An efficient methodology to estimate probabilistic seismic damage curves

1 An efficient methodology to estimate probabilistic seismic damage curves Yeudy F. Vargas1, Luis G. Pujades2, Alex H. Barbat3 and Jorge E. Hurtado4 1 Universidad Politécnica de Cataluña, Department of Civil and Environmental Engineering, Jordi Girona 1-3, Barcelona, Spain. Researcher. 1 Universidad Politécnica de Cataluña, Department of Civil and Environmental Engineering, Jordi Girona 1-3, Barcelona, Spain. Full profesor. 2 Universidad Politécnica de Cataluña, Department of Civil and Environmental Engineering, Jordi Girona 1-3, Barcelona, Spain. Full profesor. 3 Universidad Nacional de Colombia, Department of Civil Engineering, Manizales, Colombia. Full profesor Abstract The incremental dynamic analysis is a powerful methodology which can be easily extended for calculating probabilistic seismic damage curves. These curves are metadata to assess the seismic risk of structures. Although this methodology requires a relevant computational effort, it should be the reference to correctly estimate the seismic risk of structures. Nevertheless, it would be of high practical interest to have a simpler methodology, based for instance on the pushover analysis, to obtain similar results to those based on the incremental dynamic analysis. In this article, the pushover analysis is used to obtain probabilistic seismic damage curves from the stiffness degradation and the energy of the nonlinear part of the capacity curve. A fully probabilistic methodology is tackled by means of Monte Carlo simulations, with the purpose of establishing that the results based on the simplified proposed approach are compatible with those obtained with the incremental dynamic analysis. Comparisons between the results of both approaches are included for a low-to-mid-rise reinforced concrete building. The proposed methodology significantly reduces the computational effort when calculating probabilistic seismic damage curves. 1 Introduction In the last decades, several approaches have been developed for estimating the seismic risk of structures. One of these is based on the Vulnerability Index Method. In this method, the seismic hazard is considered 2 by means of macroseismic intensities, Egozcue et al. 1991, and the structural behavior by means of vulnerability indices, Lantada et al. 2010, Barbat et al. 2011, Lagomarsino et al. 2006. Another widely invoked method is based on the capacity spectrum method, CSM, developed by Freeman et al. 1975, Freeman 1998, and further developed by Fajfar and Gaspersic 1996, Chopra and Goel 1999, and Fajfar 1999, among others. In this method, the seismic action is considered by means of the elastic response spectrum, while the building capacity is represented by means of a capacity curve or a capacity spectrum. The capacity curve is calculated with an incremental nonlinear static analysis, commonly known as Pushover Analysis (PA in the sequel); it is also assumed that the structural response is dominated by the first mode of vibration. However, it is important to note that the capacity spectrum method has been further modified in order to include the effects of higher modes of vibration, Chopra and Goel 2002, Chopra et al 2004, Kreslin and Fajfar 2012.. More recently, the Incremental Dynamic Analysis method, IDA, is becoming a powerful tool for assessing the seismic behavior of buildings which allow to take into consideration the following aspects: 1) the nonlinear dynamic response of the structure; 2) the increments of the seismic intensity; and 3) the uncertainties related to the structural properties and to the seismic hazard. These aspects allow obtaining probabilistic damage curves, PDC, which are metadata for estimating seismic risk scenarios at urban level. Notice that to correctly estimate seismic risk scenarios, the uncertainties involved in the seismic hazard action and in the mechanical and geometrical properties of the structures should be considered. Nonetheless, the computational effort involved to carry out IDA can be very large, and the use of simplified methods is an urgent need of practical interest. In this article a simplified method for calculating PDC, with an efficient approach based on the stiffness degradation and on the energy of the nonlinear part of the capacity curve, is proposed. The PDC obtained with this simplified approach fit with sufficient accuracy the PDC calculated via IDA method. The effectiveness of the simplified presented approach is tested for a low-to-mid-rise reinforced concrete framed building. The Monte Carlo method is used to perform the probabilistic calculations presented herein. 