# FE Simulation of Model Tests on Vibratory Pile Driving in Saturated Sand

## Abstract

The present study reports the extensive comparison of model tests with numerical simulations of vibro-driven pile installation in saturated sand. The purpose of the study is to validate existing simulation techniques and to investigate the ability of those to reproduce effects experimentally observed during pile installation. A limited number of cycles has been considered and the focus is placed on the cyclic evolution of soil deformations and stresses. Two axisymmetric FE models have been developed for the simulation of model tests. In the first, the pile-soil interaction is modeled in a simplified way by applying a sinusoidal displacement boundary condition at the soil-pile interface close to the pile toe. The second model simulates the performed model tests more realistically by including the pile-oscillator system. The \(\mathbf {u}\)-*p* formulation has been adopted in both models for the dynamic analysis of fluid-saturated solids with nonzero permeability. A hypoplastic constitutive model with intergranular strain has been selected to describe the mechanical behavior of the soil. The soil displacements and the evolution of pile resistance are compared. The good agreement between the results confirms that the pile installation process can be satisfactory reproduced numerically.

### Keywords

Saturated sand Vibratory pile driving Model tests## 1 Introduction

Vibratory pile driving involves substantial stress redistribution in the underground. Especially in water-saturated soil, the installation process can lead to a significant reduction of the effective stresses and thus to soil shear stiffness degradation around the pile. Adjacent foundations and structures might therefore be subjected to inadmissible deformations and also the load-bearing behavior of a vibro-driven pile is strongly influenced by the installation process. Hence, there is a strong need for the development of appropriate calculation approaches and numerical methods to quantify the mentioned phenomena, not only from a researchers perspective but also from a practical point of view.

Difficulties concerning a realistic numerical simulation of vibratory pile penetration arise from the description of pile-soil interaction and the complicated deformation process in the soil around the pile, including large monotonic as well as cyclic deformations with alternating phases of very high and very low stress regime. The interaction of soil and pore fluid in water-saturated conditions leads to considerable additional challenges. Therefore, suitable modeling techniques and constitutive models have to fulfill very high requirements. Although there are several numerical studies simulating the pile driving process in saturated soil [6, 7, 8, 12, 21], most of them rely on simplifications and rough assumptions. Additionally, due to the fact that a validation of these methods has not been carried out, a reliable prediction of the effects of pile penetration is currently difficult and questionable.

A key issue of the simulation of vibratory pile driving below the groundwater table is the dynamic analysis of fluid-saturated solids with nonzero permeability. However, for example the commonly used finite-element program Abaqus does not provide a built-in procedure for these types of simulations. Consequently, vibratory pile driving is often simulated in dry sand although in situ the penetration depths mostly reach the groundwater. If the presence of pore water is considered, its influence is either reduced to the buoyancy force [6, 7, 8] or locally undrained conditions are assumed [21]. In order to tackle this problem, a user-defined finite element has been used in the present study. The element is based on the \(\mathbf {u}\)-*p* formulation and has been proposed and validated in [4].

The simplified model has been developed in order to validate a simplified simulation technique, proposed in recent numerical studies [4, 15, 21]. Therein, the pile-soil interaction is modeled by introducing a prescribed sinusoidal displacement boundary condition at the soil-pile interface. The maximum displacement amplitude is applied at the pile toe, while the amplitude vanishes along the pile shaft. The dynamic BVP is solved with zero and finite soil permeability. Firstly, the influence of soil permeability on the solution is investigated based on the numerical results. Subsequently, experimental and numerical results are compared based on the occurring soil displacements and displacement amplitudes.

The enhanced model overcomes some of the limitations of the first model by including the pile-oscillator system and a contact definition between soil and pile. Pile and oscillator are modeled as rigid bodies and vibrate due to a prescribed dynamic load at the oscillator. This setup enables the comparison of the penetration behavior of the pile between experiment and simulation. A detailed comparison of cyclic soil deformations is also provided in this paper. Accounting for the experiences with the simplified model, the second dynamic BVP is solved only for finite soil permeability.

## 2 Model Test Concept

The model pile has an almost half-circular cross-section with a diameter of 33 mm and a flat-ending pile toe. The front and side view of the pile are shown in Fig. 2. The front part of the pile is covered by a combination of felt and PTFE (Teflon)-stripes, in order to minimize the friction between pile and observation window. Assuming that the friction between sand and wall is also low, the simplified consideration of a radially symmetric geometry of the model seems to be justified.

