# Dynamic Modification and Damage Propagation of a Two-Storey Full-Scale Masonry Building

Authors: Leonidas Alexandros S. Kouris; Andrea Penna; Guido Magenes
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

The progressive change of modal characteristics due to accumulated damage on an unreinforced masonry (URM) building is investigated. The stone URM building, submitted to five consecutive shakings, has been experimentally studied on the shaking table of EUCENTRE laboratory (Pavia, Italy). The dynamic characteristics of the test specimen are analytically estimated using frequency and state-space modal identification from ambient vibration stationary tests carried out before the strong motion transient tests at various levels of damage. A singular value (SV) decomposition of the cross-correlation matrix of the acceleration response in the frequency domain is applied to determine the modal characteristics. In the time domain, the subspace state-space system identification is performed. Modal characteristics evolve from the initial linear state up to the ultimate collapse state in correlation with accumulated damage. Modal frequencies shorten with increasing intensity, whereas modal damping ratios are enhanced. Modal shapes also change with increasing level of accumulated damage. Comparing the evolution of modal characteristics, it is concluded that modal damping ratio shift can be better correlated with the system’s actual performance giving a better representation of damage than that of natural frequency shift ratio or the modes difference.


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## Body

## 1. Introduction

During strong seismic vibrations, the dynamic characteristics of structures undergo changes. Three properties of a structure, i.e., mass, damping, and stiffness, influence its dynamic parameters which are natural frequencies, modal damping, and mode shapes. Hence, a certain variation of a specific physical feature will cause a respective change in the natural vibratory properties. During earthquake shaking, structural damage leads to modification of the modal stiffness and damping and mode shapes and consequently to the dynamic characteristics. This process can take place even due to low to moderate shaking of a structure [1, 2]. Such an accumulating damage will downgrade the seismic capacity of the structure [3]. Thus, a dynamic identification at intermittent moments is necessary to monitor the current structural state of a building, track down damage initiation, and capture the evolution of crack propagation [4, 5]. Investigations of the fundamental frequency shift ratio due to damage have been experimentally conducted for reinforced concrete structures [6, 7].A refined approach for modal identification is the state-space identification (SSI) which is capable of distinguishing noise in the measurements. Often, masonry structures are structural systems with distributed mass without stiff diaphragms needing a large number of degrees of freedom (DOFs) for the adequate description of the response. In such cases, the application of SSI may be quite cumbersome. On the other hand, the dynamic identification in the frequency domain using, for example, the frequency domain decomposition (FDD) [8, 9] or the peak-picking method (PP) can be sufficient to detect the dynamic properties through global modal analysis.The interaction between the structural system and its components is the main tool to quantify and assess damage status and cracking localisation and has been applied widely in terms of frequency and modal variations [10–13]. System identification is based on a linear filter model such as the extended Kalman filter [14–16] for the estimation of dynamic characteristics. Stochastic state-space models in the time domain provide the possibility of modelling noise, and the Markov chain properties result in clear formulation. Several algorithms for the realization of the system have been proposed such as the N4SID [17] and the Eigen realization algorithms [18]. Despite its computational cost, state-space formulation has the merit that it is not likely to be unstable or ill conditioned [19].A shaking table experiment has been performed applying an incremental and consecutive procedure: five time-histories of a scaled real recording are used to shake the structure up to a near collapse damage state. In this series of tests, damage is accumulated influencing the natural properties. The data used to estimate the dynamic properties of the structure are those from ambient vibration tests which come before each strong motion test. Hence, changes in resonant frequencies, damping ratios, and mode shapes signify damage propagation and entail nonlinear behaviour of the preceding transient response.Two methods are applied for detecting the dynamic characteristics using ambient vibrations. Moreover, the current investigation aims at studying the evolution of the natural characteristics in terms of several macro-indices of the shaking intensity: (a) the peak ground acceleration (PGA), (b) the spectral acceleration, (c) the power of the accelerogram (Arias intensity), (d) the Housner intensity, (e) the root mean square (RMS) acceleration, and (e) the displacement ductility. Displacement ductility is not a direct intensity measure but rather characterises the response of the structure. In this sense, it is similar to the macroseismic intensity which is defined in terms of the response and the observed damage of the buildings. The frequency shift ratio, the damping shift ratio, and the modal shape difference from test phase-to-test phase are compared.


