Seismic Interferometry of a Soil-Structure Interaction Model

with Coupled Horizontal and Rocking Response

by Maria I. Todorovska

Abstract This article presents a system identification analysis of a soil-structureinteraction model with coupled horizontal and rocking response based on a combina-tion of Fourier analysis, wave travel-time analysis, and a relationship between fixed-base, rigid-body, and system frequencies. The study provides insight into the couplingof the structural and soil vibrations useful for interpretation of seismic recordings instructures. The structural model captures one-dimensional shear-wave propagation inthe structure. The analysis shows that the system functions with respect to foundationhorizontal motion are those of the coupled soil-structure system, which differs fromconclusions of earlier studies based on a model without foundation rocking. Theenergy of the system vibrational response is concentrated around the frequencies ofvibration of the system, which depend on the properties of the structure, soil, andfoundation. The analysis shows that the structural fundamental fixed-base (uncoupled)frequency f1 is related to the wave travel time τ (from the base to the top) by f1 �1=�4τ� and that accurate measurement of τ , unaffected by soil-structure interaction,can be obtained from impulse response functions, provided that the data are suffi-ciently broadband. This is an important result for structural health monitoring becauseit shows that structural parameters unaffected by soil-structure interaction (τ , as wellas f1 for structures deforming primarily in shear) can be estimated from seismic mon-itoring data with minimum instrumentation (two horizontal sensors, one at the baseand one at the top). This extends the usability of old strong-motion data in buildings,most of which have not been extensively instrumented, and lessons that can be learnedfor development and validation of structural health monitoring methodologies. Thepresented results correspond to a model of the north–south response of the MillikanLibrary in Pasadena, California, which has become a classical case study for soil-structure interaction.

Introduction

The traditional approach to system identification of vi-brating structures is to determine the frequencies of vibrationand corresponding mode shapes, which are characteristics ofthe frequency domain representation of the response. In thefrequency domain, the structural response can be representedas a superposition of the modal responses. An alternative ap-proach is to identify time domain characteristics, such aswave travel times through the structure (Kanai, 1965; Şafak,1998, 1999; Ivanović et al., 2001; Kawakami and Oyunchi-meg, 2003, 2004; Oyunchimeg and Kawakami, 2003; Snie-der and Şafak, 2006; Kohler et al., 2007; Todorovska andTrifunac, 2008a,b; Trifunac et al., 2008). In the time domain,the structural response can be represented as a superpositionof waves that enter the structure and waves that reflect fromits external boundaries and from internal boundaries of im-pedance contrast (Kanai, 1965).

Snieder and Şafak (2006) consider a one-dimensionalwave propagation model of a building, which they view asanother soil layer, and show the equivalence between the twoapproaches. Further, they analyze deconvolved north–south(NS) low amplitude earthquake response of the MillikanLibrary in Pasadena, California, recorded during the 2002Yorba Linda earthquake. Deconvolution of the response withthe motion of the base gives the system impulse responsefunction, which is a time domain representation of the systemfunction, and physically represents the response of the build-ing to an input impulse. From the frequency and amplitudesof the oscillatory part of the impulse response functions, theyidentify the modal frequency (1.72 Hz), average shear-wavevelocity between the base and the top (330 m=sec, identifiedfrom the modal frequency), and attenuation factor Q (20).They conclude that these parameters are those of the uncou-

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Bulletin of the Seismological Society of America, Vol. 99, No. 2A, pp. 611–625, April 2009, doi: 10.1785/0120080191

pled (from the soil) building response, based on analysis of amodel of vertically propagating shear waves in a horizontallylayered medium, in which the building is considered as an-other soil layer. Such a model, however, does not capture thefoundation rocking, which is an important aspect of the soil-structure interaction, in particular for this building, which hasbecome a classical case study for this phenomenon. Analysisof deformation patterns during forced vibrations have shownthat, for its NS vibrations, as much as about 30% of the roofresponse can be accounted for by rigid-body rocking (Foutchet al., 1975; Luco et al., 1988; Wong et al., 1988). In thisarticle, a more realistic soil-structure interaction model isanalyzed that includes foundation rocking, which leads toconclusions about the identified parameters that differ fromthose in Snieder and Şafak (2006).

In this article, a simple two-dimensional soil-foundation-structure interaction model with coupled hori-zontal and rocking response is analyzed in the frequency andtime domains, and the identified parameters in both domainsare related. The structure is represented by a shear beam sup-ported by a rigid foundation embedded in a half-space andexcited by in-plane motion. Such a model may be appropri-ate for the NS response of the Millikan Library for which thefoundation acts as rigid, being stiffened by the two externalshear walls of the building (Foutch et al., 1975; Luco et al.,1988; Wong et al., 1988). The impulse response functionsare computed by inverse Fourier transform of transfer func-tions with respect to the horizontal driving motion or the re-sultant foundation horizontal motion, computed numericallyfor this model. The analysis shows the following:

(1) Because of the rocking of the base, the impulse responsefunctions are not completely uncoupled from the motionof the soil; the modal frequency identified from the os-cillatory part of the impulse response function is not thatof the uncoupled building response (Snieder and Şafak,2006). The wave travel times, however, as measuredfrom the first arrivals of the input impulse, are uncoupledfrom the soil motion.

(2) The quantity 1=�4τ�, where τ is the impulse travel timefrom ground level to the top, is the fixed-base frequencyof the structure, which is that of the uncoupled structuralresponse. The fact that this frequency can be extractedonly from two records of horizontal motion, one atground level and the other one at the top, is an importantresult for structural system identification and healthmonitoring because most buildings that have recordedsignificant earthquake response have not been exten-sively instrumented. Transfer functions from recordedhorizontal response give the system frequencies, whichchange not only because of damage but also because ofchanges in the soil. Until recently, it was believed thatextracting the fixed-base frequency required more exten-sive instrumentation (see, e.g., Wong et al. [1988]).

(3) The foundation rocking frequency can also be estimated,based on its relationship to the apparent system fre-

quency (estimated from system transfer functions) andthe fundamental fixed-base frequency. Being able to iso-late this frequency and monitor its changes is importantfor analyses of soil-structure interaction and changes inthe foundation system during strong earthquakes.

