Rocking Motion Analysis Using Structural Identification Tools

Abstract

This research investigates the convenience of structural identification tools to detect the rocking motion tendency, using as input the structural response to ambient vibrations. The rocking ratio and rocking spectrum are proposed as original tools to highlight the rocking motion and its frequency content. The proposed procedure allows the detection and quantification of rocking using only building vertical motion records in both cases of ambient vibration and earthquake. First, three-dimensional finite element models of reinforced concrete buildings are adopted to simulate the structural response to white noise vibration. Different low- and high-rise buildings are studied, having framed structure and frame–wall system, regular and irregular structure, shallow foundation and underground floors. The structural response obtained numerically is analyzed using different signal processing tools to obtain the dynamic features of buildings, and the rocking motion tendency is identified by comparison with a reference fixed base condition. Then, the reliability of the proposed methodology to detect rocking motion attitude, using only the structural motion, is verified and quantified using the proposed tools. Finally, the same approach is applied to real structural motion records of a high-rise reinforced concrete building.

1. Introduction

During a measurement campaign for structural identification in existing buildings, the presence of soil–structure interaction and rocking motion tendency can be detected to characterize the dynamic behavior of buildings and to control their health state if the procedure is repeated over time.

Veletsos and Meek [1] and Chopra and Gutierrez [2], using analytical simplified methods, show that the rocking motion induces a lengthening of the building fundamental period, compared with the fixed base condition, because the structure is more flexible at its base. Using simplified numerical models, Jennings and Bielak [3] show that the rocking effects on the seismic response of buildings predominantly occur in the direction of the fundamental mode shape. Stewart et al. [4] studied the natural tendency of 57 buildings in California to experience soil–structure interaction (SSI) effects by using an analytical approach and observed that SSI is directly proportional to the structure-to-soil stiffness ratio. Stewart et al. [5] introduced the concept of aspect ratio (building effective height to foundation width) as a parameter influencing the rocking effect.

According to Saez et al. [6], SSI effects are detected in a numerical analysis when the seismic response obtained by solving the dynamic equilibrium problem, applied to the assembly of soil domain and framed structure, is significantly different from that obtained by imposing the free-field motion at the base of the fixed base (FB) structure. Nevertheless, during a measurement campaign, when only identification methods are used, the only available datum, without a reference FB condition for comparison, is the set of records for the existing building.

In this research, a criterion to highlight and estimate the rocking effect is defined using a set of structural motion time histories. The rocking ratio and rocking spectrum are proposed as the original parameters to measure a rocking motion and its frequency content. The proposed approach is inspired by the work of Dunand [7], adopted for the detection of torsional motion. Three-dimensional (3D) numerical models in a finite element (FE) scheme of different multi-story multi-span reinforced concrete (RC) buildings are used to numerically generate the structural response. The advantage of using 3D numerical models is to obtain a big set of signals that generally is not achievable due to the reduced number of available sensors during a measurement campaign. Moreover, different conditions can be compared, such as the building aspect ratio, structure type and regularity. The structural response is analyzed using signal processing tools and operational modal analysis (OMA) to identify the rocking motion tendency by comparison with a reference FB configuration. Then, the reliability of the proposed methodology is verified. The rocking motion of the Nice prefecture building in southeastern France is investigated using both ambient vibration and seismic records.

The numerical modeling approach is presented in Section 2. The proposed signal processing tools are defined in Section 3. The analyzed representative buildings and input signals are described in Section 4. The results of the structural identification are discussed in Section 5. The rocking motion estimation is discussed in Section 6 for numerical signals and in Section 7 for records. The conclusions are developed in Section 8.

2. Direct Analysis of the Structural Response

A 3D soil domain, assumed as being horizontally layered, with a building at the surface, is modeled in a FE scheme to investigate the rocking motion during ground vibrations. The continuity and homogeneity of materials are assumed for the structure and each soil layer, as well as a linear elastic behavior coherent with the imposed low-amplitude loading. The vertical propagation of shear and compressional waves from the top of the underlying elastic bedrock to the soil surface is numerically simulated, as well as the dynamic response of the building at the soil surface. Site effects, due to soil stratigraphy, are directly coupled with dynamic features of the superstructure, foundation deformability and ground motion.

2.1. Dynamic Features of Building and Soil

The dynamic properties of the FB structure are obtained in the FE scheme by solving the following eigenvalue problem:

$$K_{FB}\hspace{0.17em}\Phi =\hspace{0.17em}{M}_{FB}\hspace{0.17em}\Phi \hspace{0.17em}{\Omega }^2$$

(1)

where $M_{FB}$ and $K_{FB}$ are the mass and stiffness matrices, respectively, of the FB building. The terms of the diagonal matrix $\Omega$ are the angular natural frequencies of the building. Natural frequencies are the obtained real non-zero values. The modal matrix $\Phi$ is orthonormalized with respect to the mass matrix, such as ${\Phi }^T\hspace{0.17em}{M}_{FB}\hspace{0.17em}\Phi =I$, where $I$ is the identity matrix. The columns of the modal matrix represent the mode shapes of the building. The terms of each column give the displacement associated with each degree of freedom in the mode shape. The effective mass associated with each mode shape is estimated as ${\mathsf{\epsilon }}_i=100\hspace{0.17em}\left(p_i^2/M_{FB}\right)\%$, where $M_{FB}=p^T\hspace{0.17em}p$ is the total mass of the fixed base building, $p_i$ is the participation factor—one of the terms of the participation vector $p={\Phi }^T\hspace{0.17em}{M}_{FB}\hspace{0.17em}\tau$, where $\tau$ is the influence vector of dynamic loading.