## 2. Shaking Table Test Procedure and Response of the Building

A full-scale two-storey stone unreinforced masonry (URM) building with flexible diaphragms (Figure1(a)) is tested on the single-degree-of-freedom (DOF) shaking table of EUCENTRE in Pavia (Italy) [20] to investigate the nonlinear behaviour and the failure mechanism of existing URM structures. The building is made of double-leaf stone masonry walls and timber floor and roof and has in-plan dimensions 4.70 × 6.60 m and total height 6 m including the 1 m high double-pitched roof. A series of characterisation tests for the double-leaf stone masonry have been carried out; the compressive strength is equal to 3.3 MPa and the modulus of elasticity 25.4 GPa, while the density was estimated 2.3 ton/m3 and the total mass 83 ton approximately [21–23]. A series of five shaking tests for a consecutively increasing seismic intensity is performed by scaling up the selected ground motion (natural accelerogram recorded at the Ulcinj-Hotel Albatros station during the 1979 Montenegro earthquake) applied along the longitudinal direction of the specimen. The test structure during the fifth phase exhibited a local failure mechanism characterised by rocking response of a part of the north second floor. Figure 1(b) presents the sequence of the test phases; each strong motion test is accompanied by a random stationary vibration test.Figure 1
(a) The building under testing and (b) sequence of the strong seismic inputs.
(a)
(b)Acceleration sensors are installed near the key structural points of the unreinforced stone masonry building to measure its response during ambient vibration and strong motion oscillation (Figure2).Figure 2
Location of the capacitive accelerometers (red) on the critical zones of the building. (a) First floor. (b) Roof level.
(a)
(b)The building suffered minor cracks during the first three tests performed at PGAs from 0.07 to 0.32 g, as shown in Figure3. During the fourth test at a PGA = 0.51 g, damage was extensive.Figure 3
Crack pattern during test phases 1, 2, and 3. (a) North facade. (b) West facade. (c) South Facade. (d) East Facade.
(a)
(b)
(c)
(d)Almost collapse was reached during the fifth phase at a PGA = 0.63 g as shown in Figure4. The collapse mechanism involved the upper part of the north wall which tended to overturn out-of-plane, the two spandrel beams and the wooden lintels in the east and west walls, which rotated and slid on the lintel support, and the gable wall of the south façade, which also tended to overturn following the displacement of the north wall due to their connection through the ridge beam (Figure 4). The collapse mechanism has already been formed during the fourth test phase during which the west wall displayed stepped diagonal cracks at its top and bottom of the second floor north pier [24]. In the final sixth test, the structure was mildly reinforced with ties to keep the out-of-plane facades together with the in-plane ones. The ties have diameter 18 mm and were inserted in preexisting pipes of 30 mm diameter. Their characteristic yielding strength is 670 MPa.Figure 4
Snapshot of the building during the fifth strong motion test.The ambient vibration tests come before the actual strong motion test. Hence, six ambient tests have been performed. In this investigation only those measurements have been used for the dynamic identification. Henceforth, the reference to the test phases is always for the ambient vibration tests.


## 3. Dynamic Identification in the Frequency Domain

Modal identification is performed in both the frequency and time domains using the acceleration data obtained from the stationary vibration tests.


### 3.1. Estimation of Modal Frequencies

Frequency domain decomposition (FDD) [8] is applied to the ambient vibration stationary responses at the key points of the structure (Figure 2). The frequency variations of the building under consecutive oscillations with gradually increasing intensity are relevant to the nonlinear response of the building.The position of the sensors (Figure2) aims at an adequate representation of the expected response of the structure. The timber floor and roof act as flexible diaphragms whose stiffnesses decrease with increasing intensity and damage. Therefore, the response should at least be detected at the intersections and the middle of the walls and the middle of the diaphragms. Moreover, the response of the pitched roof is measured by three sensors at the edges and the middle of the ridge beam and the excitation at the base by four more sensors at the base level of the building and on the shaking table.FDD is able to decompose modes, even when they are very close and coupled [8]. The method has been extensively applied for operational modal analysis showing reliable results [25, 26]. The first step of this method is to estimate the power spectral density (PSD) matrices Pij between i and j locations of each test phase. The rank of PSD matrices Pijjω is 26 × 26 on each discrete frequency ωk given that 26 acceleration measurements record simultaneously the response of the structure in two directions. Hence, PSD matrices Pijjω are three dimensional complex arrays whose third dimension is the frequency jωk. The sampling frequency Fs of the ambient vibration tests is 240 Hz. The recordings have a length of N = 36,000 measurements. The PSD functions Pijjω are estimated using the Welch method: (i) each segment has a finite length of m = 213 measurements, (ii) multiplied by a hamming window, (iii) 50% overlapped, (iv) and finally averaged over the total length. Hence, the frequency resolution is k = 1:4097 discrete frequencies ωk given the Nyquist frequency. The singular value decomposition (SVD) of the estimated PSD matrices Pijjω at each frequency is then performed:(1)P^ijjωk=UkSkUk¯T.MatrixS includes the singular values λk, which by definition are unique, and matrix U includes the singular vectors {φk}. The peaks of the singular values coincide with modal frequencies. The first singular vector is proportional to the modal shapes. Therefore, k singular matrices Sk and Uk are found at each discrete frequency ωk. The first value of the singular matrix is U1,1 is plotted against the respective frequency ωk.The identification of natural vibration frequency is based on the PP method in the framework of the SDOF approach. A more rational procedure would be as follows: (i) the first step is to identify the bell-shaped regions which constitute regions of resonance with the natural frequencies of the structure; (ii) each such region is isolated and a normal Gaussian function is fitted using the least square fitting procedure; (iii) the mean and the variance of the normal distribution are obtained which give the natural frequency; and (iv) the success of the fitting with the normal distribution is verified with the chi-square goodness-of-fit test. The comparison of the classical PP method and the normalised procedure [9] is shown in Figure 5 where resonant frequencies are identified. Comparing the two methodologies, the better accuracy of the normalised procedure is verified.Figure 5
Identification of natural frequencies using the FDD and the peak-picking method (blue circles) and the normalised procedure (green circles) for test phases 1 to 6.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.2. Estimation of Modal Shapes

