Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 916820, 22 pages

http://dx.doi.org/10.1155/2014/916820

## Seismic-Parameter-Based Statistical Procedures for the Approximate Assessment of Structural Damage

Department of Civil Engineering, Institute of Structural Mechanics and Earthquake Engineering, Democritus University of Thrace, Vas. Sofias 12, 67100 Xanthi, Greece

Received 9 August 2013; Revised 2 January 2014; Accepted 6 January 2014; Published 23 February 2014

Academic Editor: Yuri Petryna

Copyright © 2014 A. Elenas. 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.

#### Abstract

This study describes two statistical methodologies to estimate the postseismic damage status of structures based on seismic parameters as novel combined procedures in earthquake engineering. Thus, a multilinear regression analysis and discriminant analysis are utilized considering twenty seismic parameters. Overall damage indices describe the postseismic damage status. Nonlinear dynamic analyses furnish the damage indices, which are considered as exact indices and references for the subsequent study. The aim is to approximate the postseismic damage indices or the damage grade of buildings using statistical methods, thus avoiding complex nonlinear dynamic analyses. The multilinear regression procedure evaluates the damage indices explicitly, and the discriminant analysis furnishes the damage grade of the structures. The proposed methods are applied to a frame structure. A set of 400 natural accelerograms is used for the training phase of the models. The quality of the models is tested initially by the same set of natural accelerograms and then by a blind prediction using a second set of synthetic accelerograms. The results of both proposed methods have shown a correct classification percentage ranging from 87.75% to 97.50% and from 70% to 90% for the sets of the natural and synthetic accelerograms, respectively.

#### 1. Introduction

Several seismic parameters have been presented in the literature during the last several decades. These parameters can be used to express the intensity of the seismic excitations and to simplify their description. Postseismic field observations and numerical investigations have indicated the interdependency between the seismic parameters and damage status of buildings after earthquakes [1, 2]. The latter can be expressed by proper damage indices, whereas the interdependency between the considered quantities can be quantified numerically by appropriate correlation coefficients.

This paper expands on the above, providing alternative statistical procedures for the approximate assessment of the structural Postseismic damage grade. Thus, the first step in the proposed new methodology is to choose a set of seismic parameters that represents the properties of the accelerograms. Next, the intensity parameters chosen in the first step are evaluated for a set of natural or artificial accelerograms. The damage indices for all considered accelerograms are subsequently calculated for the building of interest. Statistical analyses are applied to the resulting data from the previous steps to extract a statistical model for the prediction of the Postseismic damage status of structures. Finally, the new methodology is verified by a blind prediction.

As mentioned previously, the proposed statistical methods are used to estimate the Postseismic structural damage status. Approximation methods are well known and broadly used in earthquake engineering. The evaluation of the fundamental period of a building from its geometric dimensions is such an approximation provided by several antiseismic codes. Thus, the fundamental period can be approximated using a simple formula depending on the type of structure (e.g., steel, reinforced concrete, masonry, frame, or wall) and the geometric dimensions of the examined building instead of needing to solve a complicated eigenvalue problem that depends on mass and stiffness matrices.

The proposed methodologies can be used, for instance, by public authorities for a quick and accurate estimation of the Postseismic damage status or for estimating damage scenarios for specific buildings. The proposed techniques are simpler to apply than the more complicated and time-consuming nonlinear dynamic analyses. The novel application of the proposed statistical procedures in the specific field of earthquake engineering is also feasible for nonspecialists, in contrast to nonlinear dynamic analyses, which can only be performed by expert engineers.

#### 2. Seismic Intensity Parameters

In general, the intensity parameters can be classified into peak, spectral, and energy parameters. Twenty parameters have been selected to represent the seismic excitation. They have been chosen from all three seismic parameter categories. The following seismic parameters are considered: peak ground acceleration (PGA); peak ground velocity (PGV); the ratio PGA/PGV; Arias’ intensity (); root mean square acceleration (RMS_{a}); the strong motion duration of Trifunac/Brady (); seismic power (); the spectral intensities of Housner (SI_{H}), Kappos (SI_{K}), and Martinez-Rueda (SI_{MR}); effective peak acceleration (EPA); maximum effective peak acceleration (); seismic energy input (); cumulative absolute velocity (CAV); the seismic damage potential of Araya/Saragoni (); central period (CP); spectral acceleration (SA); spectral velocity (SV); spectral displacement (SD); and the intensity of Fajfar/Vidic/Fischinger (). Table 1 provides an overview of the used parameters and the literature references.

