Soil-structure interaction simulations of 2D & 3D

block model with spectral element methods.

Javed Iqbal, Emmanuel Chaljub, Philippe Guéguen and Pierre-

SUMMARY:

For a macroscopic description of soil-structure interaction effects, a real building can be represented by a block
model consisting of piecewise homogeneous, visco-elastic material. Such a model can be relevant only if its
global dynamic characteristics reproduce those of the original structure. Here we are presenting 2D and 3D block
models of a scaled RC structure which has been used for a number of experiments, including pull-out tests, at the
Euroseistest Volvi site in Greece, and the dynamic properties of which are well known through published
literature. A spectral element method was used to perform a numerical simulation of the pull out test with 2D as
well as 3D block models. Comprehensive parameter studies have revealed that shear wave velocity in the
building block model plays dominant role in the tuning process, while other factors like stiffness contrast
between soil and building and even the dimensionality of the model also play their role to certain extent,
especially for damping characteristics.

Keywords: Soil-Structure-Interaction, Spectral element method, 2D and 3D effects

1. INTRODUCTION

As reported in (Guéguen et al., 2000), a pull-out test ("POT" experiment was performed on a scaled
RC structure at the Volvi test site in Greece within the framework of the EURO-SEISTEST project
(see Fig. 1c for the location). In this experiment, vibrations were measured in and around the structure
by a dense network of three component seismic instruments, while geological and geotechnical
information about the site have been thoroughly surveyed and are available in the published literature
(Jongmans et al., 1998). We have therefore selected this RC structure for testing a macroscopic
modelling of soil-structure interaction phenomena through numerical simulation (as used for instance
in Wirgin & Bard, 1996; Tsih-gka & Wirgin, 2003; Boutin & Roussillon, 2004; Kham et al., 2006;
Semblat et al., 2008; Padron et al., 2009).

The spectral element method has become a popular technique of numerical simulation of ground
motion from local to global scale (Chaljub et al., 2007). Here we have used SPECFEM2D and
SPECFEM3D computer codes (http://www.geodynamics.org) for simulating the propagation of elastic
waves in visco-elastic materials representing both the RC structure block model and the underlying
soil profile. Soil profile underneath the block model is assumed to be same as presented in Table 1
(Guéguen et al, 2000).

The RC structure used in POT experiment is represented here by a block consisting of visco-elastic
materials with the same dimensions as those of the original structure. The visco-elastic properties of
the block model are tuned so as to obtain the same dynamic characteristics of the block model as those
actually observed from the POT experiment recordings. This tuning process led us to investigate in
detail the individual and combined effects of shear wave velocity, density and Q factor on the system
natural frequency, its damping ratio and its rocking ratio. It also gave rise to an analysis of the
differences between 2D and 3D models with similar elastic properties, which provided some hints on
the explanation of the "damping paradox" in 2D vs 3D models (Meek & Wolf, 1992).

2. POT EXPERIMENT

A full description of the Volvi pull out test is provided in Guéguen et al, 2000. It is only briefly
summarized here, together with the sketch showing the main features of the structure and of the
instrumental layout as displayed in Fig. 1a. This structure was pulled out by a cable, which was
anchored on a surface concrete block at some distance from the basement, see Fig. 1b. The cable was
released suddenly to allow the structure to vibrate freely. The ground motion generated by the waves
propagating away from the foundation of the freely vibrating RC structure was recorded by a network
of seismic instruments installed in and around the structure, along longitudinal and transverse profiles
as displayed in Fig. 1c. POT was performed in the two L and T directions. The present document
considers only the POT in the L direction because of 2D code limitation to in-plane motion.

c) Site and instrument locations

Figure 1 : Sketch of the Pull out test (POT) experiment. (a) structure of the five-story scaled RC building (b) pull
out scheme in longitudinal and transverse directions (c) Location of site and instrumental layout (Guéguen et al,
2000).

