experimental and numerical analysis of soil motions caused by

1464

Bulletin of the Seismological Society of America, 90, 6, pp. 1464–1479, December 2000

Experimental and Numerical Analysis of Soil Motions Caused by

Free Vibrations of a Building Model

by Philippe Gueguen, Pierre-Yves Bard, and Carlos S. Oliveira

Abstract Effects of vibrating structure on the free-field motion are presentedthrough a field experiment at the Euro-Seistest and a corresponding numerical simu-lation. Ground motion has been recorded by a dense temporary three-component(3C), two-dimensional (2D) seismic network installed at increasing distances from amodel building forced into vibration by pull-out tests. The building is a five-storyRC structure at a one-third scale, resting on the soil through surface square footing.A traction force, F0, applied at the building top and suddenly released forced it intovibrations. Two sequences of a pull-out test (POT) have been performed, each onemade of two vibration tests in the two horizontal directions of the building (longi-tudinal and transverse).

The experimental data are then compared to the results of the numerical simulation.The soil-structure system is modeled by a three degree of freedom (3DOF) system.The soil-structure interaction (SSI) is accounted for through the help of impedancefunctions, and the motion induced by POT are estimated together with the base forceand moment developed at the soil-structure interface. By representing the base forcesby surface point seismic sources, the induced wavefield radiated in the surroundingfree field is then computed by numerical Green’s functions. The results presentedhere do validate the numerical computation method, which gave distant motions inclose relation with the experimental data, from a qualitative and quantitative pointof view. The spectral analysis does also exhibit surface waves trapped in the topmostlayer. Results confirm the significant contamination of ground motion due to buildingvibration.

Introduction

The effects of soil-structure interaction (SSI) have beenanalyzed for a very long time in order to study the seismicbehavior of civil engineering structures and to assess thepotential damage to the structures themselves in case of astrong or moderate seismic event. For example, Merrit andHousner (1954) showed with a simple numerical model thatthe fundamental frequency of a building resting on soft soilmight be lowered. This phenomenon has received numerousexperimental evidences, as for instance the measured shiftof fundamental frequency from the fixed-base, f l , to theflexible-base, f s, structure (Jennings and Bielak, 1968;Stewart et al., 1999a), or the significant part of rocking mo-tion in recorded building motion (Bard, 1988; Bard et al.,1992), also supported by numerical computations (Paolucci,1993; Gueguen, 1995; Bard et al., 1996; Stewart et al.,1999a) considering the effects of soil and building charac-teristics (e.g., shear-wave velocity, building mass, footingradius . . . ) on building behavior. On the other hand, thescattering of incident waves from the foundation has beenstudied through experimental approaches (Lee et al., 1982;

Moslem and Trifunac, 1987) and by simple analytical mod-els (Trifunac, 1972; Wong and Trifunac, 1975). The conclu-sion was that this aspect of SSI might contribute to modifi-cation and amplification of recorded motions near thefoundations of large buildings and also contribute in suchcase to modify the response of neighboring buildings (Wongand Trifunac, 1975).

Furthermore, it is also known that the dissipation of thebuilding vibration energy takes place essentially through thesoil-foundation contact by producing a wave field that isradiated back into the ground. Such assertion has been re-ported by Jennings (1970) during vibration tests of the Mil-likan Library, a nine-story building on Caltech campus: aninduced horizontal ground motion was recorded by seis-mographs located at distances up to 6 km from the building.A station at around 13 km also simultaneously recorded thevertical ground motion (around 0.02‰ of the initial motionat building top), showing that the surrounding as well as thedistant free-field motion could be influenced by the multi-story building motion. Kanamori et al. (1991) also con-

Experimental and Numerical Analysis of Soil Motions Caused by Free Vibrations of a Building Model 1465

Figure 1. (a) Five-story RC-building model (one-third scale) of the Euro-Seistest project (1995), withthe equivalent to sixth story added mass, made of sixconcrete girders. (b) POT procedures in the L (POT-L) and T (POT-T) directions. F0 corresponds to thetraction force applied between the building top andthe close free-field through a steel cable.

firmed this assertion in southern California. They attributedthe origin of a pulse recorded on seismic stations to a seismicP wave generated by the motion of high-rise buildings indowntown Los Angeles, which had themselves been excitedby the shock wave produced by the re-entry into the atmo-sphere of the Columbia space shuttle.

Related (although somewhat different) observationshave also been reported in Sweden (Erlingsson and Bodare,1996) during two rock music concerts in the Ullevi stadium.Considerable vibrations of the ground and of the structurehave been felt when the large audience at the outdoor sta-dium started to jump in phase with the music. These effectshave been explained by the presence underneath the stadiumof a soft clay deposit having a fundamental frequency closeto the “jumping” frequency. The vibrations were then trans-mitted to the superstructure through piles foundation andproduced strongly felt motion because the fundamental fre-quency of the superstructure was close to those of the beat.

Oscillating structure effects on the surrounding free-field motion have also been evaluated by recent numericalstudies carried out by Gueguen (1995), Wirgin and Bard(1996), and Bard et al. (1996) using 2D and/or 3D simplifiedmodels. Nonnegligible modifications of the free-field mo-tion, due to large-building vibrations effects, have been com-puted at distances up to several hundred meters from thebuilding base in the case of very soft soil. However, thepractical reliability of these numerical results is still un-known because of the lack of appropriate experimental re-sults: the part of SSI energy in the ground motion is generallyshrouded in the incident wave field, stacked to the radiatedmotion from the multitude of close buildings.

To fully evaluate the effects of SSI on recorded free-field motion, both scattering of incident waves from thefoundation and wave field radiated back into the half-space,in relation with building response have then to be consid-ered. Nevertheless, the present article focuses only on thelatter aspect of the SSI phenomenon.

The Volvi test site (Euro-Seistest, 1995), located on theVolvi sedimentary basin near Thessaloniki (Greece), pro-vided an ideal framework for these investigations. A multi-story RC-structure model with surface square foundation(Manos et al., 1995) has been erected at a one-third scale.Moreover, comprehensive geophysical and geotechnical sur-veys led to a detailed knowledge of the geotechnical andstructural characteristics of the soil (Euro-Seistest, 1995).