The Arias intensity [3] is a measure of the total energy content of seismic excitation and is defined by the following relation:

Here, is the Arias intensity, is the total seismic duration, and is the seismic ground acceleration.

The root mean square acceleration RMS_{a} [4] is a measure of the mean energy content calculated by the following relation:

The Husid diagram [5] is the time history of the seismic energy content scaled to the total energy content. It is defined by the following relation: where is the Husid diagram as a function of time .

The strong motion duration of Trifunac/Brady is defined as the time elapsed between 5% and 95% of the Husid diagram [6] and is defined by the following relation: where is the strong motion duration and and are the time elapsed at 95% and 5% of the Husid diagram, respectively.

The power [7] is a measure of the energy content per time unit of the seismic excitation and is defined by the following relation:

Here, is the power of the seismic excitation, and the nonnormalized energy level, at 95% and 5% of the Husid diagram, respectively, and is the strong motion duration of Trifunac/Brady.

According to Housner [8], the spectrum intensity is given by the relation where PSV is the pseudovelocity curve, is the natural period of an single-degree-of-freedom (SDOF) system, and is the damping coefficient. In the present work, the integral in (6) is evaluated for a damping ratio equal to 2% of the critical damping, as recommended by Housner when he introduced this quantity [8]. However, it is now generally accepted that the Housner spectral intensity is valid for any level of damping (e.g., 5% of critical damping) [7].

In the definition of the spectrum intensity after Kappos [9]: the integration limits are dependent on the fundamental period of the structure and the periodintervals and . Kappos suggested .

Martinez-Rueda [10] introduced a spectral intensity definition accounting for the dynamic properties of the structure which is defined as where the hardening period and is the yield period, which are computed using the tangent stiffness of the hardening branch and using the tangent stiffness after the appearance of the first plastic hinge of the idealized lateral response of the structure, respectively. The required idealized lateral response of the structure for the evaluation of and has been performed by a nonlinear pushover analysis as suggested by Martinez-Rueda [10].

The effective peak acceleration (EPA) [11, 12] is the average of the spectral ordinates of the elastic acceleration response spectrum (for 5% critical damping) in the period interval [0.1 s, 0.5 s], divided by a standard value 2.5. It is defined by the following relation:

The maximum EPA () is a variation of the EPA definition. Here, the period interval is not constant (as in the previous case from 0.1 to 0.5 s) but is a sliding window of 0.4 s, which sweeps the entire elastic acceleration response spectrum for 5% critical damping and provides the maximum of all possible EPA values calculated in this manner.

The input seismic energy [13] is defined by the following relation:

Here, is the input seismic energy of a SDOF system, is the total seismic duration, is the mass, is the displacement of the system relative to the ground, and is the ground displacement.

The cumulative absolute velocity (CAV) [1] is defined as the area under the absolute accelerogram:

Araya and Saragoni [14] introduced the destructiveness potential () of a seismic excitation, defined as where is the Arias intensity and is the intensity of the seismic acceleration zero-crossings. is defined as the number of intersections of the acceleration ordinates with the time axis (points with zero acceleration) per time unit in a seismic acceleration time-history (accelerogram).

The central period (CP) is the reciprocal value of the number of positive zero-crossings (points where the sign of the considered function changes from negative to positive) of the seismic acceleration per time unit [15]. The CP is a frequency content indicator and not an authentic seismic intensity parameter. However, this parameter has been used to characterize the seismic damage potential of seismic excitations (together with the term PGA/PGV and the strong motion duration of Trifunac/Brady ) [16]. For this reason, the CP has been included in the set of the seismic intensity parameters used in the present study.

The response spectra describe the maximum response of a SDOF system to a particular input motion as a function of the natural period (or natural frequency) and the damping ratio of the SDOF system [17]. For each ground motion measurement there are displacement (SD), velocity (SV), and acceleration (SA) response spectra, respectively.