3. 2D BLOCK MODEL FOR SIMULATION

In view of simulating the ground motion in the vicinity of a freely vibrating RC structure as in a POT
experiment, a block of visco-elastic material with the same dimensions as those of the actual RC
structure, is considered and assumed to be resting with full elastic contact on the soil profile presented
by Guéguen et al. (2000), as shown in Fig. 2. It consists of three homogeneous parts labelled A, B and
C, representing the added-mass, the main structure and the foundation slab, respectively. The unit
mass of these three sub-blocks is derived from the total mass of the corresponding part of the actual
RC structure, and the sub-block volume. A pull out inclined force is applied on the block at a height of
5.4m above the ground surface, similar to what was applied in the real POT experiment. Ground
motion time histories are then computed at the right locations shown in Fig. 1.

Table 1. Soil profile characteristics at Euroseistest site volvi, in Greece presented in (Guéguen et al, 2000).
Depth (m) Density Vp (m/sec) Vs (m/sec) Qp Qs
ρ (kg/m3)
0.0 1816 225 130 50 10
3.0 2116 261 151 50 15
4.5 2250 364 210 50 15
8.0 1815 369 213 50 30
17.0 2250 376 217 50 30
21.0 1932 540 312 80 40
25.0 1816 560 323 80 40
45.0 1932 580 335 100 50
50.0 2065 797 460 100 50
65.0 1997 876 506 100 50
120.0 1900 1143 660 100 50
175.0 2000 1576 910 100 50
240 2400 3200 1850 100 50

Figure 2. Block model used for 2D block simulation, Figure 3. Block model used for 3D block
where A, B and C represents added mass, main structure simulations. The soil structure has the same layering
and foundation slab of RC structure. D-F and E-G as in the 2D case.
represents the top two soil layers.

A satisfactory numerical simulation requires to reproduce the same natural frequency and damping as
actually observed in the in-situ POT experiment. As the actual RC structure was very stiff in the
vertical direction, a constant, high P-wave velocity was assumed for the material of block model. The
tuning was achieved by varying the shear stiffness of structure, i.e., the shear wave velocity, and the
damping values in the visco-elastic material of the structural blocks. The structural frequency and
damping values of the simulation model were derived from the analysis of the computed time histories
at building top, and their Fourier spectra, using peak picking and logarithmic decrement techniques,
and matching the computed time response to damped oscillations with ad-hoc frequency and damping.

The first step of the 2D tuning process considers a "fixed-base" structure, replacing the soil by a very
stiff rock having the properties of the deep underlying bedrock: it is simply intended to get the right
range for the structural block stiffness (especially block B), around which it may be varied when
considering the actual, softer and deeper soil. The observed natural frequency of 4.9 Hz is obtained
with a Vs value of 305 m/s. A second step taking into account the actual soil profile as presented in
Table 1, led to significantly increase the structural stiffness to keep the same natural frequency for the
soil-structure system: the best agreement was reached for Vs=420 m/s in block B.

The end results of this 2D tuning process are presented in Table 2, for two different structural damping
values (a quality factor Q of 200 corresponds to a material damping of 0.25%, while a Q of 30
corresponds to a 1.67% damping). The overall damping of the 2D soil-structure system turns out to be
much higher than the actually observed one, even for an unrealistic, quasi – zero damping within the
structure. This is due to the leak of energy into the soft soil ("radiative damping"), witnessed by the
very high value of the rocking ratio (representing the percentage of rigid rocking in the roof motion).

4. 3D SIMULATION OF BLOCK MODEL

The same block model was assumed for the 3D POT simulation, consisting in three parts, i.e.
foundation slab, main body and added top mass as shown in Figures 2 and 3. Each part has the same
density values as were used in the 2D case. The original soil profile as given in Table 1, presented by
Guéguen et al. (2000) is used here too. A horizontal Dirac force is applied at a 5.4 m height (see Fig.
1b) to simulate the POT and force into vibration the block model (this is a slight change compared to
the 2D case where an inclined force was used: present limitations in the SPECFEM3D code forced to
use a horizontal force.

A similar tuning process was performed for the 3D model, starting with the initial stiffness values
obtained at the end of the 2D tuning process. The dynamic characteristics of the 3D block model
proved indeed to be very different from those of the 2D block model: identical visco-elastic properties
result in a slight increase for the natural frequency, and a large reduction for the damping ratio.