This article is organized as follows. It first presents thesoil-structure interaction model, with an outline of the solu-tion for its response in the frequency domain. The modelparameters, chosen for illustrative purposes, are the same asthose in Todorovska and Al Rjoub (2006b, 2008) that areused to approximately represent the NS response of the Milli-kan Library. Numerical results are presented for varioustransfer functions of the foundation and building responses,along with their inverse Fourier transform, which gives thecorresponding impulse response functions. These numericalresults are used to prove claims (1)–(3). In particular, fromthe transfer functions, the system and rigid-body frequenciesare read (while the fixed-base frequency is an input modelparameter) and their relationship is verified. The analysisalso provides insight into the foundation rocking impulseresponse, along with an explanation for some of the second-ary ripples in the roof and foundation horizontal impulseresponses.

Other identification studies of building vibrational char-acteristics using impulse response functions include Kohleret al. (2007), in which the procedure of Snieder and Şafak(2006) is applied to small amplitude earthquake response of a17-story steel frame building, and Todorovska and Trifunac(2008a,b), in which changes in wave travel times are used todetect damage in two reinforced concrete buildings (six-storyand seven-story) damaged by earthquakes. Todorovska andTrifunac (2008a,b) acknowledge that the impulse responsefunctions are affected by the foundation rocking, but theyassume that the error is only in the reading of the time ofthe first arrival and is small. Further, they compute the build-ing fixed-base (i.e., the uncoupled) frequency as 1=�4τ�, theycompare these frequencies with the soil-structure systemfrequencies determined from time frequency analysis (es-sentially windowed Fourier analysis), and they concludethat their relationship is consistent with the interpretationof 1=�4τ� as fixed-base frequency. This article shows that1=�4τ� indeed gives the fixed-base frequency if the buildingdeformation is primarily in shear.

Kawakami and Oyunchimeg (2004) estimate wavetravel times in buildings using an optimization scheme(known by the acronym NIOM), which essentially givesregularized impulse response functions, and relate the wavetravel times to their dynamic properties for a numericalexample of a ten-story frame. They compare 1=�4τ� withthe fixed-base frequency obtained by modal analysis ofthe structural frame, which shows systematic differences de-pending on how the stiffness is distributed along the height(for the same equivalent stiffness). Their analysis shows thatthe modal (fixed-base) frequency is approximately equal to1=�4τ� for uniform distribution of stiffness, is higher for

612 M. I. Todorovska

decreasing stiffness with increasing height, which is com-mon in buildings, and is smaller for increasing stiffness withheight. For uniformly decreasing stiffness and first-storystiffness 3.7 times larger than the tenth-story stiffness, themodal fixed-base frequency is 20% higher than 1=�4τ�. Theyalso plot 1=�4τ� versus published modal frequencies for thesame buildings by other authors and state that the discrep-ancy is smaller than expected (from nonuniform distributionof stiffness), which may be explained by taking soil-structureinteraction into consideration; however, they do not provideany further explanation or discuss any of these trends.

Model

In-Plane Soil-Structure Interaction Model

The soil-structure interaction model considered in thisanalysis is shown in Figure 1. It is a simple two-dimensionalmodel (infinite in the y direction) for in-plane motions (in thex-z plane) in which the structure is represented as a shearbeam supported by a circular rigid foundation embeddedin a homogeneous and isotropic elastic half-space. The shearbeam has heightH, widthW, and mass per unit lengthmb (inthe y direction). The foundation has width 2a, depth h, andmass per unit length mfnd. The foundation has three degreesof freedom: horizontal and vertical translations and rotationin the x-z plane, described via displacements of point O, Δ,and V, along with angle φ (positive clockwise). The buildingmoves as a rigid body, with translations Δ and V, as well asrotation φ; it also experiences relative elastic deformations(Fig. 1). The horizontal displacement at the top of the build-ing due to its elastic deformation is urelb and the correspond-ing vertical displacement is vrelb (not shown in Fig. 1). Theshear-wave and compressional-wave velocities in the build-ing, VS;b and VP;b, are complex valued to account for thestructural damping. The excitation can be a plane P orSV wave, with incident angle θ0, a (surface) Rayleigh wave,or foundation driving motion as a simplification of the waveexcitation that neglects the kinematic scattering of the in-cident waves from the foundation. The half-space has den-

sity ρ, shear modulus μ, and Poisson’s ratio ν. This modeland its linearized solution, based on wave function expansionof the motion in the soil, have been described in detail inTodorovska (1993a,b) and its generalization to poroelasticmedium in Todorovska and Al Rjoub (2006a,b, 2008). Inthe linearized solution, the horizontal and rocking motionsare coupled while the vertical motion is uncoupled. Thismodel for in-plane excitation is a generalization of the modelin Luco (1969) and Trifunac (1972) for incident SH waves.

Frequency Domain Solution

The motion on the surface of the half-space withoutstructure and excavation is referred to as free-field motion.The effective motion at the base of the building differs fromthe free-field motion because of (i) scattering and diffractionof the incident waves from the excavation in the soil (kine-matic interaction), (ii) displacements due to feedback forcesfrom the structure (and foundation) acting on the soil (dy-namic or inertial interaction). For the linear problem and arigid foundation, the two problems can be solved separatelyand their effects superimposed. The system frequency anddamping are mostly affected by the dynamic interaction.Therefore, for simplicity, the effects of the kinematic in-teraction are neglected and the excitation is represented byprescribed foundation driving motion in the analysis inthis article.

Solution is obtained by first decomposing the probleminto the half-space with an excavation, foundation, and struc-ture. These three parts are shown in Figure 2, along with theforces with which they act onto each other. In this figure, fx,fz, and M0 indicate horizontal force, vertical force, and mo-ment about point 0; superscripts (s) and (b) indicate forcesbetween the soil and the foundation and between the buildingand the foundation. In the following, generalized force vectorF is introduced for the triplet F � ffz; fx;M0=agT and gen-eralized displacement vector Δ � fV;Δ;φagT, which arereferred to as force and displacement. As can be seen fromFigure 2, positive F�b� is the force with which the structureacts onto the foundation and positive F�s� is the force withwhich the foundation acts onto the soil. Further, harmonicmotion is assumed, Δ � Δ0e

�iωt, where Δ0 is the complexamplitude, ω is the circular frequency, and t is the time.

