Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Theoretical model for estimating the soil particular velocity due to a heavy structure
fall
Alain Pecker1, Charles Fernandez2*, Mickael Metzger3
Geodynamique et Structure, 157 rue des Blains, Bagneux, 92220, France
2
GDF SUEZ/CRIGEN, 361 av. du President Wilson, 93211, Saint-Denis, France
3
Immeuble BORA, 6 rue Raoul Nordling, 92270, France
Email: [email protected] (*corresponding author), [email protected],
[email protected]
1
ABSTRACT: Over the past few years, extensive construction of wind turbines has taken place all around the world in areas
where many steel pipelines are already buried in the ground. The possible fall of these heavy machines may induce damageable
vibrations to the pipeline. Therefore a theoretical predictive model has been developed in order to assess the particle velocity
near a buried pipeline, may a wind-turbine fall nearby. A fall of a 1.5 t mass was performed and velocities were measured in
order to update the model. An experimental soil characterization through MASW tests was performed in a representative soil in
order to obtain relevant input parameters for a non linear FE model. The FE model allowed updating the theoretical model for
very heavy structures as wind-turbines. The updated model is now part of the RAMCES software which has been developed for
more than a decade at CRIGEN and is widely used in France by pipeline transportation operators.
KEY WORDS: Modeling; Soil particle velocity estimation; Vibrations; Heavy structure fall.
1
INTRODUCTION
This model presents a simplified analytical model for
estimating the peak particle velocity (PPV) at a depth into the
ground and at a distance from the impact of a heavy structure
fall. That model is used by oil and gas operators to estimate
safety distances from their buried pipelines and the
implantation of heavy structures nearby.
Over the past few years, different models have been used,
from very simplified ones based on simple wave equations
and considering that all the energy of the falling structure was
transmitted to the wave near the buried pipeline. The wave
equation was stated in simple terms of u(x,t)  U0sin(ωt +
kx) functions. These models were overconservative because
they did not take into account the energy dispersion into the
ground and due to the plastification of the soil with the
craterization phenomenon. Then, empirical models were
developed thanks to extensive tests campaigns (like the
Mayne model presented in [1] or the Menard model from [2]
(presented below) where the peak particle velocity is
estimated from the fall of masses for dynamic compaction
techniques).
bound for the velocity under an elastic impact. An efficiency
coefficient is then introduced to take into account the
plastification phenomena. This coefficient is updated in a
fourth part thanks to vibrations measurements and MASW
tests used as input parameters in a 3D dynamic non linear
finite element model.
The terms in bold are vectors.
2
2.1
EVALUATION OF THE VIBRATIONS IN THE
GROUND
Problem and basic assumptions
The problem considered in this paper is the fall of a tall and
heavy structure in the gravity field g with an initial velocity V0
and an incident angle θ from a height h producing an impact
at a distance d from the base of structure. The fall produces
vibrations nearby a buried pipeline located at a length r from
the impact and a depth z from the surface soil (see Figure 1).
But once again, the Mayne PPV model was not adapted for all
the field configurations met by a gas operator. Indeed, the
masses at play in ground compaction are an order of
magnitude less than the masses of the wind-turbines and the
energy dissipation into the ground does not follow a linear law
in function of the falling mass.
In a first part, the problem is introduced. In a second part, the
velocity due to the Rayleigh waves is obtained for some
loading. The aim of this section is to show that Rayleigh
waves are preponderant in the phenomenon of interest. In a
third part, under elastic assumptions, the impact force is
estimated. The aim of this section is to provide an upper
Figure 1. Problem modeled.
Velocity vz at the impact point located at a distance d like in
Figure 1, is written as,
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
where τ = t cT /
.
(1)
If θ = π/2, vz is maximal but due to the configuration of the
wind-turbine nacelle, this value is highly unlikely to appear.
Moreover, a variation of the θ angle from 0° to 30° on the
horizontal gives a very small variation of the impact velocity,
thus the angle is considered to be null and therefore, the
velocity at the impact point is written as,
.
2.2
 r
w
Q
0,75
P
(2)
,
(3)
where σ is the stress tensor, Q is the force amplitude and δ
is the Kronecker symbol. The Heaviside function , H(t) is null
for t<0 and equal to 1 for t>0.