Finally, the seismic intensity of Fajfar/Vidic/Fischinger [18] is defined as where is the peak ground velocity (PGV) of the ground motion and is the strong motion duration of Trifunac/Brady.

#### 3. Seismic Acceleration Time Histories

The expected damage potential of a seismic excitation on a particular structure is the prime consideration for the selection of the accelerograms for the presented methodology. Seismic excitations that generate a wide spectrum of damage, from negligible to severe, are considered for statistical reasons. Thus, the present investigation utilized 400 worldwide natural acceleration records with strong seismic activity. Table 2 provides data for all the utilized seismic events (event name, country, date, record station, component, and PGA). Figure 1 is a scatter plot of standard (Richter) magnitude versus epicentral distance of all utilised seismic excitations. Furthermore, Tables 3 and 4 provide the number of accelerograms used per country and per PGA range. All of the above-mentioned seismic acceleration time histories have been investigated by a computer-supported evaluation of their seismic parameters, as presented in the previous section and in Table 1.

The number of the used accelerograms is essential for the accuracy of the statistical models. In addition, the utilized accelerograms must generate a whole spectrum of damage (low, medium, large, and total) for statistical completeness. However, regional (locally possible) seismic records do not exist in sufficient quantity for most of the seismically active areas. Therefore, a set of worldwide natural acceleration records has been utilized for the present study. Alternatively, artificial accelerograms compatible with the design spectrum, combined with a scaling procedure (to generate a whole spectrum of damage), could be employed. In that case, the variability of the frequency content is limited.

#### 4. Damage Indices

Among the many structural response parameters, attention is focused on those that can best describe seismic damage. The attention is focused on overall structural damage indices (OSDIs) because these parameters summarily lump all existing damage in columns and beams into a single value, which can then be easily interrelated with the single-value seismic parameters. For this purpose, the modified overall damage index of Park and Ang [19, 20] is used in the present study. In this model, the global damage is obtained as a weighted average of the local damage at the ends of each element, with the dissipated energy as the weighting function. The local damage index according to Park and Ang must be calculated using the following equation: where is the maximum rotation during the load history, is the ultimate rotation capacity of the section, is the recoverable rotation at unloading, is a constant parameter (0.1–0.15 for nominal strength deterioration [21]), is the yield moment of the section, and is the dissipated hysteretic energy.

In this study, the numerical value of parameter in (14) is equal to 0.1. This value corresponds to nominal strength degradation [21]. However, the numerical value of can vary in the range between 0 (no strength degradation) and 0.4 (severe deterioration) [21]. A low value corresponds to well-detailed reinforced concrete or steel members [22]. In contrast, a high value corresponds to poorly detailed reinforced concrete members or to unreinforced masonry [20].

The OSDI of Park and Ang is defined by the following equation: where is the energy dissipated at location and is the number of locations at which the local damage is computed.

The maximum interstory drift ratio (MISDR) has been selected as a second OSDI. It is defined as the maximum interstory drift () normalized by the story height (), as given by the relation

The maximum softening index of DiPasquale and Cakmak [23] has been selected as a third OSDI, which is based on the vibration parameters of the structure. It is given by the following expression: where is the maximum softening, is the fundamental period, and is the maximum natural period of the examined structure during the excitation. For the evaluation of the instantaneous natural period is required to be evaluated, which is accompanied by the actual time-dependent (instantaneous) tangent stiffness matrix. The latter varies with time due to nonlinear phenomena. The natural period computed for each time step of a nonlinear dynamic analysis presents high variability. The duration of these maximum values is very short, and therefore, their influence in the natural period of an equivalent linear system is not significant. Thus, a more meaningful indication of the change in the natural period can be obtained by observing a moving average of the instantaneous natural period using a sliding time window for the smoothness of the time-fundamental period curve [23, 24]. The time-fundamental period curve can be evaluated by a nonlinear dynamic procedure that calculates the fundamental period of the structure by considering the stiffness degradation in every time step. is the initial, fundamental period of the structure that corresponds to the initial stiffness of the structure.