Next, generalized force vector F�b� is expressed in termsof the displacement vectorΔ by solving the one-dimensionalwave equation of motion of the beam with zero-stress con-dition on the top and moving boundary conditions at thebase. Similarly, generalized force vector F�s� is expressedin terms of Δ and the input motion. This is done by repre-sentation of the total motion in the soil in cylindrical wavefunctions, applying the displacement compatibility conditionat the contact between the foundation and the soil, computingthe stresses along the contact surface, and integrating thesestresses to obtain F�s�. Finally, the only remaining unknown,Δ, is determined from the dynamic equilibrium condition ofthe foundation.Figure 1. The model.

Seismic Interferometry of Soil-Structure Interaction Model with Coupled Horizontal and Rocking Response 613

To understand the feedback mechanisms in this prob-lem, it is necessary to deliberate on the constituents of themotion in soil and resulting forces between the foundationand the soil. The motion in the soil can be viewed as a super-position of the free-field motion, waves scattered from theexcavation in the half-space, which is kept at rest (the kine-matic part of the interaction), and waves radiated by the vi-brating foundation in the absence of any incident waves (thedynamic part of the interaction). Consequently, F�s� can berepresented as the sum

F�s� � F�s�ff � F�s�

scat � F�s�Δ ; (1)

where F�s�ff and F�s�

scat are integrals of the stresses in the soildue to the free-field motion and scattered waves; F�s�

Δ isthe integral of the stresses due to deformation of the soilby the moving foundation. Positive �F�s�

ff � F�s�scat� is the ex-

ternal force needed to keep the foundation at rest under theaction of the incident waves and resulting scattered waves,while its negative is the effective force that drives the foun-dation. Positive F�s�

Δ is the external force that would producedisplacement of the soil Δ.

The displacements of the foundation and the causativeor resulting forces are related by the foundation stiffness,which is a 3 × 3 matrix in the generalized force and dis-placement vector formalism. In terms of such a matrix

F�s�Δ � 2μ�K�s��Δ; (2)

where the 3 × 3 matrix �K�s�� is the dimensionless complexstiffness matrix of the foundation, the real part of whichrepresents the foundation stiffness and the imaginary partrepresenting the damping due to radiation of energy in thesemi-infinite soil medium. Further, if the excitation is repre-sented as some driving foundation displacement, Δdriv, thecorresponding driving force

F�s�driv � F�s�

ff � F�s�scat � �2μ�K�s��Δdriv: (3)

It is noted that the minus sign in equation (3) is due to thesign convention of the forces and displacements in Figure 2,which implies that positive �F�s�

ff � F�s�scat� is the force required

to keep the foundation at rest under the influence of the in-cident and scattered waves.

Driving and Feedback Displacements

In this article, the resultant displacement of the founda-tion, Δ, is viewed as a sum of the driving displacement,Δdriv, and additional displacement due to the interaction be-tween the building, foundation, and soil, which is referred toas feedback displacement, Δfb. Further, without loss of gen-erality in the problem formulation and solution, and for con-venience in the interpretation of the numerical results, thedriving displacement is only horizontal motion with constantamplitude at all frequencies, Δdriv � f0;Δdriv; 0gT . Such as-sumption implies that the ironing effect of the rigid founda-tion onto the higher frequencies of the input motion isneglected. Consequently, the resultant motion of the founda-tion has components

Δ � Δdriv �Δfb; (4a)

V � 0; (4b)

φ � φfb; φinp � 0; (4c)

where Δfb, Vfb, and φfb are the “feedback” translations androtation.

Figure 2. Free body diagram of the structure, foundation,and soil.

614 M. I. Todorovska

Motions of the Structure

The building horizontal displacement u�ξ�, as functionof the height ξ measured from ground level (Fig. 1), is a sumof three terms,

u�ξ� � Δ� φξ � urel�ξ�; (5)

where the first two terms are from the translation and rotationas a rigid body (due to rotation of the foundation) and thethird term is the relative displacement due to deformation.The damage in the building will depend only on urel�ξ�. Forthe shear beam, u�ξ� can be computed as a solution of thewave equation for moving boundary conditions (Todorov-ska, 1993b). It can be represented as

u�ξ� � uΔ�ξ� � uφ�ξ�; (6)

where uΔ�ξ� is the displacement due to translation of thebase only and uφ�ξ� is the displacement due to rotation ofthe base only. For harmonic excitation with circular fre-quency ω,

uΔ�ξ� � Δcos kS�H � ξ�

cos kSH; (7a)

uφ�ξ� �φkS

sin kSξcos kSH

; (7b)

where kS � ω=VS and VS ��������������μb=ρb

pis the shear-wave ve-

locity in the building. Equations (7a) and (7b), reflecting theinterference conditions in the building, imply a fundamentalfixed-base frequency of the structure f1 � VS=�4H� andovertones at fn � �2n � 1�VS=�4H�, n > 1. If τ is the timeit takes for a wave to propagate from the base (at ξ � 0) tothe top (at ξ � H), the interference conditions in the shearbeam imply

f1 � 1=�4τ�: (8)

Similarly, for the vertical response

v�ξ� � Vcos kP�H � ξ�

cos kPH; (9)

where kP � ω=VP and VP is the compressional-wave ve-locity of the beam. The fixed-base frequencies are fn ��2n � 1�VP=�4H�, n ≥ 1. This article only considers thelateral response.

