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

-is 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.-e parameter sensitivity analysis included the coefficient of variation and scale of fluctuation of soil shear modulus. -e 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.


Introduction
It is now well recognized that natural soil properties exhibit spatial variability because of depositional and postdepositional processes. e inherent variability in soil properties has found its place in geotechnical design and has been extensively incorporated in the analysis of slope stability [1][2][3][4][5], foundation bearing capacity [6,7], foundation settlement [8][9][10], and liquefaction [11][12][13]. A lognormal distribution has been generally accepted in a geotechnical reliability analysis [14][15][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. e finite difference mesh configuration is shown in Figure 1. e 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. e 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. e Finn model can consider variations of the volumetric strain and display the increase in excess pore pressure. e relationship between variations of volumetric strain increment (Δε vd ) and cycle shearing strains (r) was defined as follows: where C 1 ,C 2 , C 3 , and C 4 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. e free-field boundaries were adopted for the right and left boundaries to simulate the half-space condition. e seismic loading in the horizontal direction was applied at the bottom boundary which was assumed to be rigid. In the stochastic analyses, the shear modulus G was assumed to be random variable and generated with the spectral representation method, which was recently developed by Shu et al. [7]. ree probabilistic distributions, including lognormal, Beta, and Gamma, were applied to model the spatial variability of G, with a mean μ G � 20 MPa and CoV G � 0.1, 0.3, and 0.5. A 2D exponential correlation function [22] was adopted with the horizontal and vertical spatial correlation lengths δ x � 6 and 60 m and δ y � 6 m, respectively. 200 sets of Monte Carlo realizations were undertaken for each combination of a distribution type, a scale of fluctuation, and a CoV G . Figure 2 plots the mean of A 80 (μ A 80 ) varying with the time history curve with different stochastic distributions. Following Popescu et al. [23], A 80 (t) from one finite difference computation is defined as

Area of Liquefied Zone A 80 .
where σ v0 ′ and μ(t) are the initial effective stress at a specific location and the excess pore water pressure at the time instant t after the earthquake, respectively. For the set of δ x in this study, the difference between the distribution types merely led to a minor diversity of the peak μ A 80 by the random shear modulus CoV G � 0.1. However, the peak μ A 80 by the Beta distribution was smaller than that by the Lognormal and Gamma distributions with CoV G � 0.3 and CoV G � 0.5 (Figures 2(b)-2(f )). In addition, the greatest peak μ A 80 was correlated with the G conformed to the Lognormal distribution in these cases.
From the perspective of the decreasing rate of A 80 , the residual of A 80 , and the sensitivity of A 80 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 CoV G . Figure 2 presents that the irregular dynamic liquefaction results from different probability distributions, in terms of the residual A 80 and the response of A 80 to instantaneous seismic loading. Additionally, the increase in CoV G can amplify the impact from the probability distribution.

Excess Pore Water Pressure Ratios Q.
e liquefication index was calculated from the mean excess pore water pressure ratio in the horizontal direction and of the form for one simulation: 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; σ v0 ′ is the initial effective stress in the vertical direction; n was set to be 40 in this study, which represents the element number of   Advances in Civil Engineering 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. Figure 3 presents the time history curves of the mean excess pore water pressure ratio (μ Q ) at z � 7.25 m with different probabilistic distributions. In the case with CoV G � 0.1 and δ x � 6 m or 60 m, μ Q 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 CoV G , the rebound amplitude of μ Q with different probabilistic distributions gradually declined (Figures 3(b)-3(f )). In addition, μ Q with the Beta distribution was slightly smaller than that with the Lognormal and Gamma distributions when the CoV G 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 μ Q curve) was smaller than that with the Lognormal distribution and Gamma distribution. Figure 3 also shows that μ Q dissipated after the occurrence of the peak μ Q appeared around t � 4 s. Table 2 summarizes μ Q 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 μ Q time history curve. In general, the dissipation rate of μ Q increased with the increase in CoV G , and this increasing trend was affected by the distribution type. For instance, the amplification was 16.11% for the Beta distribution as CoV G extended from 0.1 to 0.5, which was greater compared with that with the Gamma and Lognormal distributions.

Ground Displacement D.
In this section, the maximum surface ground horizontal movement (D x (t) max ) in Equation (4) and settlement (D z (t) max ) in Equation (5) were taken to evaluate the influence of liquefaction by earthquake, respectively: where D x,z�0 (t) and D x,z�10 (t) represent the horizontal displacements at surface and bottom at the horizontal coordinate x in the soil domain and D z (t) max and D z (t) min represent the maximum and minimum settlement at t-th second after the earthquake. Figure 4 plots the time history curve of mean ground horizontal displacement (μ D x ) with different probabilistic distributions. Similar time history curves of μ D x for different distributions were obtained provided CoV G � 0.1, indicating that the distribution types had little influence on the ground horizontal displacement (Figures 4(a) and 4(d)). e differences between μ D x became pronounced with the increase in CoV G . It was worth noting that μ D x with the Beta distribution was always the largest, while μ D x with Lognormal distribution was the smallest.
As expected, the probability distributions also had certain impact on the standard deviation of the horizontal displacement (μ D x ) ( Figure 5). e difference of μ D x gradually increased with the increase in CoV G , referring that the effect of different distribution on μ D x enhanced with the increase in CoV G . If CoV G is 0.3 or 0.5, it was obvious that μ D x with the Beta distribution was always greater than that with the Lognormal and Gamma distributions. Advances in Civil Engineering Figure 6 shows the impact of CoV G on μ D x with different distributions at t � 35 s. In general, an increase in CoV G was corresponding with the increase in μ D x with different distributions. e obtained μ D x with Beta distribution was greater than that with the Gamma and Lognormal distributions. Moreover, a larger δ x correlated with a greater μ D x, max provided with the same distribution and CoV G (Figures 6(a) and 6(b)). is 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. e time history curves of mean and standard deviation of settlement (μ D z and σ D z ) are shown in Figures 7 and 8, respectively. e influences of the distributions of G on μ D z and σ D z were similar to those on μ D x and σ D x . e differences between μ D z and σ D z with different distributions were insignificant if CoV G � 0.1, which implied that the distributions had small impact on the settlement. Nonetheless, the difference was positively correlated with CoV G . For example,  .     Advances in Civil Engineering with a given δ x of 6 m and CoV G � 0.3, μ D z with the Beta distribution was 7.81% and 7.28% larger than that with Gamma distribution and Lognormal distribution. If CoV G was 0.5, the results enhanced 20.51% and 35.89%, respectively. It addresses the conclusion that the influence of the distributions on μ D z and σ D z increased with the increase in CoV G . e 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 9 shows the relationship between μ D z , max and CoV G . e distribution type had a similar impact on μ D z , max , which is consistent with the findings in Figure 6. A greater μ D X, max was corresponding with a greater CoV G for all three distributions. Comparing between

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. e following conclusions can be drawn: (1) e 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. e pore water pressure dissipation rate accelerated with the increase in CoV G 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. e sensitivity of the ground horizontal displacement and Advances in Civil Engineering 13 settlement to CoV G decreased successively for Beta, Gamma, and Lognormal distribution.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.