In order to study the effects of the structure on the free-field motion, a field experiment has thus been performed,which consisted of installing a series of 3C seismic stationsin the immediate vicinity of the structure. The motion in-duced by building free oscillations, resulting from a pull-outtest (POT), were then recorded at the free-field ground sur-face in the three main directions of the structure (longitu-dinal, transverse, and vertical). A numerical computationhas also been done, which consisted of comparing the soil-building system to a three degree of freedom model (3DOF)

and in computing the induced radiated wavefield at increas-ing distances from the building.

This article successively presents the experimental ob-servations, the numerical model, the resulting computation,and their comparison with observed data.

Experimental Results

The Experiment

Since 1993, a European test site for engineering seis-mology, earthquake engineering, and seismology (Euro-Seistest, 1995) has been set up in the Volvi sedimentarybasin, near Thessaloniki (Greece). One goal of the Euro-Seistest project was the construction of a multistory structuremodel at a one-third scale (3.5 � 3.5 � 5.4 m) in order toinvestigate its behavior under loading. It consists of a RCframe, with masonry infill panels, resting on surface squarefooting (0.40 m thick) (Fig. 1a). Detailed information about

1466 P. Gueguen, P.-Y. Bard, and C. S. Oliveira

Table 1Characteristics of the Temporary Network Installed in the Vicinity of the RC Building Model to Record the Distant Motion

from the Building Vibration Test

Sequence 1: POT-T(1) and POT-L(1)

SeL01 SeL02 SeL03 SeL04 SeL05 SeL06 SeL07 SeL08 SeL09 SeL10 SeL11 SeL12 SeL13 SeL14Sensor CMG5 CMG40 L22 CMG40 CMG40 L22 CMG40 CMG40 CMG40 CMG40 L22 L22 L22 L22

Distance 1.00 2.20 3.85 5.00 7.00 8.00 9.20 12.00 14.00 16.00 17.40 19.00 23.20 26.75

Sequence 2: POT-L(2) and POT-T(2)

SeL01 SeL02 SeL03 SeL04 SeL05 SeL06 SeL07 SeL08 SeL09 SeL10 SeL11 SeL12 SeL13 SeL14Sensor CMG5 CMG40 L22 CMG40 CMG40 L22 CMG40 CMG40 CMG40 L22 CMG40 L22 L22 L22

Distance 1.00 3.00 3.85 6.00 7.00 8.00 9.20 13.00 15.00 17.40 17.90 21.00 25.00 26.75

Sequence 1 and 2

SeT01 SeT02 SeT03 SeT04 SeT05 SeD01 SeD02 TST11Sensor LE1 LE1 LE1 LE1 LE1 LE1 LE02 CMG5

Distance 2.15 4.12 6.00 7.45 14.00 8.45 14.00 0.00

The distance corresponds to the distance of the sensors from the building base edge.

the material and the construction sequences are shown in thefinal scientific report (Euro-Seistest, 1995; Manos et al.,1995). It was densely instrumented with accelerometers inorder to record and to analyze its seismic behavior in caseof a strong or moderate seismic event. The seismic instru-mentation consisted of a series of permanent accelerometersensors placed at each story (Euro-Seistest, 1995). To ana-lyze its vibration characteristics at low strains, the RC modelwas forced into vibration by POT sequences, performed bya traction force, F0, applied at the top of the RC structurethrough a steel cable anchored at nearby ground point, lo-cated at 10.4 and 8.5 m from the building base for the POTperformed in the L and the T direction, respectively (Fig.1b). After being prestressed, the cable was suddenly releasedand the free vibrations of the structure were recorded. De-tailed results about first studies are shown in the final sci-entific report (Euro-Seistest, 1995; Manos et al., 1995). Withrespect to these early tests, the model building has beenslightly modified to simulate a sixth story. An added masswas installed over the building top, made of six concretegirders with the support of four steel columns in order tosimulate the mass and the position of the sixth story (Fig.1a). The total building-footing system height, h, is then mod-ified from 5.4 m to 6.4 m. But the anchoring point at thebuilding top was not changed.

In order to record the ground motion induced by struc-ture vibrations, a dense temporary survey made of 21 three-component seismic stations (Table 1) has been installed inthe close vicinity of the RC structure (Fig. 2).

1. 14 Reftek stations along the longitudinal axis (SeL), con-nected to one CMG5 accelerometer (Guralp products),four 2.0-Hz L22 sensors (Mark products), and nine 0.05Hz broadband CMG40 sensors (Guralp products)

2. Five Lennartz stations along the transverse axis (SeT)connected to 1.0-Hz Lennartz sensors (LE01);

3. Two Lennartz stations along the diagonal direction (SeD)connected to one 1.0-Hz (LE01) and one 0.2-Hz (LE02)Lennartz sensors.

All the sensors have been oriented according to the maindirections of the RC building (Fig. 2): the longitudinal com-ponent is oriented positive outward from the building (i.e.,east direction), the transverse component is 90� counter-clockwise from the longitudinal (i.e., north direction), andthe vertical component is positive in the upward direction.Two testing sequences have been performed; both consistedin two POT along the T (POT-T) and the L (POT-L) directions,respectively. The sensor position spread along the L axischanged between sequence 1 and 2, as shown in Table 1.The TST11 station (Fig. 2) has been installed at the upperleft corner of the basement in order to know the base motioninduced by the amplitude value F0 of the traction force.

Time Domain Traces

About 80% and 90% of SeL stations, 60% and 70% ofSeT stations, and 75% and 100% of SeD stations workedduring the sequences 1 and 2, respectively. For sake of sim-plicity, only the free-field velocities recorded during onePOT in each direction are shown (Fig. 3). In the following,each trace is referred by the index of the recorded componentof the motion, the axis of the station spreading, and the di-rection of the POT (e.g., the ZLT component corresponds tothe Z component of the SeL stations during POT-T). Becauseof the high noise due to the bad atmospheric conditions, thetraces have been filtered in the 1–10 Hz range with a but-terworth filter, given in time a high signal-to-noise ratio. Allthe traces have been normalized with respect to the highestvelocity of each series of components, except for the TST11station. The maximal velocity (in mm/sec) is indicated at theupper right side of each trace (Fig. 3).