3 2 Incremental dynamic analysis A nonlinear dynamic analysis, NLDA, allows to simulate the time history response of a structure subjected to an earthquake. This analysis also allows to calculate the maxima of structural response variables like the displacement at the roof, the global damage index, among others. A method which uses NLDA as structural solver is the Incremental Dynamic Analysis, IDA, Vamvatsikos and Cornell 2002. IDA is generally performed with several earthquakes records increasingly scaled. The IDA method may be extended to include uncertainties in the structural properties, Vamvatsikos and Fragiadakis 2010, Vargas et al 2012, ensuing a very good approach to adequately estimate PDC. Groups of PDC are generally used to simulate seismic risk scenarios at urban areas. However, if uncertainties are included when employing IDA, it generally turns into a computationally costly method, Vargas et al. 2015. In spite of this, this section is devoted to explaining how IDA can be applied for obtaining PDC. A low-to-mid-rise reinforced building is considered as a testbed. 2.1 Description of the studied building A reinforced concrete framed building has been designed and used as a testbed in order to perform the structural calculations proposed in this article. A sketch of this building is shown in Figure 1a. This figure also displays the geometrical dimensions of the building. Notice that, due to its symmetry, the building can be modeled as a two-dimensional structure by using a single frame (Figure 1b). 4 Figure 1 a) Building studied in this article and b) 2D model The main characteristics of the beams and columns of the framed structure are given in Table 1. The inelastic behavior of beams and columns elements follow the concept of the Gilberson one-component model, which allows hinges at one or both ends of the elastic central length of the member. In this model, the length of plastic hinges was assumed 5% of the length, L, of the element; that is 0.05 L. To consider the hysteretic cycle for flexure-failing mechanism in beams and columns, the modified Takeda hysteresis law, Otani 1974, has been considered. Yielding surfaces are defined by the bending moment-axial load interaction diagram for columns and bending moment-curvature for beams. The applied loads follow the recommendations given by the Eurocode 2, CEN 2004, for reinforced concrete structures. Notice that nonlinearities due to shear forces are not allowed because it has been assumed that the structural elements are well confined. Columns Beams Story b (m) h (m) ρ b (m) h (m) ρ 1 0.5 0.5 0.03 0.45 0.6 0.0066 2 0.5 0.5 0.02 0.45 0.6 0.0066 3 0.45 0.45 0.015 0.45 0.6 0.0066 4 0.4 0.4 0.015 0.45 0.6 0.0066 Table 1 Characteristics of the elements of the studied building (Figure 1). b, h and ρ denote base, height and steel percentage of the cross sections of the structural elements, respectively. 2.2 Mechanical properties of the materials as random variables Generally, the characteristic mechanical properties of concrete and steel are the values used in the design of reinforced concrete buildings. Design standards require characteristic strength values for the materials 5 obtained during the quality control process, from compression and tension tests in concrete and steel samples, respectively. In order to meet the probabilistic approach of this article, the concrete compressive strength, fc, and the steel elastic modulus, Es, will be modeled as random variables. Table 2 presents the assumed mean, µ, standard deviation, σ and coefficient of variation (cov) of these random variables. Truncated normal distributions are considered, being the limits the mean value minus and plus 2.5 standard deviations. Thus, negative samples are not allowed for variables in Table 2. The coefficient of variation of the concrete compressive strength, fc, may vary from building to building, in the range 0.07-0.2; the average value of this cov is about 0.15, Melchers 1999. However, this value depends on the quality control carried out during construction. For instance, a highly controlled concrete tends to exhibit a cov value close to 0.1, Melchers 1999. For a mid-controlled concrete, the expected cov is about 0.15. In this research, the cov for the concrete compressive strength has been assumed 0.15. The coefficient of variation of the elastic modulus of the steel, Es, has been set to 0.1, Melchers 1999. Variable µ σ cov fc (kPa) 2.1E04 3.15 E03 0.15 Es (kPa) 2 E08 2 E07 0.1 Table 2 Characteristics of the input random variables. µ, σ and cov represent the mean, the standard deviation and the coefficient of variation of the random variables, respectively Other likely uncertainties as those related to cracking and crushing of concrete, strain hardening and ultimate strength of steel, effects such as slab participation or uncertainties related to the model options, can be also included in the probabilistic structural analysis. It is worth noting that, if new random variables were considered, they would affect both, dynamic and static results. In this article, the random variables mostly affecting the uncertainties of the response in an explicit way were considered. Nonetheless, notice that the uncertainties of several structural variables are taken into account indirectly as they are calculated as functions of the explicitly considered random variables. For instance, the elastic modulus of the concrete, 6 Ec, is related to fc by the equation Ec=4500√fc and, for the calculation of the yield strength, fy, the expression fy=0.0021Es is used. 2.3 Seismic hazard to perform IDA One of the most important sources of uncertainty, when estimating the seismic risk of structures, is the random variability of the ground-motion. Its influence has been studied by Bommer & Crowley Bommer and Crowley 2006. The forecasting of the ground-motion parameters has been studied by Abrahamson et al. 1991, Bommer et al. 2007 and by Arroyo and Ordaz 2011. Having said that and according to the probabilistic simulation approach of this article, it is necessary to model the seismic hazard as a random variable. To do that, 10 earthquakes have been selected from the European database, Ambraseys et al. 2004, by using a criterion for controlling the dispersion of the data. Use is made of a method proposed by Vargas et.al. 2013a for obtaining a set of earthquake records from a database. Such records fit a target response spectrum, under a mean square metrics. The response spectral function selected in this study corresponds to the type 1 of the Eurocode 8 (EC8), CEN 2004, that has a surface-wave magnitude greater than 5.5. Several methods to establish the optimal number of accelerograms, required to perform the inelastic dynamic analyses, have been studied by Hancock et al. 2008 and they conclude that the exact number depends on the damage measures which are considered and also on their predictability. Most of these methods are based on the magnitude and on the spectral shape. Figure 2 depicts the spectra of the selected earthquakes, their average spectrum, and the target spectrum. In Figure 2 can be seen that the procedure used for selecting earthquake records allows a good fit. The list of these earthquakes and of their main characteristics are given in Table 3. 7 Figure 2 Response spectra calculated from the procedure based on the mean spectrum The selection of the earthquake records according to the method proposed by Vargas et.al. 2013a is based on the mean squared error calculated between the mean of a group of response spectra, previously selected from a sorted database criterion, and the target spectrum function. The criterion for sorting the database is based on the mean squared errors between actual spectra and a target spectrum. The earthquake records are sorted according to such errors in ascending order; thus, the record whose response spectrum is the most similar to the target spectrum is ranked as the first one; and so on. It is worth noting that this selection procedure avoids the sensitivity of the method to single records, as the sorting procedure itself discards records leading to likely significant differences in the dynamic analyses as well as in the computations of the performance points. It was also found that the number of records that minimizes the error function as defined in Vargas et al. (2013a) takes values between 10 and 30. Ten records have been used to take into account the randomness of the seismic actions in this study. However, an anonymous reviewer noted that the way the seismic actions have been selected could underestimate the record-to-record variability, thus enhancing the variability in the response due to the randomness of the mechanical properties. Eads et al. (2013) quantified the uncertainty in the collapse fragility curves and mean annual frequency of collapse as a function of the number of ground motions used in calculations, showing that using a small number of 8 ground motions can lead to unreliable estimates of a structure collapse risk. It is recognized the importance of this issue in seismic expected damage and seismic risk assessments and should