At the identified natural frequenciesωn, the singular matrices Un are selected whose first column U(:,1) includes the singular vectors {φn} of the structure. The identified modes are complex due to the presence of noise and damping. For common damping ratios and high signal to noise ratios, the angle of the complex mode representing the phase lag is near 0 ± α·π, where α is a scalar. Thus, the cosine is close to ±1 and specifies the sign of the modal amplitude. The modal shapes estimated using the singular matrices of equation (1) are illustrated in Figure 6. For each test phase, the first four modal shapes are depicted. The first mode for the first test phase is translational modal across the longitudinal X-axis. The second mode is translational in the two axes. The third mode is a rotational modal shape. The fourth one includes the out-of-plane response of the north and south walls. Modal shapes progressively change.Figure 6
Modal shapes applying FDD for the test phases 1 to 6: in each row the first four modal shapes of each test phase.In Table1 the modal participation factors (MPFs) in x and y axes for the first four modes are presented. In y direction, the second mode is the prevailing one with increasing MPF. This should be attributed to the increased intensity. In x direction, there is variation of the MPFs during the test phases. In phases 4 and 5, the mode related to the out-of-plane response of the walls appears very influencing.Table 1
Modal participation factors (MPFs) inx and y directions of the six test phases.
MPF inx direction
MPF iny direction
1
2
3
4
1
2
3
4
Phase
1
0.67
0.13
0.11
0.05
0.00
0.61
0.18
0.16
2
0.65
0.12
0.12
0.06
0.00
0.69
0.10
0.15
3
0.59
0.17
0.14
0.05
0.07
0.69
0.03
0.14
4
0.52
0.17
0.01
0.23
0.06
0.86
0.03
0.01
5
0.42
0.01
0.01
0.46
0.00
0.86
0.01
0.06
6
0.52
0.13
0.24
0.06
0.00
0.89
0.04
0.01


### 3.3. Estimation of Modal Damping Ratios

The modal damping ratios are estimated applying the inverse Fourier transform on the PSD functionPijjω. Since Pijjω is proportional to the transfer function, the inverse Fourier transform of Pijjω reflects the impulse response function of the structure (Figure 7).Figure 7
The impulse response functions of the first mode of the structure for test phases 1 to 6.
(a)
(b)
(c)
(d)
(e)
(f)The modal damping ratioζn of the n mode representing the viscous and hysteretic damping can be estimated from the logarithmic decrement δ of the impulse response using the following empirical formula:(2)ζn=δnδn2+4π2.The application of equation (2) is performed in a statistical sense because for each n mode, the logarithmic decrease δn is not constant along the duration as shown in Figure 7. The average value of the logarithmic decrease δn is used to estimate the modal damping ratio ζn. The evolution of the first natural frequency and modal damping ratio is presented in Table 2.Table 2
Natural frequencies and modal damping ratios for the structure during ambient test phases 1 to 6.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
f (Hz)
7.47
7.36
7.15
6.85
5.85
6.85
ζ (%)
6.38
6.45
6.73
7.11
11.65
7.97


## 3.1. Estimation of Modal Frequencies

Frequency domain decomposition (FDD) [8] is applied to the ambient vibration stationary responses at the key points of the structure (Figure 2). The frequency variations of the building under consecutive oscillations with gradually increasing intensity are relevant to the nonlinear response of the building.The position of the sensors (Figure2) aims at an adequate representation of the expected response of the structure. The timber floor and roof act as flexible diaphragms whose stiffnesses decrease with increasing intensity and damage. Therefore, the response should at least be detected at the intersections and the middle of the walls and the middle of the diaphragms. Moreover, the response of the pitched roof is measured by three sensors at the edges and the middle of the ridge beam and the excitation at the base by four more sensors at the base level of the building and on the shaking table.FDD is able to decompose modes, even when they are very close and coupled [8]. The method has been extensively applied for operational modal analysis showing reliable results [25, 26]. The first step of this method is to estimate the power spectral density (PSD) matrices Pij between i and j locations of each test phase. The rank of PSD matrices Pijjω is 26 × 26 on each discrete frequency ωk given that 26 acceleration measurements record simultaneously the response of the structure in two directions. Hence, PSD matrices Pijjω are three dimensional complex arrays whose third dimension is the frequency jωk. The sampling frequency Fs of the ambient vibration tests is 240 Hz. The recordings have a length of N = 36,000 measurements. The PSD functions Pijjω are estimated using the Welch method: (i) each segment has a finite length of m = 213 measurements, (ii) multiplied by a hamming window, (iii) 50% overlapped, (iv) and finally averaged over the total length. Hence, the frequency resolution is k = 1:4097 discrete frequencies ωk given the Nyquist frequency. The singular value decomposition (SVD) of the estimated PSD matrices Pijjω at each frequency is then performed:(1)P^ijjωk=UkSkUk¯T.MatrixS includes the singular values λk, which by definition are unique, and matrix U includes the singular vectors {φk}. The peaks of the singular values coincide with modal frequencies. The first singular vector is proportional to the modal shapes. Therefore, k singular matrices Sk and Uk are found at each discrete frequency ωk. The first value of the singular matrix is U1,1 is plotted against the respective frequency ωk.The identification of natural vibration frequency is based on the PP method in the framework of the SDOF approach. A more rational procedure would be as follows: (i) the first step is to identify the bell-shaped regions which constitute regions of resonance with the natural frequencies of the structure; (ii) each such region is isolated and a normal Gaussian function is fitted using the least square fitting procedure; (iii) the mean and the variance of the normal distribution are obtained which give the natural frequency; and (iv) the success of the fitting with the normal distribution is verified with the chi-square goodness-of-fit test. The comparison of the classical PP method and the normalised procedure [9] is shown in Figure 5 where resonant frequencies are identified. Comparing the two methodologies, the better accuracy of the normalised procedure is verified.Figure 5
Identification of natural frequencies using the FDD and the peak-picking method (blue circles) and the normalised procedure (green circles) for test phases 1 to 6.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.2. Estimation of Modal Shapes