Fixed-Base, Rigid-Body, and SystemFrequencies of Vibration

Two extreme cases of the model in Figure 1 are (1) flex-ible structure on rigid soil (fixed-base solution) and (2) rigidstructure on flexible soil (rigid-body solution). In case 1,the foundation will move exactly as the driving motion

(Δ � Δdriv) and the building will vibrate with its fixed-basefrequencies: fn � �2n � 1�VS=�4H�, n ≥ 1 for horizontalmotions and fn � �2n � 1�VS=�4H�, n ≥ 1 for vertical mo-tions. In case 2, the soil acts as a spring (one for each degreeof freedom of the foundation) and the structure will also vi-brate with its rigid-body frequencies of vibration: fRB for thecoupled horizontal and rocking motion and fV for the ver-tical motions. If both structure and soil are flexible, the fre-quencies of the vibrations will be those of the system, fn;sys,n ≥ 1, which are different from the fixed-base frequencies.The fundamental building mode is most affected by the cou-pling; the following approximate relation applies betweenthe first mode system frequency, f1;sys, and the fundamentalfixed-base frequency, f1, (Luco et al., 1987):

1

f21;sys≈ 1

f2H� 1

f2R� 1

f21; (10)

where fH and fR are the frequencies related to the horizontaland rocking foundation stiffness coefficients (see equation 2)referred to as horizontal and rocking rigid-body frequencies.The rigid-body frequency of the coupled horizontal androcking motion is

1

f2RB≈ 1

f2H� 1

f2R: (11)

Equations (10) and (11) imply f1;sys < min�f1; fH; fR�and f1;sys < min�f1; fRB�, that is, f1;sys is lower than thesmallest of f1 and the rigid-body frequencies. They also im-ply the following facts that will be useful in the interpreta-tion of the model results. If f1 → ∞ (very stiff building),f1;sys → fRB. Further, if fH → ∞ (foundation very stifffor horizontal motions), then fRB → fR; if both fH → ∞and f1 → ∞, then f1;sys → fR.

The energy of the response of the coupled system is con-centrated around the system frequencies fn;sys, n ≥ 1, whichare in turn straightforward to measure as the frequencies ofthe peaks of the response transfer function. If f1 can also beestimated (e.g., from wave travel times), then fRB can be es-timated based on equations (10) and (11). Further, if fRB andfR are known, then fH can be estimated from equation (11).

System Transfer Functions and ImpulseResponse Functions

The model response in the frequency domain can beused to compute various transfer functions, that is, transferfunctions for different input-output relationships, and corre-sponding impulse response functions by transforming themfrom the frequency to the time domain. As pointed out bySnieder and Şafak (2006), depending on the choice of theinput (which does not need to be the physical input motion),a transfer function may correspond to different displacementboundary conditions.

Seismic Interferometry of Soil-Structure Interaction Model with Coupled Horizontal and Rocking Response 615

Let u�ξ� be the displacements of the structure at level ξfrom the ground and let uref be the reference motion (input)for computing the transfer function. Then the transfer func-tion is

h�ω; ξ� � u�ω; ξ�=uref�ω�; (12)

where the hat symbol indicates the Fourier transform. Thecorresponding impulse response function is

h�t; ξ� � FT�1fu�ω; ξ�=uref�ω�g�t�; (13)

where FT indicates Fourier transform. Equations (12) and(13) imply that, at the point of the input (where uref is mea-sured), h�ω� � 1 and h�t� � δ�t�, where δ�t� is the Diracdelta function. Hence, h�t; ξ� physically represents responseto an impulse [initial displacement δ�t�] applied at the refer-ence point.

Because δ�t� is zero at all t except at t � 0, then theresulting system function corresponds to zero horizontal dis-placement condition at the reference point, which may differfrom the physical boundary condition (Snieder and Şafak,2006). In this article, three cases of reference motions areconsidered: (1) uref � Δdriv (the input horizontal motion),(2) uref � Δ � u�0� (the actual foundation horizontal mo-tion, equal to the input motion modified due to dynamicsoil-structure interaction), and (3) uref � u�H� (the roof mo-tion), which give system functions for three different bound-ary conditions:

(1) When uref � Δdriv, both Δ ≠ 0 and φ ≠ 0, along withthe interaction between the structure and the soil (re-sulting in system softening and in radiation damping)occur through both of these two degrees of freedom(effectively fH < ∞ and fR < ∞, along with ςH > 0

and ςR > 0, where ςH and ςR are the radiation dampingratios for the rigid-body horizontal motion and rocking,respectively).

(2) Case uref � Δ � u�0� corresponds to boundary condi-tion Δ � 0 and φ ≠ 0. Because the foundation does notmove horizontally, the effective fH → ∞ but fR < ∞;ςH � 0 but ςR > 0. In this case, fRB � fR. Conse-quently, this is not a true fixed-base condition, contraryto a model without rocking (Snieder and Şafak, 2006).

(3) uref � u�H� corresponds to zero displacement condition,in addition to the natural zero-stress condition, at the topof the structure. Because this is not possible to achieveexcept for the trivial solution, u�H�will be different fromzero only at t � 0, when the input is applied, and all re-flections from the roof will be effectively suppressed,which will simplify the impulse response functionsand their interpretations, as discussed in Snieder andŞafak (2006).

The theoretical resonance condition for the shear beamgives fundamental fixed-base frequency f1 � VS=�4H�. If τ

is the time for the input pulse to travel distanceH through theshear beam, then 1=�4τ� gives exactly f1, which is the un-coupled (from the motion of the soil) fundamental modal fre-quency of the structure. Hence, f1 can be determined bymeasuring the pulse travel time between ξ � 0 and ξ � Hin the impulse response function. This travel time physicallydepends only on the properties of the structure, while thepredominant frequencies in the system function may dependon the soil.

Finally, for band-limited data (jωj ≤ ωmax < ∞), the in-put impulse will have finite width, which will increase withdecreasing ωmax. Hence, the bandwidth of the data will limitthe resolution of reading the pulse arrival time. The theo-retical limit for ωmax is the Nyquist frequency of the data;however, the actual limit depends on the bandwidth of theexcitation and the number of modes that are excited and con-tribute to the response.

All of the effects discussed in this section will be dem-onstrated in the next section on numerical examples. Thesewere computed using FORTRAN computer codes written bythe author.