In the tests, the pile is pre-installed in a certain depth before it is subjected to vibrations for a short time period of a few seconds. During the test preparation, the dry sand is pluviated into water while the pile is already fixed in its initial position. A description of the test preparation methods and a discussion on sample homogeneity are given in [19, 20]. It is intended to keep the pile penetration as low as possible. Therefore, the pile-oscillator system is suspended by a compensation spring that reduces the static driving force. The stress level in the soil due to gravity is very low and might be problematic for the calibration parameters used in the numerical simulation of the model tests. A moderately elevated stress level is achieved by application of a uniformly distributed load with a magnitude of 2 kN/m\(^2\) on the ground surface.

During the model tests, the region of interest around the pile toe is filmed with a high-speed camera providing approximately 15 images per cycle of vibration. The video files are then decomposed into individual images. The occurring deformations are evaluated using the Particle Image Velocimetry software JPIV [18] and a subsequent summation and strain calculation procedure [19, 20]. In the test sequence evaluated here, the other measurements concern the pile head force and the global penetration. They are recorded with a frequency of 2400 Hz. A detailed description of the measurement methods and the evaluation procedures can be found in [20].

## 3 Test Material, Constitutive Model and Soil Parameters

Properties of Karlsruhe Sand

Mean grain size | d\(_{50}\) | [mm] | 0,55 |

Coefficient of uniformity | U | [-] | 1,5 |

Critical friction angle | \(\varphi _c\) | [\(^{\circ }\)] | 32,8 |

Min. void ratio | e\(_{min}\) | [-] | 0,549 |

Max. void ratio | e\(_{max}\) | [-] | 0,851 |

(a) Constitutive parameters of Karlsruhe Sand and (b) additional constitutive parameters of the extended hypoplastic model with intergranular strain

*k*was determined by laboratory tests [2, 20] and can be estimated for a given porosity

*n*using Eq. 1, which corresponds to the Kozeny/Carman equation [3, 11].

## 4 Simplified Finite Element Model

*f*the vibration frequency and

*t*the time from the beginning of vibration. This displacement boundary condition is imposed with the maximum amplitude \(u_{ampl}\) at the pile bottom and 5 mm upwards along the pile shaft. Subsequently, follows a 20 mm long transition zone in which the amplitude decreases linearly down to zero, see Fig. 3(c). The horizontal displacement at the soil-pile interface is set to zero. Concerning the pore water boundary conditions, the pore water pressure at the top surface is set to zero, while the other boundaries are taken to be impermeable.

The dynamic BVP has been solved with the FE-Software Abaqus/Standard both with \(k=0\) and \(k=1.5 \cdot 10^{-3}~\text {m/s}\) soil permeability, the last chosen accordingly to Eq. 1. As in [15], the simulation with \(k=0\) is performed using CAX4 elements available in Abaqus/Standard by introducing the pore water pressure as an internal variable in the constitutive model. For the dynamic analysis of fluid-saturated solids with nonzero permeability the axisymmetric *u*8*p*4 \(\texttt {UEL}\), presented and validated in [4], is used. The dynamic calculation is carried out with the implicit HHT integration schema [9] for the duration of 1 s. The time increment is taken to be constant and equal to \(10^{-4}\) s, which corresponds to approximately 425 increments per cycle of vibration. In case of finite permeability, additional viscous stresses have been used for better numerical treatment of the dynamic step, as proposed in [4], with the values of viscosity coefficients \(\lambda \) and \(\mu \) (see Eq. 5 in [4]), chosen small enough in order not to strongly influence the numerical solution. Furthermore, in order to avoid positive (tensile) mean effective stress in the soil, the small-stress correction according to [4] has been used.

Parameters for the simulation of the model test

\(h_{sand}\) (m) | \(e_{0}\) \((-)\) | \(I_{D,0}\) \((\%)\) | \(u_{ampl}\) (mm) | \(v_m\) (mm/s) |
(Hz) | K\(_0\) (-) | K\(_f\) (GPa) | S\(_r\) (-) | k (m/s) |
---|---|---|---|---|---|---|---|---|---|

0.85 | 0.691 | 53 | 0.5 | −11 | 23.5 | 0.4 | 2.2 | 1 | 0/0.0015 |

### 4.1 Observations from FE Simulations

In the present section, the interest is focused on the influence of soil permeability on the solution of the boundary value problem. The main target is placed on the cyclic evolution of stresses in the vicinity of the pile toe due to the pile vibration. Since the instrumentation of model tests does not allow stress measurements in the soil, a validation of these observations based on the present model experiment is not possible. Consequently, in this section only numerical simulation results will be presented.

Figures 4 and 5 show the distributions of mean effective stress near the pile toe after 10 and 23.5 cycles (at the end of simulation, \(t=1\) s), obtained from the solution of the numerical simulation calculated with \(k = 0\) (Fig. 4) and \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) (Fig. 5). The red area in the figures corresponds to a low-stress zone, in which the mean effective stress lies between 0 and −0.5 kPa, i.e. does not exceed 15\(\%\) of the initial value at the pile toe level.

In the locally undrained case (\(k = 0\)), the mean effective stress is reduced close to the pile toe to nearly zero after several cycles of vibration. The low-stress zone is approximately circular and extends over an area of about two pile diameters after 1 s of vibration. For \(k = 1.5 \cdot 10^{-3}~\text {m/s}\), a low-stress zone is only generated at the pile shaft above the pile shoulder. Throughout the entire simulation time, the area beneath the pile toe is subjected to a quasi-stationary change between large effective stress during pile penetration and small effective stress during upward pile motion.

In the case of a finite permeability, the dynamic excitation is accompanied by a consolidation process in the soil. Figure 8 shows the calculated distribution of void ratio after 1 s of vibration. The initial void ratio is \(e_0=0.691\) (\(I_{D,0}=0.53\)) and corresponds to the light blue color in the figure. As can be seen from Fig. 8, there is a narrow zone with dilative behavior along the pile shaft and directly beneath the pile toe. Here, the soil behavior is governed by the large monotonic deformation due to the displacement trend downwards. Outside of this area, the cyclic deformation predominates, leading to an approximately circular soil compaction zone. The strongest densification occurs beneath the pile toe.

### 4.2 Comparison of FE- and Experimental Results

Beneath the pile toe, the horizontal displacements concentrate in a zone located below the pile shoulder in all three cases, Fig. 9(a)–(c). The soil is pushed outwards due to the imposed downward displacement at the pile-soil interface. The maximum values reach about 2 mm which corresponds to ca. 17% of the vertical pile displacement (12 mm after 1 s of vibration) resp. about 6% of the pile diameter (33 mm). The zone that is horizontally displaced is considerably larger in the case of undrained conditions. The experimental results lie somewhere in between the two FEM solutions but resemble more the drained solution. Above the pile toe, neither the experimental results nor the FEM solution with \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) show significant horizontal displacements. Contrarily, for undrained conditions a clear motion towards the pile shaft is observed there.

The vertical displacements below the pile toe near to the symmetry axis are similar, Fig. 9(d)–(f). They remain greater than 50\(\%\) of the vertical pile displacement up to about one pile diameter below the pile toe. However again, the undrained solution appears to slightly overestimate the occurring displacements for larger depths. Although a satisfactory agreement between the three solutions is observed beneath the pile toe, the existence of the zone with rotational displacement paths in the locally undrained case leads to a zone beside the pile toe with upward soil motion. As can be seen in Fig. 9(e) and (f), the model test results and the FEM results with \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) show a global trend for settlements.

The largest horizontal amplitudes occur below the pile shoulder and reach values up to 0.05 mm. The vertical amplitudes are about one magnitude larger, with 0.4 mm in the direct pile toe vicinity. The amplitude field calculated for partially drained conditions (\(k = 1.5 \cdot 10^{-3}~\text {m/s}\)) is in almost perfect agreement with the experimental results. The occurring amplitudes as well as the affected soil region are very similar, except for the very small vertical amplitudes in the outer regions. The FE solution obtained with undrained soil conditions shows a significant overestimation of the soil region subjected to vibrations. Moreover, the large displacement amplitudes above the pile toe and in the outer zones cannot be identified in the other results. It can be stated that the assumption of locally undrained conditions leads to a qualitative disagreement with experimental observations and numerical results with finite soil permeability.

Although very satisfactory results are obtained from the solution with \(k = 1.5 \cdot 10^{-3}~\text {m/s}\), the simplified FE model demands a knowledge of the experimental results at the pile-soil interface. Consequently, only limited conclusions can be drawn for the vibratory pile driving process, particularly for in situ conditions. Furthermore, important aspects such as the evolution of pile resistance during the pile installation process cannot be investigated. For a better understanding of the process, a second, enhanced FE-model is described in the following section.

## 5 Enhanced FE-Model

Figure 11(b) shows the FE mesh in the pile toe vicinity. The FE mesh in the soil is similar to the one of the simplified FE model, except directly beside and under the pile toe. The first row of elements in this region is rather coarse (size of elements about 6 mm) in order to reduce mesh distortion problems during the simulation of pile penetration. A frictionless node-to-surface contact formulation [5] has been used for the pile-soil interaction. A separation of the contact in the normal direction is excluded, since the opening of a gap filled with void (air) is physically not justified. Like in the simplified model, a constant distributed load with 2 kN/m\(^2\) magnitude is applied to the ground surface and the normal displacement and the shear stresses at the outer boundaries are taken to be zero. At the pile toe and along the pile shaft the contact to the pile forms the boundary condition for the soil. The hydraulic boundary conditions as well as the initial conditions and material parameters for the soil body are the same as those in the simplified FE model.