A number of iterations was thus needed in view to obtain an optimal match of the target frequency and
damping values. In each case, the natural frequency, damping ratio and rocking ratio are evaluated as
for the 2D case. The final dynamic characteristics of the tuned 3D block model are presented in Table
2 along with those of the 2D block model and those derived from the in-situ POT experiment. The
"best" stiffness and quality factor values are Vs = 360m/s and Q = 30. As the radiative damping is
much lower in the 3D case than in the 2D case, the material damping in block B significantly affects
the damping ratio of the whole soil-structure 3D system.

Table 2. Dynamic characteristics identified for RC structure in POT experiment and those of 2D and 3D block
models on the soil profile presented in Guéguen et al. (2000).
Vs in B block Q-value Natural frequency % Damping Rocking Ratio
(m/s) (Hz) Ratio
Observed in POT
4.9 1.5
Experiment
2D Block Model 420 200 4.91 3.96 0.36
2D Block Model 400 30 4.99 5.33 0.40
3D Block Model 360 200 4.79 0.75 0.127
3D Block Model 360 30 4.86 1.61 0.131

5. COMPARISON OF 2D AND 3D BLOCK MODELS OF RC STRUCTURE USED IN POT

The main differences between the 2D and 3D cases concern the damping and rocking values. They are
to be interpreted in relation to the role of soil-structure interaction: the high 2D damping value is
associated with a high rocking ratio (40%), while the much smaller 3D damping value is associated
with a much lower rocking ratio (10%). The 2D block model actually can be viewed as a 3D wall-like
structure – as displayed in Fig. 4 -, while the 3D block model is limited to a square block with finite
structure – as displayed in Fig. 4 -, while the 3D block model is limited to a square block with finite
dimension compared to soil profile: as a consequence, the soil mass involved by the foundation motion
dimension compared to soil profile: as a consequence, the soil mass involved by the foundation motion
is much larger in the 2D case, which affects the frequency (larger mass results in a shift to lower
is much larger in the 2D case, which affects the frequency (larger mass results in a shift to lower
frequency), and the couple (damping, rocking ratio) with a much more efficient downward leakage of
frequency), and the couple (damping, rocking ratio) with a much more efficient downward leakage of
energy intointo
energy thethe
soil.
soil.

In order to better
In order understand
to better understandthe thetransition
transitionbetween
between 2D2D and 3D behaviours,
and 3D behaviours,aaseries
seriesofofintermediate
intermediate3D3D
models were considered, with rectangular instead of square structures having a fixed
models were considered, with rectangular instead of square structures having a fixed width W (equal width W (equal
to the actual
to the horizontal
actual size
horizontal sizeofofthe
thereal
realstructure),
structure), and
and a length
length LL varying
varyingfrom
from2 2WWtoto2020WW (quasi
(quasi 2D2D
case). TheThe
case). considered length-to-width
considered length-to-widthratios
ratiosare
are2,
2, 3,
3, 4,
4, 6, 8, 12
12 and
and20,
20,asaslisted
listedininTable
Table 3. 3.

!
Figure
Figure 4. Long,
4. Long, wallwall
likelike structure
structure in in
3D,3D, whichisisassumed
which assumedtotobe
be equivalent
equivalent to
to the
the2D
2Dblock
blockmodel.
model.The
Theblue
blue
coloured
coloured arrows
arrows schematically
schematically illustratethe
illustrate thecase
caseof
ofmultiple
multiple synchronous
synchronous sources,
sources,while
whilethe
theredred
arrow illustrate
arrow illustrate
thesingle
the single source
source case.
case.