The dynamic properties of the free-field (FF) soil are obtained by solving the same eigenvalue problem in Equation (1), using the mass and stiffness matrices obtained for a unit area soil column. The choice of soil domain area for the building–soil system is checked by evaluating the building base to bedrock transfer function and ensuring that the first natural frequency corresponding to the peak of this function matches the soil column fundamental frequency in the FF case. A similar building-to-soil domain length ratio is adopted for both horizontal directions.

2.2. Dynamic Equilibrium Equation

The discrete dynamic equilibrium equation is solved directly for the assembly of soil domain and structure, including the compatibility conditions, the 3D linear constitutive relation and the imposed boundary conditions. It is expressed in the matrix form as

$$M\hspace{0.17em}\Delta a+C\hspace{0.17em}\Delta v+K\hspace{0.17em}\Delta d=\Delta F$$

(2)

where $M$, $C$ and $K$ are the mass, damping and stiffness matrices, respectively, for the FB building or building–soil system. Velocity and acceleration vectors, $v$ and $a$, respectively, are the first and second time derivatives of the displacement vector $d$. The load vector $F$ derives from the adopted absorbing boundary condition (see Appendix A.1) when the building–soil system is modeled. It corresponds, in an FB model, to the imposed inertial load $F=-M\hspace{0.17em}\tau \hspace{0.17em}{a}_g$, where $\tau$ is the influence vector and $a_g$ is the acceleration at the ground level.

The dynamic process is solved step-by-step by the implicit Newmark algorithm (Hughes [8]). The integration parameters $\mathsf{\beta }=0.25$ and $\mathsf{\gamma }=0.5$ guarantee unconditional numerical stability of the time integration scheme and zero numerical damping. This choice is considered suitable in the linear elastic regime. The time step used for the input signal sampling is adopted for the analysis.

3. Inverse Analysis of the Structural Response

The analysis of building dynamic features is first undertaken in terms of natural frequency and shape mode variation. Afterwards, the rocking ratio and spectrum are introduced as efficient signal-processing tools to highlight the existence of the rocking motion. All input and output signals in the numerical simulation of structural response to white noise vibration are filtered using a 4-pole Butterworth bandpass filter in the frequency range $0.1–15\hspace{0.17em}\mathrm{Hz}$ because it includes the most relevant frequency content of the buildings and the quality of FE mesh is assured.

3.1. Frequency Domain Analysis

The Fourier spectrum of the structural response is obtained using the Fast Fourier Transform (FFT) algorithm. In the case of seismic input, the FFT is applied to the total duration of the motion time history. On the contrary, in the case of white noise, the FFT is applied to a sliding time window of $30\hspace{0.17em}\mathrm{s}$ along the whole signal length, using a Hanning tapering function, and after an average Fourier spectrum is calculated over all the $30\hspace{0.17em}\mathrm{s}$ time windows. In both cases, the Fourier spectrum is smoothed to facilitate the detection of peaks related to the structure modes, according to the procedure proposed by Konno and Ohmachi [9], using a bandwidth coefficient $b=40$.

The transfer function (TF) is defined as the ratio of the FFT at two different heights. The frequencies corresponding to the peaks of the building top-to-bottom TF, obtained for each component of motion, are the building natural frequencies related to mode shapes in the same direction of motion. By proceeding like this, it is easy to distinguish natural frequencies associated with translational mode shapes for each direction and, eventually, torsion modes if a peak is present at the same frequency for both directions.

The Frequency Domain Decomposition (FDD) is an output-only system identification technique (Brincker et al. [10]) based on the modal transformation $d=\Phi \hspace{0.17em}q\hspace{0.17em},$ where $d$ and $q$ are the displacement vector in nodal and modal coordinates, respectively, and $\Phi$ is the modal matrix whose columns contain the mode shapes. The power spectral density matrix of structural response $G_d\left(f\right)$ is estimated, and a singular value (SV) decomposition is applied under the assumption of uncorrelated modal coordinates (Brincker and Ventura [11]):

$$G_d\left(f\right)=\hspace{0.17em}\Phi \hspace{0.17em}{G}_q\left(f\right)\hspace{0.17em}{\Phi }^T$$

(3)

The terms of the diagonal matrix $G_q\left(f\right)$ are interpreted as power spectral densities of the modal coordinates. Natural frequencies of the system are obtained by picking the peaks in the plot of the first three SV spectra and reading the related frequency $f$ on the horizontal axis. In the following analysis, the SV spectra, represented in terms of decibels $\left(S{V}_{dB}=10\hspace{0.17em}\log \left[SV\right]\right)$, are obtained using the signals in the three directions of motion. Consequently, the direction of motion associated with each peak is not given directly, but it is deduced from the related mode shape. Moreover, if a peak is observed in the first two SV spectra at the same position, it means that two mode shapes exist and are associated with very close natural frequencies. In that case, it may be difficult to differentiate these two natural frequencies.

3.2. Rocking Estimation

When a rocking motion occurs, the building base moves from side to side, turning around a horizontal rocking axis. The vertical displacement of two distant points, located from either side of the rocking axis, are in phase with anti-correlated amplitudes (phase shift of 180°). On the contrary, in the case of pure vertical translation, these two points present the same vertical displacement. When calculating the difference between the displacements of the two points, a high difference is obtained in the case of rocking and a negligible difference is obtained in the case of pure vertical translation.

