Nonlinear Response of Soil–Structure Systems using ...

Journal of Earthquake Engineering

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Nonlinear Response of Soil–Structure Systems using Dynamic Centrifuge Experiments Johanes Chandra & Philippe Guéguen To cite this article: Johanes Chandra & Philippe Guéguen (2017): Nonlinear Response of Soil–Structure Systems using Dynamic Centrifuge Experiments, Journal of Earthquake Engineering To link to this article: https://doi.org/10.1080/13632469.2017.1398692

Published online: 27 Nov 2017.

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Date: 27 November 2017, At: 08:54

JOURNAL OF EARTHQUAKE ENGINEERING https://doi.org/10.1080/13632469.2017.1398692

Nonlinear Response of Soil–Structure Systems using Dynamic Centrifuge Experiments Johanes Chandra and Philippe Guéguen ISTerre, Université Grenoble Alpes, CNRS/IFSTTAR, Grenoble, France ARTICLE HISTORY

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ABSTRACT

This study presents centrifuge tests analysis of nonlinear soil–structure interactions and the reciprocal influence of the soil and the structure. Weak and strong motions were considered and applied to several centrifuge setups, implemented “with” or “without” buildings. For each setup, the response of the site was analyzed using arrays of accelerometers located in the soil and buildings, using time and frequency domain methods. We conclude that the presence of buildings may strongly modify the nonlinear soil response in the free field and the simplified relationships linking the contribution of each motion to the overall soil–structure system response were discussed.

Received 25 February 2017 Revised 10 September 2017 Accepted 30 September 2017 KEYWORDS

Centrifuge Modeling; Soil–Structure Interaction; Nonlinear Response; Earthquakes

1. Introduction During earthquakes, the presence of surface heterogeneities in urban areas, such as structures and their foundations, produces interactions between the soil and the structures [Kausel, 2010]. The coupling results from (a) the stiffness impedance between the foundations and the soil, which may cause scattering of the incident waves [kinematic soil–structure interaction (SSI)], and (b) the vibrating characteristics controlled by the dynamic properties of the structure (inertia soil–structure interaction). The SSI is expressed by the rocking and horizontal motion of the foundation [Luco et al., 1987] [Bard, 1988] [Guéguen and Bard, 2005] and the dynamic response of the soil–structure system differs from that of the structure, moving from a fixed-base to a flexible-base response in the presence of soft soil. Furthermore, kinematic and inertia SSI results in contamination of the seismic ground motion in dense urban areas, confirmed by experimental observations (e.g., [Guéguen et al., 2000] [Kim et al., 2001] [Cornou et al., 2004] [Guéguen and Bard, 2005]) and numerical modeling (e.g., [Kham et al., 2006] [Isbiliroglu et al., 2015]), with consequences on the assessment of seismic site effects [Guéguen et al., 2002], on the stand-alone assumption considered for building response analysis [Kitada et al., 1999] [Chazelas et al., 2003], and on the relevance of the term “freefield” (FF) for seismic ground motion observed or predicted in urban areas. Experimental soil–structure analysis cannot easily capture the shear-induced deformation localized below shallow foundations. However, a soil-induced nonlinear response can be expected when buildings are strongly forced into vibration and cyclic stress increases in the uppermost layers below the foundations. For example, Bray and Dashti [2014] recently used centrifuge tests to analyze the induced effect of building vibrations on liquefaction localized immediately underneath the foundations. Dynamic centrifuge tests have also proved to be CONTACT Philippe Guéguen © 2017 Taylor & Francis Group, LLC

[email protected]

ISTerre, BP 53, 38041 Grenoble cedex 9.

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reliable for the analysis of wave propagation into the soil [Semblat and Luong, 1998] and the soil–structure and structure–soil–structure coupling under seismic excitation [Ghosh and Madabhushi, 2007] [Mason et al., 2013] [Chazelas et al., 2003] [Trombetta et al., 2013; 2014]. A conventional frequency-based approach to determine SSI consists in separating each parameter of the SSI system contributing to total motion (e.g., [Paolucci, 1993] [Meli et al., 1998] [Guéguen and Bard, 2005]). From the total response observed, the flexible-base, fixedbase, or rocking motions of the system are identified. Based on this model, Luco et al. [1988] proposed analytic equations to relate the contributing factors of SSI to the total response, in terms of frequency and damping characteristics. These values are based on linear concepts and are rarely discussed using experimental data. An alternative time-based approach exists, related to the travel time of the deformation through the height of the building (e.g. [Kanai, 1965] [Safak, 1998, 1999] [Snieder and Şafak, 2006] [Kohler et al., 2007]). In this case, the response of the structure is represented as a superposition of waves propagating in the structure with the same transmission and reflection properties as those of the soil. Using this assumption, Snieder and Şafak [2006] considered the 1D wave propagation models in both the building and the soil as equivalent. Recently, Chandra et al. [2015] and Guéguen [2016] evaluated the nonlinear response of the soil using the wave propagation model with a vertical geotechnical array, and Petrovic and Parolai [2016] applied this approach to recordings of sensors installed in buildings and nearby boreholes to investigate wave propagation through the building soil layers, and the soil–structure interactions. Period lengthening and increased system damping are currently accounted in seismic design but a lack of studies and observations exists to account for nonlinear SSI in practice. The nonlinear responses of the soil and the building are generally analyzed independently and the nonlinear coupling, or reciprocal effect, of the building and soil motion is rarely discussed. In this work, several centrifuge setups were implemented using different types of stand-alone buildings with shallow foundations resting on homogenous soil. The aims of this study are, firstly, to assess how dynamic SSI may influence the nonlinear free field response of the soil; secondly, to compare SSI assessment with frequency and time domain methods; and finally, to investigate the effect that the nonlinear process may have on the soil, the structure and the SSI. After describing the centrifuge tests, the methods applied to the soil and structures are described and finally the results in terms of nonlinear SSI observation are discussed.

