Pseudo-Static Simplified Analysis Method of the Pile-Liquefiable Soil Interaction considering Rate-Dependent Characteristics

The lateral pressure generated by liquefied soil on pile is a critical parameter in the analysis of soil-pile interaction in liquefaction-susceptible sites. Previous studies have shown that liquefied sand behaves like a non-Newton fluid, and its effect on piles has rate-dependent properties. In this study, a simplified pseudo-static method for liquefiable soil-pile interaction analysis is proposed by treating the liquefied soil as a thixotropic fluid, which considers the rate-dependent behavior. The viscous shear force generated by the relative movement between the viscous fluid (whose viscosity coefficient varies with excess pore pressure and shear strain rate) and the pile was assumed to be the lateral load on the pile. The results from the simplified analysis show that the distribution of bending moment is in good agreement with experiments data. Besides, the effects of various parameters, including relative density, thickness ratio of nonliquefiable layer to liquefiable layer, and frequency of input ground motion, on the pile-soil rate-dependent interaction were discussed in detail.


Introduction
Pile foundation has been the preferred foundation form for structure founded in the lique able eld [1]. However, the failures of pile foundations caused by saturated sand liquefaction were observed in many earthquakes around the world (e.g., Alaska earthquake (1964) [2], Niigata earthquake (1964), Edgecumbe earthquake (1987) [2], Indonesia earthquake (2018) [2], Wenchuan earthquake (2008) [3], Tohoku earthquake (2011) [4], etc.). Accordingly, the performance of pile foundations during liquefaction remains an area of active research [5][6][7]. e soil-structure interaction is a complicated and pseudo-statically procedure driven by the lateral displacement of the eld [8]. e soil-pile interaction in lique able sand is signi cantly a ected by relative density, strength degradation, excess pore pressure, prior displacement history, and loading rate [9]. However, the mechanical mechanism of lique ed soil-pile interaction is still uncertain, and it is di cult to quantify and evaluate the in uence of various factors on the soil-pile interaction.
In general, the studies on seismic performance of pilesoil system are carried out by the following methods: model or in situ test [6,[10][11][12], simpli ed analysis methods (such as p-y curve, p is the force soil acts on the unit length of pile, and y is the lateral displacement), nite element [13][14][15], and theoretical analysis [7,16]. In practice, the shake table test and in situ test have disadvantages such as high cost and long time-consuming. e application of nite element numerical analysis method in engineering practice is limited due to the di culty of modeling, low calculation e ciency, and di cult parameter calibration. e simpli ed calculation method is usually preferred in practice due to the advantages of simplicity, high e ciency, and easy engineering implementation [17]. e p-y curve method based on Winkler foundation beam assumption is widely applied to model pile-soil laterally interaction, whereby the soil-structure interaction is modeled employing p-y curves, as shown in Figure 1 [18]. At present, the simpli ed models based on p-y curve method include Penzien model [19], Novak model [20], Nogami model [21], Naggar model [22], and Boulanger model [23]. e majority of simpli ed methods for pile-soil interaction regard lique ed soil as solid, which is contradictory to the de nition of liquefaction by the American Society of Civil Engineers, that is, the process of soil transforms from solid to liquid. ese methods considered the pile-soil interaction as the relationship between lateral load on the pile and soil deformation. Moreover, the in uence of liquefaction on pile-soil interaction was considered using the strength attenuation method [18], such as adjusting soil spring sti ness, modifying p-y curve (i.e., residual strength method, zero strength method [24], p-multiplier(α), and C u -factor [25]), or recalculating soil modulus according to e ective stress. e p-y curve method is well used in the analysis of pilesoil interaction in nonlique ed foundations; however, it is not completely suitable for pile-soil interaction analysis in lique ed soil without the reliable p-y curves for lique ed soils [26]. erefore, the appropriate description of the lateral pressure of liquefaction soil on piles is a key problem in pile-soil interaction simpli ed analysis.
Proper assessment of the characteristics of lique ed soil is necessary for the analysis of the lique able soil-pile interaction. Various studies have shown that the lique ed soil behaves like uid [3,[27][28][29][30][31][32] and exhibits rate-dependent characteristics [13,33,34], which are caused by viscous nature [34]. Wang et al. [35] considered the soil fabric as a uid net-type structure and presented a uni ed thixotropic uid model describing the behavior of the lique able soil in response to cyclic loading. However, the previous simple analysis methods of pile-soil interaction only considered the strength reduction of soil and the uid characteristics, i.e., the rate e ect [36] of lique ed sand were ignored. Ignoring the rate-dependent e ect in the analysis of lique able soil-pile interaction may underestimate the dynamic response of pile foundation, resulting in misjudgment of the seismic failure mechanism of pile foundation in the liquefaction eld.
In the present study, a simpli ed analysis method based on the hydrodynamics theory was proposed to investigate the rate-dependent interaction of lique ed soil and pile. In this model, the lique ed soil is treated as a thixotropic uid, and the viscous shear force related to the shear strain rate and apparent viscosity of lique ed soil is applied to replace the conventional soil pressure acting on the pile. Based on these assumptions, the simpli ed analysis method of pilesoil rate-dependent interaction is described in detail. en, based on two cases with di erent soil pro les, the validity of the simpli ed analysis method is demonstrated by comparing the results of the simpli ed analysis method with that of experiments and other methods. Finally, the e ects of the relative density of foundation soil, the thickness ratio of crust nonlique able layer to lique able layer, and the ground motion frequency on the bending moment and lateral displacement of pile are discussed in detail.