Table 5 presents the range limits of the four damage classes using the different damage indices [25]. The four levels (low, medium, large, and total) correspond to undamaged or minor damage, repairable damage, irreparable damage, and partial or total collapse of the structure.

#### 5. Application

The proposed procedures have been applied to a reinforced concrete structure, shown in Figure 2, and designed in agreement with the rules of the recent Eurocodes for structural concrete and aseismic structures, EC2 and EC8 [26, 27]. The fundamental period of the frame was 1.20 s. The following loads have been considered: self-weight, seismic loads, snow, wind, and live loads. The cross-sections of the beams are treated as T-beams with a 40 cm width, 20 cm plate thickness, 50 cm total beam height, and 1.45 m effective plate width. The distances between each frame of the structure have been chosen to be 6 m. According to the EC8 Eurocode, the structure shown in Figure 2 is considered an “importance class II, ductility class M” structure with category B subsoil. The ordinate of the used elastic response spectrum of EC8 was 0.24 g.

After designing and detailing the reinforced concrete frame structure, nonlinear dynamic analyses were carried out to evaluate the structural seismic response. For this purpose, the computer program IDARC [21] has been used. The hysteretic behavior of the beams and columns has been specified at both ends of each member using a three-parameter Park model [21]. This hysteretic model incorporates stiffness degradation, strength deterioration, nonsymmetric response, slip-lock, and a trilinear monotonic envelope. The parameter values that specify the above degrading parameters have been chosen from the cyclic force-deformation characteristics from experimental results of the typical components of the studied structure [20]. Thus, nominal parameters for stiffness degradation and strength deterioration have been chosen. In contrast, no pinching has been considered. Realistic hysteretic material models result in realistic proposed statistical models. On the other hand, simplified assumptions for the hysteretic material models, like ignoring the pinching effect (default option in the utilized computer program IDARC [21]), do not affect on principle the value of the proposed statistical models that are in any case procedures for the approximate assessment of structural damage. Of the several response parameters, the focus is on the OSDIs. Consequently, the OSDIs of Park and Ang and of DiPasquale and Cakmak (maximum softening index) and the MISDR have been calculated for all of the accelerograms corresponding to the excitations given in Table 2, which were used as the seismic input for the nonlinear dynamic analyses. For the evaluation of the global damage index of DiPasquale and Cakmak according to (17), a sliding time window of 2.5 s is used to smooth the time-fundamental period curve. This window size is between 2 and 5.5 times the fundamental period of the examined structure ( = 1.20 s), as suggested by Rodriguez-Gómez/Cakmak [24]. The maximum value of the smoothed time-fundamental period curve is the maximum natural period of the examined structure during each seismic excitation considered ( in (17)). Finally, the hardening period and yield period needed for the assessment of the SI_{MR} in (8) are evaluated by a nonlinear pushover analysis, as described in Section 2. Their numerical values are s and s.

#### 6. Multilinear Regression Analysis

In general, a multilinear regression analysis [28] is used to express a dependent variable () by ()-independent variables (, with ) and a constant value () minimizing the error term (), as shown by the relation

In the present study, the dependent variable () represents each used OSDI (of Park and Ang, of DiPasquale and Cakmak (maximum softening index), and the MISDR), and the independent variables () are the twenty seismic parameters () shown in Table 1.

The analyses have been carried out using the statistical software STATGRAPHICS [29]. The regression results for the OSDIs after Park and Ang, for the MISDR, and for the maximum softening index after DiPasquale and Cakmak are given in Tables 6–9 and will be presented in detail subsequently. Thus, the result for the OSDI after Park and Ang is given in Table 6. The equation of the fitted multilinear model is