The calculation is performed in Abaqus/Standard with \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) and consists of three steps: In the first step, pile and oscillator are fixed and the initial geostatic equilibrium is calculated. In the second step, pile and oscillator are set free in vertical direction and move slightly downwards due to their own weight without the existence of inertia forces. The last step is a dynamic calculation step, where the excitation is applied as a prescribed dynamic excitation of the oscillator. The evolution of the load is shown in Fig. 11(c). In the experiment, the oscillator reaches the full vibration frequency and thus, the oscillating force after about two vibration periods. This behavior is approximated in the FE model. The dynamic calculation step was carried out according the implicit HHT integration schema with a constant time increment of \(10^{-4}\) s. Similar to the simplified model, the small stress correction and small viscous stresses have been used.

Parameters for the simulation of the model test with the enhanced model

\(h_{sand}\) (m) | \(e_{0}\) \((-)\) | \(I_{D,0}\) \((\%)\) | \(m_{pile}\) (kg) | \(m_{oscillator}\) (kg) | \(c_{connector}\) (kN/mm) |
(Hz) | K\(_0\) (-) | K\(_f\) (GPa) | S\(_r\) (-) | k (m/s) |
---|---|---|---|---|---|---|---|---|---|---|

0.85 | 0.691 | 53 | 2.25 | 11.4 | 173 | 23.5 | 0.4 | 2.2 | 1 | 0.0015 |

### 5.1 Comparison of Numerical and Experimental Results

The pile penetration behavior is compared in Fig. 12. Figure 12(a) shows the evolution of measured and calculated pile displacements. Figure 12(b) depicts in detail the time period from 0.4 to 0.62 s of Fig. 12(a). The cycle at about 0.5 s is selected as a representative cycle for the comparison of the results and is marked with five points. The measured and calculated related soil reaction force during pile penetration is compared in Fig. 12(c) and (d).

It should be noted that by modeling the dashpot, it is not intended to propose a universal model to substitute the shaft friction. Shaft friction depends on the normal stresses acting on the shaft and the ultimate shaft friction is usually not constant throughout the process. However, it has been experimentally observed that system friction predominates in the present case and that its limit value is relatively constant. This system friction influences the penetration behavior of the pile. The purpose of the dashpot feature here is to achieve a better comparability between experiment and simulation by introducing effects that are not incorporated in the numerical model.

Figure 15 shows the comparison of experimental and numerical results like Fig. 12, but with the dashpot. The penetration behavior in the simulation is now more similar to the experiment, indicating a beneficial performance of the dashpot. Also the evolution of penetration resistance is reproduced better. However, the strong increase of pile resistance between the points 2 and 3 is still missing in the simulation. The reason for this discrepancy will be discussed later on.

Open image in new window There is a good agreement between numerical and experimental results. Both reveal soil displacements directed away from pile toe in a spherical area of about two pile diameters. However, compared with the model test the numerical simulation overestimates the soil displacements.

Open image in new window The soil displacements are qualitatively similar to Phase \(1\div 2\). In the experiment, a deeper and larger zone of the soil is affected compared to the first phase of penetration. The numerical results do not exhibit significant differences to the preceding phase.

Open image in new window Due to the sharp decrease of tip pressure after the reversal of pile motion, a slight uplift and horizontal soil movement towards the pile is observed. The vertical displacement field is similar to the previous phase \(2\div 3\), but inversed. Model test and simulation results are in good agreement.

Open image in new window The deformation mechanism corresponds to the Phase \(1\div 2\) in opposite direction. Directly below the pile toe, the soil is moved to the symmetry axis following the vertical motion of the pile. The simulation reproduces this mechanism qualitatively and quantitatively with almost perfect agreement.

For the sake of brevity, the incremental displacement fields obtained from the simulations without the dashpot feature are not presented here. However, it should be noted that they are also in very good accordance with the experimental results.

Experiment and simulation show a considerable oscillation of volumetric strain, which proves the presence of at least partially drained conditions in the soil. At element A, the largest amplitude of volumetric strain is observed. In the model test, the peak-to-peak amplitude amounts 1.7% at element A resp. only 0.5% at element C. The numerical simulation clearly underestimates the changes of volumetric strain (0.5% at element A and 0.25% at C).