Two kind of traces are observed in Figure 3, depending


Figure 2. Location of the dense temporarynetwork composed of 21 3C-seismic stations.Each station (open circle) was oriented accord-ing to the three main building directions (L, T,and Z direction).

on the observed component with respect to the POT direc-tion, and showing the symmetry of the experimental data,obtained by permutation of the T and L index: (1) a contin-uous time decrease of the ground motion for the LLL, ZLL,and LTL components, as well as for the symmetrical TTT,ZTT, and TLT components, and (2) a spindle-shape envelope(i.e., wave packet) of the time decrease for the ZLT and LLTcomponents, as well as for the symmetrical components, thatis, the ZTL and TTL components.

The highest motion has been recorded (1) on the hori-zontal components parallel to the POT direction, namely, theTTT (Fig. 3A) and LLL (Fig. 3B) components; and (2) onthe vertical direction along the array parallel to the POT di-rection (Rayleigh waves), namely, the ZTT (Fig. 3A) andZLL (Fig. 3B) components. Unfortunately, no information isavailable concerning the amplitude of the POT forces. Nev-ertheless, as observed on the ground velocity records, POT-T (Fig. 3A) exciting force seems to be about two timeshigher than the POT-L (Fig. 3B), as especially shown on theTST11 records, for which ZTT and TTT values are 0.362 and0.194 mm/sec, respectively, while ZLL and LLL amplitudesare 0.191 and 0.118 mm/sec, respectively (i.e., ZTT/ZLL �1.9 and TTT/LLL � 1.7).

The TDT and LDL components are consistent with theobservations, as well as the vertical ZDT and ZDL compo-nents, which shows the same wave shape independently ofthe POT direction. However, the TDL and LDT componentsseems to show an intermediate wave shape between perpen-dicular- and parallel-to-POT component wave shape of theSeL and SeT stations (Fig. 3). Smaller motions were re-corded along horizontal components perpendicular to thePOT direction (i.e., LLT, LTT, TLL, and TTL) and along ver-tical component along arrays perpendicular to the POT di-

rection (i.e., ZLT and ZTL) (Fig. 3). They show a wavepacket with a spindle-shape envelope, also numerically ob-served by Gueguen (1995) and Bard et al. (1996) at distanceslarger than 500 m, in the case of a more realistic buildingmodel resting on a very soft soil layer. They explained it bythe coupled effect between propagation and trapping of sur-face waves diffracted by the foundation into the topmostsuperficial layer. Nevertheless, since the primary energy ofvibration is here provided at the top of the building, in thecase of pure longitudinal or transverse excitation, compo-nents should have information only on those which are inthe same direction that the POT (i.e., the LLL, ZLL, LTL, andTTT, ZTT, TLT components). Since the motion in LLT andLTT (Fig. 3A) and in TLL and TTL (Fig. 3B) directions arenot zero, some explanations should be looked for. It may becaused by some interaction between L and T fundamentalbuilding mode, and/or by the torsion motion. Moreover,since the steel cable is anchored at the surface close point(Fig. 1), the vertical mode of vibration of the superstructuremay be also excited. Finally, the sudden release of the forceat the ground anchor may also generate waves in the groundthat interfere with those radiated from the building.

This is consistent with the existence of two appliedforces, suddenly released, one on top of the building andanother at soil surface. We observe first an impulsive waveon various recorded free-field motion, on the vertical (e.g.,ZTL and ZLT) and parallel-to-POT (e.g., LLL ad TTT) com-ponents, with higher amplitude close to the anchoring point(e.g., SeT03 and SeT04, component TTT).

Time Domain Decrease and Soil-Structure Damping

The components parallel to the POT direction (i.e., theLLL, LTL, TTT, and TLT components) show a time-decrease


Figure 3. SeL, SeT and SeD velocity records during the first sequence of POT. (A) POT inthe T direction and (B) POT in the L direction. The traces represent the surface velocity (in mm/sec), and the right number is the maximal velocity. The flat traces represent failed components.


Figure 4. Time decrease of the absolute amplitude value of the velocities recordedby the T (left) and L (right) component of the SeL stations, respectively, during thePOT-T and POT-L of the first sequence of test. The dashed lines represent the recordsof the TST11 station.

which seems to be directly dependent on the apparent build-ing-soil system damping ratio, fs (Fig. 3A, B). This one canbe easily defined by the free-oscillations x(t) of the building-soil system, which are expressed as:

�f x ts sx(t) � cos x t •e (1)s

where xs is the apparent frequency of the building soil sys-tem. fs can be measured by the slope of the envelope of thefunction (Clough and Penzien, 1975):

ln|x(t)| � �f x t, (2)s s

which leads in the present case to fs � 1.5% for the fourPOT (for xs � 2pf s, with f s � 4.9 Hz, as defined in thefollowing part).

In this sense, the time decrease observed at each SeLstations (Fig. 4) shows the same downward slope and thenconfirms that the building-soil system damping mainly con-trols the time decrease of the free-field records. Moreover,the frequency of the wave train seems also to be only dom-inated by the first-mode frequency, f s, of the system (Fig.5C). In fact, the maximal spectral amplitude (MSA), com-puted by FFT for the SeL and SeT stations, corresponds re-spectively to around 4.761 Hz and 4.944 Hz for the POT-T(Fig. 5A) and POT-L (Fig. 5B) of both sequences, althoughthe induced motion from the two POT performed in the samedirection varied by a factor of 2. We also note that the free-oscillation frequency differs between the T (4.761 Hz) andL (4.944 Hz) direction of the vibrations, which suppose someslight asymmetries of the structure as already mentioned byManos et al. (1995) and Euro-Seistest (1995). However, itmay also be induced by different SSI characteristics due tosoil inhomogeneity beneath the foundation. The close T andL frequencies may induce quite strong coupling effects be-

tween transverse and longitudinal modes and may then con-tribute to explain the spindle shape envelope of the free-fieldsurface motion records. Strong coupling effects betweenhorizontal and torsion response were previously observed byseveral authors, including Bard et al. (1992) and Trifunac etal. (1999), which could result from inhomogeneities of thesoil below the foundation. Unfortunately, no information de-tailed enough about soil inhomogeneity at the soil-footinginterface is available to explain the dissymetry. However,this issue is not the scope of the present article.