At the identified natural frequenciesωn, the singular matrices Un are selected whose first column U(:,1) includes the singular vectors {φn} of the structure. The identified modes are complex due to the presence of noise and damping. For common damping ratios and high signal to noise ratios, the angle of the complex mode representing the phase lag is near 0 ± α·π, where α is a scalar. Thus, the cosine is close to ±1 and specifies the sign of the modal amplitude. The modal shapes estimated using the singular matrices of equation (1) are illustrated in Figure 6. For each test phase, the first four modal shapes are depicted. The first mode for the first test phase is translational modal across the longitudinal X-axis. The second mode is translational in the two axes. The third mode is a rotational modal shape. The fourth one includes the out-of-plane response of the north and south walls. Modal shapes progressively change.Figure 6
Modal shapes applying FDD for the test phases 1 to 6: in each row the first four modal shapes of each test phase.In Table1 the modal participation factors (MPFs) in x and y axes for the first four modes are presented. In y direction, the second mode is the prevailing one with increasing MPF. This should be attributed to the increased intensity. In x direction, there is variation of the MPFs during the test phases. In phases 4 and 5, the mode related to the out-of-plane response of the walls appears very influencing.Table 1
Modal participation factors (MPFs) inx and y directions of the six test phases.
MPF inx direction
MPF iny direction
1
2
3
4
1
2
3
4
Phase
1
0.67
0.13
0.11
0.05
0.00
0.61
0.18
0.16
2
0.65
0.12
0.12
0.06
0.00
0.69
0.10
0.15
3
0.59
0.17
0.14
0.05
0.07
0.69
0.03
0.14
4
0.52
0.17
0.01
0.23
0.06
0.86
0.03
0.01
5
0.42
0.01
0.01
0.46
0.00
0.86
0.01
0.06
6
0.52
0.13
0.24
0.06
0.00
0.89
0.04
0.01


## 3.3. Estimation of Modal Damping Ratios

The modal damping ratios are estimated applying the inverse Fourier transform on the PSD functionPijjω. Since Pijjω is proportional to the transfer function, the inverse Fourier transform of Pijjω reflects the impulse response function of the structure (Figure 7).Figure 7
The impulse response functions of the first mode of the structure for test phases 1 to 6.
(a)
(b)
(c)
(d)
(e)
(f)The modal damping ratioζn of the n mode representing the viscous and hysteretic damping can be estimated from the logarithmic decrement δ of the impulse response using the following empirical formula:(2)ζn=δnδn2+4π2.The application of equation (2) is performed in a statistical sense because for each n mode, the logarithmic decrease δn is not constant along the duration as shown in Figure 7. The average value of the logarithmic decrease δn is used to estimate the modal damping ratio ζn. The evolution of the first natural frequency and modal damping ratio is presented in Table 2.Table 2
Natural frequencies and modal damping ratios for the structure during ambient test phases 1 to 6.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
f (Hz)
7.47
7.36
7.15
6.85
5.85
6.85
ζ (%)
6.38
6.45
6.73
7.11
11.65
7.97