These
These simulations
simulations werecarried
were carriedout
outwith
with two
two types
types of
of excitation
excitation sources:
sources:the
thefirst
firstone
oneis isa single
a single
horizontal Dirac force applied in the centre of the wall at a 5.4 m height above the
horizontal Dirac force applied in the centre of the wall at a 5.4 m height above the ground ground level, while
level, while
the second one is a series of synchronous horizontal forces uniformly distributed along the stretch of
the second one is a series of synchronous horizontal forces uniformly distributed along the stretch of
the wall at the same height. The corresponding results in terms of time histories and spectra of the
the wall at the same height. The corresponding results in terms of time histories and spectra of the
computed motion at the wall top in its centre top are displayed in Fig. 5 for the single source case, and
computed
Fig.6 formotion at the wall
the multiple, top in itssource
synchronous centrecase.
top are displayed in Fig. 5 for the single source case, and
Fig.6 for the multiple, synchronous source case.
The single source results exhibit several modes below 10 Hz, when the stretch of the wall exceeds 6
Thetimes
singlethesource results
slab width. exhibit
Indeed, several
higher modes
modes also below 10smaller
exist for Hz, when
aspecttheratios,
stretch
butofthey
the correspond
wall exceedsto 6
times
frequencies larger than 10 Hz : their signature can be detected on time histories down to L = 3W. to
the slab width. Indeed, higher modes also exist for smaller aspect ratios, but they correspond
frequencies
These higherlarger than 10modes
transverse Hz : are
their
notsignature can
seen in the be detected
multiple on time
source case (Fog.histories
6). Theydown to Lto=the
are linked 3W.
These higher transverse
propagation of wavesmodes
along are
the not
wallseen in the
from the multiple source
point where the case (Fig.
single 6). isThey
force are linked
applied, to the
and their
propagation
subsequentofreverberation
waves alongonthe eachwall from
edge. the modes
These point where
cannot the single force
be excited is applied,
for uniformly and their
distributed,
synchronous
subsequent forces, since
reverberation onthey
eachareedge.
associated
Thesewith
modesan increasing
cannot benumber
excitedoffornodes and out-of-phase
uniformly distributed,
motion along
synchronous the longitudinal
forces, since they direction.
are associated with an increasing number of nodes and out-of-phase
motion along the longitudinal direction.
The overall dynamic characteristics of these wall like structures are listed in Table 3 for both the
Thesingle anddynamic
overall multiple characteristics
source cases. They exhibitwall
of these a clear
likeincrease of the
structures aredamping
listed invalue
Tableand3decrease
for bothofthe
the rocking ratio with increasing wall length. The increase rate however significantly
single and multiple source cases. They exhibit a clear increase of the damping value and decrease decays for wall of
lengths beyond 4W, so that damping and rocking ratios do not reach their 2D values, even at aspect
the rocking ratio with increasing wall length. The increase rate however significantly decays for wall
ratios L/W as large as 20. This phenomenon has also been observed by various authors (e.g., Meek &
lengths beyond 4W, so that damping and rocking ratios do not reach their 2D values, even at aspect
Wolf, 1992, or Adam et al., 2000): some of them have proposed solutions to eliminate this discrepancy
rations
(WolfL/W as large
& Meek, as 20.
1994), butThis phenomenon
could has also
never completely been observed
eliminate this 2D-3D byinconsistency
various authors (e.g.,
(Wolf, Meek &
2004).
Wolf, 1992, or Adam et al. 2000): some of them have proposed solutions to eliminate this discrepancy
(Wolf & Meek, 1994), but could never completely eliminate this 2D-3D inconsistency (Wolf, 2004).
Figure 5. Time histories and spectra of record at the centre-top of the wall model when single Dirac force is
applied in the centre of the stretch of the wall model

Figure 6. Time histories and spectra of record at the centre-top of the wall model when uniformly distribute
multiple Dirac forces uniformly distributed through out the stretch of the wall
 Table 3. Comparison of simulation results between 2D and 3D block and wall models of different aspect ratio
L/W for excitation cases (single source applied in the centre of the stretch of the wall, uniformly distributed
multiple sources). Vs in the top two soil layers below the block model is increased to 200m/s to simulate the
rocking ratio 0.25 reported for RC structure.
Natural Frequency
% Damping Rocking Ratio
Vs (m/s) Q value (Hz)
Model
in model in model Single Multiple Single Multiple Single Multiple
Source Source Source Source Source Source
2D Block Model 380 200 4.88 3.86 0.25
3D Block Model 380 200 5.12 0.77 0.098
3D wall, L = 2W 380 200 4.98 4.89 1.06 0.93 0.195 0.164
3D wall, L = 3W 380 200 4.84 4.86 1.83 1.38 0.210 0.195
3D wall, L = 4W 380 200 4.83 4.84 2.17 1.80 0.144 0.209
3D wall, L = 6W 380 200 4.82 4.83 2.61 2.16 0.173 0.216
3D wall, L = 8W 380 200 4.82 4.74 2.70 2.70 0.198 0.202
3D wall, L = 12W 380 200 4.81 4.85 2.73 2.69 0.176 0.227
3D wall, L = 20W 380 200 5.0 4.83 4.48 2.75 0.2 0.227

