# Investigation on the Effect of Probability Distribution on the Dynamic Response of Liquefiable Soils

Authors: Juncheng Wang; Li Zhou; Wenzhi Song; Houle Zhang; Yongxin Wu
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

This study investigated the effect of different probabilistic distributions (Lognormal, Gamma, and Beta) to characterize the spatial variability of shear modulus on the soil liquefiable response. The parameter sensitivity analysis included the coefficient of variation and scale of fluctuation of soil shear modulus. The results revealed that the distribution type had no significant influence on the liquefication zone. In particular, the estimation with Beta distribution is the worst scenario. It illuminated that the estimation with Beta distribution can provide a conservative design if site investigation is absent.


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## Body

## 1. Introduction

It is now well recognized that natural soil properties exhibit spatial variability because of depositional and postdepositional processes. The inherent variability in soil properties has found its place in geotechnical design and has been extensively incorporated in the analysis of slope stability [1–5], foundation bearing capacity [6, 7], foundation settlement [8–10], and liquefaction [11–13]. A lognormal distribution has been generally accepted in a geotechnical reliability analysis [14–16] because of its capability to model the randomness of positive soil parameters.Recent studies proved that different distributions impacted the stochastic properties of soil. Popescu et al. [17] and Jimenez and Sitar [18] performed a series of random finite element analyses with different probability distributions of soil parameters, which has significant effects on the foundation settlement and bearing capacity. Most recently, Wu et al. [19] applied the random finite element method to investigate the effect of different probabilistic distributions on the tunnel convergence and demonstrated the mechanisms of tunnel convergence and the probability of exceeding liquefaction thresholds with different probabilistic distribution types. To date, publications on the application of random field theory to soil dynamic behavior are limited and the impact of probabilistic distribution on the soil liquefiable response has not been clearly defined. Wang et al. [20] investigated the liquefaction response of soil using the spatial variability of the shear modulus by considering different values of the coefficient of variation and the horizontal scale of fluctuation.In this study, we performed the nonlinear dynamic simulation of the liquefiable response of a sand layer with the water table 1 m below the ground level under a seismic load using the finite difference program FLAC3D. The finite difference mesh configuration is shown in Figure1. The soil domain had a length of 40 m and a height of 10 m, a liquefiable layer of 9 m, and a nonliquefiable layer of 1 m. The Mohr–Coulomb model and the Finn model were used to simulate the nonlinear soil behavior and the accumulation of the pore pressure in sand before the liquefaction triggered by a dynamic load, respectively. The Finn model can consider variations of the volumetric strain and display the increase in excess pore pressure. The relationship between variations of volumetric strain increment (Δεvd) and cycle shearing strains (r) was defined as follows:(1)Δεvd=C1r−C2εvd+C3εvd2r+C4εvd,where C1,C2, C3, and C4 are constant coefficients that can be obtained from cyclic triaxial tests. Following the study of Azadi and Hosseini [21], the four values were selected as 0.79, 0.52, 0.2, and 0.5, respectively. Table 1 summarizes the soil parameters in the constitutive model for the deterministic analysis. In the dynamic analysis, the boundaries were used to absorb the reflected waves and enforce the discrete half-space conditions of the numerical model. The free-field boundaries were adopted for the right and left boundaries to simulate the half-space condition. The seismic loading in the horizontal direction was applied at the bottom boundary which was assumed to be rigid.Figure 1
Mesh used in finite difference analysis.Table 1
Summary of soil parameters.
ParametersValueShear modulus,G: MPa20Total unit weight,γ: kN/m326.6Poisson’s ratio,ν0.35Permeability coefficient,k: m/s2.64 × 10−4Porosity,N0.435In the stochastic analyses, the shear modulusG was assumed to be random variable and generated with the spectral representation method, which was recently developed by Shu et al. [7]. Three probabilistic distributions, including lognormal, Beta, and Gamma, were applied to model the spatial variability of G, with a mean μG = 20 MPa and CoVG = 0.1, 0.3, and 0.5. A 2D exponential correlation function [22] was adopted with the horizontal and vertical spatial correlation lengths δx = 6 and 60 m and δy = 6 m, respectively. 200 sets of Monte Carlo realizations were undertaken for each combination of a distribution type, a scale of fluctuation, and a CoVG.