## 4. Dynamic Identification in the Time Domain

The state-space formulation applied in the discrete (digitised) time domain is compared to FDD. This method is based on the transformation of the second-order differential system equation into the form of a first-order matrix differential equation. It is then possible to solve the latter and represent the system by a model which can be used to retrieve the dynamic response of any general input excitation. Therefore, state-space formulation is a technique performed in the time domain, yet applicable only for linear systems. It should be emphasized that the shaking table tests involve a highly nonlinear (NL) response as the strong motion tests proceed to larger intensities, but ambient vibration tests are performed in very low magnitude to measure the “linear” response of the gradually damaged structure. Therefore, the analysis corresponds to snapshots of the dynamic properties instantly appearing as damage is accumulated.The advantage of a state-space formulation is that describes all signals simultaneously establishing the multi-input multi-output (MIMO) system as a time-varying system readily. However, it should be noted that state-space formulation demands a high burden of computational work for setting up a complex system. The number of state variables is defined as the minimal set of variables needed to determine a possible future behaviour of the system from the system’s inputs.The DOFs of the structure under investigation are equivalent to the number of the accelerometers. The state variables of the system are determined converting the second-order differential equation into a set of two first-order differential equations. For a MIMO stochastic system affected by noise withn state variables, p inputs, and q outputs, they are as follows:(3a)x¯˙t=A⋅xt+B⋅ut+wkt,(3b)yt=C⋅xt+D⋅ut+vkt.In equations (3a) and (3b), x¯t is the state vector, yt is the output, ut is the input which is measurable, and wkt and vkt are noise vectors which are not measurable. Moreover, A is called the state (dynamic) matrix, B is called the input matrix, C is called the output (sensor) matrix, and D is called the feed through matrix.For this experiment, where the data are retrieved from discretised time measurements, theZ-transform of equations (3a) and (3b) will result in the following formula:(4)Hz=CzI−A−1B+D.Having determined the frequency response functionHz from the state-space model of the system, the poles λk of the frequency response function Hω can be estimated along with the residue ajr for the modal amplitude {Φk} at frequencies ωk and the modal damping ratios ζk using the following equations where k is the order of the system:(5a)Hω=∑k=12Nakjrζkωk+jω−ωk1−ζk2+akjr∗ζkωk+jω+ωk1−ζk2,(5b)λk=−ζkωk±jωk1−ζk2.The pole-zero plot of the identified discrete dynamic system for phases 1 and 2 in the complexz-plane is presented in Figure 8. This figure shows the location in the complex plane of the poles indicated by an (×) and that there are no zeros (○) in the transfer function. The systems are stable and underdamped. Comparing the two systems, changes can be tracked down. Similar figures are derived for the identified systems of the next phases.Figure 8
The pole-zero maps of the phases (a) 1 and (b) 2; no zeroes detected.
(a)
(b)The physical representation of the system∑A,B,C,D is obtained using a similarity transformation [27] since the direct interpretation of the state-space matrices is only possible in limited ideal cases, e.g., a simply supported beam in bending [28]. The SV decomposition of the discrete state matrix A is applied as follows [29]:(6)Α=ΨΛΨ−1.In equation (6), Λ is a uniquely determined diagonal matrix containing the complex singular values {λk} of A which represent the eigenvalues of the system and Ψ is a square matrix. The eigenvectors Φ = {φi} of the system are obtained as follows:(7)Φ=CΨ.It should be emphasized that the eigenvaluesλk of the discrete system should be transformed to the continuous-time model. Let λi denote the eigenvalues of the continuous-time system and Ac the continuous-time state matrix. Then, it can be easily found that [30, 31](8)A=e1/FsA¯c,λi=Fslnλk,ζi=Reλiλi.It is noted here that the transformation of equation (8) is not unique as any ±κ · 2π, κ∈Z can be added to the natural logarithm [30].The stochastic subspace identification (SSI) is applied in Matlab using the N4SID algorithm [32] to obtain the eigenvalues and the SVD implemented in the system identification toolbox [33]. The method is not iterative and thus numerically stable, and the estimation of the state matrices is optimal.The identified natural frequency and the respective modal damping for each test phase are presented in Table3 and all modal frequencies vs. the model order are plotted in the stabilisation diagram of Figure 9 where the stabilised frequencies are denoted with a red star.Table 3
Natural frequencies and modal damping ratios for the structure during ambient test phases 1 to 5.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
f (Hz)
7.38
7.20
7.03
6.42
5.68
6.78
ζ (%)
6.07
6.48
6.71
7.36
11.21
7.74Figure 9
Natural frequencies and damping vs. model order applying using the subspace state-space system identification approach for the 5 test phases. (a) Stabilisation diagram: phase 1. (b) Stabilisation diagram: phase 2. (c) Stabilisation diagram: phase 3. (d) Stabilisation diagram: phase 4. (e) Stabilisation diagram: phase 5. (f) Stabilisation diagram: phase 6.
(a)
(b)
(c)
(d)
(e)
(f)From equation (5a), it is obvious that the identified modes exist always in pairs and are complex due to the presence of noise and damping. Only half of the 2N modes of the system are taken into account since the rest half is the same modes in the negative axis. For common damping ratios and high signal to noise ratios, the angle of the complex mode representing the phase lag is near 0 ± κ · π (κ∈Z), and thus the cosine is close to ±1 specifying the sign of the modal amplitude. However, not all of these N modes are true modes but only those that are stabilised in terms of frequency and damping ratio assuming a tolerance error 2% for natural frequencies and 5% for damping ratios [34].The identified modal shapes using equation (7) are presented in Figure 10 arranged in six lines corresponding to the five phases of the test as well as the original state before any shaking.Figure 10
Modal shapes applying subspace state-space N4SID modelling for the 5 test phases: in the first row, the first four modal shapes of the 1st phase; in the second row, the first modal shape of the next phases.


