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

  • Stylianos Chrisopoulos
  • Jakob Vogelsang
  • Theodoros Triantafyllidis
Chapter
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 82)

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].

In the present study, model tests on vibro-driven pile installation in saturated sand are compared with numerical simulations. The model validation is carried out based on the back-calculation of an exemplary test sequence that has already been extensively evaluated [19]. In this first validation step, a limited number of cycles has been considered, in order to justify the application of an implicit dynamic FE-formulation in combination with a hypoplastic constitutive soil model. Two axisymmetric FE models have been developed for the simulation of model tests, see Fig. 1.
Fig. 1.

(a) Simplified and (b) enhanced modeling technique

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 experimental set-up and the instrumentation for the model tests was developed by Vogelsang et al. [19], see also [20]. It is illustrated schematically in Fig. 2. The test container has a half cylindrical base area and a plane acrylic glass front sheet, which is used as an observation window for the measurement of the soil displacements around the pile using Digital Image Correlation (DIC) technique. A similar test set-up is used by Savidis et al. [16], Tehrani et al. [17] and Arshad et al. [1].
Fig. 2.

Set-up of the experiments with detail photos of the pile-oscillator system and the pile toe

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

A poorly graded medium quartz sand with sub-rounded grains has been used in the model tests. It is referred to in the literature as Karlsruhe Sand. Some important properties of the sand are given in Table 1. An extensive characterization of the material can be found in [20]. In the numerical simulations, the hypoplasticity constitutive model according to von Wolffersdorff [25] extended by the intergranular strain concept proposed by Niemunis and Herle [14] has been used to describe the granular soil behavior. The hypoplastic parameters of Karlsruhe sand, used in the present study are depicted in Table 2. The parameter set has been proposed for the test sand in [24] and has also been used for numerical simulations of vibro-penetration in [2, 21].
Table 1.

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

Table 2.

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

The permeability coefficient 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].
$$\begin{aligned} { k(n) = \frac{1}{308} \frac{\gamma _w}{\eta _w} \frac{n^3}{(1-n)^2} d^{2}_{w} } \end{aligned}$$
(1)
with \(\gamma _w\) the specific weight and \(\eta _w=1.37 \cdot 10^{-3}~\text {kN s/m}^2\) the dynamic viscosity of the water and \(d_{w}=0.5\) mm the effective grain size.

4 Simplified Finite Element Model

The radially symmetric FE-model with the initial and boundary conditions is shown in Fig. 3. The dimensions of the model are selected to coincide with the geometry of the model test. Figure 3(c) shows a detail of the FE-mesh at the pile toe vicinity. The size of the finite elements amounts about 1.5 mm near the pile and 25 mm at the outer boundary of the model. As in the experiment, a constant distributed load with a magnitude of 2 kN/m\(^2\) is applied to the ground surface in order to avoid inadmissible tensile effective stresses and ensure better numerical stability. In order to approximate the influence of the side walls on the model test, no normal displacement and no shear stresses are selected at the outer boundary of the numerical model. A possibly occurring friction between sand and outer wall is thus neglected. As previously mentioned, the soil displacements occurring in the model test are evaluated near the pile toe with the Digital Image Correlation (DIC). The measured values are imposed as a time-varying boundary condition at the soil-pile interface of the FE model (see Fig. 3d), corresponding to the vibration of the pile. Consequently, the vertical displacement \(u_{y}\) is prescribed by means of a composite function, which decomposes into a cyclic part due to pile vibration and a trend due to pile penetration:
$$\begin{aligned} { u_{y}(t)=v_m~t- u_{ampl}\left[ 1-\cos (2\pi ft)\right] ,} \end{aligned}$$
(2)
with \(v_{m}\) the mean penetration velocity, \(u_{ampl}\) the pile displacement amplitude, 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.
Fig. 3.