With the assumption that the soil Poisson’s coefficient ν =
0.25, the Lame’s coefficients, λ and μ, which characterize the
elastic behavior of the half-space are equal. The vertical
displacement w(t,r), at a distance r from the impact point and
time t is written as,
for
R
0,25
0,00
The original formulation for the normal point-source load
over a half-space was established by Lamb for a harmonic
load. The soil surface displacements for a point-source impact
load is given by Pekeris (see [3]). The displacement is
reproduced below for a load with a Heaviside time
dependency; the derivation of this solution with modern
techniques in the complex plane is available in reference [4].
At the surface of the half-space, the boundary conditions are
written as,
,
S
0,50
Point-source impact, general solution
for
is the scaled time with cT the celerity of
the shear waves,
the time of arrival of the
Rayleigh wave which propagates with celerity cR. These
expressions are plotted at Figure 2.
;
,
-0,25
-0,50
cT t r
-0,75
0,0
0,5
1,0
1,5
2,0
2,5
Figure 2. Vertical surface displacement (Pekeris, 1955), from
equations (4). Amplitude of the displacement in vertical axis,
dimensionless time in horizontal axis. P, S and R mean arrival
of P-waves, S-waves and Rayleigh waves respectively.
2.3
Vibrations induced by surface waves, point-source
impact,
Equations (4) show that the Rayleigh wave gives the larger
perturbation at the soil surface; these expressions show
moreover a geometric attenuation when r increases. The
following assumption is then made, which is endorsed by
seismic observations: the perturbations propagating with the
Rayleigh waves give larger amplitudes of vibrations which are
of interest in this paper. The problem is then considerably
simplified because the displacement of interest can be
obtained by solving the equation in the vicinity of the pole
associated to the Rayleigh waves. That solution was
calculated by Chao et al. in ref. [5] (see also [4]). These
expressions are valid for times close to the arrival time of the
Rayleigh waves and for depths z/r (see Figure 1) which are
small. They are written as,
horizontal displacement
for
,
(5)
(4)
;
vertical displacement
for
,
;
,
with the following notations,
900
(6)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
;
1,5E-01
;
(7)
1,0E-01
,
5,0E-02
with
,
and
are respectively the real
part and the imaginary part of a complex number Z. These
expressions are plotted at Figure 3.
0,0E+00
du/dt
-5,0E-02
dw/dt
2,0E-02
-1,0E-01
-3,00 -2,00 -1,00
0,0E+00
1,00
2,00
3,00
Figure 4. Velocity due to the Rayleigh waves. Horizontal axis,
the scaled time (cT t – τR r) / z ; vertical axis, the scaled radial
-2,0E-02
-4,0E-02
-6,0E-02
u
-8,0E-02
w
-1,0E-01
-3,00 -2,00 -1,00
0,00
0,00
1,00
2,00
(
) velocity components. This
figure shows the amplitudes of the velocity components due to
the Rayleigh waves.
3,00
2.4
Figure 3. Displacements due to the Rayleigh waves (Chao et
al. 1961). Horizontal axis, the scaled time (cT t – τR r) / z ;
vertical axis, the scaled radial (
) and vertical (
) and vertical (
)
displacement components. This figure shows the amplitudes
of the displacement components due to the Rayleigh waves.
Vibrations induced by surface waves, impulse load
The case of an impulse load can be calculated with previous
equations, considering at time t an impact of amplitude + Q
and at time t + dt, where dt is finite, an impact of amplitude Q, if dt is small enough, the solution is obtained by
differentiation and superposition of equations (8) and (9),
which is written as,
Expressions (5) to (7) show that the displacements depend
only on the dimensionless time τ and that they decrease in
function of 1/
compared to the other components of the
displacement which decrease in function of 1/r. Therefore for
high distances from the impact, z/r << 1, the Rayleigh wave
contribution is predominant.
The velocity is obtained from the displacement by
differentiation, which is written as,
(8)
,
(10)
.
(11)
Considering equations (8) and (9), equations (10) and (11)
are written as,
;
.
