in [k_{{7_{\hbox{min} } }} ,k_{7} ]} \right\} = \int\limits_{{k_{{7_{\hbox{min} } }} }}^{{k_{7} }} {f_{{K_{7} }} (t){\text{d}}t}$$

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$$k_{{7_{\hbox{min} } }} \le k_{7} \le k_{{7_{\hbox{max} } }} .$$