## 2. Results and Discussion

### 2.1. Area of Liquefied ZoneA80

Figure2 plots the mean of A80 (μA80) varying with the time history curve with different stochastic distributions. Following Popescu et al. [23], A80(t) from one finite difference computation is defined as(2)A80t=areaut/σν0>0.8areatotal,where σv0′ and μt are the initial effective stress at a specific location and the excess pore water pressure at the time instant t after the earthquake, respectively.Figure 2
Time history curve of mean liquefaction range with different distributions: (a)δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)For the set ofδx in this study, the difference between the distribution types merely led to a minor diversity of the peak μA80 by the random shear modulus CoVG = 0.1. However, the peak μA80 by the Beta distribution was smaller than that by the Lognormal and Gamma distributions with CoVG = 0.3 and CoVG = 0.5 (Figures 2(b)–2(f)). In addition, the greatest peak μA80 was correlated with the G conformed to the Lognormal distribution in these cases.From the perspective of the decreasing rate ofA80, the residual of A80, and the sensitivity of A80 to instantaneous seismic loading, it was found that the influence of the different probability distributions on the dynamic liquefaction results was irregular, considering the results of the non-Gaussian probability distributions. However, in general, the differences among the calculated results from the three probability distributions became more obvious with the increase in CoVG. Figure 2 presents that the irregular dynamic liquefaction results from different probability distributions, in terms of the residual A80 and the response of A80 to instantaneous seismic loading. Additionally, the increase in CoVG can amplify the impact from the probability distribution.


### 2.2. Excess Pore Water Pressure RatiosQ

The liquefication index was calculated from the mean excess pore water pressure ratio in the horizontal direction and of the form for one simulation:(3)Qz,t=1n∑xrx,z,t,(4)rx,z,t=ux,z,tσν0',where x and z are the horizontal and vertical coordinates of the central point of one finite difference element, respectively; r(x, z, t) is the excess pore water pressure ratio at the central point (x, z) at t-th second after the earthquake; σv0′ is the initial effective stress in the vertical direction; n was set to be 40 in this study, which represents the element number of the finite difference model in the horizontal direction; and Q(z, t) reflects the average excess pore water pressure ratio at the depth of z at t-th second after the earthquake. Due to the fact that large accumulation of pore water pressure might occur in the deep soil layer, this study paid close attention to the variation of Q at z = 7.25 m.Figure3 presents the time history curves of the mean excess pore water pressure ratio (μQ) at z = 7.25 m with different probabilistic distributions. In the case with CoVG = 0.1 and δx = 6 m or 60 m, μQ with varying types of distribution presents a similar trend with the seismic load imposed to the soil layer (Figures 3(a) and 3(d)). However, with the increased CoVG, the rebound amplitude of μQ with different probabilistic distributions gradually declined (Figures 3(b)–3(f)). In addition, μQ with the Beta distribution was slightly smaller than that with the Lognormal and Gamma distributions when the CoVG is 0.3 and 0.5, and its dissipation rate of excess pore water pressure (in terms of the slope of the descending part of μQ curve) was smaller than that with the Lognormal distribution and Gamma distribution.Figure 3
Time history curves ofμQ at (z) = 7.25 m with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure3 also shows that μQ dissipated after the occurrence of the peak μQ appeared around t = 4 s. Table 2 summarizes μQ with shear modulus by different distributions after 7 s and 35 s occurrence of the earthquake. Table 3 tabulates the dissipation rate of pore water pressure, which was depicted from the descending section of the μQ time history curve. In general, the dissipation rate of μQ increased with the increase in CoVG, and this increasing trend was affected by the distribution type. For instance, the amplification was 16.11% for the Beta distribution as CoVG extended from 0.1 to 0.5, which was greater compared with that with the Gamma and Lognormal distributions.Table 2
Summary ofμQ at z = 7.25 m and t = 7 and 35 s (10−2).
δx (m)Probabilistic distributionCoVG = 0.1CoVG = 0.3CoVG = 0.5t = 7 st = 35 st = 7 st = 35 st = 7 st = 35 s6Lognormal73.9642.7675.7637.8376.0035.72Gamma74.2543.0475.7038.0775.6535.13Beta74.0042.3775.6037.2274.5834.4560Lognormal74.2342.7274.7539.1374.2038.10Gamma73.8841.9674.7539.0274.1238.23Beta73.9242.9774.4838.5273.4237.49Table 3
Variation of dissipation rate ofμQ (10−3 × s−1) at z = 7.25 m.
δx (m)Probabilistic distributionCoVGAmplification (%)0.10.30.56Lognormal11.1413.5514.3929.17Gamma11.1513.4414.4729.78Beta11.3013.7114.3326.8160Lognormal11.2512.7212.8914.58Gamma11.4012.7612.8212.46Beta11.0512.8412.8316.11