Principle and Process of Simplified
Analysis Method

e General Process of Simpli ed Analysis
Method. e analytical model proposed in this study is shown in Figure 2. It is assumed that the soil is homogeneous and isotropic. e simpli ed analytical model is composed of two main components: (a) a free eld system that is not a ected by pile to achieve the ground displacement and (b) a lumped mass model of the pile. In this method, the lique ed soil is assumed as a liquid and its lateral loading applied on the pile is related to the shear strain rate and apparent viscosity, which is independent of the ground displacement. Whereas the lateral loading of the nonlique able soil layer is dependent on the displacement and strength of soil, and the relative displacement is applied to the pile system through soil springs to model the pile-soil interaction in the nonlique able soil. e ideas and steps of the pseudo-static simpli ed analysis method to describe the pile-soil ratedependent interaction proposed in this study are as follows: (1) e acquisitions of the response of each soil layer. A one-dimensional free-eld response analysis is carried out to obtain shear strain, acceleration, lateral displacement of each soil layer based on soil pro le characteristics, and a given base acceleration time history. e maximum acceleration and soil displacement along the pile length during shaking can be obtained. (2) [37]. e lateral inertia force is applied to the concentrated mass points. (5) e bending moment and lateral displacement of pile response are calculated using the static method.

ixotropic Fluid Constitutive Model of Lique able Soil.
Mao et al. [36] found that the lique ed soil acting on the pile behaves like a viscous uid, the bending moment of the pile is proportional to soil velocities (i.e., shear strain rate), and the lateral force of lique ed soil acting on the pile has ratedependent characteristics. To present the rate-dependent nature in this simpli ed analysis method of pile-soil interaction, the thixotropic constitutive model describing the uid characteristics of saturated sand was established by Wang et al. [35], which is used in this study. e in uence of excess pore pressure and apparent viscosity representing the deformation resistance of lique ed sand on the ow e ect is considered in the thixotropic constitutive model, and the expression of the model is as follows: where τ and _ c are the shear stress and shear strain rate of the soil, respectively, r u is the excess pore pressure ratio, and η e and η ∞ are the viscosity coe cient in the initial equilibrium state (r u 0) and limit equilibrium state (r u 1), these two parameters are related to the e ective con ning pressure and relative density, which can be found in reference [35,38].