Spatial Domain Decay

The spatial decrease of the maximal spectral amplitude(MSA), normalized with respect to the MSA of the TST11station, is displayed in Figure 6 as function of distance rbetween building base and free-field sensors. Although theforce amplitude, F0, was slightly different for each POT, asconfirmed by the velocities amplitude of the TST11 station,the spatial decay of the normalized MSA values is similarfor the three components of the SeL and SeT stations. Duringboth sequences, the ZLL and ZTT components do show MSAvalues of the same order, from around 25% of the base mo-tion at distance around 7–8 m (i.e., two times the buildingbase size) to 5% at 10 times the building base size. The Land T components also provide similar observations, theMSA values being also of the same order by comparing theL to the T components, and indicate the isotropic feature ofEuro-Seistest ground behavior. Despite the small size of thestructure, these values are very significant. Jennings (1970)recorded ground motion also dominated by the first mode ofvibration of the nine-story Millikan Library building (around20 � 22 m in plan) that had about 22% to 11% of thebuilding base motion at around 20 and 30 m from the centerof the foundation, respectively (i.e., around 9 and 18 m fromthe edge of the foundation, namely 1.0 and 1.5 times the


Figure 5. Frequency of the maximal spectral am-plitude (MSA) of the free-field recordings, for the (A)POT-T and (B) POT-L of both sequences (the recordsdone by the parallel-to-POT stations are only shownand the average frequency (fm) of each series of re-cords is computed) and (C) examples of spectra com-puted by fast Fourier transform (FFT) for the threecomponents of the SeL stations.

building base size) with a range between 1/r and 1/r3/2 ofthe spatial decay. Close to the structure, spatial decay ex-hibits a 1/r2 dependence that could be related to the near-field terms. By modeling, for a less realistic 2D SH model,Wirgin and Bard (1996) computed around 20% of the build-ing top motion at around six times the building base size,while Gueguen (1995) and Bard et al. (1996), for a 3Dmodel of the Mexico City case, computed around 30% and8% of the building base motion at around 2 times and 10times the building base size. The ground motion observedin Volvi is therefore in satisfactory agreement with thosepreviously observed experimentally or computed numeri-cally.

Furthermore, the spatial decay of the MSA values alsoshows a slope break at around 7 m from the foundation slab(i.e., two times the building base size) on the Z componentand around 10–12 meters (i.e., 3–3.5 times the building basesize) on the T component (and to some degree on the Lcomponent), respectively. While at a short distance, the spa-tial decay is more rapid, with a decay rate close to the 1/rrate of body waves (or 1/r2 corresponding to the near-fieldterms), it becomes slower at longer distances and becomes

closer to the 1/r decay of surface waves, this observationbeing independent of the observation direction (SeL or SeTstations). Thus, the free-field records seem to provide thetransition distance at which the diffracted wavefield inducedby the free oscillations of the building becomes dominatedby surface waves, which may be either Rayleigh waves (LLor TT) or Love waves (TL or LT).

Numerical Computation

Generalities

In order to model the effects of the structure on the closefree field, the soil and building properties have to be accu-rately known, including the soil–foundation interface. A3DOF (Fig. 7A) has been used in a first step to simplify thenumerical computation of the SSI effects. This model hasbeen widely employed in previous studies (e.g., Jenningsand Bielak, 1973; Todorovska and Trifunac, 1992; Wolf,1985; Paolucci, 1993; Gueguen, 1995; Bard et al., 1996).The soil is represented by a linear, isotropic, elastic, stratifiedhalf-space, and the soil-foundation system is approximatedby discrete springs and dashpots, which are depending uponsoil-foundation system configurations. The spring and dash-pot properties must be frequency dependent (Hsieh, 1962).As commonly defined in structural analysis, the building it-self is modeled by a linear, viscous damped oscillator. More-over, the foundation is assumed to move as a rigid body,without uplift between the base and the foundation. Recentstudies (e.g., Trifunac et al., 1999) showed experimental evi-dence for the flexibility of a building foundation and gen-erally focused on rocking mode (Liou and Huang, 1994).Nevertheless, the common assumption, which considers thefoundation as rigid, reduces the number of degrees of free-dom of the model and gives good approximation for a longwavelength relative to the foundation dimensions (Lee,1979). The SSI effects of the vertical and torsional motionof the structure are considered negligible with respect tothe translation and rocking (Paolucci, 1993), as well as forthe mode shapes other than the fundamental (Jennings andBielak, 1973).

As commonly considered in earthquake engineeringstudies, the soil-structure system is subjected to a seismicmotion represented by only the horizontal ground motion(xg). With such assumptions, the total motion, xt, of thebuilding mass, m1, concentrated at the height H is related to(1) the horizontal displacement, x1, due to internal defor-mation; (2) the horizontal displacement, x0, of the base mass,m0, relative to the free-field motion xg; and (3) the rotationalmotion � of the basement (Fig. 7B) (Paolucci, 1993; Gue-guen, 1995; Bard et al., 1996). H is interpreted as the dis-tance from the ground surface to the centroid of the inertialforces associated with the first mode of vibration. The totalhorizontal motion, xt, of the structure relative to the soil isthen the sum of the 3DOF (xt � x0 � x1 � H1�), which arederived from the following classical equation of motion:


Figure 6. Spatial dependence of the MSAin the L and the T direction, recorded duringthe sequence 1 and 2. For sake of simplicity,only the main components are shown here.Left: vertical components of the parallel-to-POT stations (i.e., ZLL and ZTT components);Middle: parallel-to-POT components of theparallel-to-POT stations (i.e., LLL and TTTcomponents); Right: parallel-to-POT compo-nents of the perpendicular-to-POT stations (i.e.,LTL and TLT components.