Simplified FE model: (a) Geometry and boundary conditions, (b) initial conditions, (c) detail of the FE-mesh in the vicinity of pile toe and (d) prescribed displacement boundary condition at pile toe

The initial soil stresses are considered to be geostatic with the coefficient of earth pressure at rest \(K_0\), calculated according to Jaky [10]:
$$\begin{aligned} K_0=1-sin(\varphi '_{p})=0.4 \end{aligned}$$
(3)
with \(\varphi ^{'}_{p}=37^\circ \) corresponding approximately to the peak friction angle of the test sand for the given density. The initial distribution of the pore water pressure is assumed to be hydrostatic with the water level lying at the ground surface. The initial distribution of pore fluid pressure and effective stress components are illustrated schematically in Fig. 3b. The soil is considered to be fully saturated (\(S_r=1\)) with the compression modulus of the fluid taken to be constant and equal to the bulk modulus of the pure water, \(K_f=2.2\) GPa. The initial density is taken from the model test, with \(e_0=0.691\) corresponding to a medium dense sand with \(I_{D,0}=0.53\). The initial intergranular strain tensor components are set to zero.

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 u8p4 \(\texttt {UEL}\), 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.

Some important parameters of the simulation are summarized in Table 3.
Table 3.

Parameters for the simulation of the model test

\(h_{sand}\)

(m)

\(e_{0}\)

\((-)\)

\(I_{D,0}\)

\((\%)\)

\(u_{ampl}\)

(mm)

\(v_m\)

(mm/s)

f

(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.

This behavior is demonstrated in Fig. 6(a), where the evolution of mean effective stress at Point A (see Figs. 4 and 5) is plotted for the two different permeabilities. Point A is located about two pile diameters below the pile toe at a horizontal distance of a half pile radius from the symmetry axis. For \(k = 1.5 \cdot 10^{-3}~\text {m/s}\), the mean effective stress decreases slightly and oscillates with an amplitude of circa 1 kPa, while it vanishes after several cycles of vibration in the case of \(k = 0~\text {m/s}\). For \(k = 1.5 \cdot 10^{-3}~\text {m/s}\), due to the consolidation process, the average pore water pressure does not increase during the vibration but oscillates slightly about its initial value, see Fig. 6(b). In the case of zero permeability, the displacement trend downwards leads to a gradual increase of the mean pore water pressure during the vibration without reaching an asymptotic state. For a great number of cycles, the accumulation of pore water pressure during pile penetration can lead to unrealistic pore pressure gradients.
Fig. 4.

Mean effective stress near the pile toe after 5 cycles (left) and 23.5 cycles (right) for \(k=0\)

Fig. 5.

Mean effective stress near the pile toe after 5 cycles (left) and 23.5 cycles (right) for \(k = 1.5 \cdot 10^{-3}~\text {m/s}\)

Fig. 6.

(a) Mean effective stress (\(\sigma \)) and (b) pore pressure (p) at Point A (see Figs. 4 and 5) as functions of time for \(k = 0~\text {m/s}\) and for \(k = 1.5 \cdot 10^{-3}~\text {m/s}\)

Figure 7 shows the calculated displacement vectors in the soil after 1 s of vibration in the vicinity of the pile toe for the two investigated soil permeabilities. For \(k=0\), a significant zone with circular displacement paths about a stationary point, located in the low-stress zone, is observed. Similar to the observations in [15], the circular zone shows a significant accumulation of permanent displacements. Its existence is related to the assumption of locally undrained conditions in the soil. The rotation disappears for the solution with high soil permeability, as can be seen in Fig. 7 (right). For \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) the direction of the soil displacements is almost gravitational. A comparable behavior occurs in the model tests, where no rotational zones were observed and the direction of the soil displacements is found to be almost in the gravitational direction (see Fig. 9 in the following section).
Fig. 7.

Displacement vectors near to the pile toe after 1 s of vibration for \(k=0\) (left) and \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) (right)

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.

Figure 8 proves rather significant volume changes and thus, partly drained conditions in the soil. Based on these observations, the assumption of locally undrained conditions with almost constant void ratio cannot be justified. This consequence is also supported by the experimental observation of free particles that move cyclically within the soil skeleton due to water flow.
Fig. 8.

Void ratio near to the pile toe after 1 s of vibration for \(k = 1.5 \cdot 10^{-3}~\text {m/s}\)

4.2 Comparison of FE- and Experimental Results

In this paragraph, the experimental results are compared with the simulations using the simplified FE model. Figure 9 shows the isolines of horizontal (a–c) and vertical (d–f) displacements after 1 s of vibration. The experimental results are depicted in the left column, the central column corresponds to the FE solution with undrained conditions and on the right hand side, the FE solution with \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) is shown.
Fig. 9.