Seismic Response Analysis of Free Field.
To describe the soil-pile interaction, the dynamic shear strain and lateral displacement response of the soil need to be obtained. e e ect of the pile on soil movement was not considered to simplify the analysis. erefore, the lique able site was regarded as a free site, and its acceleration, displacement, and shear strain dynamic responses can be obtained through one-dimensional equivalent-linear site response analyses. e equivalent-linear site response software, Proshake, which is widely used in the site seismic response analysis, was utilized to generate the dynamic response of soil layer. e variation curve of shear modulus G and damping ratio λ with the shear strain c expressing the nonlinear characteristics of soil layer, as well as the initial shear modulus (maximum shear modulus G 0 ), initial damping ratio λ 0 , and shear velocity v s of each soil layer should be given in Proshake software.
Based on the wave theory, the shear wave velocity of soil layer can be calculated by the following formula: where v s is the shear velocity of soil and ρ is the soil mass density. As shown in Figure 3, the variation curves of G/G 0 -c a , λ-c a recommended by Seed and Idriss [39,40] were adopted in this study. In addition, the initial shear modulus G 0 of each soil layer needs to be determined in one-dimensional equivalent linear analysis. e value of G 0 can be decided in two methods, one is the empirical formula established by the laboratory test [41] and the other is to use the eld shear wave velocity test results. In this study, the empirical formula of the maximum dynamic shear modulus G 0 of sand proposed by Seed   Shock and Vibration 3 where K max is a parameter determined by the relative density D r of soil, K max 61[1 + 0.01(D r -75)], P a is the atmospheric pressure, and σ c ′ is the e ective con ning pressure. e backbone curve of soil will decline due to the development of excess pore pressure for cohesionless liqueable soil. is shape characteristics of the backbone curve can be described according to the attenuation of G with the pore pressure ratio. Matasović and Vucetic [42] established an empirical formula for the attenuation of backbone curve with the development of pore pressure ratio according to the cyclic load test results of saturated sand.
where G * 0 is the maximum shear modulus at the excess pore pressure of r * u . n is the parameter, n ≈ 0.5 for sand. e soil layer is divided into layers with equal heights of 0.5 m. e maximum shear modulus of soil layer under di erent pore pressure ratios, relative density, and con ning pressure (related to soil depth) can be calculated through (2) and (3).

Lumped Mass Model of Pile Foundation.
e pile is modeled using the lumped mass model proposed by Penzien et al. [19] in this study. Lumped mass model, which was also called multiparticle model or Mindlin static model, can simply and e ciently handle the relationship between mass, sti ness, and damping in dynamic analysis [43]. e pile foundation was simpli ed as a series particle system that was the mass of the pile body was dispersed on each particle, and the particles were connected by massless beams. e pile was equally divided into several parts along the length, and the length of each part was l e .

Viscosity Shear Force on Pile Section in Lique ed Soil.
As shown in Figure 4, the viscous shear force of soil acting on the pile body is equivalently distributed in the circumference of pile-soil interface. e horizontal lateral loading of the soil acting on the pile can be expressed as Equation (1) is substituted into (5): where ΔP z is the lateral load of the soil on the pile body with a length of ΔL at depth of Z, d is the diameter of pile, η ∞ and η e are the viscosity coe cient of soil when the excess pore pressure ratio r u is equal to 0 and 1, respectively, and _ c z is the shear strain rate of soil at depth of z. e values of η ∞ and η e could be referred to Wang et al. [35].

Lateral Force on Pile in Nonlique able Soil.
e lateral force of the nonlique able soil layer acting to pile foundation is the product of soil spring coe cient (subgrade bed coe cient or reaction coe cient) and soil lateral deformation. e maximum relative horizontal displacement at di erent depths in the free eld is applied as external boundary conditions to one end of the soil spring element that relates to the pile model. In this study, the soil spring coe cient along the depth was determined by Mindlin's solution [44][45][46] commonly used in the pilesoil interaction analysis. e soil spring coe cient of each node is calculated by (7) where k z is the coe cient of soil spring at the depth of z, is equivalent elastic modulus of soil at the depth of z, r 0.5d is the radius of pile, and l e is the length of pile element, which is 0.5 m in this study.