2. Centrifuge Tests and Data Centrifuge experiments provide an effective means of observing the coupling of buildings and soil during seismic excitation, testing the effects of the amplitude of seismic loading on the soil–structure system’s response as well as comparing the soil’s response with or without the presence of buildings. Authors used the centrifuge facility of the Institut Français des Sciences et Technologies des Transports, de l’Aménagement et des Réseaux (IFSTTAR, Nantes, France), at the 60g scale (g = gravity acceleration). The scaling from the model to the real (prototype) scale respects the factors given in Chandra et al. [2016], proportional to 60g and this manuscript refers, in priority, to the real scale. If not, model scale is specified in the rest of the manuscript. Chazelas [2010] and Chandra et al. [2016] described these experiments in detail. In this section, a brief description of the setups is summarized to facilitate understanding.

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Figure 1a presents the plan view and cross section of the container used for the centrifuge test and the positions of the accelerometric sensors spread along the soil column and on the surface. Equivalent shear-beam (ESB) containers, 800 × 350 × 416 mm in dimension (model scale), were used to limit container edge effects on wave propagation [Zeng and Schofield, 1996]. Horizontal and vertical piezoelectric accelerometers (PCB type 200 A1 and Bruel & Kjaer type 4317) were used, attached to a thin plate to control orientation and position. Soil–structure models were excited by a shaker driven in displacement at the bottom of the soil column. Two different input signals, representing weak and strong excitation, were considered (Fig. 2). The first signal was a synthetic accelerogram representing strong ground motion with a realistic phase, corresponding to a Mw 5.5 earthquake at 15 km [Chazelas, 2010] and having a peak ground acceleration (PGA) equal to 0.43g (dominant frequency close to 1.86 Hz). The second signal was a moderate seismic ground motion recorded during the Mw 7.3 earthquake in 2007 in Martinique by one of the French Accelerometric Network’s stations [Péquegnat et al., 2008]. PGA was 0.07g with a dominant frequency close to 2.7 Hz. In the rest of the manuscript, strong and weak response will refer to the response of the soil and structure models under strong and weak excitation, respectively. For each setup, we prioritized experiment control and reproducibility of the trials [Chandra et al., 2016] rather than multiplication of the input signal. Moreover, during experiments, slight settlements usually occur that may slightly change the position of the sensor or the elastic properties of the sand. Without measurements of the sensor position and density during the experiments, the experimental sequence was therefore performed with two different con-

Figure 1 Schematic view of the centrifuge tests (a) Cross section and plan view of the container and the localization of the sensors in the soil and the structure. Gray vertical bands correspond to the laminated ESB boundary. The gray square indicates the position of the building at the surface of the containers. (b) Schematic view of the building models tested with the position of the mass and of the shallow square foundation. Horizontal and vertical arrows indicate the position and the direction of the accelerometric sensors. The numbers in parenthesis refer to the sensor number mentioned in the text [modified from Chazelas, 2010]. Unit in model scale: mm.

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Figure 2 Accelerometric time histories and Fourier transform of the weak (a) and strong (b) input motion used for the centrifuge modeling.

tainers and between three and six trials were done for each [building/container/input signal] setup. Chandra et al. [2015] showed the efficiency and stability of centrifuge tests, in terms of limitation of wave reflections on the container edges, equivalent response of the soil column between containers and reproducibility of the soil motion generated by the shaker. The soil was constituted of uniform, homogeneous, fine Fontainebleau sand (reference N234, emin = 0.55; emax = 0.86; γd = 15.42 kN/m3), poured into the container by dry pluviation to attain a relative density, Dr, equal to 57%. Chandra et al. [2015] computed the shear wave velocity (Vs) along the FF soil profile by applying the seismic interferometry by deconvolution method to the accelerogram recorded along the soil profile. According to Chandra et al. [2015], Vs values between the top and the bottom of the container equal to 242 m/s and 197 m/s were recorded under weak and strong excitation, respectively, the same as values provided by Li et al. [2013] for the same container configuration. The response of the FF container was tested first, and then the building setups (Fig. 1b). These models were designed to represent a low rigid structure (LRS), a rigid structure (RS), a flexible structure (FS), and a 2-DOF flexible structure (2DOFS). A partially embedded steel mass represents the LRS, steel–aluminum–oak materials compose the RS, while FS and 2DOFS are made of composite steel–aluminum materials. The fixed-based fundamental frequency of the structures was designed to have a resonance frequency ranging from 1 to 3 Hz and the dimensions of the RS were defined so that its center of gravity was at the same height as those of the FS with respect to the soil surface. All structures had shallow foundations, made from two aluminum and pinewood (laminated) plates. The mass of the foundations at the model scale was 0.156 kg, the LRS was 0.980 kg, the RS was 0.976 kg, the FS was 1.072 kg, and the 2DOFS was 1.018 kg.

3. Signal Processing 3.1. Pre-conditioning Process Before analysis, all the signals were conditioned to limit the effect of signal processing on the results and their interpretation. After removing mean and trend, the signals were zero-

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Figure 3 Examples of a wave propagating from bottom to the top of the soil column (a) and from the bottom to the top of the building (b). z indicates the vertical position of the sensor from the soil surface, distances given in meter (prototype scale).

padded, as proposed by Boore [2005], before integration, differentiation or time to frequency domain transformation. The signals were tapered using a 5% Tukey window, and filtered using a third order of Butterworth filter between 0.3 and 13.8 Hz, where the maximum energy is concentrated [Chandra et al., 2016]. An example of a strong propagating signal from the bottom of the container to the top of the building model for the 2DOFS setup is shown in Fig. 3. We observe amplification of the signal level from the bottom to the top of the soil column and the building response. After loading, freeoscillation is clearly observed at the top of the building for the 2DOFS setup. In the Fourier domain (Figs. 5–9), the Fast Fourier Transform algorithm was used and the Konno and Ohmachi [1998] smoothing function (b = 40) was applied. For each setup, the mean spectrum (± standard deviation) was computed with data from all trials and the two containers used for the same soil–structure setups.

3.2. Transfer Function of the Soil Column and the Building In the time domain, the 1D soil response can be represented as a superposition of incoming waves and waves reflected from the FF surface and from internal impedance, with reflection and transmission conditions at each interface [Thomson, 1950] [Haskell, 1953] [Kennett, 1974]. In vibrating buildings, the conventional approach to system identification is to determine the resonance frequencies, damping and corresponding mode shapes that characterize the frequency response of the soil–structure system.