;
(12)
.
(13)
(9)
These expressions are plotted at Figure 4.
These expressions are plotted at Figure 5.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
,
4,0E-01
3,0E-01
2,0E-01
1,0E-01
0,0E+00
-1,0E-01
-2,0E-01
-3,0E-01
-4,0E-01
-5,0E-01
-6,0E-01
-3,00 -2,00 -1,00
(18)
.
3
3.1
du/dt
dw/dt
0,00
1,00
2,00
3,00
Figure 5. Velocity due to the Rayleigh waves and an impulse
force. Horizontal axis, the scaled time (cT t – τR r) / z ; vertical
axis, the scaled radial (
) and vertical (
)
velocity components. This figure shows the amplitudes of the
velocity components due to the Rayleigh waves and an
impulse load Qdt.
The graphs presented in Figure 5 show that, for any time t,
IMPACT FORCE DETERMINATION
Circular foundation
Under the assumption that the soil behavior remains elastic
when impacted, the wind-turbine nacelle is assumed to be a
circular foundation (mass m) animated with an initial velocity
equal to the impact velocity given in equation (2).
In that case, the foundation can be modeled by a spring and
a damper (assumed to be frequency independent for the sake
of simplicity). The values of the spring and the damper used
to model the impedance of a circular foundation over a
homogenous, isotropic half-space are given by Gazetas (see
reference [6]). For a foundation of radius r0 over a half-space
with behavior parameters μ and ν (=0.25, like previously), the
expressions are written as,
,
(14)
.
2.5
With
Vibrations induced by surface waves, for any load
Any given load can be modeled by a superposition of timeshifted impulse loads. The velocity due to that load and
generated by the Rayleigh waves is given by:
and
,
(19)
.
(20)
, the response of the 1-DOF
oscillator is immediate and the displacement and vertical
velocity are written as,
(15)
,
(21)
,
(22)
;
with vz is the velocity of the missile (i.e. wind-turbine nacelle)
at the impact and where the following notations are used,
(16)
,
(23)
.
(24)
,
where T = N dt is the total duration of the impact force
application, Φ and are functions not used below. When dt
becomes close to 0, the last member of the equations (15) and
(16) tends to the total impulse contact load, which is written
as,
The impact force can then be written as,
=
.
(17)
With equation (17), equations (14) to (16) provide the
following equations,
902
.
(25)
As an example, Figure 6 depicts the variation of the missile
displacement and velocity after the impact, and the developed
contact force. This graph was plotted for a mass m = 50.103 kg
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
and r0 = 2 m, falling from 80 m over a soil with the following
characteristics, cT= 300 m.s-1, μ = 162 .106 Pa.
Force
600 (MN)
550
500
450
400
350
300
250
200
150
100
50
0
-50
-100
0
parameters above and the 3 fundamental terms). The
following dimensionless parameters are chosen and written as,
D. (mm) ,
V. (m.s-1)
0,02
t(s)
120
110
Force
100
90
Displacement
80
70
Velocity
60
50
40
30
20
10
0
-10
-20
0,04
0,06
,
(26)
.
(27)
and the relation is written as,
The dimensionless parameters
for i=1 to 3 are computed
with the values shown in Table 1. Figure 7 and Figure 8 show
the variation of
as a function of
and .
3,50
3,00
Figure 6. Impact characteristics. Horizontal axis, time t; left
vertical axis, Force (106 N) in dotted line, right vertical axis,
displacement (mm) continuous line and velocity (m.s-1) in
discontinuous line.
Force diagram can be seen as a triangular force diagram with
maximum value at 390 MN, duration 0.013 s and total
impulse 2.55 MN.s. The maximum force is obtained for t = 0
and equal to C vz. After 0.012 s, the force becomes negative,
which means, taking into account the previous assumptions
that there is a traction force from the missile to the soil; this
phenomenon cannot be possibly physical and thus, after 0.012
s the mass lifts off and previous equations are not valid any
more.
2,50
2,00
1,50
1,00
0,50
0,00
0
3,50
In this section, the impulse load
(see eq. (18)) is
computed numerically on the interval [0,T]; that corresponds
to a positive force and the parameters used vary like in the
following Table 1.