The proposed tools for rocking motion detection and estimation can be applied to both earthquake and ambient vibration records. When a three-component seismic motion is analyzed, the vertical motion can be a mix of rocking oscillations with anti-correlated amplitudes and a pure vertical translation due to the compressional wave propagation. As it is observed that the rocking motion is mostly related to the building oscillation due to shear wave propagation (especially if surface waves are neglected), the rocking motion can then be highlighted by filtering the vertical displacement around the frequency of the first building translational mode shape $f_b$ in the associated direction.

The approach defined by Dunand [7] to measure the torsional effect at the top of a building, using the same horizontal component of motion recorded in two distant points, inspired the definition of the following parameters to characterize the rocking motion at the base of buildings. The rocking ratio $RR$ is defined as the time history of the difference in vertical displacement between two opposite corners $A$ and $B$ (Figure 1) at the building base, normalized by the average peak displacement, and it is expressed as

$$RR\left(t\right)=\frac{\left|d_{vA}\left(t\right)-d_{vB}\left(t\right)\right|}{\mathrm{mean}\left(\max \left(\left|d_{vA}\right|\right),\max \left(\left|d_{vB}\right|\right)\right)}$$

(4)

where $d_{vA}\left(t\right)$ and $d_{vB}\left(t\right)$ are the vertical component of simultaneous motion in the two points $A$ and $B$. The maximum displacement is peaked in a sliding time window. The rocking ratio $RR$ highlights the phase shift in vertical motion between two selected locations $A$ and $B$. When the phase of signals is opposed $\left(d_{vB}\left(t\right)=-d_{vA}\left(t\right)\right)$, as in Figure 1, $RR$ is about two. In the case of pure vertical translation $\left(d_{vB}\left(t\right)\cong d_{vA}\left(t\right)\right)$, $RR$ exhibits low amplitudes. After estimating the rocking ratio $RR$ time history, its envelope is represented by taking only the local maxima, and the same result is compared for two orthogonal directions to highlight the rocking motion direction.

The horizontal motion at the top of the building is mainly due to lateral deflection under horizontal loading and, eventually, to rocking motion. A rocking amplitude ratio $\left(RA\right)$ is adopted to quantify the part of rocking motion on the whole horizontal motion at the top of the building. It is defined as

$$RA\left(\%\right)=\left(\frac{\Delta d_v\hspace{0.17em}H}{l_1}\right)\hspace{0.17em}\frac{1}{\Delta d_h}\hspace{0.17em}100$$

(5)

where $\Delta d_v=\max \left|d_{vB}\left(t\right)-d_{vA}\left(t\right)\right|$ is the maximum vertical displacement shift between two aligned corners $A$ and $B$ at the building base (Figure 1), $H$ is the building effective height and $l_1$ is the building length in the direction of the first mode shape. Considering the rigid body rotation due to rocking motion (Figure 1), the consequent horizontal displacement at the building top is given by $\left(\Delta d_v\hspace{0.17em}H/l_1\right)$. It is normalized with respect to the horizontal relative displacement $\Delta d_h$ at the building top (with respect to the ground surface level) in the direction of the first mode shape.

When seismic records are used, the rocking spectrum $RS$, illustrating the frequency content of rocking motion, is defined as

$$RS\left(f\right)=\frac{FFT\left[a_{vA}\left(t\right)-a_{vB}\left(t\right)\right]}{\max \left(\mathrm{mean}\left(FFT\left[a_{vA}\right],FFT\left[a_{vB}\right]\right)\right)}$$

(6)

where the Fourier spectrum of the difference between the vertical acceleration time history in two locations $A$ and $B$ at the building base (Figure 1) is normalized by the average spectrum peak. When the phase of signals is opposed, the rocking spectrum $RS$ is approximately two. The frequencies corresponding to the peaks of the Fourier spectrum are the building natural frequencies related to the rocking motion. If these frequencies are excited by the ground motion frequency content, the rocking motion is amplified.

In the case of a three-component stationary random excitation, as ambient motion, the following is proposed: first, extract the random decrement (RD) function (Cole [12], Rodrigues and Brincker [13]) associated with the building fundamental frequency, and then calculate the equivalent rocking ratio $RR$ using the RD function instead of the vertical displacement. The RD function provides an estimation of the free response of the building at a given resonance frequency. The stacking of a large number of time windows with equivalent initial conditions cancels the random part of the signal. The random decrement function of the filtered vertical acceleration $a_v$ (filtered around the building natural frequency) is defined by the expression

$$RD\left(t\right)=\frac{1}{N}\hspace{0.17em}{\displaystyle \sum _{i=1}^N{a}_v\left(t_i+t\right)}$$

(7)

where $N$ is the number of averaged time windows. The adopted triggering condition to select $a_v\left(t_i\right)$ is a zero crossing with a positive slope. The adopted time window duration is $30\hspace{0.17em}\mathrm{s}$.

In the case of using RD functions, the rocking spectrum $RS$ is estimated as

$$RS\left(f\right)=\frac{FFT\left[R{D}_A\left(t\right)-R{D}_B\left(t\right)\right]}{\mathrm{mean}\left(FFT\left[R{D}_A\right],FFT\left[R{D}_B\right]\right)}$$

(8)

Equation (6) is modified as ambient vibration has more peaks observed with a similar amplitude. Consequently, in the case of ambient vibration, the $RS$ is estimated in a sliding time window.