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Assuming an equivalent continuous beam response, an alternative approach consists in characterizing the system response in the time domain, based on wave travel time through the structure [Şafak, 1998; 1999] [Snieder and Safak, 2006]. Named seismic interferometry by deconvolution method by Snieder and Safak [2006], they considered the building as a 1D equivalent layer and showed that the impulse response of the system was obtained by deconvoluting the total response with the input seismic motion recorded at the base of the building. Therefore, the impulse response of the 1D soil column and the 1D structure can be computed by seismic deconvolution, and transformation in the frequency domain gives their transfer function. Mehta et al. [2007], Nakata and Snieder [2012], Chandra et al. [2015], and Guéguen [2016] applied the deconvolution method to data from earthquakes recorded along Japanese, Californian, or Greek vertical arrays (boreholes) and estimated Vs as the distance between sensors divided by the time-delay of the pulse travel time. Chandra et al. [2015] and Guéguen [2016] reported shift of Vs values with shaking levels correlated with the nonlinear response of the soil. This method was also applied successfully to the FF setup of the centrifuge experiment [Chandra et al., 2016] for nonlinear analysis of the soil. In this study, deconvolution was performed using the waterlevel regularization technique proposed by Clayton and Wiggins [1976] (see Appendix A). Once the transfer function is known, the resonance frequency corresponds to the peak value of this transfer function and damping is computed using the half power bandwidth method, i.e. related to the width at –3dB of the frequency peak amplitude of the transfer function. The waterlevel coefficient k is fixed at 10% to avoid instabilities in the deconvolution process.

3.3. System Identification of the Soil–Structure System One-dimensional soil–foundation–structure structural models (Fig. 4) are commonly employed in SSI, considering different input/output pairs [Luco et al., 1988]. Under seismic excitation, the total horizontal motion of the system xtot is expressed as: xtot ¼ xg þ x0 þ Hϕ þ x1

(1)

m0 €x0 þ ch x_ 0 þ kh x0 # c1 x_ 1 # k1 x1 ¼ #m0 €xg € þ cr ϕ_ þ kr ϕ # Hc1 x_ 1 # Hk1 x1 ¼ 0 J0 ϕ

(2)

with xg the ground displacement, x0 the relative horizontal displacement of the foundations, Hø (H: height of the building) the horizontal displacement at the top induced by the rocking of the foundations around the axis perpendicular to the horizontal direction, and x1 is the displacement of the fixed-base structure. In this equation, we assume zero vertical displacement at the top of the structure and rocking is uniform throughout the building height. A significant assumption for the system description in Eq. (1) is that the dynamic SSI response is calculated using linear dynamic models. Based on this simplified 1D model, the soil–structure model was identified by processing the input and output pairs of signals recorded in the soil–structure system [Stewart and Fenves, 1998]. Neglecting the coupling effect between horizontal translation and rocking motion and assuming no FF ground rotation, the equations of motion are [Guéguen and Bard, 2005]: ! " m1 €x1 þ €x0 þ H ;€y þ c1 x_ 1 þ k1 x1 ¼ #m1 €xg

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Figure 4 One-dimensional soil–structure interaction model used in this article.

Figure 5 Transfer function of the soil H(f) computed by deconvolution between signals recorded at the top (sensor 19) and the bottom (sensor 2) of the soil column considering weak (black) and strong (gray) input motion and different set-ups (FF: Free-Field; LRS: Low Rigid; RS: Rigid; FS: Flexible; 2DOFS: Two degree-of-freedom flexible). fs and Ds represent the system frequency of the soil and the damping value of the transfer function. Under weak motion, fs1 is the fundamental frequency and fs2 is the second harmonic frequency. Solid lines are the average transfer function considering all the trials and the two containers and dashed lines are mean +/- standard deviation.

with m1, m0, and J0 the mass of the building, of the foundation and the rotation moment of inertia perpendicular to the x-axis of the structure, c1 and k1 the damping and stiffness of the fixed-base structure and ch, kh, cr, kr the damping and stiffness coefficients of the foundation impedance for the translation and rocking motions. Stewart and Fenves [1998] proposed a solution of Eq. (2) by transforming the timedependent functions to the Laplace domain, considering several configurations. If the structure and the soil are flexible, the fundamental frequency of vibrations observed at

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Figure 6 Spectral response of the horizontal motion x0 for the rigid (RS), flexible (FS) and two DOF flexible (2DOFS) setups. Same legend as Fig. 5.

Figure 7 Spectral response of the system motion x1+xH+hø (b) for the rigid (RS), flexible (FS) and two DOF flexible (2DOFS) structures. Same legend as Fig. 5.

Figure 8 Spectral response of the rocking motion hø (a) and the pseudo-flexible motion x1+hø (b) for the rigid (RS), flexible (FS), and two DOF flexible (2DOFS) structures. Same legend as Fig. 5.

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Figure 9. (a) Spectral response of the fixed-base motion x1 for the rigid (RS), flexible (FS) and two DOF flexible (2DOFS) structures (Same legend as Fig. 5). The vertical dashed lines show the fixed-base frequency values given by Chazelas [2010]. (b) Comparison of the fixed-based frequency computed using the time delay relationship (Eq. (10)) and the data by deconvolution according to Paolucci [1993] (Eq. (11)). Continuous line is 1:1 slope.

the top of the building corresponds to the system response fsys. This frequency is different from the fixed-base frequency f1 which corresponds to a flexible structure on rigid soil. On rigid soil, the system and the fixed-base structure vibrate at the same frequency, i.e. fsys = f1, the anchoring condition at the soil-foundation boundary is perfectly clamped and the ground and the foundations move in the same way. Luco et al. [1988] proposed approximate equations relating the components of the soil– structure system: 1 fsys

2

¼

1 1 1 þ þ f12 fϕ2 f02

(3)

with f0 and fϕ the frequencies related to the horizontal translation and rocking motion of the foundation relative to the soil. Equation (3) indicates that system parameters are dependent on the fixed-based parameters and the foundation impedance in translation and rocking. This equation implies that if the building is very stiff (f1 = ∞), the system frequency fsys is equivalent to the rigid-body frequency. If the soil is very stiff, the system frequency fsys is therefore equal to the fixed-base frequency f1. An exceptional situation also exists for a condition of partial base flexibility, representing base rocking only, i.e. assuming very stiff foundations in the horizontal direction. This pseudo-flexible body frequency is given: 1 fapp

2

¼

1 1 þ 2 2 f1 fϕ

(4)

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Table 1. Input and output signals for system identification procedures [Stewart and Fenves, 1998].

