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Case histories have shown that the liquefaction-induced soil lateral spreading is one of the main causes of damage to pile foundations subjected to seismic loading. Post-liquefaction soil behaves similarly to a viscous fluid. This study investigated the effect of soil lateral spreading on a single pile based on fluid mechanics in which the liquefied soils were treated as Newtonian fluids. A numerical simulation on a single pile embedded in a fully saturated sandy foundation was conducted and compared with shake table tests. The lateral flow effect and the effect of shear strain rate were discussed. After liquefaction, the acceleration of the foundation shows that there are no obvious spikes and finally reaches a stable state. The presented method can predict the pile response better than p-y curve method. A parametric study was performed to explore the effect of several influence factors on pile behaviors. The results show that the pile head displacement decreases and the maximum bending moment at pile bottom increases with the increase of bending stiffness. With the same pile bending stiffness, the displacement and bending moment of pile increase with the increase of soil viscosity and acceleration amplitude.

Pile foundations have been widely used to support bridges, ports, and harbor facilities that are located on liquefiable soils near a waterfront structure. In recent years, liquefaction resulting from seismic events has become a major concern due to its impact on structures, buildings, and other infrastructure during and after an earthquake. Almost all major earthquakes are accompanied by soil liquefaction, as evidenced by the 1976 Tangshan earthquake in China [

To date, shake table experiments [

Liquefaction hazards are associated with substantial economic and personnel life losses. The large-scale deformation induced by liquefied soil is more serious and harmful to the subsurface structure than soil liquefaction itself. Previous studies of soil liquefaction have mainly focused on influencing factors, initial conditions, and liquefaction predictions. Recently, more attention has turned to the importance of the large-scale flow deformation associated with liquefaction. These studies were generally based on conventional solid mechanics and assumed a relatively limited maximum shear strain [

In the past few years, a series of laboratory tests conducted by different researchers have shown that liquefied soil behaves similarly to a viscous fluid [

In this paper, the liquefied sand was regarded as an incompressible Newtonian fluid. The pile-soil interaction was simulated and analyzed based on fluid-structure coupling theory, and the results were compared with the shake table model tests to verify the correctness of this method. Besides, the p-y curves method was adopted to compare with the numerical simulation results of pile response and discuss the lateral flow effect and the effect of the shear strain rates. Next, a parametric study was performed using the proposed method. Finally, the results were utilized to draw insights and conclusions, and future works were proposed to study the pile-soil interaction in laterally spreading ground.

In this study, liquefied sand is regarded as an incompressible fluid, and the numerical modeling involves the solution of the Navier-Stokes equations, which is based on the assumption of the conservation of mass and momentum. The conservation of mass is described by the following equation:

∂ ρ ∂ t + ∇ ( ρ u ) = 0 (1)

and the conservation of momentum is described as:

∂ ρ u ∂ t + ∇ ( ρ u u ) = ρ f v + ∇ T (2)

The fluid is assumed to follow the generalized Newton’s Law; the constitutive model can be described as follows:

T = − p I + 2 η ( S − 1 3 ∇ u ) (3)

Applying the constitutive model Equation (3) to Equation (2) results in the Navier-Stokes equation:

∂ ρ u ∂ t + ∇ ( ρ u u ) = ρ f v − ∇ p + 4 3 η Δ u (4)

In Equation (1) to Equation (4), ρ is the fluid density, S is the strain rate tensor, f v is the volume force, u is the velocity tensor, η is the viscosity, T is the stress tensor, and p is the static pressure.

To solve the Navier-Stokes equations, the PISO algorithm [

Based on the above principle, the pile-soil interaction problem was analyzed by the iterative coupling method based on ADINA finite element software, and the stress distribution and displacement of pile foundation were calculated and analyzed. To investigate the dynamic response of pile foundations in the liquefied ground, results from the shake table test conducted by Le Su [

In the FE model, the saturated soil was simulated as an incompressible fluid. The steel pipe pile was modeled as solid elements with the properties of an elastic section. A fluid-solid coupling surface was set up between the pile and liquefied soil. Due to different modules, the discretization of the model is illustrated in

The pile is considered as a linear elastic material. The constitutive behavior of the soil is captured by the fluid material since it is assumed as a fluid. Material parameters are shown in

Physical quantity | Pile | Liquefied Soil | Water |
---|---|---|---|

Modulus of elasticity, E (GPa) | 190 | / | / |

Density, ρ (kg·m^{−3}) | 2500 | 1800 | 1000 |

Possion’s ratio, ν | 0.29 | / | / |

Viscosity, η (kPa·s) | / | 20 | 0.001 |

The analysis was carried out in three steps: 1) the soil and pipe pile mesh were built. The pile and soil were assigned elastic properties and fluid, respectively, and the static gravity of the model was applied to establish the initial stress states in the soil. The values of stress in this stage were used as initial values for the next stage of loading; 2) the spring elements were incorporated into the model; 3) the dynamic analysis was performed by application of the input motion to the model base. The transient dynamic nonlinear analysis was performed with 1000 steps of 0.001 s. The Newmark method was used to integrate the dynamic response with α = 0.25 and δ = 0.5 for convergence. Besides, the modified Newton-Raphson algorithm was used to solve system equations.