### 2.3. Ground DisplacementD

In this section, the maximum surface ground horizontal movement (Dx(t)max) in Equation (4) and settlement (Dz(t)max) in Equation (5) were taken to evaluate the influence of liquefaction by earthquake, respectively:(5)Dxtmax=Dx,z=0t−Dx,z=10tmax,(6)Dztmax=Dztmax−Dztminmax,where Dx,z=0(t) and Dx,z=10(t) represent the horizontal displacements at surface and bottom at the horizontal coordinate x in the soil domain and Dz(t)max and Dz(t)min represent the maximum and minimum settlement at t-th second after the earthquake.Figure4 plots the time history curve of mean ground horizontal displacement (μDx) with different probabilistic distributions. Similar time history curves of μDx for different distributions were obtained provided CoVG = 0.1, indicating that the distribution types had little influence on the ground horizontal displacement (Figures 4(a) and 4(d)). The differences between μDx became pronounced with the increase in CoVG. It was worth noting that μDx with the Beta distribution was always the largest, while μDx with Lognormal distribution was the smallest.Figure 4
Time history curve of mean ground horizontal displacement (μDx) with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)As expected, the probability distributions also had certain impact on the standard deviation of the horizontal displacement (μDx) (Figure 5). The difference of μDx gradually increased with the increase in CoVG, referring that the effect of different distribution on μDx enhanced with the increase in CoVG. If CoVG is 0.3 or 0.5, it was obvious that μDx with the Beta distribution was always greater than that with the Lognormal and Gamma distributions.Figure 5
Time history curve of standard deviation of ground horizontal displacement (μDx) with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure6 shows the impact of CoVG on μDx with different distributions at t = 35 s. In general, an increase in CoVG was corresponding with the increase in μDx with different distributions. The obtained μDx with Beta distribution was greater than that with the Gamma and Lognormal distributions. Moreover, a larger δx correlated with a greater μDx,max provided with the same distribution and CoVG (Figures 6(a) and 6(b)). This finding highlighted that the worst scenario was covered by the shear modulus with Beta distribution. It illuminated that the estimation with Beta distribution is capable to provide a conservative evaluation if site investigation is absent.Figure 6
Variation ofμDx,max as a function of CoVG: (a) δx = 6 m and (b) δx = 60 m.
(a)(b)The time history curves of mean and standard deviation of settlement (μDz and σDz) are shown in Figures 7 and 8, respectively. The influences of the distributions of G on μDz and σDz were similar to those on μDx and σDx. The differences between μDz and σDz with different distributions were insignificant if CoVG = 0.1, which implied that the distributions had small impact on the settlement. Nonetheless, the difference was positively correlated with CoVG. For example, with a given δx of 6 m and CoVG = 0.3, μDz with the Beta distribution was 7.81% and 7.28% larger than that with Gamma distribution and Lognormal distribution. If CoVG was 0.5, the results enhanced 20.51% and 35.89%, respectively. It addresses the conclusion that the influence of the distributions on μDz and σDz increased with the increase in CoVG. The settlement obtained from the random fields obeying Beta distribution was greater and more dispersive than that calculated by the Lognormal distribution and Gamma distribution.Figure 7
Time history curve ofμDz with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure 8
Time history curve ofσDz with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure9 shows the relationship between μDz,max and CoVG. The distribution type had a similar impact on μDz,max, which is consistent with the findings in Figure 6. A greater μDX,max was corresponding with a greater CoVG for all three distributions. Comparing between Figures 9(a) and 9(b), it can be observed that small δx corresponded to large μDX,max, except for the case of CoVG = 0.3 under the Beta distribution.Figure 9
Variation ofμDz,max as a function of CoVG: (a) δx = 6 m and (b) δx = 60 m.
(a)(b)


