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3rd International Conference on Natural Hazards & Infrastructure
5-7 July 2022, Athens, Greece
Vibration reduction effects of pile group foundations during the passage of
nearby moving surface loads
Georgia Efthymiou, Dipl.-Ing., Christos Vrettos, Prof. Dr.-Ing. habil.
Technische Universität Kaiserslautern
ABSTRACT
The potential reduction of vibrations caused by moving loads in the vicinity of pile group foundations is
investigated for different layout geometries by means of finite-element analyses in the time-domain. First, the
piles are indirectly excited by the wave field induced by a distant, stationary harmonic load. For given soil
properties, the effects of the excitation frequency, the distance from the source, and the pile rigidity are
assessed. Then, a moving point load of constant or time-harmonic magnitude acting on the soil surface is
considered. Modelling details have been optimized via verification of the results against known explicit
solutions. The influence of the moving load in terms of speed and oscillation frequency on the pile group
response is investigated and compared to the free-field response, demonstrating the significant vibration
reduction potential. Considering a pile group along with the supporting soil as a composite metamaterial, a
rectangular grid as well as a “zig-zag” configuration are examined in terms of shielding efficiency.
Keywords: piles, kinematic interaction, moving load, vibration reduction, composite metamaterials
INTRODUCTION
Pile groups are typically selected as a foundation solution in less competent soils. In the field of vibration
protection, the engineering practice is interested in how a pile group foundation modifies the induced wave
field in case of a dynamic excitation. However, due to the various parameters involved such as pile group
geometry, distance from the vibrating source as well as excitation frequency, the dynamic interaction of piles
becomes far more complicated in comparison with footings. The frequency characteristics of the excitation as
well as the soil properties are of great importance. In case of a direct excitation of the foundation (e.g. machine
foundations), the dynamic response is determined through complex-valued, frequency-dependent dynamic
stiffnesses. In case of indirect dynamic loading (e.g. traffic vibrations), the kinematic interaction between the
pile group and the induced wave field in the soil needs to be taken into account. The present study addresses
this subject.
While for dynamic stiffnesses a plethora of diagrams for various configurations as well as simple methods can
be found in the literature, e.g. Gazetas (1991), the kinematic interaction has been considered mostly for the
case of seismic excitation by vertically propagating shear waves (Fan et al., 1991). Such tasks are solved almost
exclusively with specialised, powerful computer programs that are not generally accessible. Numerical
continuum methods such as the finite-element method (FEM) are rarely used in engineering practice due to
the complex modelling and the long computational time. The latter is due to the fine discretization required in
order to properly capture the small wavelengths (high frequencies) in combination with the large dimensions
of the investigated models.
For vibration protection tasks in practice a two-step method is used: firstly, the free-field vibrations to be
expected at the location of interest are either obtained directly by field measurements (on the surface of the
site and/or in a borehole) or predicted using suitable analytical/numerical methods based on the source
characteristics (Vrettos, 2009). This time-history is then applied as a stationary excitation on a FE-model of
the building, with the foundation compliance being abstracted by suitable springs and dashpots. Alternatively,
the foundation is embedded in a soil continuum. Depending on availability, two- or three-dimensional (2D or
3D) models are employed.
By contrast, the shielding efficiency of individual piles or pile groups has been the subject of only a few studies.
The knowledge gained so far from measurements does not yet allow a reliable quantification of the associated
effects. Regarding numerical methods, the work of Kaynia & Novak (1992) and Makris (1994) on the dynamic
response of a single pile to the wave field of a Rayleigh wave should be mentioned. In the former, a versatile
numerical continuum model for stratified soil is used. In the latter, a Winkler model with frequency-dependent
springs and dashpots is employed for the calculation of the soil-pile interaction, and an explicit solution of the
free-field – pile transfer function is derived. The shielding efficiency of a row of piles has been also examined
numerically (Avilés & Sánchez-Sesma, 1988; Kattis et al., 1999; Auersch, 2010).
Furthermore, the type of excitation plays an important role on the dynamic response. The studies to date mainly
concern stationary loads of variable frequency. Loads from rail-bound traffic are, however, location-variable.