3,00
Table 1. Parameters variation for the computation of the total
impulse load IQ.
1,50
Parameters
Min.
value
Symbol
Max.
value
Equivalent
radius for
0.5
≤ r0 ≤
5m
impact area
(m)
Nacelle mass
25
≤ m≤
70 t
(t 103 kg)
Fall height
50
≤ h≤
100 m
(m)
Shear waves
celerity in the
150
≤ cT ≤
1000 m.s-1
soil
(m.s-1)
The total impulse IQ depends a priori on the following
terms: m, vz, r0, ρ, cT, where ρ=μ/cT2 is the soil density. These
parameters can be expressed with the fundamental terms L
(length), T (time) and M (mass). The Vashy-Buckingham
dimensional analysis theorem states that it exists a relation
between 3 dimensionless parameters (relating the 6
40
60
Figure 7. Computation of parameters (vertical axis) versus
(horizontal axis).
Maximum velocity induced by an elastic impact
3.2
20
2,50
2,00
1,00
0,50
0,00
0
10
20
30
40
Figure 8. Computation of parameters (vertical axis) versus
(horizontal axis).
Figure 7 and Figure 8 show that the variation of
versus
and
is relatively small. A simplified average value can
be retained and is written as,
.
(28)
For a short duration impact, the elastic forces linked to the
stiffness K (see eq. (19)) become negligible in front of the
damping force from C (see eq. (20)) and thus the impulse is
simplified and its value becomes
i.e.
. In the
following, that value will be taken as a reasonable
approximation for the impact computation. The chosen upper
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
bounds for the radial and vertical velocities induced by an
elastic impact and propagated with Rayleigh waves are then
written as,
3.4
,
(29)
.
3.3
Application to a wind-turbine fall case
Under the assumptions of section 2, which are recalled below,
 Rayleigh waves predominance over the general
movement;
 Linear elastic behavior for the soil at the impact
point and in the medium,
equations (29) give an evaluation for the particle velocity in
the soil as a function of the soil characteristics (
), the
mass of the impact (m), the velocity at the impact ( ) and the
distance to the impact point ( ). The depth ( ) is arbitrarily
chosen to be equal to 5 m (the linear elastic soil behavior
leads to infinite displacement and velocity at the soil surface).
The velocity is estimated far enough (z/r << 1) from the
impact point.
Impulse: 1 MN.s
2 MN.s
3 MN.s
0.3 MN.s
Experimental dynamic compaction
1,0E+01
1,0E+00
(29) with
1,0E-01
1,0E-02
is plotted so as to compare with the model of eq.
= 0.3 MN.s (m ≈ 20 t and h ≈ 10 m).
Discussion and efficiency coefficient
When comparing the experimental results with the predicted
model, it appears that the model assumptions lead to an
overestimation of the peak particle velocity for dynamic
compaction impacts.
The overestimation is due to the fact that the impact from real
masses, like in dynamic compaction, is far from being elastic.
Moreover, the wave propagation takes place in a nonlinear
medium. For these reasons, an efficiency coefficient e is
introduced, which accounts for the plasticity and non
linearities.
Figure 9 shows that the efficiency coefficient e can be
estimated, for the lengths of interest, to
.
(30)
The assumptions for using the empirical Menard model (from
[2]) are the following, the product mh should be included
between 200 and 300 t.m (for example a fall of a 20 t-mass
from 10 to 15 m). The assumptions from [1] are mh ≤ 1000
t.m (for example 50 t from 20 m). As a reminder, the aim of
this paper is to estimate the in-ground velocity due to a heavy
structure fall like wind-turbines. For wind-turbines, the
product mh can be as large as 25000 t.m (a 250 t-nacelle
falling from 100 m) and much more for concrete-mast windturbines (on-shore).
The efficiency coefficient found for the dynamic compaction
tests is thus not assumed to be valid for the fall of a windturbine.