Figure 7. (MDOF) building model including thesoil-structure interaction for numerical computation(A) and its associated motion (B) induced by anequivalent horizontal POT force .HF0

[M]x � [C]x � [K]x � �{p}x (3)t t t g

for which [M], [C], and [K] are the mass, damping, andstiffness matrix of the soil-foundation-structure system, re-spectively, and {p} is the centroid mass vector, such as:

0 0m m H •m k1 1 1 1u uh[M] � 0 m 0 , [K] � �k k k ,0 1 0 0� � � �hu h0 0 J �H •k0 1 k k0 0

0 0 mc 11u uh[C] � �c c c , and {p} � m ,1 0 0 0� � � �hu h�H •c1 c c 00 0

where J0 is the rotational moment of inertia of the founda-tion, and k and c are dynamic stiffness and viscous dampingratio for the building (i � 1) and for the soil-foundationsystem (i � 0), the equilibrium equation of motion account-

ing for translational (j � u), rotational (j � h), and coupled(j � uh) modes for the soil-foundation system.

In case of soil-structure system subjected to an externaltraction force applied at the RC structure top, the horizontaland angular motions of the structure relative to the soil arethen derived from equation 3 as:

[M]x � [C]x � [K]x � 0 (4)t t t

This equation can be easily solved in time domain forthe case of a free oscillating damped system and then give:

0xx 110 �f x ts sx (t) � x cos(x t)e (5)0 0 s� � � �0� �

where are initial displacements of the soil foun-0 0 0 T{x x � }1 0

dation structure system, and xs and fs are the circular fre-quency (xs � 2p/Ts) and the damping ratio of the flexible-base structure, respectively (see further).

Initial conditions are derived from the position of thesystem just before the POT (i.e., t � 0) and are obtained byapplying the equilibrium formulation to the structure (massm1) and to the foundation system (mass m0) for the transla-tion and the rotation modes, such as:

for t � 00 Hmass m k x � F � 01 stat 1 0s

(translation) (6a)

0 u 0 uh 0mass m k x � k x � k � � 00 stat 1 stat0 0 stat0s

(translation) (6b)

0 hu 0 h 0Hk x � k x � k � � 0 (rotation) (6c)stat 1 stat0 0 stat0s

For the rotation equilibrium, the effect of foundationthickness is neglected with regard to the rotation of the struc-ture (Paolucci, 1993). is the horizontal component of theHF0

pull-out force, and and are the static stiffness ofjk kstat stat0s

the structure and of the soil-footing systems, respectively.


As aforementioned, impedance functions model the dy-namic stiffness and damping characteristics and are complexvalued and frequency dependent. Therefore, static stiffnessmay be deduced from impedance functions at null frequency(x � 0), as follows:

1. The impedance function, Ks, of the soil-structure systemis defined as Ks � ks � ixcs, where ks and cs are theelastic stiffness and viscous damping (proportional to thestiffness) coefficients corresponding to the fundamentalmode of vibration, respectively, and are given as functionof the fundamental period Ts (�2p/xs) and the dampingratio fs of the flexible-base structure as following:

24p m 4pm f1 1 sk � and c � . (7)s s2T Ts s

The static stiffness is then derived from equation (7), thatis, � ks.kstats

2. The impedance function, Kj, of the soil-foundation sys-tem is expressed for the translational , rota-u u(k � ixc )0 0

tional , and coupled modes ofh h uh uh(k � ixc ) (k � ixc )0 0 0 0

soil-structure interaction. In this study, the Kj frequency-dependent coefficients are provided in the handbook ofimpedance functions (Sieffert and Cevaer, 1992), whichreviewed all impedance functions of surface footingspublished in the relevant literature, so as to provide, atleast in a first step, the adequate help in the majority ofcases (e.g., Veletsos and Wei, 1971; Luco and Westmann,1971; Kausel, 1974; Luco, 1974; Gazetas, 1983; Apseland Luco, 1987). Impedance functions are formally ex-pressed as:

j j jK � k k (a ) � ia c (a ) (8)j stat0 0 0 0 0 0

in which represents the static stiffness of the soil-jkstat0

foundation system as a function of the soil characteristics(e.g., shear modulus and Poisson ratio, as well as topmostlayer thickness) and foundation geometry (e.g., radiusand embedment values), and and represent the di-j jk c0 0

mensionless impedance functions that can be interpretedas the stiffness and the viscous damping of frequency-dependent spring and dashpot, respectively (Hsieh,1962). They are real and given in relation to the dimen-sionless circular frequency a0 � x Re/b, where x standsfor the circular frequency of excitation, b stands for theshear-waves velocity into the soil, and Re is the equivalentradius of the footing. At null frequency (x � 0), theimpedance function is equivalent to the static stiffness,that is, (a0) � 1 and (a0) � 0. In case of rectangularj jk c0 0

footings, we approximate the equivalent radius, Re, giv-ing the same surface

1/24BLR �e� � � �p

and the same inertial moment

1/4 1/43 316B L 16L BR � or R �e e� � � � � �3p 3p

as a circular footing, for the translation and the rotationmodes, respectively. As consequence of reciprocity the-orem, the coupling terms Kuh and Khu of the impedancefunctions are equal. They should be considered in case ofdeep foundations where the coupling effect could be im-portant (Gazetas, 1991), but remain small for surfacefootings in comparison with those of the diagonal termsKu and Kh (Luco, 1969; Veletsos and Wei, 1969) and theywill thus be neglected in the following.