## 2.1. Area of Liquefied ZoneA80

Figure2 plots the mean of A80 (μA80) varying with the time history curve with different stochastic distributions. Following Popescu et al. [23], A80(t) from one finite difference computation is defined as(2)A80t=areaut/σν0>0.8areatotal,where σv0′ and μt are the initial effective stress at a specific location and the excess pore water pressure at the time instant t after the earthquake, respectively.Figure 2
Time history curve of mean liquefaction range with different distributions: (a)δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)For the set ofδx in this study, the difference between the distribution types merely led to a minor diversity of the peak μA80 by the random shear modulus CoVG = 0.1. However, the peak μA80 by the Beta distribution was smaller than that by the Lognormal and Gamma distributions with CoVG = 0.3 and CoVG = 0.5 (Figures 2(b)–2(f)). In addition, the greatest peak μA80 was correlated with the G conformed to the Lognormal distribution in these cases.From the perspective of the decreasing rate ofA80, the residual of A80, and the sensitivity of A80 to instantaneous seismic loading, it was found that the influence of the different probability distributions on the dynamic liquefaction results was irregular, considering the results of the non-Gaussian probability distributions. However, in general, the differences among the calculated results from the three probability distributions became more obvious with the increase in CoVG. Figure 2 presents that the irregular dynamic liquefaction results from different probability distributions, in terms of the residual A80 and the response of A80 to instantaneous seismic loading. Additionally, the increase in CoVG can amplify the impact from the probability distribution.


## 2.2. Excess Pore Water Pressure RatiosQ

The liquefication index was calculated from the mean excess pore water pressure ratio in the horizontal direction and of the form for one simulation:(3)Qz,t=1n∑xrx,z,t,(4)rx,z,t=ux,z,tσν0',where x and z are the horizontal and vertical coordinates of the central point of one finite difference element, respectively; r(x, z, t) is the excess pore water pressure ratio at the central point (x, z) at t-th second after the earthquake; σv0′ is the initial effective stress in the vertical direction; n was set to be 40 in this study, which represents the element number of the finite difference model in the horizontal direction; and Q(z, t) reflects the average excess pore water pressure ratio at the depth of z at t-th second after the earthquake. Due to the fact that large accumulation of pore water pressure might occur in the deep soil layer, this study paid close attention to the variation of Q at z = 7.25 m.Figure3 presents the time history curves of the mean excess pore water pressure ratio (μQ) at z = 7.25 m with different probabilistic distributions. In the case with CoVG = 0.1 and δx = 6 m or 60 m, μQ with varying types of distribution presents a similar trend with the seismic load imposed to the soil layer (Figures 3(a) and 3(d)). However, with the increased CoVG, the rebound amplitude of μQ with different probabilistic distributions gradually declined (Figures 3(b)–3(f)). In addition, μQ with the Beta distribution was slightly smaller than that with the Lognormal and Gamma distributions when the CoVG is 0.3 and 0.5, and its dissipation rate of excess pore water pressure (in terms of the slope of the descending part of μQ curve) was smaller than that with the Lognormal distribution and Gamma distribution.Figure 3
Time history curves ofμQ at (z) = 7.25 m with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure3 also shows that μQ dissipated after the occurrence of the peak μQ appeared around t = 4 s. Table 2 summarizes μQ with shear modulus by different distributions after 7 s and 35 s occurrence of the earthquake. Table 3 tabulates the dissipation rate of pore water pressure, which was depicted from the descending section of the μQ time history curve. In general, the dissipation rate of μQ increased with the increase in CoVG, and this increasing trend was affected by the distribution type. For instance, the amplification was 16.11% for the Beta distribution as CoVG extended from 0.1 to 0.5, which was greater compared with that with the Gamma and Lognormal distributions.Table 2
Summary ofμQ at z = 7.25 m and t = 7 and 35 s (10−2).
δx (m)Probabilistic distributionCoVG = 0.1CoVG = 0.3CoVG = 0.5t = 7 st = 35 st = 7 st = 35 st = 7 st = 35 s6Lognormal73.9642.7675.7637.8376.0035.72Gamma74.2543.0475.7038.0775.6535.13Beta74.0042.3775.6037.2274.5834.4560Lognormal74.2342.7274.7539.1374.2038.10Gamma73.8841.9674.7539.0274.1238.23Beta73.9242.9774.4838.5273.4237.49Table 3
Variation of dissipation rate ofμQ (10−3 × s−1) at z = 7.25 m.
δx (m)Probabilistic distributionCoVGAmplification (%)0.10.30.56Lognormal11.1413.5514.3929.17Gamma11.1513.4414.4729.78Beta11.3013.7114.3326.8160Lognormal11.2512.7212.8914.58Gamma11.4012.7612.8212.46Beta11.0512.8412.8316.11