Horizontal Inertia Force on Pile.
e inertia forces of pile were represented as static forces applied simultaneously with lateral force. e inertia forces acting on particle i along the pile in the pseudo-static method is calculated according to the following formula: where F i is the lateral inertia force of the lumped mass particle i; a h is the lateral acceleration of the dynamic load; ξ is the reduction coe cient of seismic e ect, which is 0.25; M i is the mass of the particle i, M i l e Sρ pile ; S is the sectional area of pile; α i is dynamic distribution coe cient of inertial force of particle i, taken as 1 here; ρ pile is the density of pile, which is obtained as 2400 kg/m 3 . In this study, the software ABAQUS is used to solve the quasi-static calculation problem of lumped mass model. e linear Timoshenko beam element that can re ect the shear deformation is selected in this analysis. It is assumed that the transverse shear sti ness of the beam element is linearly elastic and remains unchanged during deformation.

Verification of Simplified Method
e results of centrifuge tests for pile foundation response in lique ed soil carried out by Abdoun et al. [12] and He et al. [16] are compared with the results of the simpli ed analysis model to verify its correctness. Moreover, the comparison of the simpli ed analysis method and the methods proposed by other literature are conducted to analyze the superiority of the simpli ed analysis method.

Cases for Veri cation.
Abdoun et al. [12] conducted a group of centrifuge shake table tests on pile foundations in liquefaction sites.
e simpli ed calculation model was established for Model 3 to calculate and analyze the response of pile foundation. Model 3 was composed of a 6 m overlying saturated sand layer and a 2 m underlying cemented sand layer, in which the relative density D r and saturation density ρ of the saturated sand layer are 40% and 1961 kg/m 3 . e length of the single pile is 8 m, the diameter d is 0.6 m, and the bending sti ness EI is 8000 kN m 2 . e experiment was conducted using a sine dynamic load with a frequency of 2 Hz and acceleration amplitude of 0.3 g in a prototype.
Besides, the simpli ed analysis method was veri ed and validated not only using results from the experiments but also using the data obtained from other methods. Model 1 conducted by He et al. [16] was calculated by using the simpli ed analysis method proposed in this study, and the bending moment and lateral deformation were compared with that obtained by other methods to validate the correctness of the simpli ed analysis method. Model 1 consisted of a single saturated sand layer with a thickness and relative density D r of 5 m and 50%, respectively. A single vertical sti pile with a length of 5 m was installed in the model, and its diameter and bending sti ness EI were 0.31 m and 14320 kN m 2 , respectively. A sine wave with a frequency of 2 Hz and the base acceleration amplitude of 0.2 g was used in this model. Details concerning two models were documented in Abdoun et al. [12] and He et al. [16,47], respectively.