1,0E-03
1,0E-04
4
1,0E-05
100
1000
4.1
Figure 9. Plot of the particle velocity
(in m.s-1,
vertical axis, log scale) in function of the distance (impact
distance + 26 m) to the base of a wind-turbine (in m,
horizontal axis, log scale) for 5 different impulse loads: big
dashed line for 1 MN.s ; very big dashed line for 2 MN.s;
continuous line for 3MN.s; dotted line for 0.3 MN.s; dashed
line for experimental dynamic compaction estimation of the
velocity.
The results plotted in Figure 9 for a soil density of 1800 kg.m3
give the variation of the maximum amplitude for the velocity
(
, computed with eq. (29)) as a function
of distance to the foot of a wind-turbine mast which is equal
to r -d (feedback studies state that the nacelle falls at most at d
= 26 m from the foot of wind-turbine mast), where r and d are
shown in Figure 1. The input parameters used are the
following impact loads
= 0.3, 1, 2 and 3 MN.s. An
empirical relation based on tests (from [2]), where
904
UPDATED EFFICIENCY COEFFICIENT FOR WINDTURBINES
Methodology of updating
In the previous section, an efficiency coefficient has been
updated for dynamic compaction phenomenon thanks to
empirical data. In the case of this paper, a test with real
masses of interest is not possible (because it is hard to find a
field to instrument and make a wind-turbine fall). Therefore a
3D finite element computational model (Dynaflow, see [9])
has been used to model the fall of the masses of interest.
This FE-model has been updated with soil parameters
measured in field experiments performed by GDF SUEZ (see
[7] and [8]) and the dynamic compaction parameters. The
updated FE-model has then been used to compute a windturbine fall, which has provided an estimation of the
efficiency coefficient for real wind-turbines.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
4.2


Field tests
The field tests used to update the FE-model have been
extensively presented in ref. [7] and [8] and only main results
are reproduced in this paper.


4.3
Figure 10. Position of the sensors for the 1.5 t-mass fall
tests from 3, 5, 10 to 15 m high and measurement of the
velocity from 3, 8, 15, 30, 50 to 80 m from the impact point.
Frequency domain measured is in [0, 30 Hz];
Impact footprint is about 0.70 cm in diameter, the
depth increased with the number of fall from 0.30 cm
up to 1.00 m;
Measured particle velocities show a non negligible
scatter;
Some sensors were occasionally saturated (150
mm.s-1).
3D Nonlinear finite element model for the soil
The Dynaflow software has been used (see [9]). The impact
simulation was used with the following parameters: 1.5 t
mass, 0.35 m radius, 0.50 m depth of impact and initial
velocity given in eq. (2).
The Prevost constitutive model used in the computation
takes into account the data gathered at Table 2 and some
parameters were taken from literature because they were not
measured (see ref. [10] and [11]).
The mesh was adapted to the waves transmission up to
30 Hz. The criteria used was 10 elements by wavelength, the
maximum mesh size
was calculated,
.
Figure 11. Position of the sensors during the MASW tests.
(31)
The mesh dimensions presented in Table 2 are compatible
with eq. (31). The non linearities were modeled up to 10.50 m
from the impact point (it is considered that the medium is
elastic beyond that radius, up to 150 m). The damping follows
a Rayleigh law and is fixed at 2% for 0.5 Hz and 20 Hz. The
FE-model has 14,523 elements, 14,886 nodes and 29,048
DOF. Nodes on the lateral boundary have only vertical DOF
and nodes on the lower boundary are fixed (see Figure 12).
Several heights of fall have been tested to quantify the
influence of plastification at the impact point; for each fall, 5
to 6 impacts have been performed (see Figure 10). A scatter of
about 2 has been observed on the results (up to twice the
minimal value for a test).
Figure 12. FE mesh.
Geotechnical data have been measured at the same field with
Multichannel Analysis Surface Wave method and were used
in the FE-model (see Figure 11) and are gathered in Table 2.
Table 2. Parameters measured from MASW test (thickness
layer, celerities) and estimated from experience (density).
Mesh dimensions for the FE-model.