6. ATTENUATION OF PEAK GROUND MOTION IN THE VICINITY OF STRUCTURE

The ground motion has been computed at ground surface in the immediate vicinity of the structure
(according to the instrumental layout shown on Fig. 1) in order to compare it to the actual in-situ POT
recordings, to quantify the energy of the radiated wavefield and analyse its attenuation rate with
distance from the structure foundation. Fig. 7 displays the decay of the recorded and simulated pga
(peak ground acceleration) as a function of distance from the edge of foundation, for the various
values of the L/W aspect ratio, and for both the vertical and radial horizontal components. Only the
multiple source case is considered here since it is better suited to investigate the 2D-3D transition.
These attenuation curves are normalised to the peak acceleration value recorded at the closest location
to the foundation slab.

The recorded peak ground motion decays according, very grossly, to a 1/r trend for the vertical
component and horizontal components. The anomalously large values of peak ground motion recorded
at 8 and 9.2 m locations are simply due to the reaction mass where the pull-out cable was anchored:
this acted as an additional point source of equal amplitude, resulting in a very localized surge in
ground motion.

The decay observed in the simulations varies from case to case, but nevertheless exhibits some
consistent features. It is systematically smaller at "large" distances (i.e., beyond 10 m) than at "short"
distances, and it (almost) systematically decreases from the full 3D case to the 2D case. Both
observations may be related to the composition of the radiated wavefield, consisting of a mixture of
body and surface waves : in the very near field (i.e., at distances comparable to the foundation size,
body waves and near-field terms are predominant, resulting in a rapid decay, while at larger distances,
surface waves are predominant. The geometrical spreading of surface waves is highly dependent on
the dimensionality: it follows a 1/√r trend in the full 3D case as they spread on a circular wavefront,
while it completely vanishes in the full 2D case, since this model is equivalent to a wall like structure
resting on strip foundation, acting as a line source in 3D, away of which surface wave propagate as a
plane wave front, without any amplitude decay.

At very short distances from the slab, there are thus no significant differences between the 2D and 3D
cases, as the source (i.e., the freely vibrating RC structure) in the 3D real world is seen almost as a line
source very similar to the 2D model source. At larger distances, there is gradual increase of decay
from 2D to 3D. It is worth to notice that whenever the wall length exceeds 6 times the slab width, the
decay with distance gets very close to the full 2D case: it can be concluded that long structures with
aspect ratios exceeding 5-6 could be simulated with 2D modelling, at least as far as macroscopic
effects of soil-structure interaction are concerned. It is to be mentioned also that the downward bend at
the end of attenuation curves (large distances) can be associated to the imperfect absorbing boundaries
used in the simulation code.

As a consequence, the agreement between observations and simulations is more satisfactory – though
by no means excellent – for the 3D case than for the 2D; or long walls cases, as one could expect.
Apart from the "anomalous" amplitudes around the reaction mass,

Figure 7. Attenuation of recorded as well as simulated peak ground motion when 3D wall models are subjected
to multiple, synchronous Dirac forces uniformly distributed along the stretch of the wall. The distance is
measured from the edge of the foundation slab. All the curves are normalised to the value at the closest location
to the foundation slab.

7. CONCLUSIONS

This example case study, though limited, allows to draw some more useful conclusions as to the taking
into account the macroscopic effects of soil-structure interaction in densely urbanized areas. The
"dimensionality" effects (i.e., 2D versus 3D models) are very significant for aspect ratios less than 3-6:
2D models are unconservative as they tend to significantly overestimate the soil-structure interaction,
and thus the radiative damping and rocking ratios. Conversely, effects of material damping within the
structure are much more important in the 3D (real) case than in 2D modelling or for elongated
structures, at least when they rest on soils that are soft enough to result in significant soil-structure
interaction.

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