Comparison with Results of Experiments.
e comparisons between the experiment results of Abdoun et al. [12] and the simpli ed analysis results under di erent pore pressure ratios are presented in Figure 5. e calculated pile bending moment distribution along the depth is consistent with the test results, and the maximum bending moment occurs at the interface of sand and clay layers. e maximum bending moments at the pile bottom obtained by the test and the simpli ed analysis method are 122 kN m and 139 kN m, respectively. e error between the simpli ed calculation results and the test results is 14%. Furthermore, the gap between the simpli ed analysis results and the test results appears to increase for the bending moment measured near the soil surface. is phenomenon may be caused by the following reason: (a) the ground surface of the experimental model was inclined at a slope of 2°, which does not consider in the simpli ed analysis method, (b) the lateral displacement of lique ed sand is largest at the foundation surface; however, this phenomenon cannot be re ected in the onedimensional equivalent linear analysis of the free eld, which leads to a larger di erence between the pile bending moment near the surface obtained from the analysis and the test. In general, the pile bending moment obtained by the simpli ed analysis method of pile-soil interaction considering ratedependence is in good agreement with the results of the test. It means that the simpli ed analysis method of using the hydrodynamics method to analyze pile-soil interaction and viscous shear force instead of the Earth pressure on pile is reasonable and feasible.
Furthermore, the rate-dependent characteristic of pile-soil interaction becomes more obvious with the development of pore pressure, and the pile response increases gradually until the pile at the interbedding of sand and clay reaches a maximum Shock and Vibration bending moment of 139 kN m at the pore water pressure ratio of 0.6. ereafter, the pile response decreases gradually with the increase of pore pressure. When the pore water pressure ratio rises to 1.0, the maximum bending moment reduces to approximately 21 kN m, only approximately 1/7 of the maximum value. is result shows that the seismic response of pile foundation in lique able soil may not reach the maximum when the soil is completely lique ed. According to the thixotropic constitutive model expressing in (1), the viscous shear stress of lique able soil on the pile is mainly determined by two factors: the shear strain rate of soil and the apparent viscosity representing the uid characteristics of the soil. When the excess pore pressure ratio of soil is low, the apparent viscosity is large, nevertheless, the uidity of foundation soil is poor, the corresponding shear strain rate is relatively small. With the increase of pore pressure, the apparent viscosity of soil decreases gradually, while the corresponding shear strain rate increases. It is noted that the development of apparent viscosity is opposite to that of the shear strain rate with the development of excess pore water pressure ratio. erefore, the apparent viscosity and shear strain rate of soil should be comprehensively considered to evaluate the rate-dependent e ect of soil on pile rather than relating to the pore pressure ratio.
It should be noted that the strongest response of the pilesoil rate-dependent interaction does not necessarily occur at the pore pressure ratio of 0.6, but possibly be related to the initial state of soil, geological conditions, soil layer distribution, dynamic load frequency, and other factors, which will be discussed in detail next. e maximum bending moment and lateral displacement evaluated by the simpli ed analysis method were almost consistent with the measured peak pile response in the experiment. In comparison with the simpli ed analysis method, the methods recommended by JRA and Dobry et al. [16]. underestimate the response of the pile, and the calculated bending moment is only about 25% of the results from centrifuge tests. As a simple approach, the simpli ed analysis method proposed in this study is found to predict both maximum bending moments and lateral displacements with su cient accuracy.

Parametric Analysis of Liquefaction Soil-Pile Interaction
Pile-soil interaction is a very complex problem, and there are many e ects (such as distributional characteristics of soil, soil properties, dynamic load, pile type, etc.). e e ects of relative density of soil, the thickness of nonlique able soil layer, and dynamic loading frequency on the response of pile foundation (i.e., the bending moment and lateral displacement) are considered in this study.

