Depth-consistent models for probabilistic liquefaction potential assessment based on shear wave velocity

Abstract

In the Chinese code models for seismic design, the shear wave velocity (V_s)-based model for liquefaction potential assessment is built based on the depth consistency assumption, i.e., the seismic demand is a non-decreasing function of the soil depth. It provides an alternative way to construct empirical models based on past case histories. However, the current V_s-based Chinese code model is deterministic in nature. In this paper, the depth consistency assumption for V_s data is validated with models built within the cyclic stress ratio (CSR) framework. Then, two V_s-based probabilistic models for liquefaction potential assessment are built based on the depth consistency assumption, i.e., a model with explicit consideration of the measurement uncertainty and a model without explicit consideration of the measurement uncertainty. It is found that the performances of the two suggested models are comparable with those of two frequently used models built within the CSR framework and are superior to the Chinese code model. For a site where the measurement uncertainty is not available, the model without explicit consideration of the measurement uncertainty should be used. For a site where measurement uncertainty is available, it is suggested that the model with explicit consideration of the measurement uncertainty should be used as it is more conservative.

Appendices

Appendix A. Derivation procedure of the (V _s)_req—d _s relationship for a model built within the CSR—V _s1 framework

The model presented in Kayen et al. (2013) is taken as an example to demonstrate how to derive the (V_s)_req - d_s relationship. The cyclic stress ratio (CSR) can be calculated as follows (Seed et al. 1985):

$${\text{CSR}} = 0.65\frac{{a_{\max } }}{g}\frac{{\sigma_{v} }}{{\sigma^{\prime}_{v} }}r_{d}$$

(23)

where a_max is peak ground acceleration in g and g is the gravity of acceleration; σ_v and σ'_v are the total and effective overburden stress in kPa, respectively; r_d is the stress reduction coefficient whose average values can be estimated using the equation suggested by Liao and Whitman (1986) as follows:

$$r_{d} { = }\left\{ {\begin{array}{*{20}l} {{1} - 0.00{765}d_{s} } \hfill & {\left( {d_{s} \le 9.15{\text{m}}} \right)} \hfill \\ {{1}{\text{.174}} - 0.0267d_{s} } \hfill & {\left( {9.15{\text{m < }}d_{s} \le 23{\text{m}}} \right)} \hfill \\ \end{array} } \right.$$

(24)

  σ_v and σ'_v can be estimated through d_s and d_w using the following equations:

$$\sigma_{v} = \gamma_{s} d_{s}$$

(25)

$$\sigma^{\prime}_{v} = \gamma_{s} d_{s} - \gamma_{w} (d_{s} - d_{w} )$$

(26)

where d_s is the soil depth of interest in meters; d_w is the groundwater depth in meters; γ_s is the unit weight of the soil which is assumed to be 19 kN/m³ in this study; γ_w is the unit weight of water which is taken as10 kN/m³.

  Substituting Eqs. (24)-(26) into Eq. (23), the relationship of CSR - d_s can be obtained.

  For the model presented in Kayen et al. (2013), the cyclic resistance ratio (CRR) is calculated as follows:

$$CRR = \exp \left\{ {\frac{{\left[ {\left( {0.0073 \cdot V_{s1} } \right)^{2.8011} - 2.6168 \cdot \ln \left( {M_{w} } \right) - 0.0099 \cdot \ln \left( {\sigma_{v}^{^{\prime}} } \right) + 0.0028 \cdot FC - 0.4809 \cdot \Phi^{ - 1} \left( {P_{L} } \right)} \right]}}{1.946}} \right\}$$

(27)

where FC is the fine content of soil in percentage. Equation (27) can be regarded as a relationship of CRR - V_s1 when other descriptive parameters are given.

  Because the triggering boundary between liquefied domain and non-liquefied domain indicates a relationship of CSR = CRR, the relationship of (V_s1)_req - d_s can be defined by equalizing CSR and CRR. Then (V_s1)_req should be changed into (V_s)_req using Eq. (2) such that the relationship of (V_s)_req - d_s can be obtained.

  The relationship of (V_s)_req - d_s derived from the model presented in Andrus and Stokoe (2000) can also be obtained via a similar process in Appendix A.

Appendix B. Derivation procedure for the mean and standard deviation of ln(V), ln(S), ln(W), ln(M), and A

The ln(V), ln(S), ln(W), ln(M), and A can be expanded in a Taylor series and truncated the series above the linear terms, thereby giving a first-order approximation of the second moment at the mean value point (i.e., MFOSM), respectively (e.g., Cetin 2000; Moss 2020; Wong 1985). Thus, the means of ln(V), ln(S), ln(W), ln(M), and A [i.e., μ_ln(V), μ_ln(S), μ_ln(W), μ_ln(M) and μ_A] are given as follows:

$$\left\{ \begin{gathered} \mu_{\ln \left( V \right)} \approx \ln \left( {\hat{V}} \right){ = }\ln \left( {\hat{V}_{s} } \right) \hfill \\ \mu_{\ln \left( S \right)} \approx \ln \left( {\hat{S}} \right){ = }\ln \left[ {\exp \left( { - {{\theta_{1} } \mathord{\left/ {\vphantom {{\theta_{1} } {\hat{d}_{s} }}} \right. \kern-\nulldelimiterspace} {\hat{d}_{s} }}} \right) + \theta_{2} } \right] \hfill \\ \mu_{\ln \left( W \right)} \approx \ln \left( {\hat{W}} \right){ = }\ln \left[ {1 + \exp (\theta_{3} ) \cdot \hat{d}_{w} } \right] \hfill \\ \mu_{\ln \left( M \right)} \approx \ln \left( {\hat{M}} \right){ = }\ln \left( {{{\hat{M}_{w} } \mathord{\left/ {\vphantom {{\hat{M}_{w} } {7.5}}} \right. \kern-\nulldelimiterspace} {7.5}}} \right) \hfill \\ \mu_{A} \approx \hat{A}{ = }{1 \mathord{\left/ {\vphantom {1 {\hat{a}_{\max } }}} \right. \kern-\nulldelimiterspace} {\hat{a}_{\max } }} \hfill \\ \end{gathered} \right.$$

(28)

where $\widehat{\text{V}}$ _s, $\widehat{\text{d}}$ _s, $\widehat{\text{d}}$ _w, $\widehat{\text{M}}$ _w, and $\widehat{\text{a}}$ _max are the nominal values of x = {V_s, d_s, d_w, M_w, a_max}, the nominal values herein are mean values of descriptive parameters; ln($\widehat{\text{V}}$), ln($\widehat{\text{S}}$), ln($\widehat{\text{W}}$), ln($\widehat{\text{M}}$), and $\widehat{\text{A}}$ denote the nominal value of the function of x, respectively.

  The standard deviations of ln(V), ln(S), ln(W), ln(M), and A [σ_ln(V), σ_ln(S), σ_ln(W), σ_ln(M), and σ_A] can be estimated using the following derivation procedure:

$$\sigma_{Y} \approx \sqrt {\sigma_{X}^{2} \left( {\left. {\frac{\partial Y}{{\partial X}}} \right|_{{X{ = }\hat{X}}} } \right)^{2} }$$

(29)

where σ_Y denotes the standard deviation of Y; Y denotes the function of each descriptive parameter [i.e., ln(V), ln(S), ln(W), ln(M), and A]; X denotes descriptive parameters (i.e., V_s, d_s, d_w, M_w, a_max); $\frac{\partial {\text{Y}}}{\partial {\text{X}}}$ is the derivative of Y with respect to X.

  V_s is taken as an instance; the standard deviation of ln(V) [i.e., σ_ln(V)] can be estimated as

$$\sigma_{\ln \left( V \right)} \approx \sqrt {\sigma_{Vs}^{2} \left[ {\left. {\frac{{\partial \left[ {\ln \left( V \right)} \right]}}{{\partial V_{s} }}} \right|_{{V_{s} = \hat{V}_{s} }} } \right]^{2} } = \sqrt {\sigma_{Vs}^{2} \left[ {\left. {\frac{{\partial \left[ {\ln \left( {V_{s} } \right)} \right]}}{{\partial V_{s} }}} \right|_{{V_{s} = \hat{V}_{s} }} } \right]^{2} } = \sqrt {\sigma_{Vs}^{2} /\hat{V}_{s}^{2} } = \delta_{Vs}$$

(30)

where δ denotes the coefficient of variation; then, σ_ln(S), σ_ln(W), σ_ln(M), and σ_A can be estimated in the same way given as

$$\sigma_{\ln \left( S \right)} \approx \sqrt {\frac{{\theta_{1}^{2} \exp \left( { - 2\theta_{1} /\hat{d}_{s} } \right)}}{{\hat{d}_{s}^{{4}} \left[ {\exp \left( { - \theta_{1} /\hat{d}_{s} } \right) + \theta_{2} } \right]^{2} }}\sigma_{{d_{s} }}^{2} }$$

(31)

$$\sigma_{\ln \left( W \right)} \approx \sqrt {\frac{{\exp \left( {2\theta_{3} } \right)}}{{\left[ {1 + \exp \left( {\theta_{3} } \right)\hat{d}_{w} } \right]^{2} }}\sigma_{{d_{w} }}^{2} }$$

(32)

$$\sigma_{\ln \left( M \right)} \approx \delta_{{M_{w} }}$$

(33)

$$\sigma_{A} \approx \hat{a}_{\max }^{2} \sigma_{{a_{\max } }}$$

(34)

  Assuming that ln(V), ln(S), ln(W), ln(M), and A are statistically independent. Let σ_εc denotes the standard deviation of total measurement uncertainties associated with a given case, which can be formulated based on Eq. (6) as

$$\sigma_{\varepsilon c} = \sqrt {\sigma_{\ln \left( V \right)}^{2} + \sigma_{\ln \left( S \right)}^{2} + \sigma_{\ln \left( W \right)}^{2} + \theta_{4}^{2} \sigma_{\ln \left( M \right)}^{2} + \theta_{5}^{2} \sigma_{A}^{2} }$$

(35)