Consequently, the speed of a moving load constitutes a further parameter that needs to be considered (Auersch,
2008). In combination with the excitation frequency and the distance of the pile group from the moving
vibrating source, this leads to alternations in the induced wave field. Recently, Efthymiou & Vrettos (2022)
presented results on the shielding efficiency of pile groups and single piles in the wave field of stationary or
moving loads. This paper extends the aforementioned work under the scope of metamaterials. The latter are
defined as natural or artificial materials, or structures in periodic patterns able to control wave propagation.
Based on this concept, the piles embedded in the viscoelastic soil form a periodic composite pile – soil system,
which can provide wave shielding.
PILE GROUP UNDER EXTERNAL STATIONARY HARMONIC LOAD
Problem statement and finite-element modelling
The dynamic response of i) the free-field, ii) a single pile, and iii) a 3x3 pile group is investigated in the wave
field emanating from a vertical harmonic point load acting on the soil surface. Emphasis is placed on the
vertical component of the response. The FEM software PLAXIS 3D is used for the analyses, which are
performed in the time-domain. Taking advantage of symmetry, only half of the problem is modelled. The finite
element model is shown in Fig. 1(a). The vertical boundary is set at 20 m depth from the soil surface. In order
to simulate a half-space, appropriate build-in viscous boundaries are placed along the periphery and at the base
of the model. For the simulation of a soil stratum, the base of the model is fixed in all three directions.
Figure 1. Stationary harmonic loading: (a) 3D FE-model; (b) top view of the pile group with the reference
piles highlighted in grey.
The soil is assumed linear-elastic with a shear modulus G = 30.5 MPa, density ρ = 1.89 Mg/m3, and Poisson’s
ratio v = 0.4, yielding a shear wave velocity cS = 127 m/s and a Rayleigh wave velocity cR = 120 m/s. A
Rayleigh-type damping has been implemented corresponding to a hysteretic damping of ξ = 1 % in the
frequency range 15 to 45 Hz. For the piles, which are modelled with linear-elastic solid elements, the following
values have been selected for all analyses: l/d = 15 with length l = 10 m and diameter d = 0.67 m; Young’s
modulus Ep = 30 GPa; Poisson’s ratio vp = 0.2; unit weight γp = 25 kN/m3. For the basic configuration this
yields Ep/E = 350 and ρp/ρ=1.35.
A top view of the 3 x 3 pile group is given in Fig. 1(b). The centre-to-centre pile spacing is s = 3d = 2 m. The
distance of the central pile in the front row (in relation to the direction of the wave propagation) from the point
load is equal to 12 m, while from the central pile in the back row the distance is 16 m. The piles are either free-
headed or connected to a rigid, massless plate.
The frequency of the harmonic vertical point load Q acting on the surface is taken as f = 30 Hz in the basic
configuration, whereas a range of frequencies between 20 and 40 Hz is also considered. The finest possible
discretization in conjunction with the model size is implemented. In the main configuration, the finite element
size is set approximately equal to 0.35 m, thus corresponding to 11 finite elements per Rayleigh wavelength
λR.
Since the analyses have been performed in the time-domain, a minimum number of excitation cycles had to be
completed in order to reach the steady-state response. At 30 Hz this is achieved after approximately 15
excitation cycles.
Verification of the free-field response
In a first step, the piles have been removed from the 3D model and replaced by soil elements. The response of
the free-field due to a stationary harmonic load of f = 30 Hz has been then computed. The resulting
displacement field up to a distance of 30 m has been also derived from analyses with an axisymmetric model
using PLAXIS 2D and compared to known analytical solutions for a half-space (Vrettos, 1991) and for a soil
stratum, showing very good agreement. In addition, displacement profiles with depth have been obtained after
the system has reached a steady-state response at a point of time, where the harmonic displacement at the
surface reaches its maximum. The far-field, where Rayleigh waves dominate over the entire depth, is shown
to prevail at distances larger than 30 m. Details are given in Efthymiou & Vrettos (2022).