The proposed tools for rocking motion assessment use only the time history of vertical motion at the structure base and horizontal displacement at the top, obtained during a measurement campaign for structural identification. The procedure can be repeated over time as a health monitoring tool to control the damage state of the structure. The proposed signal processing approach is independent of the construction material, structural system and ground type.

4. Input Data

Four RC buildings characterized by their structure and height are analyzed. Three low-rise buildings, having the same height, are selected to investigate the effect of the structural system: the ST building is conceived as a shear-type building, having a framed structure, regular in plan and elevation; the RBT building represents a bending-type building, having a frame–wall system, regular in plan and elevation; the IBT building is a bending-type building, having a frame–wall system regular in elevation, but irregular in plan. The HR building is a high-rise wall system, behaving as a bending-type building, irregular in elevation but regular in plan.

4.1. Buildings

The column orientation and the floor plan of the analyzed buildings are shown in Figure 2. Dimensions and dynamic features of the soil–building system are listed in

Table 1. Dimensions and dynamic features of the soil–building system.

Building	Floors	H	L	W	Foundation	Soil Area	$\mathit{H}/{\mathit{l}}_1$	${\mathit{f}}_{\mathit{b}}/{\mathit{f}}_{\mathit{s}}$
		(m)	(m)	(m)	(m × m)	(m × m)
ST	4	12	15	9	16 × 10	40 × 24	0.8	1.00
RBT	4	12	15	9	16 × 10	40 × 24	1.3	1.82
IBT	4	12	15	9	16 × 10	40 × 24	1.3	1.24
HR	20	60	17	8	56 × 24	70 × 50	7.5	0.56

H, L and W are the building effective height, length and width; $l_1$ is the building length in the direction of the first mode shape; $f_b/f_s$ is the FB building to soil frequency ratio.

. The numerical models in SSI condition are obtained by assembling the four buildings with the $30\hspace{0.17em}\mathrm{m}$ deep soil profile described in

Table 2. Soil profiles and mechanical properties of soil layers.

Depth	Thickness	$\mathit{\rho }$	vp	vs
(m)	(m)	(kg/m3)	(m/s)	(m/s)
5	5	1930	1330	200
10	10	1930	1400	240
30	15	1926.4	1500	300
>30		2000	2449.5	1000

$\mathsf{\rho }$, $v_p$ and $v_s$ are the soil density, compressional and shear wave velocities.

, using the soil domain area given in Table 1 for each building. The foundation of all the analyzed buildings is a 1 m deep RC slab embedded in the soil, whose plan dimensions are given in Table 1. The finite element model of buildings in FB and SSI conditions is displayed in Figure 3. Other details about the building models are given in (Appendix A.2 and Appendix A.3).

The analyzed HR building (Table 1) is the Nice Prefecture building in southeastern France. It is a high RC building with 20 floors in elevation and two underground levels. The structure is composed of two symmetric RC towers connected by a box girder (Figure 2e). Cantilever RC shells are fixed to the towers on the higher floors. Thin columns, not connected to the foundation but laid on the box girder, connect the floors between them to limit deflection. The effective structure height is $H=60\hspace{0.17em}\mathrm{m}$ (Table 1), and the length of each RC tower in the direction of the first mode shape is $W=8\hspace{0.17em}\mathrm{m}$ (Figure 2d). The depth of the underground part of the towers is 7.5 m, and it is rigidly fixed in the FB model. This building has a high aspect ratio (height to width) compared to the others (Table 1), which is considered related to a rocking effect.

In the following discussion, longitudinal $\left(\mathrm{L}\right)$ and transversal $\left(\mathrm{T}\right)$ directions of buildings refer to the longest and shortest side of the bearing structure, respectively. Accordingly, in the case of the HR building (Figure 2), the longitudinal direction is considered along the longer side of the towers.

The Nice Prefecture building has been continuously monitored since June 2010 through 24 accelerometric sensors operated by the RESIF-RAP French accelerometric network (DOI: https://doi.org/10.15778/RESIF.FR, accessed on 4 June 2023). Fernandez Lorenzo et al. [14] developed the finite element model of this building in FB condition, using structural motion records for calibration and considering the important rocking effects that are observed at the base of the building. In this research, the model has been updated considering the building assembled to the soil domain, taking into account the local stratigraphy, the foundation and the RC piles. After modeling the building–soil system, the building model is calibrated to reproduce the natural frequencies obtained by FDD using recorded signals (1.21, 1.22 and $1.60\hspace{0.17em}\mathrm{Hz}$, according to Fernandez Lorenzo et al. [14]). The updated building model of the Nice Prefecture is the HR building used in this analysis, assembled with the same soil domain adopted for the other buildings (Figure 3).

4.2. Soil Profile

The mechanical properties of the $30\hspace{0.17em}\mathrm{m}$ deep soil profile and underlying bedrock are given in Table 2, including density $\mathsf{\rho }$ and compressional and shear wave velocity in the medium $v_p$ and $v_s$, respectively. It is assumed that the average position of the water table is deeper than; consequently, a total stress analysis is undertaken. The soil domain area $A$ (see Table 1) is selected by checking that both dimensions are enough to guarantee that evaluating the soil surface to bedrock TF for the building–soil model, the obtained soil fundamental frequency is equivalent to the FF case. The incident motion is imposed at the soil–bedrock interface.