, which indicates that the model as fitted explains 85.9% of the variability in the Park and Ang OSDI. In addition, Table 6 presents the 95% confidence intervals for the coefficients in the multilinear model. Each variable coefficient in (19a) and in Table 6, or generally in (18), is interpreted as the change in the response based on a one-unit change in the corresponding explanatory variable keeping all other variables fixed. This interpretation is fictitious because it is not possible in a seismic accelerogram to change only one of the seismic parameters. Furthermore, in some explanatory variables (seismic parameters), a one-unit change can be easily observed (such as the SMD after Trifunac/Brady), whereas such a change cannot be easily observed in others (such as the SD). In addition, the comparison between coefficients of different explanatory variables is not possible because they are assigned quantities having different dimensions and units [30]. Finally, the constant term (“” in (18) and “−0.032” in (19a)) has a physical meaning only in the case in which all of the explanatory variables can simultaneously have zero values. Otherwise, as in the present study in which the seismic parameters with zero values are out of the observed value range, the constant term is an extrapolated value without a physical meaning. However, Table 7 presents the coefficients and their 95% confidence intervals for the multilinear regression model of the OSDI after Park and Ang, without the constant term ( = 0.900218). As indicated by , both models (with and without the constant term) explain a high percentage of the variance (85.9% and 90%, resp.) in the dependent variable.

Similarly, Table 8 presents the multilinear model for the MISDR ( = 0.799279) and Table 9 presents the model for the OSDI (maximum softening index) after DiPasquale and Cakmak ( = 0.763627), along with their 95% confidence intervals. The fitted multilinear models for these global structural damage indices are provided by

The models, represented by the regression equations, are strictly valid only within the range of the data (in the present study these data are the values of the seismic parameters employed) used to develop these regression equations (training phase). Trying to predict the outcome outside the range of the data can be seriously misleading and is not advised.

The initial set of explanatory variables (see (19a), (19b), and (19c)) contains a relative large number of seismic intensity parameters. Thus, in the present study twenty seismic intensity quantities have been used (Table 1) to avoid an a priori omission of important, independent (explanatory) seismic parameters. A regression model with only a few explanatory variables (e.g., two to three accidentally selected parameters) runs the risk of becoming an unreliable or invalid regression model. However, a statistically systematic reduction of the initial regression model is possible. This reduction has been realized in the present study by a stepwise backward elimination procedure combined with an elimination criterion.

Thus, the multilinear regression model, which contains the twenty examined seismic parameters (Model 1), can be simplified to consider fewer parameters without losing significant model quality. This simplification is realized by successive elimination of the independent variables with a value greater than or equal to 0.05 (elimination criterion) because that term is not statistically significant at the 95% or higher confidence level. This lack of significance is explained by the strong correlation between several seismic parameters.

The value is the smallest level of significance that would lead to rejection of the null hypothesis with the given data. In the case of multilinear regression analysis, the null hypothesis () and alternative hypothesis ( are and , where are the coefficients of the explanatory variables in (18) [30]. Thus, the importance to keep a seismic intensity parameter in the regression model is determined by the previously described explicit statistical procedure.

As an example, Figure 3 presents the values for all of the terms of regression equation (19a) (Table 6) in ascending order. The SI after Kappos parameter has the maximum value (0.809) which is greater than 0.05. Therefore, this specific parameter can be eliminated from the considered model. Subsequently, a new regression equation is evaluated without the Kappos SI parameter in its explanatory variables, and the corresponding values are different from those of the initial equation (19a) (with all of the seismic parameters considered initially). In the new regression equation the explanatory variable (seismic parameter), with the maximum value greater than 0.05, will be eliminated. This elimination procedure is repeatedly applied until all of the values of the explanatory variables remain less than 0.05. Applying the described procedure to (19a), the sequence of the eliminated parameters is SI after Kappos, SA, SD, SMD, , , CP, , and PGA/PGV. The result of this procedure is the reduced regression model shown in Table 10. It is not recommended to eliminate all of the explanatory variables that have a value greater than 0.05 in Figure 3 at once from the regression model in (19a) because the values are different in the case of stepwise elimination of explanatory variables and in the case of their at-once elimination.

Thus, in accordance with the proposed stepwise elimination procedure with a value greater than 0.05 criterion, Tables 10, 11, and 12 show the reduced multilinear models (Model 2) for the OSDI of Park and Ang ( = 0.856131), the MISDR ( = 0.795159), and the OSDI (maximum softening index) of DiPasquale and Cakmak ( = 0.757862) with their respective 95% confidence intervals.