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System Flexible base Pseudo-fixed base Fixed base

Input xg x g + x0 xg + x0 +Hø

Output xg + x0 + Hø + x1 xg + x0 + Hø + x1 xg + x0 + Hø + x1

herein referred to as apparent frequency fapp. This condition is important because actual flexible-base parameters are often well approximated by pseudo flexible-base parameters. According to this model, and for the case of a rigid building on soft soil, f1 is always larger than fsys. This implies that the ratio of structure-to-soil stiffness is the key factor controlling lengthening of the frequency [Bielak, 1975] [Aviles and Perez-Rocha, 1996]. For a given stiffness ratio, the period lengthening increases for the tallest structures (i.e., apex ratio between height and width greater than 1) with more overturning moment and more rocking motion simultaneously. Stewart and Fenves [1998] summarized (Table 1) the input/output pairs that actually represent the information required for system identification: we need recordings of FF, foundation, and top of building motions to enable direct derivation of the fixed- and flexible-base parameters, as schematized in Guéguen and Bard [2005]. Recently, Snieder and Şafak [2006] and Todorovska [2009] applied a deconvolution method between the data recorded at the top and at the bottom of the building. Moreover, the response by deconvolution is sensitive enough to reflect modifications in the elastic properties of the soil or the structure ([Chandra et al., 2015] [Nakata and Snieder, 2012] [Guéguen, 2016]) and under strong motion, the nonlinear process may change the stiffness ratio between soil and structure, thus modifying the relationship of the frequency model in Eqs. (3) and (4). In the rest of the article, we consider frequency and damping variations as proxies of the nonlinear effects. Transfer function mentioned in the next section refers to the function obtained by deconvolution (e.g., [Snieder and Safak 2006] [Chandra et al., 2015, 2016]). We denote u1, the input motion applied to the system (the bottom of the soil column or the building), and u2, the output motion of the system recorded at the top of the soil column or the building top. The impulse response of the system is given by X ðωÞ ¼

u1 u&2 jX2 j2 þ ε

(5)

where * denotes complex conjugate, X are displacements, and ε is a regularization coefficient used for the stabilization of the deconvolution process (e.g., [Snieder and Şafak, 2006] [Chandra et al., 2016]) and equal to 10% as proposed by Chandra et al. [2016] for this dataset. The description of the method is given in Appendix A. We computed the building motion components using the sensors described in Fig. 1 (the number of the sensor is noted in brackets): ●

system frequency, fsys, is the peak frequency of the FFT of the motion recorded at the top, reduced by the FF motion xg, assuming that the FF is not contaminated by building vibrations. Herein, system frequency will be obtained by deconvolution between the top [30] and the FF [14] signals, i.e. (capital letters mean Fourier transform):

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Xsys ðωÞ ¼

●

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(6)

apparent frequency, fapp, is the peak frequency of the transfer function obtained by deconvolution between the signal recorded at the top [30] and at the bottom [29] of the building, i.e. Xapp ðωÞ ¼

●

11

& X30 X29 jX29 j2 þ ε

(7)

rocking frequency, fø, is computed as the peak frequency of the function proposed by Paolucci [1993] and Guéguen and Bard [2005] obtained by dividing the vertical motion of the two opposite extremities of the foundations: % # $& ðV27 ðt Þ # V28 ðt ÞÞH Hϕ ¼ FFT B

(8)

where V27(t) and V28(t) are the time histories of the vertical foundation displacement, i.e. sensors [27] and [28], B is the distance between V27 and V28 and H the building height. ●

frequency of the horizontal translation, f0, is the peak frequency of the deconvolution between the horizontal signal recorded on the foundation [29] and on FF [14], i.e.

X0 ðωÞ ¼

& X29 X14 jX14 j2 þ ε

(9)

This motion reflects the kinematic interaction due to the scattering of waves at the footing and we speculate that this is the same for all the building models because of similar footings. ●

fixed-base frequency, f1, is computed with different models. First, if the building responds primarily in shear, the time delay τ of the pulse travel time along the building [Snieder and Şafak, 2006] is computed thus:

1 (10) 4τ In our case f1 is obtained by picking the time delay of the pulse between the bottom [29] and the top [30] obtained by inverse Fourier Transform of the deconvolved function (Eq. (A4), Appendix A). Second, the fixed-based frequency can also be computed as the peak of the function obtained after removing the rocking motion from the rigid-body motion, as proposed by Paolucci [1993]. This function Ψ(f) is then adapted from Paolucci [1993] considering the total motion at the top xtot [30] and the rigid-body motion of the foundations xtotRB =xg+x0+hø [29 + 27&28] (see Eq. (8)), as follows: f1 ¼

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ψð f Þ ¼

& X30 X29þH;