The responses of the soil and the pile are presented in this section and compared with the recorded data of the shake table test. To demonstrate the accuracy of the numerical model, only the results of shake table test with rotational stiffness of 120 kN·m/rad were presented here.

As shown in

_{max}) occurs near the base. However, the displacement response of pile decreases gradually along with the depth of foundation, which is contrary to the bending moment response of pile. The lateral displacement of the pile reaches the maximum at the pile head. The simulated pile displacement is slightly less than that from the experiment, and the difference between them can be corrected by adjusting the soil viscosity parameters.

The acceleration responses of soil in shake table tests could be divided into three main stages. In Stage 1, the amplitude of soil acceleration gradually increased.

The excess pore pressure, u_{e}, developed rapidly in soil foundation, and the soil reached the initial liquefaction state in this stage. In Stages 2 and 3, the acceleration attenuated significantly as the sand stratum liquefied and then remained constant at a low level, indicating that the liquefied sand had very low shear strength. But the numerical modeling does not well simulate the acceleration response of the soil layer along with the depth in Stage 1, since the fluid had a low shear strength at a high shear strain. It can compare with the experiment in Stage 3.

result in Stage 3, but the frequency of simulated soil acceleration is lower than the frequency of base excitation in Stage 1 and Stage 2. Therefore, in the process of phase transformation, the frequency of soil acceleration changes, and the acceleration response of soil reaches a stable state after liquefaction. With the increase of excess pore pressure, the soil softens and the natural frequency decreases gradually, and the shear strain rate increases with the decrease of frequency.

Currently, two simple lateral soil pressure profiles (uniform and triangular) for a single pile subjected to lateral soil flow induced by liquefaction have been proposed. The first profile recommends a uniform soil pressure profile proposed by Dobry et al. [

p = K γ z (5)

where p is the lateral soil pressure, and γ represents the total unit weight of overburden soil, K is soil pressure coefficient, and z is soil depth. JRA recommends a coefficient K with a value of 0.3.

Applying these approaches to the results of this study, the uniform soil pressure and coefficient K can be estimated based on matching the measured bending moment. In this regard, the soil pressure can be integrated readily along the depth to obtain the bending moment and pile displacement [

_{s}, and more details can be found in Le Su [

In the BNWF model, the behavior of the nonlinear p-y spring element is described by American Petroleum Institute [

p u = ( C 1 H + C 2 D ) γ H (6)

where C_{1}, C_{2} are related to the internal friction angle, ψ (see

Here, the equation proposed by American Petroleum Institute [

p = A p u tanh ( k h A p u y ) (7)

where A is the influence factor of dynamic load which is set as 0.9, h is the depth of soil, and y is the displacement of pile.

Note that both p_{u} and k must be further modified to take the liquefaction effect into account. Therefore, Li et al. [_{u} and k. Parameters p_{u} and k were modified as follows:

p u l i q = α p u (8a)

k l i q = β k (8b)

where α and β are the liquefaction effect modification factors, derived from D_{r} :

α = ( 0.026 z / l + 0.055 ) e 0.016 D r (9a)

β = ( 3.195 z / l + 0.495 ) D r − 1.45 (9b)

where e is Euler’s number, z is pile depth below the ground surface, l is pile length and D_{r} is the relative density of soil.

Based on the modified ultimate capacity and initial modulus, the equation is modified as follows:

p = A p u l i q tanh ( k l i q h A p u l i q y ) (10)

ψ | C_{1} | C_{2} |
---|---|---|

30 | 1.95 | 2.7 |

32 | 2.1 | 2.9 |

34 | 2.9 | 3.4 |

36 | 3.2 | 3.6 |

38 | 4.0 | 4.0 |

>40 | 4.6 | 4.35 |

According to the experimental configuration and the aforementioned formulas, the soil spring parameters ( p u l i q and k l i q ) in the BNWF model can be calculated.

As shown in

Moreover, the simulated pile head displacement is closer to the experimental one compared to that obtained from p-y method. It can be seen that the method of analyzing pile response under dynamic load is reasonable when the liquefied soil is regarded as fluid.