(a)–(c) Horizontal and (d)–(e) vertical displacement fields after 1 s of vibration obtained in the model test (left), by the numerical simulations with \(k =0~\text {m/s}\) (middle) and \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) (right)

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.

Figure 10 shows the fields of horizontal and vertical displacement amplitudes after 0.5 s of vibration. The figure is composed in the same way as Fig. 9.
Fig. 10.

(a)–(c) Horizontal and (d)–(e) vertical displacement amplitude fields after 0.5 s of vibration obtained in the model test (left), by the numerical simulations with \(k =0~\text {m/s}\) (middle) and \(k = 1.5 \cdot 10^{-3}~\text {m/s}\) (right)

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

The enhanced model is also radially symmetric and includes not only the soil body but also the pile-oscillator system, as can be seen in Fig. 11. The model dimensions correspond to those of the simplified model. The pile and the oscillator are assumed to be horizontally guided rigid bodies, each with its own mass defined in the related reference point. The pile is connected to the oscillator with a 1-D connector element, which represents the load cell in model test. This model setup enables the output of the force at the location of force measurement in the experiment.
Fig. 11.

Enhanced FE model: (a) Geometry and boundary conditions, (b) detail of the FE-mesh in the vicinity of pile toe and (c) evolution of the prescribed point load at oscillator

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.

For the comparison of the FE results (“full” model) and the model test results (almost “half” model), the masses of oscillator and pile as well as the force amplitude and the connector element stiffness are scaled by the ratio of the cross-sections of full- and half pile (scaling factor equal to 1.73). Thereby remains the free amplitude of the pile-oscillator system unchanged and the dynamic behavior of the total system is only slightly influenced. The results are compared based on the occurring soil reaction force. This force is the result of the pile head force subtracted by the inertia force of the pile. It also includes pore pressures effects. Comparability between the force measurements and simulation results is achieved by division by the cross-section of the pile. The result is named as related soil reaction force:
$$\begin{aligned} (F_H-m_{pile}~a_{pile})/A_b \end{aligned}$$
(4)
with \(F_H\) the pile head force, \(m_{pile}\) and \(a_{pile}\) the mass and acceleration of the pile and \(A_b\) the pile cross-section at the toe. It should already be noted here that in the experiment, \(F_H\) also includes system friction between pile and front sheet.
Some important parameters of the simulation with the enhanced model are summarized in Table 4.
Table 4.

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)

f

(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).

Apart from the first cycles a qualitatively similar behavior is observed in Fig. 12(a). After about 0.3 s, a quasi-steady state is reached with approximately constant mean penetration velocity. The oscillation amplitude is also similar, which indicates a successful transmission between the “full” simulation model and the “half” experimental model. However, the pile penetrates faster in the simulation than in model test. The penetration per cycle amounts about 0.65 mm in the simulation, while it is only 0.4 mm in the test.
Fig. 12.

(a) Evolution of measured and calculated pile displacements during the pile vibration, (b)–(d) detailed analysis of the section from 0.4 to 0.6 s: (b) evolution of pile displacement (c) related soil reaction force during pile penetration for the model test and (d) for the FE- simulation (\(m_{p}=m_{pile}\) and \(a_{p}=a_{pile}\))

The cycles of the force-displacement curves in Fig. 12(c) and (d) differ in shape and size between simulation and model test. Although the penetration per cycle is greater in the simulation, the maximum mobilized soil reaction force is only about a third of that in the experiment. Moreover, the characteristic S-shape of the evolution of penetration resistance in the test as described in [19, 20] is not evident in the simulation. The numerical model fails to reproduce the large force amplitude between the marked points 2 and 4. In the experiment, this difference is a result of the pile shaft friction and the friction between pile and front sheet. These effects are excluded in the numerical simulation. However, it can be assumed, that this simplification at least partly causes the observed differences in terms of penetration rate, see Fig. 12(a). Subtracting frictional influences from the experimental soil reaction force leads to a more realistic force amplitude compared to the simulation (see the schematic explanation Fig. 13). However, there is still a difference in the mobilized soil resistance.
Fig. 13.