The independent variables in the reduced regression model are selected by the proposed statistical procedure primarily to predict the damage indicator and not to explain it physically. Therefore, it is not unexpected that different seismic parameters are used for the adequate prediction of different damage indices (Tables 10–12).

The multilinear regression is a robust procedure, even for the case in which two or more explanatory variables are linearly dependent or highly correlated. Perfect multicollinearity occurs when two or more explanatory variables in the multilinear regression are linearly dependent. This multicollinearity type is rare. Imperfect multicollinearity occurs when two or more regressors are highly correlated. Perfect and imperfect multicollinearity does not affect the overall predictive power or reliability of the regression model. However, multicollinearity does reduce the effectiveness of a regression analysis if its primary purpose is to determine the specific effects of the various independent variables. The goal of the present study is to predict the dependent variable (damage indices) from a set of independent variables (seismic intensity parameters); therefore, multicollinearity does not affect the quality of the proposed multilinear regression model. Nevertheless, the proposed stepwise backward elimination procedure reduces the number of independent variables and consequently provides models with less multicollinearity. An a priori omission of several independent variables includes the risk of becoming an untrustworthy model.

#### 7. Testing the Multilinear Regression Model

To verify the quality of the full (Model 1) and the reduced (Model 2) regression models, in general, it is sufficient that the validation data set is a fraction of the training data set. In the present study, the validation data consists of 410 accelerograms, separated into two sets. The first one is identical to the 400 accelerograms used for the training phase of the regression model, as shown in Table 2. The second set is from an entirely different set of 10 artificial accelerograms. In their original form, the latter are compatible with the design spectrum employed. Thus, set 1 is used as internal and set 2 as external validation set. This is common validation procedure in statistical analyses. Two utilized artificial accelerograms are presented in Figure 4. Because the examined frame structure is designed using the same spectrum to which the artificial accelerograms are compatible, the expected damage grade will be low. However, the proposed statistical models must also be verified for medium, large, and total damage grades. Therefore, some of the artificial accelerograms have been scaled (the ordinates of the considered seismic accelerogram have been multiplied by a factor with a value greater than one) to achieve medium, large, and total damage grades. Thus, some of these accelerograms have been scaled to produce medium, large, and total structural damages for each of the examined damage indicators. The program SIMQKE [31] has been utilized to create the aforementioned artificial accelerograms. The method used by the program for the artificial seismic accelerogram generation is the superposition of sinusoids having random phase angles and amplitudes derived from a stationary power spectral density function of the motion. The produced signals are then enveloped in a trapezoidal shape to simulate the nonstationary characteristics of the ground motion [31]. The following input data are required for the generation of the artificial accelerograms: peak ground acceleration (PGA), total duration (TD), duration of the ascending and the descending parts of the trapezoidal envelope, and design spectrum with which the artificial accelerograms must be compatible. Another option is to use natural accelerograms instead of the artificial accelerograms to verify the proposed models. Scaling an accelerogram to a greater PGA value is a simple procedure to produce accelerograms with more energy content and damage potential. The seismic intensity parameters have been evaluated also for all the utilized scaled accelerograms because they change with the scaling process.

Table 13 presents the prediction results of all of the examined damage indicators. In this context, correct prediction means that the evaluated damage index lies within the 95% confidence interval of the multilinear regression model, as provided in Tables 6–12. Thus, the results of Model 1 display correct predictions, from 94.75% to 97.50% and from 80% to 90% for acceleration sets 1 and 2, respectively. In addition, Figure 5 shows the regression line and the 95% confidence interval of the correct versus predicted Park and Ang OSDI values. In this case the coefficient of determination = 85.9%. This indicates that the model as fitted explains 85.9% of the variability in Park and Ang damage index. Furthermore, the percentages of correct prediction for Model 2 are from 95% to 97% and from 70% to 90% for acceleration sets 1 and 2, respectively. Model 2, with its reduced number of seismic parameters in the regression model, is not of poorer quality than Model 1, and the results of acceleration set 2 are satisfactory, particularly for the OSDI of Park and Ang and the MISDR. Finally, Table 14 shows the Pearson and the Spearman rank correlation coefficients between the correct and predicted values of all examined OSDIs based on regression analyses. The numerical results show a high correlation in all cases.