Response of a pile group
The shielding efficiency of a 3x3 pile group in a half-space has been subsequently investigated. The distance
of the central pile in each row from the vibrating source is denoted with x0, and equals 12 and 16 m with respect
to the front and back row, respectively (recall Fig. 1). These two piles are selected as reference piles to monitor
the attenuation effects. These are quantified herein by the frequency-dependent transfer function wp(0)/wff(0),
where wp(0) is the vertical displacement amplitude of the pile head, and wff(0) is the respective value of the
free-field at the same location. Results are presented in Fig. 2. The response of the reference pile as an isolated
single pile, namely after removal of all its neighbours, is also given.
Figure 2. Vertical transfer function of a single pile compared to that of a pile in the centre of the front and
back row of a 3x3 pile group with s/d=3; vertical harmonic point load; v = 0.4 (Efthymiou & Vrettos, 2022).
As part of the pile group without cap, the reference pile at x0 = 12 m can have a greater amplitude in comparison
with the corresponding single pile, depending on the excitation frequency. This increase can be attributed to
reflections from the central pile row. In contrast to this, it exhibits the lowest vibration level when it is part of
a pile group with cap.
For the back central pile a reduction up to 80 % compared to the respective singe pile is observed. In this case,
the vibration reduction is due to diffraction of the wave field by the front pile rows, so that the shielding effect
is also evident in the pile group without cap (at 40 Hz, though, the response is almost identical to that of the
single pile). Consequently, the attenuation depends on the relation of the Rayleigh wavelength λR to a
characteristic length of the pile group, e.g. the pile-to-pile distance s. Note that in the parametric study, s/λR
varies between 1/3 and 2/3.
The influence of the stiffness ratio Ep/E on the response of a single pile at x0 = 16 m embedded in a half-space
or stratum is examined at the excitation frequency of 30 Hz. The resulting transfer functions are depicted in
Fig. 3, from where it is inferred that between Ep/E = 350 and 10000 there is practically no difference in the
response. When the pile becomes less stiff in relation to the surrounding soil (Ep/E = 100), the value of the
transfer function increases, as expected, indicating that the pile follows the free-field motion in a more
compliant manner.
Figure 3. Influence of the stiffness of a single pile at a distance of 16 m (=4∙λR) from the harmonic load (f =
30 Hz) on the vertical transfer function; v = 0.4.
FREE-FIELD RESPONSE UNDER MOVING LOAD OF CONSTANT MAGNITUDE
Finite-element modelling
The response of the free-field due to a vertical point load travelling on the soil surface is investigated next. The
FE-model created with PLAXIS 3D is based on the work of Galavi & Bringreve (2014), and has been modified
appropriately for the needs of this study. Taking advantage of symmetry, only half of the problem has been
modelled. The length of the model is 104 m, with a 100 m long load path. The width and depth of the model
have been taken equal to 50 and 100 m, respectively. Viscous boundaries have been applied at the periphery
and at the base of the model in order to simulate a half-space.
The vertical point load Q has a constant magnitude. Regarding the travelling speed v0, two values have been
examined: 50 and 100 m/s, which correspond to 180 and 360 km/h, respectively. Both values are typical of
high-speed trains.
For the soil, the following parameters have been assumed: G = 30.5 MPa, ρ = 1.89 Mg/m3 and v = 0.3, yielding
cS = 127 m/s and cR = 118 m/s. The observation point is located in the centre of the model at the soil surface
at a distance x0 = 6 m from the load path axis. To obtain a smooth response, a very small amount of Rayleigh
damping has been introduced in the soil.
The response of the free-field at the observation point under this type of loading attains a maximum as soon as
the load reaches the distance x0. The simulation of a load that is travelling from a very far distance requires an
extremely long model, which is computationally not feasible. The activation of the load at the beginning of the
analysis leads to a transient procedure, the effects of which appear as a disturbance in the response that ideally
should be eliminated. To do so, different ways of applying the load have been compared: i) the total magnitude
of the load is directly applied by means of a “step function”, or ii) the magnitude of the load increases linearly
0
0.25
0.5
100 1000 10000
wp(0)/wff(0)
E
p
/E
Halfspace
Soil stratum H=20m
up to its maximum value by employing a “ramp function” of constant gradient. A total time for the ramp-
function equal to 20 to 30 % of the time needed for the load to reach the end of the load path has been proven
optimal.