## 5. Comparisons

The modals shapes obtained using the FDD and the SSI are compared using the modal assurance criterion (MAC) [35]. MAC estimates the relation between two modal shapes {Φ1} and {Φ2} through the following equation where ∗T stands for the complex conjugate and transpose:(9)MACΦ1,Φ2=Φ1∗TΦ22Φ1∗TΦ1Φ2∗TΦ2.A MAC value between two modes greater than 80% is considered to refer to the same modal deformed shapes. In Figure11, the vertical axis is the MAC coefficient, whereas the modal shapes derived using the time and the frequency domain methods are across the horizontal axes.Figure 11
The MAC coefficient between the modes derived using the frequency domain decomposition (FDD) and stochastic subspace identification (SUB). (a) Phase 1. (b) Phase 2. (c) Phase 3. (d) Phase 4. (e) Phase 5. (f) Phase 6.
(a)
(b)
(c)
(d)
(e)
(f)Therefore, comparing the FDD and the SSI modal shapes, it is seen that for all the test phases, the MAC value is higher than 80% for the diagonals which represent the main modal shapes. For the first test phase, only three modes are in correspondence between the time and the frequency domain methods due to the low signal to noise ratio, but this number escalates quickly as the ratio increases. Hence, the comparison shows that the derived modal shapes are in good agreement drawing to the conclusion that FDD can indeed be a rather efficient and reliable method for modal identification of URM structures.


## 6. Modification of Modal Characteristics

In this section, the gradual variation of the modal characteristics of the building is investigated. Furthermore, a curve fitting has been attempted to model this variation; linear or higher-order polynomials and exponential or rational functions would be appropriate for regression analysis with the ascending trend of the data as will be shown below. For comparison reasons, only polynomials have been used to model the variations of the dynamic characteristics. In all regressions below, the coefficient of determination (usually denoted asR2) attains a high value 98-99% approximately. Simpler trends (linear or quadratic) have been detected for modal frequencies and higher order (cubic) for damping. Some regression equations are provided for comparison.


### 6.1. Modal Frequencies

The gradual change of the natural frequencies is mostly visible for the first three modes. This is visualised in Figure12 where it is shown in a 3D graph the change of the natural frequencies with PGA. After the fourth mode, changes in the frequency are less than 2% and for the second and the third modes are less than 10%. Hence, the investigation focuses only on the first natural modal parameters.Figure 12
Variation (%) of the modal frequencies with PGA.In order to characterise the variation of the first natural frequency, a number of intensity measures are applied: (i) the PGA (% g), (ii) the Arias intensity, i.e., the power of the acceleration time-history [36] (%), (iii) the Housner intensity (m/s), (iv) the root mean square (RMS) acceleration (%), and the attained displacement ductility (%). The ductility is defined as follows:(10)μ=Δu,topΔy,top.In equation (10) Δu,top is the ultimate horizontal relative displacement at the roof of the building and Δy,top is the yielding relative displacement at the same point. These displacements have been estimated subtracting the displacement of the base. Therefore, if δ is the absolute displacement, then Δtop = δtop − δbase. As already mentioned, the collapse mechanism of the structure involves the rocking response of the roof. Therefore, the ductility defined in equation (10) reflects the local ductility of the system rather than the whole structure.In Figure13, the variation of the first natural frequency is presented against the previous measures of intensity; with dashed lines, the curve fitting is presented for each one of the methods of dynamic identification separately (only for PGA and Arias intensity) and with black solid line the curve fitting against both.Figure 13
Degradation of natural frequency and regression models for FDD and SSI methods.
(a)
(b)
(c)
(d)
(e)As expected, the frequency is progressively decreasing with increasing seismic intensity. It is very important to note that though the visual inspection did not detect any significant crack during the first three test phases, there was a rather significant shifting of the frequency, 20% approximately. Therefore, even structures that seem undamaged after a minor shock may already have a frequency shift which deteriorates their vulnerability state. Moreover, it is interesting to note that reinforcement of structure succeeds in restoring the frequency shift to a certain extent.The ascending evolution of the frequency ratio (i.e., the elasticfel to the inelastic finel frequency) against PGA can be modelled by a second-order polynomial. The polynomial fitting results in the following equation:(11)felfinel=1+0.18PGA+0.83PGA2.It is obvious that equation (11) is developed for the specific structure excited by the specific earthquake. It is interesting to note that the rate of frequency ratio is linear, i.e., it keeps increasing linearly with PGA up to collapse. The same regression polynomial order has been applied to model the frequency shift vs. acceleration RMS, Housner intensity, and ductility.Plotting the frequency shift against Arias intensity (AI) can be well approximated by a line which implies a constant rate of change. Below follows the trend of the frequency shift after linear regression:(12)felfinel=1+1.5AI.Equation (12) has a high coefficient of determination over 98%.The application of the strengthening using tie rods succeeds in reinstating the stiffness of the structure and thus the frequency to a certain extent. Obviously, this extent depends exclusively on the strengthening measures.


### 6.2. Modal Damping Ratios

The evolution of modal damping ratio with seismic intensity is presented in Figure14. Modal damping develops in a similar trend for all intensity quantities: (a) for the first three phases, the modal damping increase is almost linear; (b) then, for next phase, there is a sharp escalation of modal damping; (c) finally, the reinforced structure presents lower damping. These findings are close to the structural state of the structure: only during phase 4, the collapse mechanism is formed. The reinforcement of the structure has exactly the expected result: the tie rods tend to restore the structure at the damage situation before the collapse mechanism. Hence, the structure returns to a cracked situation but avoiding the development of the out-of-plane collapse mechanism.Figure 14
Degradation of first modal damping for FDD and SSI methods.
(a)
(b)
(c)
(d)
(e)For all fit curve regressions, a cubic equation is used. The evolution of modal damping with ductility, useful for displacement based assessment or design, is provided below:(13)ζin0.05ζel=μ3−0.9μ2+0.7μ+19.2,μ≥1.In equation (13), μ is the ductility of the structure as defined by Equation (10).