Initial conditions are then formulated by matrix for-mulation as follows:

0x 110 0ksu 0 H0 k 0 x � F 1 , (9)0 0 0� � � � � �h 00 0 k0 � H

and then

0 �1x k 0 0 1 1/k1 s s

0 u H H ux �� 0 k 0 F 1 ��F 1/k . (10)0 0 0 0 0� � � � � � � �0 h h� 0 0 k /H 1 H/k0 0

The next step is to compute the wavefield radiated intothe ground from the building base. Because impedance func-tions are expressed in the frequency domain (equation 8),the Fourier spectral function of the 3DOF of the model arecomputed by the fast Fourier transform (FFT) method, suchas X1(x) � FFT(x1(t)), X0(x) � FFT(x0(t)) and U(x) �FFT(�(t)). Derived from the general formulation of impe-dance functions, linking the displacements to the forces(Sieffert and Cevaer, 1992), the base forces developed at thesoil-foundation interface are then derived from the relativemotion of the foundation and are computed as follows:

F K 0 Xu 0(x) � (x) (11)� � � �� M 0 K Uh

where F(x) and M(x) represent the frequency-dependenthorizontal base shear force and rocking moment, respec-tively, and where the coupling terms are again neglected.

These forces are then taken as surface seismic sources,as following:

1. The shear force, F(x), is spread uniformly along the soil-foundation interface of length 2L and represented by 2nhorizontal point forces f h(xi, x), with F(x)/2n amplitudeand applied at the xj � �i L/n abscises from the interfacemiddle

2. The rocking moment, M(x), is approximated by n cou-ples of vertical point forces f v(xi, x), with amplitude in-


Table 2Building Parameters of the Volvi RC Structure Used for the


m1 (kg) m0 (kg) J0 (kg m2) 2B (m) 2L (m) H (m) Ts (s) fs (%)

23.2 103 12.25 103 1.0 105 3.5 3.5 4.5 0.202 1.5

(Euro-Seistest, 1995)

Table 3Soil Parameters of the Volvi Sedimentary Basin Used for the


z (m) q (kg/m3) �* (m/sec) b (m/sec) Qp Qs

0.0 1816 225 130 50 103.0 2116 261 151 50 154.5 2250 364 210 50 158.0 1815 369 213 50 30

17.0 2250 376 217 50 3021.0 1932 540 312 80 4025.0 1816 560 323 80 4045.0 1932 580 335 100 5050.0 2065 797 460 100 5065.0 1997 876 506 100 50

120.0 1900 1143 660 100 50175.0 2000 1576 910 100 50240 2400 3200 1850 100 50

(Euro-Seistest, 1995)For this computation, we did not take the true � value, but assumed a

constant Poisson ratio of 0.25.

creasing linearly as function of xi and f v0(x), that is,f v(xi, x) � i f v0(x) and

M(x) 3f � .v0 L (n � 1)(2n � 1)

As the building motion energy is dissipated into the groundthrough the base forces developed beneath the foundation,and since the slab is very thin, the lateral sides play a minorrole in the interaction of the embedded foundation and willbe neglected in the following. The discrete distribution andvalues of subforces f h and f v are chosen in conformity withthe elastic and rigid foundation assumptions.

The Case of Volvi

One of the advantages to use the Volvi Euro-Seistest isthe very good knowledge of the characteristics of the soil-structure system. On the one hand, exhaustive geophysicaland geotechnical experiments have been performed as partof the Volvi Euro-Seistest project, supplying very detailedknowledge of the surface soil layers in the vicinity of theRC structure (Table 3). On the other hand, all the dynamicand static parameters of the RC structure have been identifiedto model its dynamic behavior (Table 2). Nevertheless, theadded mass placed at the building top before the POT mightproduce some variations of the previously defined dynamicfeatures of the building. For this reason, some of them (asfundamental period and damping ratio) were chosen (e.g.,Ts � 0.202 sec) to better fit the experimental observations,while the H value was chosen accounting for the sixth story(i.e., H � 2/3h).

The wavefield radiated back into the soil is then com-puted by distributing the base shear force and rocking mo-ment into 10-point sources (n � 5), equally spaced justbeneath soil-foundation interface, the intersource spacingbeing therefore 0.35 m. In case of 3D-modeling formulation,the shear force, F(x), and rocking moment, M(x), should bediscretized and distributed over the 2D whole surface of thesoil-footing interface. The comparison of the free-field mo-tion computed with forces discretized in 1 or 2 dimensionsis simultaneously shown (Fig. 8), in case of rigid foundation,resting on a stratified half-space. F(x) and M(x) have beenconsidered as those computed in the present case of theVolvi RC-model. Then, the discretization chosen does notinfluence strongly the free-field motion computed byGreen’s functions. There is a good coherency between bothwave fields resulting from 1D or 2D discretization, namelythe location of the receiver. We will therefore consider only1D force distribution in the remainder of this article.

The intersource spacing is therefore much smaller thanthe shortest wavelength, that is, around 30 m in case of Volvisoil-building system (the shear-wave velocity is 130 m/secand the building interacting frequency is around 5 Hz).Moreover, dispersion curves computed for the Volvi case(Fig. 9) show phase velocities around 200 m/sec at 5Hz,

giving a wave length of 40 m, which may be considered aslong relative to the foundation length (3.5 m). Therefore, forthis study, the assumption that the foundation is rigid is agood approximation.

The surface motion due to the radiated wave field iscomputed with the modified discrete wavenumber method(Hisada, 1994, 1995). This code computes analytically theGreen’s functions for a viscoelastic stratified half-space,even when point sources and receivers are at very closedepths.

In order to compare numerical computation results andexperimental data, the radiated wave field has been com-puted at distances from 2 to 30 meters from building basecenter (i.e., 0.25 to 28.25 m from building base), each ob-servation point separated by 2 meters. As the force amplitudewas not measured, the amplitude applied at the buildingHF0

top has been estimated to 1933N to better fit the data.The L and Z components have then been computed at

the observation points spread along the L branch (MoL andMoZ), while only the T component was computed along theT branch (MoT) (Fig. 10a). All the other components arezero because of symmetry. As aforementioned, the soilstructure system is submitted to the force and also to theHF0