Results and Analysis

The model parameters are chosen to approximately cor-respond with the NS response of the Millikan Library andare the same as in Todorovska and Al Rjoub (2006b) fordry poroelastic soil, except that the model in this article con-siders structural damping, while the one in Todorovska andAl Rjoub (2006b) does not. The choice of the building andfoundation dimensions and mass is as follows: the buildingweight is 1:05 × 108 N, the foundation weight is 0:14×108 N, the building height is 44 m, the foundation depth is4 m, and the building in plan dimensions are 21 m × 23 m,guided by Luco et al. (1986). The radius of the semicircu-lar foundation of the model was taken to be a � 12 m. Forthe half-space, the Poisson ratio was assumed to be 0.3 andthe shear-wave velocity of dry soil (i.e., elastic soil) to be300 m=sec, which roughly corresponds to the geology be-neath the building site. Starting with these parameters,Todorovska and Al Rjoub (2006b) assumed fixed-base fre-quency f1 � 2:5 Hz, chosen by trial and error so that themodel gives system frequency that is close to the observedone during small amplitude earthquake shaking.

Figure 3 shows the foundation complex stiffness matrix.The plots on the left show the real parts and those on the rightshow the imaginary parts of the terms of the complex stiff-ness matrix, where K�s�

11 , K�s�22 , and K

�s�33 represent the vertical,

horizontal, and rocking stiffness, respectively; K�s�23 � K�s�

32

represents the coupling term (all normalized by 2μ). Forthe semicircular foundation and relaxed zero-stress conditionfor the scattered waves (simplifying assumption in this solu-tion), K�s�

11 � K�s�22 .

616 M. I. Todorovska

Transfer Functions

Figures 4 and 5 show transfer-function amplitudesfor rigid and flexible building, respectively, for case 1(uref � Δdriv) and case 2 (uref � Δ) reference motions. Asdiscussed in the previous section, they correspond to differ-ent boundary conditions and will give different rigid-bodyand system frequencies.

The top plot in Figure 4 shows transfer functions for thefoundation motion for rigid building for uref � Δdriv. Thedifferent lines correspond to the resultant horizontal dis-placement Δ, the feedback displacement Δfb � Δ �Δdriv,and rotation φ as φH representing the horizontal roof dis-placement due to foundation rocking. The plot in the middleshows the corresponding transfer function of the absoluteroof displacement, u�H� � Δ� φH. These plots show thatthe transfer functions for Δfb, φH � φfbH, and u�H� allhave peaks at approximately the same frequency, which isthe rigid-body frequency fRB ≈ 1:62 Hz (see equation 11).The plot in the bottom shows transfer functions for the roofresponse u�H� and for the foundation feedback motions Δfb

and φH � φfbH for reference motion uref � Δ. It can beseen that they all have peaks near the same frequency, whichis the rigid-body rocking frequency fR ≈ 2:055 Hz. ThenfRB ≈ 1:62 Hz and fR ≈ 2:055 Hz give fH ≈ 2:63 Hz,based on equation (11).

Figure 5 shows transfer-function amplitudes when thebuilding is flexible. As in Figure 4, the plot on the top showstransfer functions for the foundation motion (Δ, Δfb, andφH) for uref � Δdriv; the plot in the middle shows transferfunctions for the absolute roof displacement u�H� and for therelative roof displacement urel�H� � u�H� �Δ � φH. It canbe seen that the transfer functions for Δfb, φH � φfbH,u�H�, and urel�H� all have their first peaks near frequencyf � f1;sys ≈ 1:37 Hz. The plot in the bottom of Figure 5shows the transfer function of u�H� and urel�H� foruref � Δ, which have peaks at frequency f � f1;app ≈1:64 Hz, referred to in this article as apparent first systemfrequency. It satisfies

1

f21;app≈ 1

f2R� 1

f21: (14)

This plot also shows the transfer functions urelΔ �H�=Δ, whereurelΔ �H� is the relative roof displacement due to base transla-tion only (equation 7a), and urelφ �H�=�φH�, where urelφ �H� isthe relative roof displacement due to base rotation only(equation 7b). It can be seen that, while urelΔ �H�=Δ andurelφ �H�=�φH� have their first peaks at the fundamental fixed-base frequencyf1 � 2:5 Hz (see equations 7a,b), the peak ofurel�H�=Δ, where urel�H� � urelΔ �H� � urelφ �H� is the totalrelative roof displacement, is at a different frequency,f � f1;app ≈ 1:64 Hz.

The numerical values for the rigid-body and system fre-quencies read from these graphs, along with the fixed-basefrequency, specified as an input parameter, can be used toverify the relationships between these frequencies, given inequations (10) and (14). Substitution of these numericalvalues in these two equations shows that the right-hand sideof equation (10) predicts f1;sys within 0.7% error; the right-hand side of equation (14) predicts f1;app within 3% error,which verifies these relations.

Impulse Response Functions

All of the results for impulse response functions havebeen normalized to unit amplitude virtual source pulse.Pulses with motion in the positive x direction are referredto as positive. Clockwise foundation rotation, which pro-duces roof motion in the positive x direction is also referredto as positive. All of the impulse response functions havebeen computed from broadband transfer functions, 0–50 Hz,which gives narrow impulses and facilitates the interpretationof these theoretical impulse response functions.

Rigid Building. Figures 6, 7, and 8 show impulse responsesfor a rigid building, respectively, for reference motionsuref � Δdriv, uref � Δ, and uref � u�H�. Impulse responsefunctions are shown for the foundation responses, Δ and

Figure 3. Foundation complex stiffness matrix coefficients for model parameters that approximately correspond to the NS response of theMillikan Library. Left: real part. Right: imaginary part.

Seismic Interferometry of Soil-Structure Interaction Model with Coupled Horizontal and Rocking Response 617

φH, the roof response, u�H�, and also for the feedback mo-tions, Δfb. These plots provide insight in the nature of thesystem response, for example, the time delay between differ-ent responses; they also provide insight in the manifestationof the feedback motions in the time domain representation ofthe system function and in the radiation of energy.

Figure 6 shows impulse responses to input Δdriv. It canbe seen that it results in Δ that is also a positive pulse, butwider, with reduced amplitude and slightly delayed, by about0.02 sec relative toΔdriv. The foundation rocking response φis a counterclockwise pulse, also delayed by about 0.02 secand resulting in negative displacement φH of the roof. Theroof total response u�H�, which is a sum of these two and is anegative pulse because jφHj > jΔj is also delayed by about0.02 sec relative toΔdriv. The feedback displacementΔfb is anegative pulse at t � 0, immediately followed by a causalpositive pulse, which explains the time shift in Δ. These

pulses in Δ, φH, u�H�, and Δfb are followed by oscillatorydecaying motion with frequency fRB � 1:62 Hz, which isthe predominant frequency in the corresponding transferfunctions (see Fig. 4, middle).