## 2.3. Ground DisplacementD

In this section, the maximum surface ground horizontal movement (Dx(t)max) in Equation (4) and settlement (Dz(t)max) in Equation (5) were taken to evaluate the influence of liquefaction by earthquake, respectively:(5)Dxtmax=Dx,z=0t−Dx,z=10tmax,(6)Dztmax=Dztmax−Dztminmax,where Dx,z=0(t) and Dx,z=10(t) represent the horizontal displacements at surface and bottom at the horizontal coordinate x in the soil domain and Dz(t)max and Dz(t)min represent the maximum and minimum settlement at t-th second after the earthquake.Figure4 plots the time history curve of mean ground horizontal displacement (μDx) with different probabilistic distributions. Similar time history curves of μDx for different distributions were obtained provided CoVG = 0.1, indicating that the distribution types had little influence on the ground horizontal displacement (Figures 4(a) and 4(d)). The differences between μDx became pronounced with the increase in CoVG. It was worth noting that μDx with the Beta distribution was always the largest, while μDx with Lognormal distribution was the smallest.Figure 4
Time history curve of mean ground horizontal displacement (μDx) with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)As expected, the probability distributions also had certain impact on the standard deviation of the horizontal displacement (μDx) (Figure 5). The difference of μDx gradually increased with the increase in CoVG, referring that the effect of different distribution on μDx enhanced with the increase in CoVG. If CoVG is 0.3 or 0.5, it was obvious that μDx with the Beta distribution was always greater than that with the Lognormal and Gamma distributions.Figure 5
Time history curve of standard deviation of ground horizontal displacement (μDx) with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure6 shows the impact of CoVG on μDx with different distributions at t = 35 s. In general, an increase in CoVG was corresponding with the increase in μDx with different distributions. The obtained μDx with Beta distribution was greater than that with the Gamma and Lognormal distributions. Moreover, a larger δx correlated with a greater μDx,max provided with the same distribution and CoVG (Figures 6(a) and 6(b)). This finding highlighted that the worst scenario was covered by the shear modulus with Beta distribution. It illuminated that the estimation with Beta distribution is capable to provide a conservative evaluation if site investigation is absent.Figure 6
Variation ofμDx,max as a function of CoVG: (a) δx = 6 m and (b) δx = 60 m.
(a)(b)The time history curves of mean and standard deviation of settlement (μDz and σDz) are shown in Figures 7 and 8, respectively. The influences of the distributions of G on μDz and σDz were similar to those on μDx and σDx. The differences between μDz and σDz with different distributions were insignificant if CoVG = 0.1, which implied that the distributions had small impact on the settlement. Nonetheless, the difference was positively correlated with CoVG. For example, with a given δx of 6 m and CoVG = 0.3, μDz with the Beta distribution was 7.81% and 7.28% larger than that with Gamma distribution and Lognormal distribution. If CoVG was 0.5, the results enhanced 20.51% and 35.89%, respectively. It addresses the conclusion that the influence of the distributions on μDz and σDz increased with the increase in CoVG. The settlement obtained from the random fields obeying Beta distribution was greater and more dispersive than that calculated by the Lognormal distribution and Gamma distribution.Figure 7
Time history curve ofμDz with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure 8
Time history curve ofσDz with different distributions: (a) δx = 6 m and CoVG = 0.1; (b) δx = 6 m and CoVG = 0.3; (c) δx = 6 m and CoVG = 0.5; (d) δx = 60 m and CoVG = 0.1; (e) δx = 60 m and CoVG = 0.3; (f) δx = 60 m and CoVG = 0.5.
(a)(b)(c)(d)(e)(f)Figure9 shows the relationship between μDz,max and CoVG. The distribution type had a similar impact on μDz,max, which is consistent with the findings in Figure 6. A greater μDX,max was corresponding with a greater CoVG for all three distributions. Comparing between Figures 9(a) and 9(b), it can be observed that small δx corresponded to large μDX,max, except for the case of CoVG = 0.3 under the Beta distribution.Figure 9
Variation ofμDz,max as a function of CoVG: (a) δx = 6 m and (b) δx = 60 m.
(a)(b)


## 3. Concluding Remarks

In this study, the influence of the probability distributions of soil shear modulus on the area of liquefaction zone, the ratio of excess pore water pressure, and the ground displacement is investigated. The following conclusions can be drawn:(1)
The probability distribution type of shear modulus had no significant influence on the reduction rate of liquefaction zone and the sensitivity of the liquefaction zone to the instantaneous seismic load.(2)
Compared with the Lognormal distribution and Gamma distribution, a smaller excess pore water pressure ratio could be observed with the Beta distribution employed. The pore water pressure dissipation rate accelerated with the increase in CoVG and was affected by the distribution type.(3)
Regarding the ground movement, the estimated horizontal displacement and settlement with Beta distribution were the worst scenario. The sensitivity of the ground horizontal displacement and settlement to CoVG decreased successively for Beta, Gamma, and Lognormal distribution.



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