According to Fares et al. [15], SSI effects and rocking are amplified for the double resonance when the fundamental frequency of the building and soil profile are close together. As weaker SSI effects are expected for the low-rise framed structure (ST building), the soil profile with a depth of shear wave velocity $v_s$ is arbitrarily selected to obtain a natural frequency of $2.4\hspace{0.17em}\mathrm{Hz}$ (close to the fundamental frequency of the ST building (see the FB building to soil frequency ratio $f_b/f_s$ in Table 1)) so that an eventual rocking motion tendency can be highlighted. The shear wave velocity in the upper $30\hspace{0.17em}\mathrm{m}$ of the soil is $v_{s30}=257\hspace{0.17em}\mathrm{m}/\mathrm{s}$, corresponding to a ground type C according to the Eurocode 8 classification [16]. The compressional wave velocity $v_p$ and the soil density $\mathsf{\rho }$ are deduced according to the following relationships discussed by Boore [17]:

$$v_p\left[\mathrm{km}/\mathrm{s}\right]=0.9409+2.094\hspace{0.17em}{v}_s-0.82\hspace{0.17em}{v}_s^2+0.2683\hspace{0.17em}{v}_s^3-0.0251\hspace{0.17em}{v}_s^4$$

(9)

$$\begin{array}{ll}{v}_p<1.5\mathrm{km}/\mathrm{s}\hfill & \mathsf{\rho }\left[\mathrm{kg}/{\mathrm{m}}^3\right]=1930\hfill \\ 1.5<v_p<6\mathrm{km}/\mathrm{s}\hfill & \mathsf{\rho }\left[\mathrm{kg}/{\mathrm{m}}^3\right]=1.74\hspace{0.17em}{v}_p^{0.25}\hfill \\ v_p>6\mathrm{km}/\mathrm{s}\hfill & \mathsf{\rho }\left[\mathrm{kg}/{\mathrm{m}}^3\right]=1.6612\hspace{0.17em}{v}_p-0.4721\hspace{0.17em}{v}_p^2+0.0671\hspace{0.17em}{v}_p^3-0.0043\hspace{0.17em}{v}_p^4+0.000106\hspace{0.17em}{v}_p^5\hfill \end{array}$$

(10)

then, the elastic shear and P-wave moduli ($G_0=\mathsf{\rho }\hspace{0.17em}{v}_s^2$ and $M_0=\mathsf{\rho }\hspace{0.17em}{v}_p^2$, respectively) are estimated for each soil layer. The Poisson ratio is evaluated as a function of the compressional to shear velocity ratio, according to the relation $\mathsf{\nu }=\left(0.5v_p^2/v_s^2-1\right)/\left(v_p^2/v_s^2-1\right)$.

4.3. Synthetic White Noise and Recorded Seismic Loading

A three-component synthetic white noise and seismic record are applied as dynamic input motion at the base of the FE models in Figure 3 to numerically obtain the structural response.

Each component of the synthetic input signal is a zero mean unit variance white noise sequence, generated with an acceleration amplitude of $1\hspace{0.17em}\mathrm{mm}$ (Figure 4a) and a time step of $dt=0.02\hspace{0.17em}\mathrm{s}$ to ensure a maximum frequency of $f_{\max }=1/\left(2\hspace{0.17em}dt\right)=25\hspace{0.17em}\mathrm{Hz}$. A synthetic white noise excitation has a flat FFT spectrum to avoid the alteration of amplitude amplification at the resonances in the frequency response of structures.

A three-component earthquake record is also used as input motion. It used the mainshock of the 6 April 2009 ${\mathrm{M}}_{\mathrm{w}}\hspace{0.17em}6.3$ L’Aquila earthquake and recorded the Sulmona (SUL) station of the Italian strong motion network, localized in the Abruzzo region (Italy), at an epicentral distance of 53.7 km. The PGA is $0.34\hspace{0.17em}\mathrm{m}/{\mathrm{s}}^2$ in the east–west direction, $0.27\hspace{0.17em}\mathrm{m}/{\mathrm{s}}^2$ in the north–south direction and $0.24\hspace{0.17em}\mathrm{m}/{\mathrm{s}}^2$ in the vertical direction (Figure 4b). The sample rate of the recorded signals is $dt=0.005\hspace{0.17em}\mathrm{s}$. SUL is a free-field station on stiff soil (having shear wave velocity higher than $1000\hspace{0.17em}\mathrm{m}/\mathrm{s}$). Consequently, the record is considered as rock outcropping motion, and it is used (halved) at the base of the horizontally multilayered soil in terms of three-component velocity. This low-amplitude input motion is selected so that the assumption of linear elastic behavior of materials in the FE model is coherent for both vibration and seismic input. Moreover, this seismic event has been selected because, according to the Fourier spectrum in Figure 4b, energy is present in a large frequency range $\left(0.3–4\hspace{0.17em}\mathrm{Hz}\right)$ and, consequently, it is able to excite the fundamental frequency of all analyzed buildings. The north–south and east–west components are applied in the longitudinal $\left(\mathrm{L}\right)$ and transversal $\left(\mathrm{T}\right)$ direction of buildings (Figure 4b), respectively, as defined in Section 4.1.

5. Identification of Building Dynamic Features

The white noise vibration in Figure 4 is applied as a three-component input motion at the base of the FE models in Figure 3 and the time history of structural response is simulated. The structural response in the three directions of motion is stored at the four corners indicated in Figure 2a, at both the ground level and at the four decks. Consequently, the set of signals is composed of 60 time histories for ST, RBT and IBT buildings. Instead, for the HR building, the time history of the structural response is stored in 72 points, yielding a set of 216 signals. The frequency range $0.5–15\hspace{0.17em}\mathrm{Hz}$ is selected for the analysis because it includes the most relevant frequency content of the buildings and because the quality of the FE mesh is assured.