#### 8. Discriminant Analysis

In the case for which it is sufficient to know the Postseismic damage grade as provided in Table 5 (low, medium, large, and total), then a statistical discriminant analysis can be performed. The purpose of discriminant analysis is to classify objects into one or more possible groups based on a set of features that describe the objects [32]. In general, we assign an object to one of a number of predetermined groups based on observations made about the object. The discriminant analysis creates an equation which will minimize the possibility of misclassifying cases into their respective groups or categories.

The principle is to find the discriminant functions for each group. The element to be classified belongs to the group with the greatest value of the discriminant functions. In the present study, four groups and twenty independent parameters are used. Thus, the discriminant functions are given by the relations where are the discriminant functions, are parameters to be specified, and are known quantities. Here, = 4 (four damage grade groups) and = 20 (twenty seismic parameters). The discriminant analysis has a direct analytical solution and therefore, calculations are very fast. The produced models are concise and easily programmed. On the other hand, the discriminant analysis is sensitive to outliers.

#### 9. Testing the Discriminant Analysis Model

To verify the quality of the discriminate analysis, the same two sets of accelerograms from the regression analysis have been used. Table 15 presents the prediction results of all of the examined damage indicators. In this context, correct prediction means that the damage grade estimate by the discriminant analysis agrees with the corresponding evaluation of the nonlinear dynamic analysis. The results display correct predictions from 87.75% to 97.50% and from 70% to 90% for acceleration sets 1 and 2, respectively. Figure 6 shows the correct versus predicted damage grade based on the Park and Ang OSDI values. The points out of the diagonal line show the incorrect classified cases. Thus, 9 cases have been classified in damage group 2 (medium damage) instead of the correct damage group 1 (low damage) and 1 case has been classified in damage group 2 (medium damage) instead of the correct damage group 4 (total damage). The points in the diagonal line show the correct classified cases (390 cases, 97.5%). The corresponding discriminating functions have been evaluated with values less than 0.05 and are statistically significant at the 95.0% confidence level. Thus, all points in Figure 6 are inside the 95.0% confidence interval.

#### 10. Remarks on the Proposed Statistical Procedures

The proposed regression models (Tables 6–12) and discriminant analysis model are clearly tailored to the examined reinforced concrete frame structure. For another structure, the nonlinear dynamic analyses should be repeated to determine the coefficients in the regression models (Tables 6–9) and in the discriminant analysis model. Although the proposed procedures are applied to a two-dimensional reinforced concrete frame as an example, they could be applied to any two-dimensional frame and expanded to three-dimensional frames for other building materials (e.g., steel, masonry, timber) or to other building types (e.g., bridges, shell structures). Furthermore, the set of twenty seismic intensity parameters considered here could be extended with additional quantities or partially or totally replaced by other parameters. However, the proposed procedure remains the same regardless of the structure examined and seismic intensity parameters employed. Thus, in the first step, a set of seismic intensity parameters is evaluated for a set of accelerograms. Then, nonlinear dynamic analyses furnish the damage indices of the considered structure. In the final step, the coefficients of the proposed statistical models (multivariate regression and discriminant analysis) are determined.