(11)

jX29þH; j2 þ ε

4. Results and Discussion

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4.1. Effect of the Presence of Buildings on Site Response Figure 5 shows the transfer function H(f) of the soil, considering weak (0.07g) and strong (0.43g) excitation, and the FF setup is compared with the setups with buildings. H(f) is computed as the deconvolution between the signal recorded at the top of the soil column [19] and the input signal at the bottom of the containers [2], and the resonance frequency of the soil, fs, is considered as the peak frequency of H(f). Damping, Ds, was estimated by the half-power bandwidth method applied to H(f). All the results show a frequency shift toward a lower value between weak and strong motion and damping increases. This characteristic reflects the classic nonlinear response of the soil observed in experimental in situ data [Bonilla et al., 2005] [Wu et al., 2009]. Under weak motion, H(f) shows a fundamental system frequency fs1 at 2.68 Hz, close to the reference value of 2.7 Hz given by Chazelas [2010], and a second mode fs2 at 7.50 Hz, which is a relatively good fit for the theoretical shear model (fk/f1 = 3, 5, 7. . ., k = 2:n modes). Under strong motion, the second mode is also observed; however, it is less acute, reflecting the increase in damping and the very minor contribution of this mode to the total response. Due to the proximity between the fundamental frequency of the soil response and the building models, only the fundamental mode will be considered hereafter (i.e., fs and Ds). At the fundamental frequency fs, H(f) amplitude is comparable for all the models. The soil was poured into the containers using the same procedure (dry pluviation) and only slight variations in soil properties may exist in the soil profile for each setup, also reported by Chandra et al. [2015]. We assume that the soil profiles are the same when comparing configurations “with” or “without” buildings. All the setups are summarized in Tables 2 and 3 for the variation between weak and strong motion (Δ), and the variation between “with” and “without” building models (Δ*), respectively. Dynamic soil–structure coupling modifies the site response. We first defined a global criterion CD based on the correlation function between H(f) without and with buildings [Fenves and Desroches, 1995] [Kim and Stewart, 2003]: Table 2. Summary of the frequency and damping of the soil models and percentage of variation (Δ = (strong-weak)/weak) between weak and strong input signals. Building setups FF weak strong Δ LRS weak strong Δ RS weak strong Δ

fs (Hz) 2.68 2.19 −22% 2.56 2.19 −14% 2.56 2.45 −4%

Ds (%) 10.01 27.08 17.0% 8.87 21.46 12.6% 8.45 19.96 11.5%

Building setups FS weak strong Δ 2DOFS weak strong Δ

fs (Hz) 2.59 2.33 −10% 2.53 2.24 −11%

Ds (%) 9.76 20.48 10.7% 8.88 19.86 11.0%

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Table 3. Summary of the variation Δ* of response of the soil considering the frequency, fs, and damping, Ds, in presence of building models. Δ* = (Building-FF)/FF, for weak and strong input motion. Δ* f Δ* D

LRS −4% 0% −1% −6%

weak strong weak strong

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CDi

leftj ¼ PN

PN

RS −4% 12% −2% −7%

k¼1 H ðkÞFF H ðkÞi j2 2 PN 2 k¼1 H ðkÞFF k¼1 H ðkÞi

FS −3% 6% 0% −7%

2DOF −6% 2% −1% −7%

(12)

with k the frequency sample, index i corresponding to the building setup (LRS, RS, FS, and 2DOFS) and N the total number of frequency samples. The FF model is defined as a reference. A CDi close to 1.0 indicates that building presence does not contaminate the site response; a CD significantly lower than 1.0 implies that there is a coupling effect between the soil and the structure which modifies the site response. With this coefficient, we evaluate the effect of the building over the whole frequency domain of interest (0.5– 13.8 Hz) and not only at the resonance frequency. Under weak motion, CDi values are 0.969, 0.960, 0.940, and 0.930 (mean value 0.95), while under strong motion, CDi values are 0.981, 0.978, 0.988, and 0.986 (mean value 0.98) for FS, LRS, RS, and 2DOFS setups, respectively, indicating a stronger impact under weak motion. Maximum softening is observed for the FF model (Table 2), with the resonance frequency moving from 2.68 to 2.19 Hz (Δ = −22%) and damping from 10% to 27% between weak and strong motion. The softening of fs is lessened in the presence of buildings, regardless of the model, compared with the FF setup. This indicates that the dynamic coupling with the building modifies the soil conditions, for instance by increasing the confining pressure underneath the foundations, and limits the nonlinear behavior of the soil by increasing the stiffness (i.e. the shear wave velocity). Simultaneously, the damping variation between weak and strong motion is smaller with buildings setups, conform to the attenuation of the nonlinear response. All the building models have the same mass. LRS is a specific situation: only kinematic interaction is present and the inertial effect is null. For LRS, the reduction of frequency is −14%, less than for FF. Kinematic interaction generally results from base-slab averaging effects related to non-vertically incident wave fields, from embedment effects that increase with the depth of the foundations, and from wave scattering effects. In our case, only shallow rectangular footings and vertically incident wave fields are considered and we speculate that only scattering may control the kinematic interaction for the LRS model. For other cases, Δ is less acute in the presence of rigid and flexible structures, with Δ equal to −4% (RS) and −11% (FS and 2DOFS), respectively. For RS resting on soft soil, the building moves as a rigid body; the induced rocking motion results in the largest increase in confining pressure and stress in the dry sand underneath the foundations. The increase in confining pressure may limit the nonlinear-induced effect. Bray and Dashti [2014] also reported the effect of dynamic coupling related to confining pressure in centrifuge tests for liquefaction analysis. This observation confirms that in the presence of buildings, the confining pressure in the uppermost layer increases and the soil conditions become stiffer, i.e. shear modulus increases [Amini, 1993], reducing