The numerical simulation reproduces the contractant soil behavior during Phase 1–2 but fails to mobilize dilatancy, that is experimentally observed towards the end of the downward pile motion (Phase 2–3). The dilatant phase in the experiment corresponds to the phase with strong increase of penetration resistance (between the points 2 and 3 in Fig. 15(c)) and results in a deeper and larger deformation mechanism (see Phase 2–3 in Fig. 16). On the other hand, the lack of dilatancy in the simulation leads to the almost linear increase of soil resistance during the downward pile motion (Fig. 15(d)) and very similar incremental displacement fields for the two penetration phases (Fig. 16). During the upward pile motion, generally a volume increase is observed, which is more pronounced in the experiment. However, in the experiment, at element C the reversal of pile motion is accompanied by a strain reversal and a slight contractant soil behavior (between points 3 and 4 in Fig. 18(d)), which is not evident in the simulation.

The inability of simulation to reproduce the effects mentioned above is related with the ineffective performance of the hypoplastic constitutive model to describe some issues of cyclic soil behavior. Important deficiencies of the applied hypoplastic model will be discussed in the following paragraph based on element tests.

### 5.2 Hypoplasticity in Cyclic Triaxial Element Tests

Triax A: alternation of monotonic triaxial compression of \(\varDelta \varepsilon _\mathrm{1}=-4\%\) followed by unloading to \(q=0\)

Triax B: repetition of cycles consisting of a monotonic triaxial compression of \(\varDelta \varepsilon _\mathrm{1}=-4\%\) and a monotonic triaxial extension of \(\varDelta \varepsilon _\mathrm{1}=2\%\)

Both experiments show a global trend to dilatancy due to the large monotonic portion of the deformation. Cyclic deformation without shear stress reversal in Triax A leads to stronger dilatancy than it is the case in Test B with shear stress reversal. The unloading phases are associated with significant contractant material behavior, which is more pronounced in Test B (with shear stress reversal). During reloading, the material continues to contract slightly before a strong mobilization of dilatancy is observed. The volumetric strain amplitudes are about 0.5% for Triax A and 1% for Test B respectively. The reloading stiffness is lower in Test B than in Triax A.

Comparison with back-calculations reveal considerable problems for hypoplasticity to realistically reproduce the test results. The most important shortcomings of the hypoplastic model are a lack of contractancy during unloading, a too long phase of contractant behavior during reloading and too weak mobilization of dilatancy subsequently. Similar shortcomings have been observed and described by Niemunis et al. [13] and Wichtmann [23]. From these deficiencies arise the major problems of the numerical simulation to reproduce the experimental observations, such as the evolution of pile resistance and the penetration velocity (see Fig. 15). Recently developed constitutive models, like neohypoplasticity [13], aim to improve the described deficiencies of the commonly used hypoplasticity model. There is reasonable hope that by implementation of a better description of soil behavior within the FE framework of the present study, enhanced simulations of vibro-penetration can be achieved.

## 6 Closing Remarks

Vibratory pile driving in saturated soil has been intensely studied by means of Finite Element simulations with help of a user-defined element in the framework of an \(\mathbf {u}\)-*p* formulation. A simplified and an enhanced model have been developed and used to back-calculate model test results.

The simplified solution technique proposed in [4, 15, 21], regardless its rough simplifications with the prescribed sinusoidal displacement boundary condition, can be used to study the soil deformations around a vibrating pile if the considered time period and the occurring pile penetration is limited. The assumption of zero soil permeability affects the numerical solution and leads to results that have not been confirmed by experimental observations, notably the formation of a zone with rotational displacement patterns around the pile toe and an unrealistic build-up of pore water pressure. Consequently, the assumption of locally undrained conditions in the soil seems to be inappropriate for the numerical simulation of vibratory pile driving. The solution with the actual soil permeability achieves to reproduce adequately the soil deformations in the vicinity of the pile during the vibratory pile driving process.

The good agreement between the results of the enhanced FE-model and the model test confirms that the pile installation process can be satisfactory reproduced numerically. The differences observed in the developed pile resistance and penetration velocity do not originate from the numerical method, but rather from the inability of the hypoplastic model to describe adequately the soil behavior for the occurring deformation paths.

As a next validation step, the developed enhanced FE model will be applied for the simulation of new vibratory tests, presented in [22] in this book. These experiments include also large pile penetration effects and provide an improved instrumentation to enable pore pressure measurements.

## Notes

### Acknowledgments

The study was financed by the Deutsche Forschungsgemeinschaft as part of the Research Unit FOR 1136 ‘Simulation of geotechnical construction processes with holistic consideration of the stress strain soil behaviour’, Subproject 6 ‘Soil deformations close to retaining walls due to vibration excitations’.

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