The proposed procedures are simple and accurate tools for the estimation of the damage index of a building by a multivariate regression analysis and of its Postseismic damage grade by discriminant analysis. These multivariate statistical techniques do not replace the exact nonlinear dynamic analyses, which are necessary to determine the coefficients of the statistical models, but they do serve as a simple utility for the Postseismic damage status estimation of a specific building based on seismic intensity parameters. Evaluation of the coefficients in statistical models, in which nonlinear dynamic analyses are obligatory, must be performed by professional engineers. If the models have been defined, then they may feasibly be applied by nonspecialists. The presumption is that the seismic parameters employed are known or estimated. The nonlinear analyses and determination of the proposed statistical models are complicated and time-consuming numerical procedures that must be performed by professional engineers (e.g., this can be realized in connection with the design procedure of a considered new building and the evaluated statistical models can be announced to the building owner and the housing authority). The statistical models are determined by the evaluation of the coefficients in (18) and (20) for the regression and discriminant models, respectively. These statistical models can be easily and quickly applied once they are resolved. They could also be used by nonspecialists under the assumption that the seismic intensity parameters for an actual seismic event are provided by the responsible authorities (e.g., seismological institutes). The proposed methods convert the complicated dynamic nonlinear models to black boxes that take the ground motion information and predict the Postseismic damage levels. Thus, as mentioned in the introduction section, the proposed procedures can be applied to damage scenarios of important buildings (e.g., hospitals, schools, bridges, silos, and hangars) by public authorities. The proposed statistical models are faster and easier to use than the nonlinear dynamic analyses. Epigrammatic, nonlinear dynamic analyses can be utilized only by specialists, while on the other hand, the proposed statistical models can be utilized also by nonspecialists. Thus, nonspecialists can approximately evaluate the Postseismic damage status of buildings, ignoring the exact damage values provided by nonlinear dynamic analysis. The exact damage value needs not to be known once the statistical models are resolved. Furthermore, the proposed methods can handle probable uncertainties in the seismic ground motion and variability in structural or material properties by parametric analyses or by their combination with fuzzy logic procedures (fuzzy regression analysis). Finally, the proposed method can also be applied to a group of buildings. In that case, the damage parameters must be appropriately tailored to describe the Postseismic damage status and the proposed statistical procedures can be applied once on the group and not repeatedly on separate buildings. A proper damage status description could be a financial damage index [25].

Alternative procedures for seismic damage potential classification are presented in the literature and are based on intelligent techniques. Thus, fuzzy logic, artificial neural network, neurofuzzy and adaptive neurofuzzy inference systems, and support vector machines techniques have been used for seismic damage potential classification. The efficiency of the intelligent techniques was expressed by the correct classification rates of a set of accelerograms that were not used during the training process. The intelligent techniques provided a correct classification rate between 71% and 97.5%. The results depend on the used procedure, the overall structural damage indices (MISDR or ), and the number of training samples [33]. The intelligent techniques in comparison with the proposed statistical procedures have no significant better correct classification rates and are not appropriate to be used by nonspecialists.

#### 11. Conclusions

Two statistical methods have been presented for the evaluation of the Postseismic damage status of buildings based on the parameters of the seismic excitation. The first is multilinear regression analysis, and the second is discriminant analysis. Twenty seismic parameters have been extracted from the accelerograms employed. The overall damage index of Park and Ang, the MISDR, and the maximum softening of DiPasquale and Cakmak have been used to describe the damage status of the structures. The multilinear regression has been used for the explicit evaluation of the damage indices. In contrast, the discriminant analysis furnished the Postseismic damage grade of the structures. These statistical procedures, combined with seismic intensity parameters, are novel methodologies in earthquake engineering and advantageous for the estimation of the damage indices and damage grades of buildings after severe seismic excitations.

The proposed statistical methods have been applied to an eight-story reinforced concrete frame structure designed in accordance with the rules of the EC2 and EC8 Eurocodes for reinforced concrete and antiseismic structures, respectively. A set of 400 natural accelerograms has been applied for the training phase of the models. In addition to the first set, a second set of synthetic accelerograms has been used to verify the statistical methods. The numerical results have shown that 94.75–97.50% and 70–90% of the first and second sets of accelerograms, respectively, were correctly classified in accordance with their damage potential for the examined structure within the confidence interval provided by the regression analysis. In addition, the regression model with a reduced number of considered seismic parameters is not of poorer quality than the model that takes all twenty seismic parameters into account. In contrast, the damage grades for 87.75–97.50% and 70–90% of the first and second sets of accelerograms, respectively, were correctly predicted by the discriminant analysis. Thus, these results led to the conclusion that the multilinear regression analysis and discriminant analysis are useful tools for the prediction of the Postseismic damage status when the seismic parameters are known. Finally, based on the regression and the discriminant analysis results of the 10 examined artificial accelerograms (set 2 in Tables 13 and 14), it can be suggested to use the Park and Ang OSDI and the maximum softening index of DiPasquale and Cakmak as appropriate Postseismic damage value and damage grade estimators of a building, respectively.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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