Initial Relative Density of Soil.
Based on Model 1 conducted by He et al. [16], three di erent initial relative densities of sand, which are 30%, 50%, and 70%, respectively, are considered in this study to investigate the in uence of the initial relative density on the pile-soil rate-dependent interaction. e distribution of bending moment and displacement of pile foundation under di erent initial relative densities are presented in Figure 7.
With the development of excess pore water pressure, the internal force and displacement response of the pile foundation can be roughly divided into two stages which are the growth stage and the attenuation stage. It is suggested that there is a threshold pore pressure that could be used to identify the stage change. When the pore pressure ratio is below the threshold pore pressure ratio, the pile response (bending moment and lateral displacement) is in the growth stage. e bending moment and lateral displacement of the pile reach the maximum values while the excess pore water pressure reached the threshold value. Once the pore pressure ratio passes this threshold, the pile response enters the fast attenuation stage.
As shown in Figure 7, the threshold pore pressure ratio of loose sand with the relative density of 30% is approximately 0.6, whereas the corresponding threshold pore pressure ratio for the relative density of 50% (medium dense sand) and 70% (dense sand) foundations are approximately 0.8 and 0.9, which indicating that with the increase of initial density of soil, the threshold pore water pressure ratio corresponding to the maximum pile response increases gradually. e reason is that the liquefaction resistance of the foundation soil increases with the increase of soil relative density. erefore, a higher pore pressure ratio is required in the foundation with higher relative density to achieve the corresponding uid characteristics, which has been conrmed in previous studies [31]. When the foundation soil is at a low pore pressure ratio level, the change of pile internal force and deformation response with the development of pore pressure ratio is not obvious, especially in the foundation with higher initial relative density. For example, in the foundation soil with a relative density of 70%, the pile bending moment distribution curve almost coincides when the pore pressure ratio increases from 0.1 to 0.7, which indicates that higher the pore pressure ratio is required to re ect the rate-dependent interaction in the foundation with higher the relative density. e maximum bending moments of the pile under relative densities of 30%, 50%, and 70% are 75 kN m, 124 kN m, and 118 kN m, respectively. Interestingly, the maximum bending moment of the pile body under the relative density of 50% case is the largest among the three cases. is result indicates that the response of pile-soil rate-dependent interaction does not monotonously develop with the initial relative density of the foundation soil. e reason may be that the apparent viscosity is smaller, and the uidity is stronger under the same pore pressure ratio in the loose foundation, consequently, the soil with lower relative density can reach a large shear strain rate under the same shear stress. In contrast, the dense sand has a larger apparent viscosity and lower uidity than that of the loose sand, which causes the corresponding shear strain rate to be smaller. erefore, the apparent viscosity and shear strain rate of soil should be comprehensively considered for the e ect of soil on the pile. It can be inferred that an "unfavorable" relative density is existence for the rate-dependent e ect of pile-soil in some cases, which generates the greatest bending moments at the piles. e relationship between shear strain rate and excess pore pressure ratio under the di erent relative densities of lique able sand is shown in Figure 8. Under the same pore pressure ratio, the denser the soil is, the smaller the shear strain rate is, which means that the shear strain rate is inversely correlated with the relative density D r of soil. When the excess pore pressure ratio developed to 1.0, the shear strain rate of soil with di erent relative densities are almost the same. e development curve of shear strain rate with pore water pressure ratio (considering D r 50% as an example) can be divided into two stages.
(i) Slow development stage: the uidity of soil increases slowly in this stage. e corresponding shear strain rate of soil with a relative density of 50% increases from 0.004 s −1 to 0.009 s −1 (increased about 2 times), while the pore pressure ratio increases from 0.1 to 0.6 ( Figure 8). However, the increase of bending moment response of pile caused by the rate-dependent e ect can be ignored (Figure 7(b)). (ii) Rapid development stage: in this stage, when the pore pressure ratio reached about 0.6, a sudden increase in the shear strain rate was observed, which means that the soil uidity developed rapidly. e shear strain rate of the soil was considerably improved with the development of excess pore pressure, with values of 0.009 s −1 at r u 0.6 and 0.13 s −1 at r u 1.0. In this stage, the pile bending moment reaches a maximum value at the excess pore pressure ratio of 0.8. For soils with di erent relative densities, the corresponding pore pressure ratios were di erent at the beginning of the rapid development stage of shear strain rate, that is, the higher the relative density, the greater the corresponding pore pressure ratio.