anchoring force, which can be divided into horizontal AFX

and vertical force, in opposition to F0 (Fig. 10b) appliedAFZ

to the ground surface at 10.4 m from the building base. AFX

amplitude corresponds to amplitude, while the am-H A�F F0 Z


Figure 8. Comparison of the wave trains ra-diated into the ground computed for differentforce F (left) and moment M (right) distribu-tions, at the soil-foundation contact. (a) 1D(thin line) and 2D (thick line) forces distribu-tion. The 1D corresponds to the substitution ofF and M by 10 horizontal subforces and fivecouples of vertical subforces, respectively, dis-tributed along the x axis of the basement. The2D corresponds to the substitution of F and Mby 50 subforces, distributed along the x and yaxis, that is, 10 horizontal subforces (y � 0.35m) along 5 x-axis lines (x � 0.50 m) and byfive couples of vertical subforces (x � 0.35)distributed along five y-axis lines (y � 0.75 m)for the F and M forces, respectively. (b) 1Dforce distribution along the x axis of the base-ment, substituted by 10 horizontal subforcesand 5 couples of vertical subforces (x � 0.35m, thin line), by 4 horizontal subforces and twocouples of vertical subforces (x � 0.70 m, me-dium line) and 1 horizontal subforce (at thecenter of the basement) and 1 couple of verticalsubforces (x � 1 m, thick line) for the F andM forces, respectively.

plitude is derived from the geometrical relation A AF � FZ X

tanh (Fig. 10b). The total computed free field is then ob-tained by adding the ground motion coming from the build-ing and from the anchoring forces (Fig. 11). The first im-pulse wave is well simulated and gives the same wave shapeas those observed in experimental data, with higher ampli-tudes close to the anchoring point. But in comparison withthe vibrating effect coming from the building, the impulseeffect is not significant for the free-field ground motion andthen, in following, anchoring forces are not considered forthe numerical computation.

For sake of simplicity, only the four normalized pairsof recorded and computed traces at around 4, 10, 16, and 28m from building center are shown (Fig. 12), as well as theTST11 recordings.

This comparison calls for several comments. First, thenumerical computation traces are in good agreement withthe experimental data, especially concerning the wave shape.The time decrease is tightly related to the model damping,fs, which gives the same time duration as the experimentaldata (Fig. 12). Such assumption can validate the initial pa-rameters used for numerical computation, as well as the HF0


Figure 9. Dispersion curves for Love (a) and Rayleigh (b) waves computed for theVolvi ground velocity profile.

Figure 10. a) Location of the three com-ponents computed by the Hisada’s code (1994,1995) in case of a POT provoked by the force

applied at the building top, in the L direc-HF0

tion. The associated experimental componentsare indicated. (b) Diagram of the POT proce-dure, with anchoring forces and loca-A AF FX Z

tions. (c) Relaxation type of POT force F0

shown in time and frequency domain.

value, which gives almost the same velocity value of exper-imental data. However, with respect to the free-field exper-imental data, the numerical results seem to systematicallyoverestimate the actual motion of ZLL and LLL components(i.e., MoZ and MoL) of about 50%–60%, and only of 30%for the TLT component (MoT). But, as aforementioned, the

experimental POT-T force was estimated to be about twotimes higher than the POT-L (Fig. 3A,B), which is in goodagreement with the numerical overestimation.

This is also confirmed by the spatial dependence of thenumerical MSA (Fig. 13), computed for the three compo-nents. The experimental (sequence 1) and numerical MSA


Figure 11. Wavefield radiated into the ground owing to anchoring forces (left),building vibration (middle), and the total radiated wavefield (right).

Figure 12. Comparison between experimental data (thick line) and numerical re-sults (thin line) at 4, 10, 16, and 28 m from the building base. The top traces correspondto the total building base motion. The traces are surface velocity (in mm/sec). Thenumber at the right of each trace is the maximal velocity.


Figure 13. Comparison of the spatial de-pendence of the experimental and numericalMSA values.

Figure 14. Attempt to model the spindle shape en-velop of the wave packet observed in some case onthe experiment data, by stacking the radiated wave-field induced by the transverse ( f � 4.761 Hz) andthe longitudinal ( fs � 4.944 Hz) building vibration.

values have been normalized by the maximal amplitude ofthe closest experimental receiver point to the source (i.e., 2m from the building base center). First, the numerical MSAvalues systematically overestimate the experimental data.The latter are 60% to 30% higher than the former on the Z,L, and T components. A constant overestimation of the MSAmight come either from an improper assessment of the soilparameters or from an underestimation of the actual forceimposed on the building, or formally from an improper as-sessment of impedance functions. But we can also note thatthe order of magnitude is respected. Furthermore, a slopebreak, similar to the one observed on the spatial dependenceof the experimental MSA values, is obtained at around 5 and10 m from building base center, for the MoZ and MoT com-ponent, respectively. As for the experiment, the MoL com-ponent does not exhibit any clear slope break and the spatialdecay of the MSA seems to be close to 1/r and 1/Zr, re-spectively.

On the other hand, the spindle shape (see Fig. 3) hasalso been modeled considering two free oscillation buildingdirections. We computed the building motion (and the in-duced reaction forces F and M), as supposed forced intovibration in the longitudinal (f s � 4.944 Hz) and transverse(f s � 4.761 Hz) directions, each one with its own naturalfrequency, f s. The two induced wave trains were thereforeradiated and stacked on the free-field motion (Fig. 14), con-sidering only the longitudinal component. A spindle-shapeenvelop of wave is then clearly shown, considering the ob-servation point at 2, 4, and 10 m from the building center.This tentative model is shown here only for qualitative in-formation, but for the numerical modeling, we consider aPOT force ratio of 2 between the in the longitudinal andHF0

the transverse direction, as suspected with the experimentaldata. The coupling between translational and longitudinalmodes of structure is then quite strong because of their veryclose soil-structure system frequencies, which also contrib-ute to the spindle-shape waves.