The building natural frequencies are first estimated by the numerical modal analysis in an SSI condition (

Table 3. Natural frequency f, direction of motion and effective mass ε of the first mode shapes obtained in FB and SSI conditions for the ST, RBT, IBT and HR buildings.

				FB					SSI
			FE		TF	FDD		FE		TF	FDD
Bldg	Mode	Dir	f (Hz)	ε (%)	f (Hz)	f (Hz)	Dir	f (Hz)	ε (%)	f (Hz)	f (Hz)
	1	TL	2.40	83.5	2.4	2.4	TL	2.19	46.0	2.4	2.2
	2	TT	2.59	84.1	2.55	2.5	TT	2.23	55.8	2.5	2.5
ST	3	T	2.98	0.0	-	-	TL	2.56	31.9	-	-
	4	TL2	7.50	10.4	6.9	7.0	TT	2.63	22.1	-	-
	5	TT2	7.96	10.5	7.3	7.1	T	2.97	0.0	-	-
	6	T	9.22	0.0	-	-	TL2	6.37	10.6	6.9	6.0
	7						TT2	6.39	9.1	7.3	6.1
	1	TT	4.36	71.1 + 4.4	4.25	4.3	TT	2.31	64.5 + 10.6	3.4	3.4
	2	TL	4.70	72.3 + 4.4	4.55	4.5	TL	3.22	65.0 + 10.7	3.45	3.5
RBT	3	T	6.55	0.1	-	6.2	TT	3.44	2.5	-	-
	4	TT2	16.04	16.2	12.4	13.4	TL	3.60	2.1	-	-
	5	TL2	19.26	13.5	13.5	16.5	T	4.99	0.0	4.8	4.8
	6						TL2	6.40	10.0	6.2	6.0
	7						TT2	6.41	10.5	6.2	6.2
	1	TT + TL	2.98	56.3 + 5.5	2.95	2.9	TT + TL	2.28	61.6 + 7.9	2.8	2.7
	2	TL + TT	7.23	58.2 + 12.6	6.8	6.7	TL + TT	2.31	67.6 + 8.7	2.8	2.8
IBT	3	TT2	9.22	4.5	9.1	8.5	TT	2.84	7.4	-	3.0
	4	T	10.45	14.7 + 11.4		9.4	TL	3.94	1.7	3.9	3.9
	5						TT2	6.16	2.6	5.95	5.9
	6						TL2	6.42	9.8	6.8	6.1
	1	TT	1.34	57.7	1.35	1.3	TT	0.97	7.2	1.0	0.96
	2	TL	1.48	63.1	1.45	1.5	TL	0.98	7.9	1.0	0.98
HR	3	T	1.86	0.0	-	-	T	1.37	0.0	-	-
	4	TT2	6.19	18.4	5.90	5.1	TT2	2.40	71.3	4.9	2.4
	5	TL2	7.10	0.0	6.10	5.2	TL2	2.42	70.6	5.3	2.5

(FE) Finite element model, (TF) transfer functions and (FDD) Frequency Domain Decomposition. (TL, TL2) first- and second-order translation in longitudinal direction; (TT, TT2) first- and second-order translation in transversal direction; (T) torsion.

, FE column). The obtained natural frequencies are related to the building–soil system. Even if the first mode shapes of the building–soil system are associated almost exclusively with the building motion, a low contribution of soil motion could have some influence. Then, the building natural frequencies are deduced from the top-to-bottom TF applied to the signals obtained by numerical simulation of structural response under white noise (Table 3, TF column) for the numerical signals in building corner 1 (Figure 2). In Figure 5a, Figure 6a, Figure 7a and Figure 8a, the top-to-bottom TF of each direction of motion is shown for each building. Finally, the natural frequencies are obtained from the FDD process using the set of three-component numerical signals and a frequency step $df=0.01\hspace{0.17em}\mathrm{Hz}$ (Table 3, FDD column). In Figure 5b, Figure 6b, Figure 7b and Figure 8b, the SV spectra obtained by FDD are shown for each building. The natural frequencies listed in Table 3, obtained with the three different approaches (FE, TF, FDD), are close together.

In Table 3, the direction of motion for each mode shape is indicated, as well as the effective mass related to each mode shape provided by the numerical modal analysis. Mode shapes are ordered by ascending natural frequency as it is generally carried out in structural dynamics and not by decreasing effective mass that is proportional to the participation factor.

In the numerical simulation, the SSI effects and rocking motion are detected by comparing FB and SSI conditions. No frequency variation is obtained for the first two mode shapes of the ST building and a negligible variation for the first mode shape of the IBT building. Instead, a reduction in the fundamental frequency is detected for the RBT building (5%) and HR building (35%). The frequency reduction reaches 45% for the second natural frequency of the HR building. This frequency decrease (period lengthening) is associated with an inertial interaction and rocking effect. The aspect ratio $H/l_1$ of the HR building (Table 1) can justify a rocking effect in soft soil.

The first mode shapes obtained by numerical modal analysis in a FE scheme and by operational modal analysis (FDD process) are displayed in Figure 5c, Figure 6c, Figure 7c and Figure 8c and Figure 5d, Figure 6d, Figure 7d and Figure 8d for both FB and SSI conditions, respectively. The reference signal used for normalization in the FDD procedure (to obtain the mode shapes) is the acceleration motion at the top slab (corner 1 in Figure 2) for all the buildings. In the SSI condition, the modal analysis in the FE schemes of ST and RBT buildings shows two additional mode shapes between first- and second-order flexural modes (Table 3). These modes are not clearly distinguished in TF and SV spectra because the associated natural frequency is too close to that of other modes.