Verification of the free-field response
The results for v0 = 100 m/s (v0/cR = 0.847) are presented in Fig. 4. Despite the long load path, after about 0.4 s
a significant disturbance is observed, which arrives at the observation point with the S-wave velocity cS. The
disturbance becomes less intense when the load is applied with the ramp-function; it does not yet disappear
completely. The system requires considerable time to reach a steady-state condition. Despite the disturbance,
though, the settlement peak, which appears when the load is located at the minimum distance from the
observation point x0, is reproduced with high accuracy. From this point on, there is a very good agreement with
the analytical solution by Barber (1994). In fact, by taking into account time-symmetry, one can reproduce the
complete solution by mirroring the response obtained after the settlement peak (Galvín & Domínguez, 2007).
The settlement derived from the Boussinesq solution, which corresponds to the static case (v0 = 0) and is shown
in Fig. 4 on the left, is equal to 43% of the maximum settlement due to the moving load.
Figure 4. Moving constant load on a half-space at v0 = 100 m/s and v0 = 50 m/s: vertical displacement at a
surface observation point at the center of the model at distance of x0 = 6 m from the travel axis; v = 0.3
(Efthymiou & Vrettos, 2022).
The results for v0 = 50 m/s (v0/cR = 0.423) are also reproduced in Fig. 4. The negligible deviation from the
Boussinesq solution confirms the static character of this type of loading in the case of a comparatively low
velocity. Shortly after the settlement peak is reached, significant deviations from the analytical solution arise.
This is attributed to the insufficient accuracy of the viscous boundaries in case of low velocities, that is to say,
as the static case is being approached. In an effort to improve the accuracy, a rigid base has been implemented
instead of the viscous boundary. A rigid base at this great depth of 100 m has a minor effect on the targeted
simulation of the half-space response. In cases where a fine mesh discretization is necessary, and therefore a
large model depth is not feasible, this constitutes an important limitation of the FEM. The verification of the
results of an analysis with a constant moving load on a soil stratum against the analytical solution from Kausel
(2018) can be found in Efthymiou & Vrettos (2022). Subsequently, for the following investigations of a single
pile or a pile group under excitation by a harmonic moving load, a soil stratum of thickness equal to 20 m is
considered.
PILE GROUP RESPONSE UNDER TIME-HARMONIC MOVING LOAD
3D Finite-element modelling
For a moving point load with a time-harmonic magnitude of frequency f = 30 Hz, the previously described
model of a soil stratum has been reduced to 40 m in width and 20 m in depth, see Fig. 5. A uniform mesh has
been applied with 5 finite elements per Rayleigh wavelength λR (along the load path the mesh is finer).
Assuming the typical railway sleeper distance of 0.6 m, the above excitation frequency corresponds to a
travelling speed v0 = 18 m/s (64.8 km/h). For the soil it is taken: G = 30.5 MPa, ρ = 1.89 Mg/m3, and v = 0.4.
The Rayleigh damping corresponds to a hysteretic damping ξ = 1%. The moving load is introduced into the
system by means of a ramp-function, the duration of which is 1.1 s. It is underlined that the use of symmetry
implies the existence of an identical, mirror pile group on the other half of the stratum that is not modelled.
This could have a small influence on the results. An analysis with a full model without use of symmetry was
computationally not possible. Indicatively, the computation on an Intel i9-10900K, 3.70 GHz processor with
64 GB RAM lasted 15 hours.
Figure 5. 3D FE model for a moving load on a soil stratum.
Shielding efficiency of a single pile
Fig. 6 shows the maximum amplitude of the vertical displacement at various distances x0 from the load path
axis. For the application is practice, it is interesting to know, whether a stationary harmonic load constitutes
an acceptable approximation for a harmonic moving load. The comparison on the left side of Fig. 6 show that
this holds, as long as the travel speed it not too high. The curves on the right side of the same figure reveal the
significant reduction of the free-field motion due to the presence of a pile at various distances. The attenuation
is stronger as the distance of the pile from the moving load axis becomes smaller. This phenomenon is
quantified by the kinematic interaction factor wp(0)/wff(0), where wp(0) is the maximum pile head displacement
over the considered time window, and wff(0) is the respective value free-field value. The ratio wp(0)/wff(0) over
the entire time-history attains values between 0.26 at x0 = 4 m and 0.38 at x0 = 16 m which are comparable to
those derived from the stationary harmonic load analyses (see Efthymiou & Vrettos, 2022).