Figure 7 shows impulse responses to inputΔ, which arevery similar to those in Figure 6, except that (i) Δ, φH, andu�H� are pulses at t � 0, while Δfb is shifted by�0:02 sec(starting as acausal) and (ii) the oscillatory part has a higherfrequency and is less damped, as it can be expected from theanalysis of the corresponding transfer functions (Fig. 4, bot-tom). The frequency of this motion is fR � 2:05 Hz.

Figure 8 shows the system response to input u�H�. Thismeans that the roof motion is fixed, except at t � 0. It can beseen that the sign of the pulses in Δ, φH, andΔfb at t � 0 isreversed compared to Figuress 6 and 7, as it can be expected,because u�H� has a negative pulse in these figures. The timedelays are as in Figure 7, that is, u�H�, Δ, and φH are allpulses at t � 0, while Δfb has a pulse at �0:02 sec. What isalso different is that there is no oscillatory motion, whichsuggests that this system does not oscillate as a pendulumfixed at the top, although the foundation can horizontallymove and rock.

Figure 4. System transfer functions for rigid structure and flex-ible soil, with respect to foundation driving motion Δdriv (top andmiddle) and foundation resultant horizontal motion Δ (bottom).

Figure 5. System transfer functions for flexible structure andflexible soil, with respect to foundation driving motion Δdriv (topand middle) and foundation resultant horizontal motionΔ (bottom).

618 M. I. Todorovska

Flexible Building. Figures 9, 10, and 11 show impulse re-sponse functions for a flexible building for reference motionsuref � Δdriv, uref � Δ, and uref � u�H�. All of these func-tions show prominent pulses related to propagation of theinput pulse through the structure and reflections from itsboundaries (ξ � 0 and ξ � H), along with other ripples,which are due to feedback motions of the foundation, asshown in the following.

Figure 9 shows impulse responses to input Δdriv. Fromtop to bottom, impulse responses are shown for u�H�, for φand Δ, which are the effective input motions for the struc-ture, and for Δfb, which is the modification of Δdriv due tothe feedback forces. On the bottom is the input impulse Δapplied by forces from the soil. The motion of the roof is

u�H� � Δ� φH � urel�H�, where Δ� φH is displace-ment due to rigid-body motion of the structure; urel�H� �f�Δ;φ� is displacement due to its elastic deformation.Consequently,Δ and φ have a twofold effect on u�H�. Theireffect through the rigid-body motion mechanism would beinstantaneous, while their effect through deformation of thestructure would be delayed by time τ , equal to the wavetravel time between ξ � 0 and ξ � H. The model input pa-rameters give τ � 0:1 sec.

The plots in Figure 9 show the same features in Δ, φH,and Δfb near t � 0 as seen in the rigid building response(Fig. 6), that is, φH has a counterclockwise pulse and Δis also modified relative to Δdriv due to feedback motionsof the foundation in response to the input motion appliedby forces from the soil. As these pulses in Δ and φH arethe effective initial motions for the structure, the entire im-pulse response for u�H� will be affected by the coupling ofthe soil.

The first arrival of the input pulse in u�H� can be seen att � 0:1 sec, which is equal to the theoretical wave traveltime τ � 0:1 sec over distance H. Its amplitude, however,is smaller than expected based on attenuation in the structureonly because of the pulse in φ at t≈ 0, which produces theopposite effect than the initial pulse in Δ. The next arrival,following reflection from the base and sign reversal, is at t �0:3 sec (� τ � 2τ ), the third one, following another reflec-tion from the base and sign reversal, is at t � 0:5 sec(� τ � 2τ � 2τ ), and so on. These pulses are marked byan open circle and a numeral 1, 2, 3, etc.; they all occurat the theoretical values of their arrival times.

Figure 6. System impulse response functions for rigid structureand flexible soil for input impulse Δdriv.

Figure 7. System impulse response functions for rigid structureand flexible soil for input impulse Δ.

Figure 8. System impulse response functions for rigid structureand flexible soil for input impulse u�H�.

Seismic Interferometry of Soil-Structure Interaction Model with Coupled Horizontal and Rocking Response 619

The impulse response function for u�H� has othersmaller pulses, showing as ripples between the main arrivals1, 2, 3, etc. These are due to the soil-structure interaction andcan be explained by the pulses inΔ and φH at t > 0. In φH,pairs of pulses can be seen, the first two positives marked by10 and 100, then two negatives marked by 20 and 200, and thenagain two positives marked by 30 and 300, etc. They occurslightly delayed, by about 0.02 sec, relative to the time ofthe reflections of the primary pulses from the top and fromthe base; they have the sign of the incident pulse. For exam-ple, pulses 10, 20, 30, etc. (also marked by closed circles) occurat times t � τ � 0:02, 3τ � 0:02, and 5τ � 0:02 sec, thatis, slightly after the reflections from the top (ξ � H); theyare clockwise rotation if the pulse at the top is positive. Sim-ilarly, pulses 100, 200, 300, etc. (also marked by triangles) occurat times t � 2τ � 0:02, 4τ � 0:02, 6τ � 0:02 sec, that is,slightly after the reflections from the bottom (ξ � 0); theyare clockwise rotation if the incident pulse onto the base ispositive. The second set of pulses can also be clearly seen inΔ andΔfb. These two sets of pulse are obviously due to feed-

back displacements of the foundation caused by forces fromthe structure. These pulses instantaneously affect u�H� viathe mechanism of rigid-body motion; however, they also actas sources of waves, which arrive at the top with delay τ.Some of the ripples in u�H�, marked by the double symboland the same numerals, 10, 20, 30, etc. and 100, 200, 300, etc. canbe explained as arrivals due to the corresponding pulses in φand Δ. Each of these (secondary) arrivals at the top willfurther generate feedback motions of the foundation, whichwill act as sources of waves, etc.