### 6.3. Modal Shapes

MAC is used to investigate changes of modes in shape during gradual seismic intensity along with the normalised modal difference (NMD) [37]. As already mentioned, MAC shows the correlation between two sets of modal shapes [Φi] and [Φj]. NMD is introduced to show instead the average difference between two shapes [Φi] and [Φj]:(14)NMDΦi,k,Φj,m=1−MACΦi,k,Φj,mMACΦi,k,Φj,m.In equation (14), i and j are the subsequent test phases (i.e., i = 1 and j = 2 for the first and the second test phases) and k and m are the mode numbers. In the current investigation, NMD focuses on the first mode and therefore k = m = 1.From Figure15, it is obvious that for the first three phases, there is a high correlation of modal shapes exceeding 80%. However, with increasing seismic intensity, the mode coincidence in shape is becoming less obvious. The reinforced structure shows rather different modal shapes than the URM building.Figure 15
Degradation of MAC coefficient between subsequent phases from FDD method.
(a)
(b)
(c)
(d)
(e)Figure16 presents the evolution from phase to phase of the modal shapes derived using SSI method where a similar trend is depicted with method FDD presented in Figure 15. However, looking at the NMDs in Figure 17, it is observed that SSI presents higher differences. Specifically, after phase 1, the difference with initial state is estimated at 25% for the first mode. FDD method presents more justified NMDs: for the first two phases, NMD is very low, and then it increases to drop again at the reinforced structure.Figure 16
Degradation of MAC coefficient between subsequent phases from subspace state-space method.
(a)
(b)
(c)
(d)
(e)Figure 17
The evolution of NMD for the subsequent phases and for FDD and SSI methods.
(a)
(b)
(c)
(d)
(e)
(f)


## 6.1. Modal Frequencies

The gradual change of the natural frequencies is mostly visible for the first three modes. This is visualised in Figure12 where it is shown in a 3D graph the change of the natural frequencies with PGA. After the fourth mode, changes in the frequency are less than 2% and for the second and the third modes are less than 10%. Hence, the investigation focuses only on the first natural modal parameters.Figure 12
Variation (%) of the modal frequencies with PGA.In order to characterise the variation of the first natural frequency, a number of intensity measures are applied: (i) the PGA (% g), (ii) the Arias intensity, i.e., the power of the acceleration time-history [36] (%), (iii) the Housner intensity (m/s), (iv) the root mean square (RMS) acceleration (%), and the attained displacement ductility (%). The ductility is defined as follows:(10)μ=Δu,topΔy,top.In equation (10) Δu,top is the ultimate horizontal relative displacement at the roof of the building and Δy,top is the yielding relative displacement at the same point. These displacements have been estimated subtracting the displacement of the base. Therefore, if δ is the absolute displacement, then Δtop = δtop − δbase. As already mentioned, the collapse mechanism of the structure involves the rocking response of the roof. Therefore, the ductility defined in equation (10) reflects the local ductility of the system rather than the whole structure.In Figure13, the variation of the first natural frequency is presented against the previous measures of intensity; with dashed lines, the curve fitting is presented for each one of the methods of dynamic identification separately (only for PGA and Arias intensity) and with black solid line the curve fitting against both.Figure 13
Degradation of natural frequency and regression models for FDD and SSI methods.
(a)
(b)
(c)
(d)
(e)As expected, the frequency is progressively decreasing with increasing seismic intensity. It is very important to note that though the visual inspection did not detect any significant crack during the first three test phases, there was a rather significant shifting of the frequency, 20% approximately. Therefore, even structures that seem undamaged after a minor shock may already have a frequency shift which deteriorates their vulnerability state. Moreover, it is interesting to note that reinforcement of structure succeeds in restoring the frequency shift to a certain extent.The ascending evolution of the frequency ratio (i.e., the elasticfel to the inelastic finel frequency) against PGA can be modelled by a second-order polynomial. The polynomial fitting results in the following equation:(11)felfinel=1+0.18PGA+0.83PGA2.It is obvious that equation (11) is developed for the specific structure excited by the specific earthquake. It is interesting to note that the rate of frequency ratio is linear, i.e., it keeps increasing linearly with PGA up to collapse. The same regression polynomial order has been applied to model the frequency shift vs. acceleration RMS, Housner intensity, and ductility.Plotting the frequency shift against Arias intensity (AI) can be well approximated by a line which implies a constant rate of change. Below follows the trend of the frequency shift after linear regression:(12)felfinel=1+1.5AI.Equation (12) has a high coefficient of determination over 98%.The application of the strengthening using tie rods succeeds in reinstating the stiffness of the structure and thus the frequency to a certain extent. Obviously, this extent depends exclusively on the strengthening measures.