A clear torsional effect is observed in all the mode shapes of the IBT building (Figure 7) that is irregular in plan, and consequently, as the torsion effects in the structural response are not negligible, they are clearly detected by the inverse analysis tools. On the contrary, torsion effects are not detected by the inverse analysis tools for buildings that are regular in plan (ST, RBT and HR buildings, Figure 5, Figure 6 and Figure 8), having equivalent inertia in both directions and uniform mass distribution on the slabs in the FE model. This yields a zero effective mass for the torsional mode. This is not the case when the structural identification is undertaken using real recordings because, in reality, mass distribution is not perfectly symmetric, and consequently, a torsional mode is present in the structural response, which can be captured. In fact, Fernandez Lorenzo et al. [14] detected the torsional mode of the Nice Prefecture building (named here HR building).

6. Rocking Motion Analysis Using Numerical Structural Response

Conversely to the previous section in which the comparison between SSI and FB conditions is needed to analyze the impact of rocking motion, in this section, a method to directly detect rocking without the use of any reference is proposed. Such a method is, therefore, more adapted to earthquake or ambient vibration recordings in real structures. The rocking ratio defined in Equation (4) and the rocking spectrum in Equations (6) and (8) are proposed with the purpose of highlighting the presence of rocking effects and detecting in which specific frequency ranges the rocking motion is emphasized.

The vertical component of building base motion is composed of a possible rocking effect due to the shear components of ground motion, mixed with a vertical translation due to compressive waves. Especially when the vertical and horizontal component of ground motion has the same amplitude level, for both ambient vibration and seismic motion, the rocking signal cannot be easily detected. It can be isolated by filtering the vertical displacement time history (at the building base) around the frequency of the first building translational mode shape (in the direction of the analyzed rocking motion). In this section, the building rocking motion is isolated by applying only one horizontal component of input motion at the base of the soil domain for the building–soil models in Figure 3 in the direction of the first mode shape of each building. This corresponds to the case of a low vertical to maximum horizontal component ratio.

The rocking motion induced by the incident shear seismic wave in Figure 4b and by the white noise applied in the horizontal direction of the first translational mode shape is investigated, and the results are represented in Figure 9 and Figure 10, respectively. In these figures, the reference corner $A$ is corner number 3 (see Figure 2) for all the buildings. Corner $B$ is placed along the direction of the first mode shape (two for the ST building and four for the RBT, IBT and HR buildings). Corner $C$ is placed along the direction of the second mode shape (four for the ST building and two for RBT, IBT and HR buildings).

In Figure 10, the vertical displacement at the base of the four buildings is illustrated in a time window, including the peak displacement. The shear seismic wave induces a vertical motion with phase inversion for all of the buildings. In the direction of the first mode shape, the rocking ratio $RR$ in Equation (4) is close to 2 over the whole time window. While in the orthogonal direction, it is negligible, except for the IBT building because of structural irregularity. The seismic shear wave induces a negligible vertical displacement for the ST building at the ground level, a low vertical displacement for RBT and IBT building and a higher vertical displacement in the HR building. For all types of buildings, the rocking ratio $RR$ confirms the phase shift in vertical displacement associated with the presence of rocking effects at the base of these buildings, but it does not quantify the effects. This suggests that an amplitude criterion is needed.

The frequency band where the rocking motion occurred is highlighted by the rocking spectrum in Equation (6). The rocking motion occurs mainly at the frequency of the first translational mode but can also be seen at higher modes. In this case, the rocking motion is induced only by the excitation of the first translational motion by the shear waves. Consequently, the rocking spectrum has an evident peak at the fundamental frequency of the building. It is negligible in the orthogonal direction, except for the IBT building (due to torsional effects).

In Figure 10, the RD functions are displayed, deduced from the filtered vertical acceleration at the base of the four buildings, when the white noise input is imposed in the horizontal direction of the first translational mode shape. Signals are filtered using a 4-pole Butterworth bandpass filter in the frequency range $f_b\pm 0.5\hspace{0.17em}\mathrm{Hz}$, around the frequency of the first building translational mode shape $f_b$ in the associated direction. At the frequency of the first mode shape, the vertical motions are in the opposite phase, indicating a rocking motion. The rocking ratio $RR$ in Equation (4) is close to 2 when two opposite corners in the direction of the first mode shape are considered. It is negligible in the orthogonal direction. According to both Figure 9 and Figure 10, the irregular structure and significant torsional motion of the IBT building imply that the two corners at the building base located on the same side $\left(A–C\right)$ with respect to the axis of rocking have slightly different vertical motions.

A fundamental frequency decrease is detected for RBT, IBT and HR buildings when comparing the structural response obtained numerically in the SSI and FB conditions associated with the rocking effect (Table 3). In particular, the HR building shows a higher discrepancy in the fundamental frequency compared with IBR-building. In this section, the rocking effect is quantified in terms of the rocking amplitude ratio $RA$ in Equation (5), which is proposed as a reliable parameter to quantify the part of rocking motion in the horizontal oscillation; the values are reported in

Table 4. Rocking amplitude ratio in the case of the 2009 L’Aquila earthquake and white noise for the shear-type (ST), regular-bending-type (RBT), irregular-bending-type (IBT) and high-rise (HR) building in SSI condition.