Schematic explanation of the characteristics of the force curve in the model test and the numerical simulation

Fig. 14.

(a) Dashpot element attached to the pile to substitute system friction effects, (b) characteristic curve and (c) related force-displacement curve of the dashpot

In order to prove the above explanation, we introduce a dashpot feature in the FE model, Fig. 14. The dashpot connects the pile to the ground, thus, the pile velocity \(v_{pile}\) corresponds to the relative velocity \(v_{rel}\). The characteristic curve of the dashpot and the resulting force-displacement relationship for the simulated test are given in Fig. 14. Due to the velocity-based definition of the characteristic curve, immediately before the reversal points of pile motion, the resulting force differs slightly from a typical friction model.
Fig. 15.

The same as in Fig. 12 for the solution with the dashpot (\(m_{p}=m_{pile}\) and \(a_{p}=a_{pile}\))

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.

Figures 16 and 17 compare the measured and calculated incremental displacement fields near the pile toe for the representative cycle (highlighted cycle in Fig. 15). Figure 16 presents the results during the penetration phase and Fig. 17 during the phase of the upward pile motion. Both phases are divided into two sections between the marked points \(1\div 2\) and \(2\div 3\) respectively \(3\div 4\) and \(4\div 5\), see Fig. 15(b). The images on the left show the horizontal displacements whereas the vertical displacements are depicted in the images on the right. The left part of each image illustrates the experimental results and the right the results obtained from the FE simulation. As can be seen in Fig. 15(b), the displacement amplitudes are almost identical and thus a direct comparison of the displacement fields is possible.
Fig. 16.

Comparison of incremental displacement fields for the representative cycle during the penetration phase \(1\div 2\) (above) and \(2\div 3\) (below). The horizontal displacements (left) and the vertical (right) are shown in the images. The left part of each image depicts the experimental results and the right the FE-simulation.

Fig. 17.

Comparison of incremental displacement fields for the representative cycle during during the upward pile motion phase \(3\div 4\) (above) and \(4\div 5\) (below). The horizontal displacements (left) and the vertical (right) are shown in the images. The left part of each image depicts the experimental results and the right the FE-simulation.

Comparing the images, the following observations can be made:
  • 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.

Figure 18 illustrates the logarithmic volumetric soil deformation \(\varepsilon _{vol}\) depending on the pile displacement for the representative cycle (see Fig. 15) for two selected locations around the pile toe (named element A and C following the nomenclature in [19]). Element A is located below the pile toe while element C lies slightly beside, see Fig. 18(a). In Fig. 18(b)–(e), the experimental results are compared with the results obtained from the numerical simulation. The procedure for the calculation of volumetric strain in the experiment has been presented in [19]. The volumetric strains are set to zero at the beginning of the highlighted cycle.
Fig. 18.

(a) Volumetric strain plotted versus the pile displacement for the representative cycle (see Fig. 15b) at three selected points: (a) Location of the selected points, (b)–(d) experimental and numerical results at the selected points

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

Two cyclic drained triaxial tests have been carried out and are recalculated in element tests via the software Abaqus/Standard. The two triaxial tests imitate typical loading paths of pile penetration. In both tests (named Triax A and Triax B), a large monotonic deformation is interrupted by phases of unloading, in Triax A without and in Triax B with shear stress reversal. The tested sand was initially medium dense and the initial void ratios similar. The conducted test paths are:
  • 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 \(\varDelta \varepsilon _\mathrm{1}=2\%\)

For the back-calculations, the same hypoplastic model and parameters as described in Chap. 3 have been used. Figure 19 shows a comparison of test and simulation results.
Fig. 19.

Back calculation of drained cyclic triaxial tests: (a), (b) Triax A (\(e_\mathrm{0}=0.654\), \(I_\mathrm{D,0}=0.66\)) and (c), (d) Triax B (\(e_\mathrm{0}=0.668\), \(I_\mathrm{D,0}=0.62\))

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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Stylianos Chrisopoulos
    • 1
  • Jakob Vogelsang
    • 1
  • Theodoros Triantafyllidis
    • 1
  1. 1.Institute of Soil Mechanics and Rock MechanicsKarlsruhe Institute of TechnologyKarlsruheGermany

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