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the soil’s ability to behave in the nonlinear domain. In fact, as confining pressure increases, the conventional shear modulus reduction curve for sandy materials moves upwards and the damping variation curve moves downwards, corresponding to a reduction of nonlinearity [Zhu et al., 2014]. This manuscript suggests that the relative importance of this effect must depend on the dynamic characteristics of the buildings (and of course of the soil, but not tested herein). Table 3 shows that under weak motion, slight modifications in the site response are observed depending on whether setups with or without buildings are considered. For example, under weak motion fs varies from 2.68 Hz (FF) to 2.56 Hz for the rigid structures (Δ* = −4% for LRS and RS), to 2.59 Hz for FS (Δ* = −3%), and to 2.53 Hz for 2DOFS (Δ* = −6%). For damping, no large differences are observed (less than 2%) and the confining pressure effect on damping seems to be limited, as confirmed by the similarity of the H(f) shapes both with and without buildings (Fig. 5). In fact, the influence of the confining pressure may be countered by the impact of different loading frequencies during testing, which is not the case in our experiment. It may also be countered by the dynamic induced loading of the building motion on the uppermost soil layers. For example, under strong motion, the resonance frequency of the FF soil column varies less than for setups with buildings. In our case, the contamination of the FF response by buildings is not linear since Δ* under weak motion is systematically different compared with strong motion, i.e. for the FF setup, fs is larger than for building setups under weak motion (Δ* < 0) and smaller under strong motion (Δ* > 0). One explanation may be the increase of the inertial effect, since under strong motion buildings vibrate at larger amplitude. In this case, the dynamic loading of the soil due to rocking increases, thus increasing dynamic induced-stress in the soil underneath. No more data are available to confirm this hypothesis but Bray and Dashti [2014] also reported an increase in induced stress resulting from shaking-induced effects. In our case, the nonlinear response of the soil with building setups is less pronounced because of the increased induced stiffness resulting from the building motion, as observed by comparing the transfer function and the resonance frequency of the soil profile. This is particularly true for the RS model, which acts as a rigid-body structure (Δ*fs = 12%; Table 3). On the contrary, this does not apply to the LRS model (Δ*fs = 0%), which shows no inertial effect. Simultaneously, and unlike what was observed under weak motion (Table 3), damping of the soil H(f) decreases to 7% for building setups compared with FF (Δ* < 0). For an induced confining stress increase, the strain of the soil immediately underneath the foundations is less and the conventional damping variation with strain, represented by the D-γ curve, can be moved downwards. 4.2. Identification of the Building Response Including SSI under Weak and Strong Motion The effect of nonlinearity on each component of motion is compared using the SSI models described above (Figs. 6–9). Chazelas [2010] defined the expected fixed-base frequency used as a reference, i.e. 2.65 Hz for FS, and 0.78 Hz and 2.99 Hz for the first two modes of the 2DOFS model. f1 for RS was calculated from the mass and stiffness of the structure, giving 4.05 Hz. Values are summarized in Table 4. Only the RS, FS, and 2DOFS setups are considered in the rest of the manuscript.

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Table 4 Summary of the frequency values (in Hz) obtained in centrifuge tests* 2 fsys (1)

RS FS

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2DOFS mode 2

Weak Strong Δ Weak Strong Δ Weak Strong Δ

2.69 2.33 −13% 2.05 1.59 −22% 2.70 2.54 −6%

3

4

f0 2.69 2.42 −10% 2.87 2.45 −15% 2.59 2.42 −7%

fØ 2.46 1.83 −26% 2.30 1.83 −20% 2.60 2.39 −8%

5 fapp (1)

2.64 2.13 −19% 1.99 1.58 −21% 2.74 2.60 −5%

6 f1 2.72 2.23 −18% 2.34 1.88 −20% 2.73 2.73 0%

7

8

fapp

fsys

1.82 1.41 −23% 1.64 1.31 −20% 1.88 1.80 −4%

1.51 1.22 −19% 1.42 1.16 −18% 1.52 1.44 −5%

(2)

(2)

9

10

Δ fapp −45% −51%

Δ fsys −78% −91%

−21% −21%

−44% −37%

−46% −44%

−78% −76%

* Columns 2 to 6 are obtained from the data as explained in Sec. 3.2. f1 is computed using Eq. (8). Columns 7 and 8 are obtained from Eqs. (3) and (4) and using columns 2 to 6. Columns 9 and 10 (gray zone) are the error between direct estimation of frequency using data and empirical equations [9 = (7–5)/7; 10 = (8–2)/8]. The columns labeled (1) are obtained using deconvolution method, and (2) are obtained from Eqs. (3) and (4). The rows labeled Δ correspond to the percentage of variation between weak and strong motion, i.e. (strong–weak)/weak.

4.2.1. Horizontal motion – x0 – Column 3 Table 4 Figure 6 presents the horizontal motion of the SSI model for the RS, FS, and 2DOFS setups. This motion represents the kinematic interaction influenced by the type of footing, giving an approximate representation of the theoretical concept of Foundation Input Motion. As expected, the functions are very similar in amplitude and shape regardless of the model. Lai and Martinelli [2013] reported that for shallow foundations and vertically propagating S-waves, the kinematic interaction is negligible. For all cases, two frequencies are observed at around 2.7 and 7.5 Hz, close to the resonance frequencies of the soil because of the effects of inertial SSI on the foundation are concentrated near the fundamental frequency of structure. By consequence, kinematic effects can be studied over the remaining frequency band. Thus, the effect of strong motion is well observed, amplitude and fundamental frequency being reduced, while high frequencies are extensively damped. The first frequency moves to around 2.4 Hz, corresponding to softening of Δf of 10% for RS, 15% for FS, and 7% for 2DOFS. This nonlinear effect is of the same order of magnitude as the nonlinear effect observed for the soil response model (Table 1). 4.2.2. Motion of the Soil–Structure System – x1+xH+hø – Column 2 Table 4 Figure 7 displays the spectra of system motion and the peak of the function corresponds to fsys. Nonlinearity is observed by (1) a shift in frequency from 2.69 to 2.33 Hz (Δ = −13%) for RS, from 2.05 to 1.59 Hz (−22%) for FS and from 2.70 to 2.54 Hz (−6%) for the second mode of the 2DOFS model; (2) an increase in damping from 13% to 17% (Δ = 4%) for RS, from 13% to 16% (3%) for FS and from 6% to 10% (4%) for the second mode of the 2DOFS model; (3) a reduction in spectral amplitude between weak and strong motion once the input signal has been removed from the total motion of the structure; (4) no effect on the magnitude of the input signal on first frequency (0.76 Hz) and damping (5.5%) of the 2DOFS model. Input motion does not impact the fundamental frequency of the 2DOFS model and the resonance effect between soil and building mainly affects the second mode. The frequency softening of the soil–structure system is positively correlated with the structure-to-soil stiffness ratio [Bielak, 1975] [Aviles and PerezRocha, 1996] [Stewart and Fenves, 1998]; at high frequency, the stiffness of the structure is