Layers
1
2
3
4
5
6
Thickness
(m)
1.5
2.5
2.0
4.0
5.0
5.0
Density
(t.m-3)
1.8
1.8
1.9
2.0
2.0
2.0
Shear
waves
celerity
(m.s-1)
Compression
waves
celerity
(m.s-1)
Mesh
height
(m)
150
220
220
330
450
450
400
400
800
800
1800
1800
0.25
0.24
0.35
0.50
0.50
0.50
Mesh
width
(m)
The numerical integration scheme is detailed in ref. [12];
the non linear system of equations is solved with a modified
Newton-Raphson algorithm. The integration step is taken
equal to 10-3 s and total analysis duration is taken equal to 1 s.
The first step of the computation, which is the initialization
of the stresses in the soil, is statically realized; after stress
initialization, displacements are then reset to 0 and the
propagation is computed.
4.4
0.35
0.35
0.35
0.35
0.35
0.35
Results
Figure 13 shows an example of result obtained with the FEmodel which gives conservative results compared to the field
measurements.
It can be said of the measurements:
905
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
CONCLUSION
100
FE-model
Field measurements
10
1
y = 141.95x-1.153
0,1
y = 71.966x-1.791
0,01
0
20
40
60
80
100
Figure 13. Radial velocity (m.s-1, log-scale, vertical axis) in
function of distance to the impact point (m, horizontal axis).
Continuous line: FE-model prediction with interpolation curve
in dashed line; Triangular symbols: field measurements with
interpolation curve in continuous line.
All comparisons between FE-computations and field
measurements have globally the same shape, which means
that the relation between the velocity and the other parameters
(see eq. (29)) is different from the relationship used with
linear elasticity assumptions multiplied by a constant
efficiency coefficient (except for distances far enough from
the impact).
In this paper, an analytical model for estimating the peak
particle velocity near a buried pipeline due to the fall of a
heavy structure like a wind turbine is presented.
In a first step, it is assumed that the impact and then the
propagation are elastic. It is also assumed that the Rayleigh
waves are predominant. These assumptions lead to a model
which can be updated with an efficiency coefficient when the
impact is far enough from the buried pipeline, the efficiency
coefficient is a simple constant scaled with empirical relations
based on field experiments.
However, an estimation of the velocity obtained with a 3D
non linear plastic finite element model in time domain,
updated with specific field test data showed that the
approximation for distances close to the impact were not
proportional to the linear elastic solution (it is still true with an
efficiency coefficient, far enough from the impact), therefore
a PPV relation based on empirical relations used in dynamic
compaction was performed. The impact energy for windturbines falls is an order of magnitude larger than for dynamic
compaction, then the PPV relation presented in this paper had
to be updated numerically for the specific values of the windturbines parameters. A specific example is presented: fall of a
70 t wind-turbine nacelle from a height of 100 m.
ACKNOWLEDGMENTS
The authors thank the GRTgaz Engineering Center in Lille
and Paris, who found the right place to perform the seismic
tests, and also INNOGEO who performed the field tests.
Therefore, a relation based on the Mayne model (
) and the Menard model (
)
[1]
is written as,
,
(32)
where,
 PPV is the Peak Particular Velocity in mm.s-1;
 r is the distance to the impact point in m;
 m the impacting mass in kg;

is the velocity at the impact in m.s-1;

is the soil density in kg.m-3;

and n depends on
.
Figure 14 shows the use of equation (32) for the fall of a 70 t
nacelle from 100 m.
500,0
50,0
5,0
0
50
100
150
Figure 14. Peak particle velocity due to the fall of windturbine nacelle of 70 t from a height of 100 m. Vertical axis,
PPV in mm.s-1(log-scale), horizontal axis, distance in m
from the point of interest to the foot of the wind-turbine mast.
906
REFERENCES
P.W. Mayne, J.S. Jones and J.C. Dumas, Ground response to dynamic
compaction, Journal of Geotechnical Engineering (109), n°16, pp757774, ASCE, 1984.
[2] Menard Sol Traitement. Vitesse particulaire resultante en function de la
distance au point d’impact. Private conversation, 2001
[3] C. L. Pekeris, The seismic surface pulse, Proceedings of the National
Academy of Sciences, (41), pp469-480, 1955.
[4] J.D. Achenbach, Wave propagation in elastic solids, North Holland
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