			Earthquake	White Noise
Building	Dir.	Mode	RA (%)	RA (%)
ST	L	1	0.6	0.55
	T	2	0.0	0.00
RBT	L	2	2.3	2.3
	T	1	19.5	20.0
IBT	L	2	19.0–6.7	4.1–0.1
	T	1	28.5	25.0–20.6
HR	L	2	0.8	0.3
	T	1	39.8	43.0

(L) longitudinal direction, (T) transversal direction.

. According to the high aspect ratio $H/l_1$ in Table 1 and the high fundamental frequency reduction in SSI condition (Table 3), HR buildings present a much higher rocking amplitude ratio compared with low-rise buildings.

7. Rocking Motion Analysis Using Recorded Structural Response

Finally, the proposed rocking motion detection tools are also applied to real data; that is, the vertical component of motion recorded by the sensors located at the base of the Nice Prefecture building during the 7 April 2014 ${\mathrm{M}}_{\mathrm{w}}\hspace{0.17em}4.9$ Barcelonnette earthquake that occurred approximately $100\hspace{0.17em}\mathrm{km}$ north of Nice (Fernandez Lorenzo [14]). The peak acceleration reached $0.036\hspace{0.17em}\mathrm{m}/{\mathrm{s}}^2$ at the base of the building, in the transversal direction of towers (direction of the first mode shape). The presence of rocking effects at the base of the Nice Prefecture building was discussed by Fernandez Lorenzo et al. [14] by observing the phase shift in the vertical displacement time history at different corners.

Figure 11a displays the vertical displacement recorded at the base of the Nice Prefecture building in the three corners in a time window that includes the peak displacement. The rocking spectra in Figure 11a exhibit a peak at the frequency of the first translational mode in each direction ($1.21\hspace{0.17em}\mathrm{Hz}$ and $1.22\hspace{0.17em}\mathrm{Hz}$, see Section 4.1). In addition, the recorded rocking motion has an energy content associated with higher frequencies.

An ambient noise recorded at the base of the building is also analyzed. The RD function of the filtered vertical acceleration is shown in Figure 11b. In both cases, the rocking ratio $RR$ (Equation (4)) is close to 2 when the vertical displacement of two opposite corners, in the direction of the first mode shape, is compared. In the orthogonal direction, it is less important during the seismic event and negligible for ambient vibration. The analysis of ambient vibration records gives a rocking amplitude ratio $RA$ equal to $34\%$ in the direction of the first mode shape and $1.7\%$ in the orthogonal direction. This result is coherent with the estimation given in Table 4 for the HR building (corresponding to the Nice Prefecture building on a simplified soil stratigraphy).

The analysis of records during the 2014 ${\mathrm{M}}_{\mathrm{w}}\hspace{0.17em}4.9$ Barcelonnette earthquake gives a much lower rocking amplitude ratio $RA$ equal to $3.6\hspace{0.17em}\%$ in the direction of the first mode shape and $1\hspace{0.17em}\%$ in the orthogonal direction. This percentage is due to a low energy content in the 2014 ${\mathrm{M}}_{\mathrm{w}}\hspace{0.17em}4.9$ Barcelonnette earthquake at the fundamental frequency of the building. On the contrary, the 2009 ${\mathrm{M}}_{\mathrm{w}}\hspace{0.17em}6.3$ L’Aquila earthquake, used as the input motion in the previous cases (Table 4), shows a high energy content at the fundamental frequency of the analyzed buildings (Figure 4).

8. Conclusions

The rocking motion can be highlighted using instrumentation set up at the base of the buildings, allowing the measure of vertical displacement at opposite corners. In this research, three parameters are defined to investigate the rocking motion:

• The rocking ratio $\left(RR\right)$, detecting the phase shift of vertical displacement time history at the corners of the building base and indicating when rocking motion occurs for a specific frequency band;
• The rocking amplitude ratio $\left(RA\right)$, an amplitude criterion proposed to estimate the part of rocking motion on the whole horizontal motion at the top of the building, allowing the estimation of an expected rocking tendency;
• The rocking spectrum $\left(RS\right)$ indicates the frequency bands in which the rocking motion is amplified because the first mode is not necessarily the only one affected by the swaying motion, and higher modes can also be affected.

The proposed tools are adapted to in situ measurements, and they can be applied to both earthquake recordings and ambient vibrations. In the case of seismic records, the rocking ratio $RR$ and rocking amplitude ratio $RA$ are estimated using the vertical displacement time history. Conversely, in the case of ambient vibration records, the use of random decrement functions is proposed.

In this research, the structural response of buildings is obtained numerically in a finite element scheme, simulating the vertical wave propagation in a limited soil domain assembled with the 3D building structure. The building dynamic features are estimated from inverse analysis of the structural response to white noise vibration. First, the rocking motion tendency is detected by comparing the dynamic features of the buildings in both cases of soil–structure interaction and fixed base condition. In agreement with previous studies, the natural frequencies of the analyzed buildings decrease when the rocking motion amplitude increases when compared with the fixed base condition. Depending on the building aspect ratio, all the analyzed low-rise buildings are less sensitive to rocking effects than high-rise buildings. The detected rocking motion tendency is verified and quantified using the proposed tools, and similar results are obtained using both white noise and seismic input. The application to structural motion records of the Nice prefecture building confirms the presence of rocking effects already detected in previous studies.