The oscillatory motion that develops in all of the im-pulse response functions can be viewed as the result of theinterference of all the generations of waves traveling throughthe structure and the related rigid-body motions. It has pre-dominant frequency f � f1;sys � 1:37, which is the soil-structure system frequency (see Fig. 5, middle, and therelated discussion), and quickly decays because of structuraland radiation damping (through foundation rocking andtranslation relative to the driving motion).

Figure 9. System impulse response functions for flexible structure and flexible soil for input impulse Δdriv.

620 M. I. Todorovska

Figure 10 shows the system impulse response to inputΔ, which is very similar to the one in Figure 9. It differs fromthe response in Figure 9 in that here: (i) Δ is zero except att � 0, (ii) the feedback pulses in φ near t � 0 are slightlyshifted by about 0.02 sec, as for the case of rigid structure,(iii) the oscillation motion has higher frequency, f �f1;app ≈ 1:64 Hz (see Fig. 5, bottom), and has larger ampli-tude due to the fact that there is no radiation of energy viafoundation translation.

Figure 11 shows impulse response functions for inputimpulse on the top. As for the case of a rigid structure, shownin Figure 8, these impulse responses correspond to the fixedboundary condition at the top, in addition to the physicalzero-stress condition. Because this is not possible, there areno reflections of pulses from the top, which simplifies theimpulse response functions. The impulse response of Δshows two pulses: one acausal at t � �0:1 sec, which corre-sponds to the physical source, and one causal at t � 0:1 sec,radiated by the virtual source at the top. The impulse re-sponse of φ shows the feedback rotation due to the physicalinput motion (from the soil) at t � �0:1 sec, then the feed-back rotation generated by the pulse at the top from the vir-

tual source at t≈ 0:02 sec, and another pulse at t � 0:1 sec,which is feedback rotation generated as the incident pulsefrom the virtual source at the top hits the foundation.

Finally, Figure 12 shows impulse response functions foru�ξ� at eight levels along the shear beam (ξ � H=8, H=4,H=2, 5H=8, 6H=8, and H), as well as of Δ � u�0� and φHat the bottom, all for input impulse Δ and for t < 1:4 sec.The pulse propagation, its reflections from ξ � 0 and ξ � H,and sign reversal at ξ � 0 can be clearly seen. It can also beseen that the pulse propagation and arrival times at differentlevels are not affected by the soil-structure interaction.

Discussion

In real life problems, the free-field motion (Δdriv) is of-ten not known, and transfer-functions are commonly com-puted with respect to the motion recorded at ground levelor in the basement (Δ in this problem). The discussion inthis section concerns impulse responses for input impulseΔ. As previously discussed, such transfer functions corre-spond to a fixed-base boundary condition for the foundationhorizontal motion but not for the rocking; the interaction be-tween the structure and soil is manifested by feedback rota-

Figure 10. System impulse response functions for flexible structure and flexible soil for input impulse Δ.

Seismic Interferometry of Soil-Structure Interaction Model with Coupled Horizontal and Rocking Response 621

tions that act as sources of waves and affect the impulse re-sponse of the structure.

For this theoretical model, the feedback rotations are de-layed by about 0.02 sec relative to the causative motions inthe structure (reflections from the bottom and top bound-aries); their manifestation in u�H� is seen with additional de-lay τ�� wave travel time from bottom to top). Consequently,the first arrival of the input pulse in u�H� (at t � τ ) is sepa-rated from the first arrival of pulses due to interaction be-tween foundation and structure (which have the same signas the first arrival of the input pulse) by time intervalτ � 0:02 sec. To resolve these motions and to accuratelyread τ , the pulses need to be narrow enough (have a widthless than τ � 0:02), which requires sufficiently broadbandinput motion that would excite many modes of vibration.

The results for this theoretical model also showed thatthe input impulse results in feedback rotation that is counter-clockwise pulse concurrent withΔ, that is, at t � 0 (Fig. 10).This pulse also acts as a source of waves arriving at the top ofthe structure at time t � τ , simultaneously with the pulsefrom the input motion, Δ, but with negative amplitude. Con-sequently, this feedback motion does not distort the shape ofthe first arrival in u�H�, the reading of τ ; however, it doesaffect its amplitude, which reflects dissipation in the structureas well as radiation of energy in the soil (via foundation rock-ing). This suggests that the time of the first arrival in u�H�from input motion Δ [hence f1 � 1=�4τ�] is not affected bythe soil-structure interaction. However, its amplitude is af-fected, and analysis of the reduction of amplitude will notgive the structural damping alone, but the damping of thecoupled system.

Figure 13 illustrates the effect of the bandwidth of thedata on the computed impulse response u�H� for input im-pulse Δ and on the readings of τ . This bandwidth would de-

pend on the nature of the earthquake excitation. For example,large distant earthquakes may not excite higher modes thanthe fundamental mode, while small local earthquakes maynot excite the fundamental mode, or the frequency of thismode may be below the cutoff frequency of the data proces-sing due to low signal-to-noise ratio (Trifunac, 1971). In thetop of Figure 13, the model transfer function is shown foru�H� with respect to Δ for frequencies 0–50 Hz. Below, im-pulse response functions are shown that are obtained fromwindowed transfer functions containing different modes ofvibration. The following windows are considered: 0–50 Hz(10 modes), 0–25 Hz (five modes), 0–10 Hz (two modes),0–3 Hz (first mode only), and 3–6 Hz (second mode only).The readings of τ from the time of the peak of the first pulseare also shown. It can be seen that the width of the pulseof the first arrival increases with decreasing maximum fre-quency, but the reading of τ is accurate even from data con-taining only the first two modes. The window 0–3 Hz,however, gives τ � 0:16 sec and 1=�4τ� � 1:56 Hz, whichis closer to the system frequency. The window 6–9 Hz gives agood reading of τ .

The theoretical impulse responses presented in this ar-ticle can be used to examine possible causes for artifactsreported by Kohler et al. (2007) in the form of delta-function-like pulses at t � 0, interpreted to be due to low

Figure 11. System impulse response functions for flexiblestructure and flexible soil for input impulse u�H�.

Figure 12. Impulse response functions for u�ξ� at differentlevels in the structure. The impulse responses of the foundationtranslation Δ (the input impulse) and rocking φH are also shownat the bottom.