Ratio of Nonlique ed Layer ickness to Lique able Layer ickness.
To investigate the in uence of the thickness of nonlique able soil layer on the response of the pile, three di erent thickness ratios of the overlying clay layer to the lique able layer were considered in this study. e distribution of the bending moment and displacement of pile under di erent thickness ratios H N /H L, which is de ned as the ratio of the thickness of nonlique able layer in the model to that of the lique able layer, are presented in Figure 9. Comparing the bending moment curves of the pile under the three di erent H N / H L conditions, it can be found that although di erent thickness ratios H N /H L lead to a large di erence bending moment of pile, the bending moment distributed curve in either condition was similar, which was distributed in an "S-shaped" along with the depth of pile, that is, there were two maxima of bending moment in the pile, i.e., one near to the center of the sand layer and the other one located at the underlying nonlique able layer respectively. In addition, the maximum bending moment of the pile in the underlying nonlique able layer was smaller than that in the sand layer, and the trend was more obvious when the H N /H L decreased. Figure 9 also shows the distribution of the displacement obtained in the analysis. e displacement Shock and Vibration 7 distribution of the pile body along the height shows the shape of "parabola." e displacement curve of the pile has a reverse in ection point near the overlying clay layer due to the lateral restraint e ect of the overlying clay layer on the pile. erefore, the lateral displacement of the pile top is signi cantly less than those of the conditions without an overlying clay layer. Further, it was found that the lateral displacement of pile decreased with the increase of thickness of the top nonlique able soil layer, which was consistent with the conclusion in [48].
When H N /H L is 1 : 2, the maximum bending moment of the pile appears in the middle of the sandy soil layer, and the maximum lateral displacement of the pile appears around the boundary between the sandy soil layer and the overlying clay layer. With the increase of H N /H L , the maximum bending moment of the pile gradually decreases, and its position gradually moves upward from the lique able layer to the boundary between the overlying nonlique able layer and the lique able layer. e corresponding in ection point of the lateral displacement curve gradually moves from the  Shock and Vibration sandy soil layer to the overlying clay layer, which is caused by the di erent uidity of the lique able layer under di erent thickness ratios. It is obvious that with the increase of H N /H L , not only the maximum bending moment of pile foundation decreases but also the threshold excess pore pressure ratio corresponding to the maximum bending moment is higher, indicating that when the lique ed soil layer is thin or deeply buried, the conditions required for the soil to re ect its ow characteristics are relatively harsh. erefore, the pile-soil ratedependent interaction is inapparent.

Frequency of Dynamic Load.
e frequency of dynamic load has a great in uence on the foundation soil response and soil-structure interaction. Six di erent frequencies were considered in this study to reveal the e ect of the frequency of dynamic load on the pile-soil rate-dependent interaction. e distributions of pile bending moment along with the depth under di erent dynamic frequency f are compared in Figure 10. e pile-soil rate-dependent interaction is signi cantly correlated with the frequency of input ground motion. e inputted ground motion frequency corresponding to the bending moment of pile reached its peak value gradually transformed from 4 Hz to 1 Hz with the soil pore pressure ratio developed from 0.1 to 1.0 (as shown in Figure 10 and Table 1). e reason for the above phenomenon is that the natural frequency of soil gradually decreases as a result of the loss of sti ness and strength associated with the excess pore water pressure generation, and the eld is increasingly sensitive to long-period dynamic loads.
For the eld with an initial relative density of 30%, when the input dynamic load period is relatively long (i.e., 1 Hz and 2 Hz), the bending moment response of pile could be divided into growth stage and attenuation stage with the development of pore pressure, i.e., there was a threshold pore pressure ratio. In contrast, the pile bending moment response decreased gradually with the development of pore water pressure when the dynamic frequency was relatively high (i.e., 3 Hz, 4 Hz, 5 Hz, and 6 Hz in this study), which indicated that the threshold did not exist in those cases. It could be because, with the development of pore pressure, the uidity was increased; however, the lter e ect of softened soil on the high-frequency ground motion was also obvious. erefore, the pile foundation response under high-frequency load decreased monotonically with the development of pore pressure. Moreover, when the excess pore pressure ratio of soil was high, the bending moment response of pile under high-frequency load is smaller than that under lowfrequency load.
By comparing the pile bending moment of di erent cases (Table 1), it should be noted that the smaller the input ground motion frequency, the higher the threshold pore pressure ratio, and the smaller the pile bending moment corresponding to the threshold pore pressure ratio. is observation indicated that whether there is a threshold pore water pressure ratio, and its value is related to relative density, soil layer distribution, dynamic load frequency, etc. e signi cance of the above observations may be better understood by reference to the relationship between the shear strain rate of soil and the excess pore pressure ratio in di erent loading frequencies, as shown in Figure 11. With the excess pore pressure developed from 0.1 to 1.0, the vibration frequency of the case corresponding to the maximum shear strain rate of soil decreases from 4 Hz to 1 Hz, which was primarily a result of the decrease of soil natural frequency caused by the increase of excess pore pressure. Furthermore, the shear strain rate of soil increased in all conditions when the pore water pressure ratio reaches 1.0, as shown in Figure 11. However, the increased amplitude of shear strain rate under the conditions with the higher frequency load (3 Hz, 4 Hz, 5 Hz, and 6 Hz) was less than that under the conditions with lower input frequencies (1 Hz and 2 Hz). is is because that the soil sti ness decreased due to Shock and Vibration the development of excess pore pressure ratio, resulting in the high-frequency motions cannot be traveled to the upper soil (the filtering effect of liquefied soil on high-frequency load), therefore the shear strain rate was relatively smaller for the cases that the model was subjected to higher frequency.