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greater than at the fundamental mode and frequency softening must therefore be greater, as observed herein. The system response of the RS model is less affected than for the FS model in terms of frequency (13% softening versus 20%), consistent with the nonlinearity observed for the soil response (Tables 1 and 2). For RS setups, nonlinear SSI may affect system motion by reducing the softening of the soil through dynamic induced loading of the underlying soil layer. The variation in system damping is much smaller than the damping of the soil observed in Table 1 (Δ > 10%), damping being ultimately less affected by soil nonlinearity. 4.2.3. Rocking motion – hø – Column 4 Table 4 Figure 8a displays the spectra of the rocking motion and the peak of the frequency corresponds to fø. We ignore the rotation of the soil, which also contributes to the total rocking of the footing. The nonlinear effect is observed between weak and strong input signals: rocking frequency moves from 2.45Hz to 1.83 Hz (Δ = −26%), 2.30 Hz to 1.83 Hz (−20%), and 2.60Hz to 2.39 Hz (−8%) for the RS, FS, and 2DOFS (second mode) models (Table 4). Simultaneously, damping increases from 10% to 21% (Δ = 11%) and 7% to 8% (1%) for RS and 2DOFS, while damping decreases for the FS model from 19% to 10% (−9%). Once again, strong motion does not affect the first mode of the 2DOFS model and the rocking amplitude is very small compared with the second mode. If we compare with the total motion of the structure (Fig. 7a), we observe that the rocking spectra are very similar in terms of shape, frequency and damping values. We can conclude on the importance of rocking in the total motion of structures with shallow foundations, as previously reported [Stewart and Fenves, 1998] [Chen et al., 2013]. Furthermore, the rocking amplitude under strong and weak motion is more pronounced for RS than for FS, which might explain the strongest nonlinear response of the soil observed in Fig. 5. Dynamic soil loading induced by SSI rocking is thus confirmed as being the main effect modifying the nonlinear response of the soil. 4.2.4. Pseudo-flexible motion – x1 + hø – Column 5 Table 4 Figure 8b displays the spectra of the pseudo-flexible motion. fapp corresponds to the case of stiff foundations in the horizontal direction. Figures 7b and 8b show equivalent functions between system and pseudo-flexible motions in terms of shape, peak frequency, and damping, allowing us to assume a small contribution of x0. From weak to strong input motion, pseudo-flexible frequencies move from 2.64 Hz (mean of the double peaks, see Fig. 7b) to 2.13 Hz (Δ = −19%), 1.99 Hz to 1.58 Hz (−21%), and 2.74 Hz to 2.6 Hz (−5%) for the RS, FS and 2DOFS (second mode) models. Damping increases from 21% to 24% (Δ = 12%), 18% to 21% (20%) and decreases from 6% to 5% (14%) for RS, FS, and 2DOFS, the variation values being of the same order of magnitude as for system motion. 4.2.5. Fixed-base motion – x1 – Column 6 Table 4 Fixed-base motion is certainly the most difficult element to assess. In this manuscript, we consider Eq. (11) to evaluate this parameter. Figure 9a shows the fixed-base response spectrum and the frequency peak of the function corresponds to f1. We observe no significant shifts in the 2DOFS model response, the fixed-base frequency for mode 1 (0.82–0.80 Hz) and mode 2 (3.04–3.07 Hz) being almost exactly the same under both weak

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and strong motion, and close to the theoretical frequencies provided by Chazelas [2010], i.e. 0.78 and 2.99 Hz. The damping variation is also insignificant in terms of nonlinearity and, finally, we can assume a perfect linear and elastic response of the fixed-base 2DOF structure. For RS, the fixed-base frequency moves from 2.72 to 2.23 Hz with equivalent damping values (about 15%). For FS, we observe a shift in frequency from 2.34 to 1.88 Hz but much more dispersion is observed in the data. Furthermore, Figure 9b compares the frequencies computed from the time delay of the pulse travel time (Eq. (9)) and by deconvolution derived from Paolucci [1993] (Eq. (10)). A bias is observed with the time delay method, and this empirical relationship does not represent the fixed-base frequency. Eq. (10) needs to be modified for buildings that do not exhibit pure shear deformation. 4.3. Discussion Table 4 provides a comparison of the frequency values extracted directly from the data using deconvolution and the model-based analytical relationships. The error is quite large, with the same order of magnitude between weak and strong shaking. For rigid buildings, the error is around 40–50% for apparent frequency (pseudo-flexible motion) and 75–90% for system frequency. These discrepancies may came from several assumptions in the analytical equation proposed by Luco et al. [1988], for example the shear response of the structure, the same rotation (rocking) at the foundations and at the top of the structure, uncoupled motion for rocking, horizontal, structural and system motion, etc. These hypotheses can be accepted in specific, actual buildings, but not in all cases, such as those tested in these centrifuge models. Moreover, these analytical equations assume a linear model for the SSI. We can also question the relevance of analytical equations and deconvolution methods in the case of strong cyclic motions. In this case, the response of the soil and the SSI varies in time. The comparison of the frequency values by these two approaches, which effectively comes down to smoothing the behavior of the soil and structures over the duration of the excitation, is then biased. The rigid and flexible building setups (RS and FS) show softening of the total frequency Δ of about 27% and 30%, respectively. Flexible buildings show a greater nonlinear response compared with rigid buildings, in terms of frequency softening for fixed-base frequency. This finding is in agreement with Nader and Astaneh [1991], who reported similar results on steelframe buildings or Chen et al. [2013] using centrifuge experiment. Moreover, the frequency shift between weak and strong motion for fixed-base and system parameters is, respectively, Δ = −18% for RS and Δ = −20% for FS; Δ = −13% for RS and Δ = −22% for FS, suggesting that RS reduces the nonlinear response of the system. This was also observed in Fig. 5 for the site response. Furthermore, total frequency variation is the smallest for RS and, assuming the same experimental conditions for FS and RS (i.e., same input signal and same soil profile), the nonlinear response of the system is controlled by the SSI. Rocking actually appears to be the most important component of SSI and it is also the parameter most sensitive to nonlinearity, compared with fixed-base and horizontal motion (Table 4). The nonlinearity of the rocking is more pronounced for the RS model (Δ = −26%) than for the other models (Δ = −20% for FS), with rocking frequency showing greater softening than the other parameters. We can speculate that rocking motion is the dominant parameter impacting the nonlinear response of the site and the soil–structure system, producing a reduction in the nonlinearity of