Figure 6. Maximum amplitude at the surface of a soil stratum without/with the presence of a single pile at
various distances x0 from the travel axis of a moving harmonic load (Efthymiou & Vrettos, 2022).
Shielding efficiency of a pile group
The response of the previously presented 3 x 3 pile group with s/d = 3 (recall Fig. 1(b)) is investigated under
the excitation of a moving point load on a soil stratum. The load path axis is parallel to the front pile row,
which is located at a distance of x0 = 12 m. A further system of 2 x 3 is also considered. It is created by
replacing the front row of piles in the 3 x 3 configuration by soil. The vibration reduction effects are quantified
through the ratio wp(0)/wff(0) for the central pile in the back row (at x0 = 16 m), which is chosen as the reference
pile.
Time-histories of the vertical displacement during the passage of the point load are depicted in Fig. 7 for the
free-field, as well as for the reference pile at x0 = 16 m; that pile is either a single isolated pile or part of a free-
headed 2 x 3 or 3 x 3 pile group. For the case of the 3 x 3 group, the influence of the pile connection to a rigid,
massless cap is additionally investigated. At selected points of time during the passage of the load three
snapshots from the analysis of the 3 x 3 pile group without cap are also shown.
Figure 7. Vertical displacements of the free-field and of the reference pile at distance x0 = 16 m as single
pile or as part of a pile group for an excitation by a harmonic moving load of f = 30 Hz and v0 = 18 m/s; soil
stratum with H = 20 m and v = 0.4.
Even the presence of the single pile leads to an important reduction of the free-field motion. The shielding
efficiency of the pile group becomes greater as the load is nearby (see highlighted time-window in Fig. 7). The
kinematic interaction factor wp(0)/wff(0) at this closest distance (calculated here between 2.75 and 2.8 s) attains
the following values: 0.46 for the single pile, 0.34 for the pile in the 2 x 3 pile group and 0.27 for the pile in
the 3 x 3 pile group. The addition of a rigid cap to the last scheme reduces dramatically this ratio to 0.09.
However, when the kinematic interaction is calculated by taking into account the complete time-history, the
following values are obtained: 0.38/0.34/0.34/0.18 for the four cases mentioned above.
Fig. 8 shows a snapshot with the load at the minimum distance from the 3 x 3 pile group without a cap. It can
be seen that the pile group acts as a barrier to the propagating waves that are visualised with contours of vertical
displacements. The vibration reduction manifests itself with the formation of a “shadow” zone at the back of
the pile group, where the displacement amplitudes are significantly reduced.
-0.25
0
0.25
1 1.5 2 2.5 3
wGx0/Q
Time [s]
free-field single pile 2x3 3x3 3x3
pile in group
without cap
pile in group
with cap
Figure 8. Snapshot of the vertical displacement field induced by a harmonic moving load (f = 30 Hz,
v0 = 18 m/s) at the minimum distance from a 3 x 3 pile group without cap; soil stratum with H = 20 m and
v = 0.4.
- Influence of layout geometry
Motivated by a possible application in the context of metamaterials, a “zig-zag” pile group layout, depicted in
Fig. 9, has also been examined under the scope of vibration reduction. In specific, a free-head 2 x 3 and a 3 x 3
configuration have been considered. The central pile at the back row at x0 = 16 m from the load axis is still
used as reference. The obtained time-histories of its response are contrasted in Fig. 9 to the free-field motion
at the same location. The kinematic interaction factor wp(0)/wff(0) determined from the entire time-history is
equal to 0.32 for the reference pile in both groups and thus, approximately identical to the respective values
observed in the previous configuration of a rectangular grid. The additional attenuation offered by the pile
group appears in the time-history shortly after the load passes by the pile group; in the corresponding time-
window from 2.98 to 3.13 s the kinematic interaction factor is equal 0.46, 0.38 and 0.31 for the pile as single
pile, as part of the 2 x 3, and as part of the 3 x 3 “zig-zag” pile group, respectively.