622 M. I. Todorovska

frequency noise. They also report other low frequency pro-cessing artifacts for frequencies 0.5–2 Hz, which becomemore pronounced near the top of the building. They elimi-nate these artifacts by computing impulse response functionsfrom band-pass-filtered data between 2 and 10 Hz, whichcuts off the first two modes. They attribute these artifactsto domination of a single mode in the response when allof the locations in the building are in phase for that mode,which is seen in the deconvolved response as simultaneousarrival throughout the building. The model response in Fig-ure 13 suggests that domination of a single mode does notlead to a delta-function-like pulse at t � 0 but to a mono-chromatic oscillatory motion that begins at t � 0 and to ab-sence of clear pulses from the traveling waves through thestructure. A delta-function-like pulse at t � 0 is seen inthe impulse response for the foundation rocking φ (Fig. 10),which was interpreted as foundation feedback motion in re-sponse to the input pulse. This rocking is counterclockwiseand produces negative displacement in the structure, withamplitude jφξj that increases with increasing height. Thecontribution of these displacements to u�ξ� may be the cause

of the reported effect in Kohler et al. (2007) rather than themodal responses.

Kawakami and Oyunchimeg (2004) computed 1=�4τ�for a group of instrumented buildings and show 1=�4τ�plotted versus published modal frequencies for the samebuildings by other authors. They do not discuss the trends,except for stating that the discrepancy is smaller than ex-pected (from nonuniform distribution of stiffness), whichmay be explained by taking soil-structure interaction intoconsideration. An inspection of their plot shows that the pub-lished modal values are systematically smaller than 1=�4τ�,while the common distribution of stiffness in buildings (de-creasing stiffness with height) would imply larger modal fre-quency, according to the analysis of a model of a ten-storybuilding in their paper. This is not a contradiction, becausethe published values for instrumented buildings are those forthe system frequency, while 1=�4τ� is a measure of the fixed-base frequency. Their plot suggests that, for the cases theyconsidered, the effect of the soil-structure interaction on themodal (system) frequency prevailed over the effect of non-uniform distribution of stiffness.

Conclusions

Fourier and impulse response analysis of a soil-structureinteraction model with coupled horizontal and rocking re-sponse was presented, which provided insight into the sys-tem response that is useful for interpretation of records inbuildings. Of significance for practical applications is if andwith what accuracy the structural fixed-base frequency anddamping can be estimated from minimum instrumentation(two horizontal, transducers, one at ground level, and oneat the roof). The analysis showed the following:

(1) The transfer functions and related impulse responsefunctions, with respect to base translation Δ, are thoseof the system and depend simultaneously on the struc-tural properties and on the foundation rocking stiffnessand damping.

(2) The energy of the system response is concentrated nearthe system frequencies, not the fixed-base ones, whichis common knowledge in soil-structure interaction re-search. It was shown that these system frequenciesand the system damping are those seen in the oscillatorypart of the impulse response functions, not the fixed-baseones (which differs from the interpretation in Sniederand Sakak [2006] and Kohler et al. [2007]).

(3) For structures primarily deforming in shear, and with ap-proximately uniform distribution of mass and stiffnessalong the height, the fundamental fixed-base frequencyof the structure can be estimated as f1 � 1=�4τ�, whereτ is the wave travel time from ground level to the roof,measured from impulse response functions for horizon-tal motions (as assumed in Todorovska and Trifunac[2008a,b] and Trifunac et al. [2008]). This would requiredata from only two horizontal sensors: one at the base

Figure 13. Impulse response functions of u�H� for inputimpulse Δ computed from a different bandwidth of the transferfunction.

Seismic Interferometry of Soil-Structure Interaction Model with Coupled Horizontal and Rocking Response 623

and one at the roof. This is an important result for inter-pretation of earthquake records in buildings, as it extendsthe useful information that can be obtained from oldrecords in buildings that are not densely instrumented.

(4) Once the first apparent system frequency has been deter-mined from Fourier analysis and the f1 has been deter-mined from wave travel times, the rigid-body rockingfrequency can be estimated using equation (14).

(5) The measurement of τ from the first arrival in the im-pulse response function of the roof motion is not affectedby feedback displacements due to the interaction forces,provided the recorded data are sufficiently broadbandand result in narrower impulses. The amplitudes of thispulse are affected by the coupling with the soil and re-flect foundation rocking and loss of energy due to bothdissipation in the structure and radiation.

(6) For this simple two-dimensional model of the NSresponse of the Millikan Library, f1 � 2:5 Hz andshear-wave velocity in the soil of 300 m=sec resultedin fRB � 1:62 Hz, fR � 2:055 Hz, fH � 2:63 Hz,f1;sys � 1:37 Hz, and f1;app � 1:64 Hz. The given andobserved frequencies in the transfer functions closelysatisfied the simple relations in equations (10), (11),and (14). The chosen model value of f1 � 2:5 Hz in thisarticle implies wave travel time τ � 0:1 sec (groundlevel to roof) and wave velocity VS � 440 m=sec. Thisvalue of τ agrees well with the observed wave travel time(ground level to roof) in Snieder and Şafak (2006). Themodel value of VS disagrees with the identified value of330 m=sec in Snieder and Şafak (2006), because theycomputed it from what turns out to be the apparent sys-tem frequency (f1;app ≈ 1:64 Hz for the model in thisarticle), which also depends on the foundation rockingstiffness. The model value of f1 is also close to the valuef1 � 2:33 Hz identified by Wong et al. (1988) for thesame building from analysis of forced vibration test data.

The system identification procedure applied to the soil-structure interaction model in this article constitutes amethod that can be applied to vibrational records in build-ings. In Todorovska (2009), it is applied to four earthquakerecords in the Millikan Library between 1970 and 2002.Also, the model system transfer functions and impulse re-sponse functions are compared with those obtained fromrecorded earthquake response.

Data and Resources

No data other than those from published works cited inthe list of references are used.

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Department of Civil and Environmental EngineeringUniversity of Southern CaliforniaLos Angeles, California 90089-2531mtodorov@usc.edu

Manuscript received 4 January 2008

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