Research Article  Open Access
Leonidas Alexandros S. Kouris, Andrea Penna, Guido Magenes, "Dynamic Modification and Damage Propagation of a TwoStorey FullScale Masonry Building", Advances in Civil Engineering, vol. 2019, Article ID 2396452, 21 pages, 2019. https://doi.org/10.1155/2019/2396452
Dynamic Modification and Damage Propagation of a TwoStorey FullScale Masonry Building
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 statespace 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 crosscorrelation matrix of the acceleration response in the frequency domain is applied to determine the modal characteristics. In the time domain, the subspace statespace 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.
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 statespace 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 peakpicking 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 statespace 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, statespace 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 timehistories 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 macroindices 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 phasetotest phase are compared.
2. Shaking Table Test Procedure and Response of the Building
A fullscale twostorey stone unreinforced masonry (URM) building with flexible diaphragms (Figure 1(a)) is tested on the singledegreeoffreedom (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 doubleleaf stone masonry walls and timber floor and roof and has inplan dimensions 4.70 × 6.60 m and total height 6 m including the 1 m high doublepitched roof. A series of characterisation tests for the doubleleaf 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/m^{3} 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 UlcinjHotel 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.
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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 (Figure 2).
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The building suffered minor cracks during the first three tests performed at PGAs from 0.07 to 0.32 g, as shown in Figure 3. During the fourth test at a PGA = 0.51 g, damage was extensive.
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Almost collapse was reached during the fifth phase at a PGA = 0.63 g as shown in Figure 4. The collapse mechanism involved the upper part of the north wall which tended to overturn outofplane, 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 outofplane facades together with the inplane ones. The ties have diameter 18 mm and were inserted in preexisting pipes of 30 mm diameter. Their characteristic yielding strength is 670 MPa.
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 (Figure 2) 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 between i and j locations of each test phase. The rank of PSD matrices is 26 × 26 on each discrete frequency ω_{k} given that 26 acceleration measurements record simultaneously the response of the structure in two directions. Hence, PSD matrices are three dimensional complex arrays whose third dimension is the frequency . The sampling frequency F_{s} of the ambient vibration tests is 240 Hz. The recordings have a length of N = 36,000 measurements. The PSD functions are estimated using the Welch method: (i) each segment has a finite length of m = 2^{13} 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 ω_{k} given the Nyquist frequency. The singular value decomposition (SVD) of the estimated PSD matrices at each frequency is then performed:
Matrix S includes the singular values λ_{k}, which by definition are unique, and matrix U includes the singular vectors {φ_{k}}. The peaks of the singular values coincide with modal frequencies. The first singular vector is proportional to the modal shapes. Therefore, k singular matrices S_{k} and U_{k} are found at each discrete frequency ω_{k}. The first value of the singular matrix is U_{1,1} is plotted against the respective frequency ω_{k}.
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 bellshaped 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 chisquare goodnessoffit 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.
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3.2. Estimation of Modal Shapes
At the identified natural frequencies ω_{n}, the singular matrices U_{n} are selected whose first column U(:,1) includes the singular vectors {φ_{n}} 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 Xaxis. The second mode is translational in the two axes. The third mode is a rotational modal shape. The fourth one includes the outofplane response of the north and south walls. Modal shapes progressively change.
In Table 1 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 outofplane response of the walls appears very influencing.

3.3. Estimation of Modal Damping Ratios
The modal damping ratios are estimated applying the inverse Fourier transform on the PSD function . Since is proportional to the transfer function, the inverse Fourier transform of reflects the impulse response function of the structure (Figure 7).
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The modal damping ratio ζ_{n} 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:
The application of equation (2) is performed in a statistical sense because for each n mode, the logarithmic decrease δ_{n} is not constant along the duration as shown in Figure 7. The average value of the logarithmic decrease δ_{n} is used to estimate the modal damping ratio ζ_{n}. The evolution of the first natural frequency and modal damping ratio is presented in Table 2.

4. Dynamic Identification in the Time Domain
The statespace formulation applied in the discrete (digitised) time domain is compared to FDD. This method is based on the transformation of the secondorder differential system equation into the form of a firstorder 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, statespace 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 statespace formulation is that describes all signals simultaneously establishing the multiinput multioutput (MIMO) system as a timevarying system readily. However, it should be noted that statespace 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 secondorder differential equation into a set of two firstorder differential equations. For a MIMO stochastic system affected by noise with n state variables, p inputs, and q outputs, they are as follows:
In equations (3a) and (3b), is the state vector, is the output, is the input which is measurable, and and 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, the Ztransform of equations (3a) and (3b) will result in the following formula:
Having determined the frequency response function from the statespace model of the system, the poles λ_{k} of the frequency response function can be estimated along with the residue a_{jr} for the modal amplitude {Φ_{k}} at frequencies ω_{k} and the modal damping ratios ζ_{k} using the following equations where k is the order of the system:
The polezero plot of the identified discrete dynamic system for phases 1 and 2 in the complex zplane 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.
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The physical representation of the system is obtained using a similarity transformation [27] since the direct interpretation of the statespace 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]:
In equation (6), Λ is a uniquely determined diagonal matrix containing the complex singular values {λ_{k}} of A which represent the eigenvalues of the system and Ψ is a square matrix. The eigenvectors Φ = {φ_{i}} of the system are obtained as follows:
It should be emphasized that the eigenvalues λ_{k} of the discrete system should be transformed to the continuoustime model. Let λ_{i} denote the eigenvalues of the continuoustime system and A_{c} the continuoustime state matrix. Then, it can be easily found that [30, 31]
It is noted here that the transformation of equation (8) is not unique as any ±κ · 2π, 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 Table 3 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.

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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 (), 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.
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 {Φ_{1}} and {Φ_{2}} through the following equation where stands for the complex conjugate and transpose:
A MAC value between two modes greater than 80% is considered to refer to the same modal deformed shapes. In Figure 11, 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.
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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 higherorder 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 as R^{2}) attains a high value 9899% 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 Figure 12 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.
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 timehistory [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:
In equation (10) Δ_{u,top} is the ultimate horizontal relative displacement at the roof of the building and Δ_{y,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 Δ_{top} = δ_{top} − δ_{base}. 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 Figure 13, 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.
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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 elastic f_{el} to the inelastic f_{inel} frequency) against PGA can be modelled by a secondorder polynomial. The polynomial fitting results in the following equation:
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:
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 Figure 14. 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 outofplane collapse mechanism.
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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:
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 [Φ_{i}] and [Φ_{j}]. NMD is introduced to show instead the average difference between two shapes [Φ_{i}] and [Φ_{j}]:
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 Figure 15, 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.
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Figure 16 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.
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7. Conclusions
The results from modal identification applied for the evaluation of the equivalent linear dynamic properties of a consecutive fullscale 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 outputonly 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 outputonly 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 inputoutput 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.
Data Availability
The data used to support the findings of this study are available for research purposes upon request to the authors.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The test campaign and the elaboration of the measurements were carried with the financial support of the “Masonry Structures” project within the ReLUIS 201416 research programme funded by the Italian Department of Civil Protection.
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Copyright © 2019 Leonidas Alexandros S. Kouris et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.