At present, there are many methods for the pile-soil interaction, such as the p-y curves and m method. These methods are based on solid mechanics theory, without considering the lateral flow effect of liquefied soil, thus underestimating the soil-pile interaction, especially for the lateral expansion of fully liquefied soil. The m method cannot describe the force acting on the pile at the pile-soil interface, which ignores the effect of the soil movement during the earthquake. The p-y curve method describes the interaction of pile-soil interaction inaccurately, due to the development of pore pressure and the flow effect of liquefied soil. There is an obvious correlation between the flow effect and soil-pile interaction. The existing methods do not consider the influence of the flow effect and the excess pore pressure of the liquefied soil on the response of the pile foundation. Therefore, it is necessary to establish a new method, which can be used for the analysis of pile-soil interaction in the liquefied foundation.

The comparison of pile responses using the proposed simulation method and p-y method under different liquefaction soil flow velocities, V, is shown in

It can be seen that the flow effect of liquefied soil cannot be neglected, and the effect is more obvious when the velocity is greater. Due to the flow effect of the liquefied soil, the force will increase, resulting in additional internal stress and deformation of the pile foundation, so the pile foundation seismic design needs to consider the flow effect of the liquefied stratum. For liquefied soils with lateral spread, the existing methods must be modified to avoid underestimating the pile-soil interaction.

In the soil liquefaction process, with the increase of pore pressure, the viscosity of soil decreases, but the corresponding shear strain rate increases gradually, which leads to the increase of shear stress of pile. Loading frequency is not an important factor affecting soil dynamic strength, but it affects the accumulation of pore pressure [

^{ }as the frequency varies from 0.5 Hz to 3 Hz, respectively. From the numerical simulation results, it can be found that the difference in frequency leads to the difference in soil shear strain. The effect of shear strain rate should be considered in the analysis of pile-soil interaction during the liquefaction process.

The influence of pile modulus E on the pile response is shown in

The influence of soil viscosity, η, on the pile response is shown in

As shown in

Based on the fluid mechanics, the pile-soil interaction simulation model under the condition of liquefaction-induced large lateral spreading was established by taking liquefied soil as a Newtonian fluid. The simulation results were compared with the shake table test and p-y method to study the behavior of piles subjected to the flow of liquefied soils. The influencing factors of pile-soil interaction in laterally spreading ground are analyzed. The following conclusions can be drawn:

1) The behavior of liquefied soil is similar to that of viscous fluid. The pile-soil interaction can be well simulated based on fluid mechanics theory which treated liquefied soil as fluid.

2) There are no obvious spikes of soil acceleration, which indicates that the liquefied soil can bear the large shear strain and consume seismic wave energy. In the process of liquefaction, the natural frequency of soil layer decreases slightly with the phase transformation, and finally reaches a stable state. It is unreasonable to assume that the distribution of soil pressure is strictly linear along the depth direction, which may underestimate the soil pressure at the pile bottom.

3) The p-y curves method for analyzing the behavior of the pile under lateral spreading was adopted to validate the numerical simulation response. Lateral flow effects exacerbate the pile response, both in pile head displacement and bending moment. A larger frequency leads to large shear strain rates in soil, and the pile will undergo a relatively large lateral displacement and bending moment due to larger shear strain rates. The effects of lateral flow and shear strain rates are not considered by p-y curves. Compared with p-y curves, the method proposed in this study is more reasonable.

4) The results show that the pile head displacement decreases with the increase of bending stiffness, but the maximum bending moment at pile bottom increases slightly. With the same pile bending stiffness, the displacement and bending moment of pile increase with the increase of soil viscosity and acceleration amplitude.

It should be noted that more shake table tests and numerical simulations with various scenarios should be conducted to investigate more influencing factors such as time-varying viscosity, overburden pressure, and earthquake parameters, which are also significant in the seismic design of pile foundations under lateral spreading.

This work was supported by the National Natural Science Foundation of China (grant number 51678300) and Middle-aged & Young Science Leaders of Qinglan Project of Universities in Jiangsu Province, China. These financial supports are gratefully acknowledged. The contributions of anonymous reviewers and editors are also acknowledged.

The authors declare no conflicts of interest regarding the publication of this paper.

Sun, J.J., Zhang, X.L., Wang, Z.H., Gao, H.M. and Xu, L. (2021) Dynamic Analysis of Single Pile in Liquefied Soils Considered as Newtonian Fluid. Open Journal of Geology, 11, 19-37. https://doi.org/10.4236/ojg.2021.112002