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Figure 10 Rocking share in the system response of RS, FS, and 2DOFS with the nonlinear response of the soil. (a) Compared to the variation of the rocking frequency (Δ values Table 4 Column 5) and the soil response frequency (Δ values Table 2) between weak and strong motion. (b) Compared to the variation of the soil response frequency between building and FF setups (Δ* values Table 3) for weak and strong motion.

the soil and the SSI. Figure 10 gives a synthesis of this conclusion. In this figure, the predominance of the rocking motion in the system response is computed as the ratio of the transfer function Hϕ(f)/Hsys(f). Figure 10a, under strong motion the rocking is predominant for RS structure and in this case we observe the smallest reduction of the soil stiffness. This could be due to local dynamic loading of the soil, which increases the confining pressure and limits the nonlinear behavior of the soil. This phenomenon is dominant in the case of RS. Nevertheless, it is also reported for FS, where rocking is more nonlinear, thus reducing the nonlinearity of the site response in the presence of FS compared with the FF model. For example (Fig. 10b) the shift of the resonance frequency between the building and the FF setups increases with the rocking motion (for RS model) under strong and weak motion and the presence of a building having an important rocking motion reduces the nonlinear response of the soil. This observation is valid for dry sand.

5. Conclusion In this study, we evaluate the nonlinear response of a soil–structure system. These experiments illustrate the role of the nonlinear coupling between the soil and the structure during earthquakes: (1) it seems like the errors observed between the frequencies obtained by the Luco’s analytical equations and the experimental approach are all the more important as the structure is rigid (RS). In this case, the interaction increases, especially by rocking increase, and the cyclic nature of the nonlinearity in the soil and in the ISS induces a bias in the identification of the ISS. The same observation is made when the fixed-base frequency is higher, which increases the impedance contrast between the ground and the structure, and thus the effect of the ISS. The relation between the frequency errors with the amount of SSI might be confirmed using numerical and experimental approaches.

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(2) the presence of buildings changes the site response as well as its nonlinear response. This modification is shown by the reduction in the nonlinear response of the soil when buildings are present. This reduction may come from the rocking motion, which increases confining pressure and reduces soil deformation. This phenomenon is more pronounced for rigid buildings; (3) the rocking motion is dominant in the total motion of the structure and this is the component most sensitive to nonlinearity; (4) soil nonlinearity is observed to be one of the parameters affecting the nonlinear response of the system, with greater impact on the rocking motion; (5) the nonlinear SSI is more pronounced for rigid buildings than for flexible buildings. All these observations question the validity of soil models derived from FF conditions for urbanized areas, as well as the modification of site conditions after construction. The effects might be non-negligible: the presence of buildings may reduce by 2 the frequency shift in the soil transfer function between weak and strong motion (Table 2, Δ values from 22% for FF and between 4 and 12% for building setups). It may also question the reliability of the nonlinear model of soil, which is defined without taking into consideration the presence of buildings. In order to improve prediction of the building’s response for design purposes, the site response must integrate effects due to the presence of structures into the nonlinear response of the soil and the SSI. However, dynamic loading of the soil produced by the building motion itself through rocking must also be integrated, which can increase the cyclic stress in the uppermost layer. In the presence of saturated soil, the response could be different, rocking being able to increase liquefaction [Bray and Dashti, 2014]. This observation also suggests that in approaching nonlinear SSI problems, soil and structure must be considered as a single, mutually influential system, rather than separately. These observations must be improved by testing real buildings or by numerical modeling. The foundation response is complex and the relationships derived from simplified theoretical models cannot be considered as sufficient.

Acknowledgments The authors express their gratitude to Jean-Louis Chazelas and the centrifuge facility team at IFSTTAR (Institut Français des Sciences et Technologies des Transports, de l’Aménagement et des Réseaux) Nantes, France, who provided the centrifuge data within the framework of the French National Research Project, ANR ARVISE. We would also like to thank IFSTTAR for funding this research. This study is part of the URBASIS project lead by ISTerre (Université de Grenoble Alpes).

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Appendix A In this study, deconvolution was performed using the waterlevel regularization technique proposed by Clayton and Wiggins [1976]. The general model applied for building and soil motion is to consider the time history of the output of the system as: oðtÞ ¼ iðtÞ ' hðtÞ þ nðtÞ

(A1)

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where o(t) is the motion recorded at the top of the system, i(t) is the seismic input motion at the bottom of the system, h(t) is the impulse response of the structure, n(t) is the noise function affecting the signal at the top and ' denotes convolution. The impulse response of the structure is obtained as the inverse Fourier transform of the spectrum of the top motion (output O) divided by the bottom (input I) signal: e& e& e ðωÞ þ 'I ðω'Þ N e ðωÞ ~ ðωÞ ¼ OðωÞ ¼ I ð'ωÞ:I ð' ωÞ H H eI ðωÞ 'eI ðωÞ'2 'eI ðωÞ'2

(A2)

where * denotes conjugation and ~ estimated functions. In order to establish a minimum amplitude threshold for the input and to limit the importance of the noise term, the minimum input amplitude is named waterlevel, expressed as a fraction of the input. Considering k the waterlevel parameter (0 ≤ k ≤ 1) and the maximum spectral amplitude of the input signal |I(ω)max|, H(f) becomes: e& e& ~ ð f Þ ¼ I ð f Þ:I ð f Þ:H ð !f Þ þ N ð f Þ:I "ð f Þ H 2 ' ' ' '2 maxf'eI ð f Þ' ; k:'eI ð f Þ'max g

The deconvolution process will be stable if the factor

(A3) &

I ð f Þ:e I ðfÞ

' '2 ! ' ' "2 remains stable as a I ð f Þ' ; k:'e I ð f Þ' g maxf'e max

function of frequency. In the case of a building or a soil column, and assuming the signal input is known as the recorded time histories at the bottom of the system and the noise coefficient as zero, the transfer function of the system is computed as follows: 0 1 Oð f Þ B n ! " Hð f Þ ¼ @ max I ð f Þ; ðk: jI ð f Þj: jIIðð ff ÞÞj

max

oC A

(A4)