Conclusions
In this study, a simpli ed analysis method, which considering the rate-dependent e ect of the lique ed sand on the pile and models the lique ed sand with a thixotropic uid, was proposed to evaluate the behavior of free-head single pile embedded in homogeneous and layered lique able soil. e calculation results were compared with that of Abdoun's tests to verify the validity of the proposed simpli ed analysis method. en, the e ects of relative density, pore pressure ratio, the thickness of overlying clay layer, and dynamic load frequency on the ratedependent characteristic of pile-soil were discussed. e main conclusions can be drawn as follows: (1) e simpli ed analysis method based on the hydrodynamics method, which adopting viscous shear force (related to the shear strain rate and apparent viscosity of soil) instead of the soil pressure on the pile to analyze pile-soil interaction, is practical. e distribution of bending moment along height obtained by the simpli ed method in this study is in good agreement with that of tests, and the maximum bending moment occurred near to the interface of lique able-nonlique able soil layer.
(2) e pile bending moment and displacement response do not necessarily change monotonically with the development of the excess pore pressure ratio. ere exists a threshold pore water pressure ratio (i.e., when D r is 30%, 50%, and 70% and threshold pore pressure ratio r u is 0.6, 0.8, and 0.9, respectively), resulting in the pile bending moment and displacement greater than those of other pore water pressure ratios. Based on the threshold pore water pressure ratio, the response of the pile can be divided into two stages, that is, the growth stage and the attenuation stage.
(3) e maximum bending moment of the pile under the relative density of 50% case is larger than that under the other two relative densities. An "unfavorable" relative density, which maximizes the pile response under the same other conditions (excess pore pressure ratio and dynamic load), is existence for the pile-soil interaction e ect. (4) Whether there is the threshold pore water pressure ratio, and its value is related to the relative density of soil, the thickness of overlying clay layer, and the vibration frequency. Compared with the higher frequency load conditions, the threshold pore pressure ratio is more prone to occur in the conditions with relatively lower frequency. Furthermore, the higher the relative density, the thicker the overlying clay layer, and the greater the threshold pore water pressure ratio. When the soil is in the state of high pore pressure ratio, the pile foundation response under high-frequency dynamic load is less than that under low-frequency dynamic load, which is caused by the high-frequency ltering e ect of soil under low e ective stress.
Data Availability e data are contained within the article.

Conflicts of Interest
e authors declare no con ict of interest.

Authors' Contributions
Conceptualization was done by X.Z. and Z.W.; methodology was performed by X.Z. and H.G.; software was provided by X.Z.; validation was performed by H.G. and Z.W.; formal analysis was performed by W.L.; investigation was performed by H.G. and W.L.; resources were provided by Z.W.; data curation was performed by Z.J.; original draft was written by X.Z. and Z.J.; reviewing and editing were done by Z.J.; visualization was performed by Z.J.; supervision was done by X.Z. All authors have read and agreed to the published version of the manuscript.