Conclusion and Discussion

The potential importance of building effects on groundmotion is confirmed by the present experimental and nu-

merical results. On the one hand, the radiated wave traingenerated by the RC-model vibration test seems tightlylinked to the structural characteristics. While the frequencycontent is dependent on the soil-structure system frequencyup to 30 m, the time decrease may be affected more by thebuilding mass and height than by the structural dampingratio (Todorovska and Trifunac, 1992), as consequence ofthe quite soft soil of the Volvi test site. The flexible-basebuilding is characterised by a fundamental frequency (f s)around 5 Hz and a damping ratio (fs) around 1.5%. The lowS-wave velocity, b, of the uppermost layer allows us to as-sume that the fixed-base frequency (f 1) should be higher, asa consequence of SSI effects (e.g., Stewart et al., 1999a). Onthe other hand, despite the small size of the Volvi RC modeland of its surface footing, the free oscillations of the structurehave been recorded at distances up to 30 m. At twice andten times the building base size, the distant ground motioncorresponds to around 25% and 5% of the base motion. Thesame values (around 10% and 3% at around two and tentimes the base size for 3D model) have already been com-puted in recent numerical studies (Gueguen, 1995; Bard etal., 1996). Because of the dense temporary array installed in


Figure 15. Transfer function of the Volvi site,computed with the soil characteristics shown in Table3. The thick open narrow represents the position ofthe soil-structure system frequency, fs, of the VolviRC model.

the longitudinal direction, with about 2m interstation dis-tances, the MSA spatial decay has been precisely defined. Itlies in the range between 1/r and 1/Zr, while Jennings (1970)reported a range between 1/r and 1/r3/2 of the spatial decay,observed on the free field during a building vibration test. A1/r-to-1/Zr slope break has been then clearly observed onthe MSA decrease. This break distance could be thereforeassociated to the transition distance between predominantbody and surface waves.

On the other hand, Bard et al. (1996) showed that theeffects of the structure are highest when the soil-structuresystem frequency is close to the fundamental or to the firstharmonic frequency of the ground. This particularity wasalso reported by Erlingsson and Bodare (1996) during thetwo rock music concerts in the Ullevi stadium, as well as byKanamori et al. (1991) during the re-entry into the atmo-sphere of the space shuttle Columbia. The proximity of thenatural frequency of the Los Angeles Basin and the down-town building frequency enabled a very efficient radiationinto the ground. This was not the case for this experimen-tation, for which there was discrepancy between soil-struc-ture system frequency and the fundamental frequency of soil(0.7 Hz) (Fig. 15); this leads us to anticipate higher effectsfor more favorable configurations. Gueguen (1995) and Bardet al. (1996) also underlined the coupling phenomenon be-tween soil-structure interaction and trapping of surfacewaves into the topmost soft layers, which might generatewave packets with a spindle-shape envelope in the surfacemotion up to several hundred meters. However, the spindle-shape envelope observed in Volvi seems to be the conse-quence of the stacking of two main radiated wave field,generated by the transverse and the longitudinal buildingvibrations (each one with its own natural frequency), alsofavored by the nearly identical fundamental frequencies ofthe structure. Even if the value of the pull-out force used forthe numerical computation seems to overestimate the exper-imental one, the computed results fit very well the experi-mental data, from a qualitative but also from a so-calledrelative quantitative point of view, the decay of the inducedwavefield being respected.

The acceleration at the building top has been obtainedfrom xt (i.e., xt � x1 � x0 � H/�) by two derivations ofthe motion computed at top (i.e., Xt (x) � �x2 Xt (x)) andestimated equal to 10�2g, which represents an equivalentshear stress P applied at the building base equal to

m x1 tP � ,4BL

that is, around 50 N/m2. In case of more realistic building(height H � 20 m) submitted to a seismic acceleration ˜xt

equal to 0.2g, the equivalent shear stress expected at thebuilding base is estimated by P � qeq H , where qeq cor-˜xt

responds to the equivalent density of the building. In thiscase, we consider qeq equal to 300 kg/m3 and then P equalto 1.2 104 N/m2, that is, 240 times higher than in the Volvi

case: this leads us to anticipate higher amplitude of the in-duced radiated wavefield (e.g., around 2 to 5 mm/sec at 30m, i.e., corresponding to about 6 to 15 gals under linearassumptions). Linear assumption is validated by previousstudies conducted at the Volvi test site (Euro-Seistest, 1995,Manos et al., 1995), which showed identical responses ofthe structure by small earthquakes, ambient noise, or POT.

Thus, these new results confirm that, in a few cases atleast, the structure may significantly modify the groundmotion at some distance from buildings, especially for con-figurations where soil and structure have close natural fre-quencies. Such a phenomenon, extrapolated for denselyurbanized areas built on very soft soil, allows the anticipa-tion of a significant increase in ground-motion duration, withamplitude levels that may be increased or decreased de-pending on the exact location of buildings, their character-istics, and the site location, because of the very complexwavefield that may generate constructive as well as destruc-tive interferences. Since the effects of SSI on recorded free-field motion have to account also for scattering of incidentwaves from the foundation, which also contribute to modi-fication and amplification of recorded free-field motion(e.g., Trifunac, 1972; Wong and Trifunac, 1975), the resultsshown in this article give a lower-bound estimation. Then,the full SSI effect (including scattering) may be even moresignificant.

In some case of seismic risk projects, the buildings maythus be taken into account not only as victims of the seismicevent but also as part of the seismic hazard because theymay modify seismic ground motion. Future tests with one-to-one scale building structures and instrumentation of a realbuilding including a network in neighboring sites should beperformed in order to validate the present qualitative andquantitative findings.


Acknowledgments

We thank G. C. Manos for performing the POT on the Volvi-RC struc-ture. We thank S. Vidal and K. Pitilakis for the help in the field experiment,and thank Y. Hisada for providing the program to compute Green’s func-tions in the layered half-space. This work was supported by the EuropeanCommunity (ENV4-CT96-0255).

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Laboratoire de Geophysique Interne et Tectonophysique (LGIT), BP5338041 Grenoble Cedex, [email protected]

(P. G., P.-Y. B.)

Laboratoire Central des Ponts-et-Chaussees (LCPC)58 bd Lefebvre75732 Paris Cedex 15, France

(P. G., P.-Y. B.)

Instituto Superior TecnicoDepartamento de Engenharia Civil. Av. Rovisco Pais1049-001 Lisboa, Portugal

(C. S. O.)

Manuscript received 24 May 1999.

experimental and numerical analysis of soil motions caused by

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