Figure 9. “Zig-zag” layout for a 2 x 3 pile group (solid lines) and a 3 x 3 pile group (solid plus dashed
lines). Vertical displacements of the free-field and of the reference pile at distance x0 = 16 m as single pile or
as part of a pile group without cap for an excitation by a harmonic moving load with f = 30 Hz and
v0 = 18 m/s; soil stratum with H = 20 m and v = 0.4.
pile in free-headed
group
-0.25
0
0.25
1.522.533.5
wGx0/Q
Time [s]
free-field single pile 2x3 3x3
pile in group
without cap
45°
x0 = 16 m
x0 = 12 m
“Zig-zag” layout
In comparison with the regular pile group layout, the “zig-zag” scheme has been proven similarly effective in
reducing the vibrations from the moving load. In any case, the addition of a further pile row, i.e. 2 x 3 pile
group becoming a 3 x 3 one, leads to an additional attenuation of the vibration level experienced by the
reference pile at the back row. The amount of the reduction depends on the time-window selected.
- Influence of travelling speed
As already shown, the travelling speed of v0 = 18 m/s (64.8 km/h) in combination with the selected soil
properties has led to a response similar to that under stationary conditions. It would be therefore meaningful
to consider a higher travelling speed. For the sake of computational consistency, and in order to keep the same
mesh discretization, the stiffness of the soil had to be increased. The shear wave velocity of the soil is set equal
to cS = 170 m/s. This allows an analysis with a 50% higher travelling speed, namely v0 = 27 m/s (97.2 km/h),
corresponding to a frequency f = 45 Hz. The previously defined 2 x 3 and 3 x 3 rectangular grids have been
examined, and the time-histories of the response are presented in Fig. 10. The kinematic interaction factor
wp(0)/wff(0) obtained from the entire time-history is equal to 0.34, 0.32, and 0.30 for the reference pile (x0 =
16 m) as single pile, and as part of the 2 x 3 and the 3 x 3 pile group, respectively. In the time-window from
1.94 to 1.99 s, in which an additional attenuation offered by the 3 x 3 pile group is manifested, the above values
become 0.39, 0.36 and 0.27.
Figure 10. Vertical displacements of the free-field and of the reference pile at distance x0 = 16 m as single
pile or as part of a pile group without cap for an excitation due to a harmonic moving load with f = 45 Hz
and v0 = 27 m/s; soil stratum with H = 20 m and v = 0.4.
CONCLUSIONS
The kinematic interaction of single piles and pile groups was investigated by means of the finite-element
method (FEM) in the time-domain. The wave field was induced by i) a stationary time harmonic load, or ii) a
moving oscillating load.
Both the single pile and the pile group can reduce the free-field vibrations mainly due to the redirection of the
energy towards greater depths, but also as a result of the wave diffraction through the front pile rows. At higher
excitation frequencies the shielding efficiency increases.
The response of a single pile or a pile group to the wave field induced by a harmonic moving load has been
investigated. For the travelling speed of 18 m/s (30 Hz in conjunction with a railway sleeper distance of 0.6 m)
in combination with the selected soil parameters, the dynamic response of the free-field can be adequately
approximated by that due to a stationary harmonic load. A connection of the piles within a group via a rigid
cap results in a significant vibration reduction. For the examined frequency of 30 Hz and the assumed soil
-0.25
0
0.25
0.6 1 1.4 1.8 2.2
wGx0/Q
Time [s]
free-field single pile 2x3 3x3
pile in group
without cap
properties and pile geometry, the pile head displacement remains below 50 % of the free-field value. This
finding is essential for optimizing vibration protection design.
To gain insight into metamaterial aspects, a further pile row to the front of the 2 x 3 pile group without cap
(turning it into 3 x 3) resulted in an additional attenuation of the vibration level experienced by the central pile
at the back. This was observed in the cases of both a rectangular and a “zig-zag” layout, as well as of the higher
travelling speed of 27 m/s (45 Hz).
The applied FEM approach in the time-domain, with adequate modelling, enables the incorporation of non-
linear soil behavior in the analysis, which is of importance for the realistic representation of the near field.
ACKNOWLEDGEMENTS
This research has been supported by the project “INSPIRE – Innovative Ground Interface Concepts for
Structure Protection”, funded by the European Union’s Horizon 2020 research and innovation program under
the Marie Skłodowska-Curie grant agreement 813424.
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