## 6.2. Modal Damping Ratios

The evolution of modal damping ratio with seismic intensity is presented in Figure14. Modal damping develops in a similar trend for all intensity quantities: (a) for the first three phases, the modal damping increase is almost linear; (b) then, for next phase, there is a sharp escalation of modal damping; (c) finally, the reinforced structure presents lower damping. These findings are close to the structural state of the structure: only during phase 4, the collapse mechanism is formed. The reinforcement of the structure has exactly the expected result: the tie rods tend to restore the structure at the damage situation before the collapse mechanism. Hence, the structure returns to a cracked situation but avoiding the development of the out-of-plane collapse mechanism.Figure 14
Degradation of first modal damping for FDD and SSI methods.
(a)
(b)
(c)
(d)
(e)For all fit curve regressions, a cubic equation is used. The evolution of modal damping with ductility, useful for displacement based assessment or design, is provided below:(13)ζin0.05ζel=μ3−0.9μ2+0.7μ+19.2,μ≥1.In equation (13), μ is the ductility of the structure as defined by Equation (10).


## 6.3. Modal Shapes

MAC is used to investigate changes of modes in shape during gradual seismic intensity along with the normalised modal difference (NMD) [37]. As already mentioned, MAC shows the correlation between two sets of modal shapes [Φi] and [Φj]. NMD is introduced to show instead the average difference between two shapes [Φi] and [Φj]:(14)NMDΦi,k,Φj,m=1−MACΦi,k,Φj,mMACΦi,k,Φj,m.In equation (14), i and j are the subsequent test phases (i.e., i = 1 and j = 2 for the first and the second test phases) and k and m are the mode numbers. In the current investigation, NMD focuses on the first mode and therefore k = m = 1.From Figure15, it is obvious that for the first three phases, there is a high correlation of modal shapes exceeding 80%. However, with increasing seismic intensity, the mode coincidence in shape is becoming less obvious. The reinforced structure shows rather different modal shapes than the URM building.Figure 15
Degradation of MAC coefficient between subsequent phases from FDD method.
(a)
(b)
(c)
(d)
(e)Figure16 presents the evolution from phase to phase of the modal shapes derived using SSI method where a similar trend is depicted with method FDD presented in Figure 15. However, looking at the NMDs in Figure 17, it is observed that SSI presents higher differences. Specifically, after phase 1, the difference with initial state is estimated at 25% for the first mode. FDD method presents more justified NMDs: for the first two phases, NMD is very low, and then it increases to drop again at the reinforced structure.Figure 16
Degradation of MAC coefficient between subsequent phases from subspace state-space method.
(a)
(b)
(c)
(d)
(e)Figure 17
The evolution of NMD for the subsequent phases and for FDD and SSI methods.
(a)
(b)
(c)
(d)
(e)
(f)


## 7. Conclusions

The results from modal identification applied for the evaluation of the equivalent linear dynamic properties of a consecutive full-scale shaking table test at various levels of damage suggest a frequency and damping shifting and a gradual change of the modal shapes. Two different approaches are applied using the ambient vibration acceleration measurements for extracting the modal characteristics: (i) firstly the FDD, an output-only method, in the frequency domain and then (ii) the SSI technique in the time domain using the input measurements as well. FDD based on the SDOF approach is a standard methodology for output-only measurements which provides intuitive justification of the results and requires less computational effort. Therefore, its application is always an effective tool with limited computational effort since the most demanding calculations are those of PSD and SVD. In the framework of FDD, a modified rationalised procedure for selecting the discrete natural frequencies is applied: a normal based fitting is proposed to estimate the natural frequencies. The proposed modification of the standard procedure succeeds in achieving higher accuracy in the selection of the natural frequencies and reliable estimates of damping as compared to SSI results. Modal damping ratios are estimated from the impulse response function applying the inverse Fourier transform of the response function. At a second step of the current investigation, modal analysis is performed in the time domain. The SSI technique is carried out using a combined input-output algorithm. A large number of calculations are necessary to obtain a stabilised system.Comparisons of the estimated modal characteristics show that both methods are in agreement for modal frequencies and damping ratios. Using the MAC coefficient, the modal shapes derived from the two methodologies are compared and high consistency is achieved especially from the third to the last phases. The signal to noise ratio affects only the first two phases. Hence, it is shown that FDD can be effectively employed in modal identification displaying robustness and accuracy similar to that obtained by SSI.The analyses show a degradation of the modal characteristics with increasing shaking intensity and accumulated damage. Under strong shaking, the structure presented extended nonlinear behaviour resulting in a frequency and damping shifting and modal shapes modification. A linear variation of the natural frequency shift ratio is observed when the intensity of the vibration is expressed in terms of Arias intensity and a quadratic one when expressed in terms of PGA, Housner intensity, RMS, or displacement ductility. The damping shift ratio instead can be expressed using a cubic function, implying a sharper rise in the fourth and fifth phases. NMD is almost equal for the first two phases and then increases linearly. The damping shift ratio shows a better correlation with the actual damage which appeared in the structure during the fourth and fifth phase than frequency shift ratio or NMD.



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*Source: 2396452-2019-06-02.xml*