be faced carefully in future research. The record selection method used here needs to be checked against other ground motion selection methods as, for instance the one proposed by Lin et al (2013). Station name Date Epicentre (degrees) Depth (km) Magnitude (Ms) Local geology Epicentral distance (km) N E Arquata del Tronto 19.09.1979 42.76 13.02 4 5.84 Rock 22 San Rocco 15.09.1976 46.32 13.16 12 5.98 Stiff soil 17 Kotor NasRakit 24.05.1979 42.23 18.76 5 6.34 Rock 21 Auleta 23.11.1980 40.78 15.33 16 6.87 Rock 25 Ponte Corvo 07.05.1984 41.73 13.90 8 5.79 Rock 31 Matelica 26.09.1997 43.03 12.86 6 5.9 Rock 20 Tricarico 05.05.1990 40.65 15.92 12 5.6 Rock 20 Izmit-M- Istasyonu 13.09.1999 40.70 30.02 13 5.9 Stiff soil 13 Bolu- Bayindirlik 12.11.1999 40.76 31.14 14 7.3 Stiff soil 39 Athens- Papagos 07.09.1999 38.13 23.54 9 5.6 Rock 26 Table 3 Main characteristics of the selected earthquakes. km11=μ Depth , km3.8=σ Depth , km22.9=μ ED , km7.65 =σ ED , 6.13=μ Ms , 0.55=σ Ms It is important to note that, in addition of finding earthquakes records compatible with a target spectrum, the method proposed by Vargas et.al. 2013a also has the ability to find and to select records corresponding to similar earthquake features, including depth and epicentral distance, local geology and size of the 9 earthquakes, according to the great (MS >5.5) and moderate (MS<=5.5) earthquakes, as considered in Eurocode EC8. At the end of Table 3, statistics of these variables are shown. Mainly shallow great near earthquakes, recorded at rock or stiff soils, were automatically selected. Note also that dispersions around the central values are relatively small. Noticeably, this method for selecting the accelerograms also guarantees some record-to-record variability; in fact, for the entire range of periods, the coefficients of variation (cov) of the spectral acceleration values are in the range between 0.16 and 0.2 and, as no matching techniques are used, the frequency content, amplitudes and phases are retained. Lin et al (2013 a, b) analyzed the crucial problem of ground motion selection based on a target spectrum, concluding that the use of uniform hazard spectrum including variable uncertainties at different periods, results in overestimations of structural response hazard, whereas the conditional mean spectrum results in underestimation. Thus, the uncertainties due to seismic actions could be underestimated in this study. 2.4 Probabilistic calculation based on IDA To take into account the uncertainties related to the variables described in Table 2, the Monte Carlo method is used. Nonetheless, an important issue related to the spatial variability should be treated before starting the calculation. It is well known that the spatial variability of the mechanical features of the structural elements may influence greatly the results, Franchin et al 2010. For instance, in order to consider the likely correlation that may exhibit the strength of concrete for the elements of a specific story, it is necessary to assume a correlation hypothesis between the random generated samples. Thus, it is supposed that the correlation between the strength of columns and beams of the same story decreases with distance. To do this, for the 2D frame depicted in Figure 1, the column of the first story numbered as k=1, will be the first from left to right, the column numbered as k=2, of the first story, will be the second one going from left to right and so on. Thus, the correlation matrix, ji, ρ , for the concrete strength samples of one story is made in the following way: 16 automatic solution of these two important concerns. Probably the main limitation of this procedure is the time consuming, as it requires evaluating the next pushing pattern in each iteration of the pushover analysis. A detailed description of this procedure can be found in the RUAUMOKO’s manuals, Carr 2000. This structural program has been used in this study for calculating static and dynamic nonlinear structural analysis. According to the values exposed in Table 1, 100 groups of Gaussian input data samples are generated, according to the correlation hypotheses exposed above (see section 2.4); then, the adaptive pushover analyses are performed. Figure 5 depicts the 100 capacity curves obtained which are characterized by random variables such as the elastic stiffness, the maximum displacement, the maximum base shear, etc. Notice that the 100 capacity curves shown in Figure 5 can be easily transformed into 100 capacity spectra, HAZUS 1999, ATC-40 1996. The right and top axes in Figure 5 serve for appreciating this transformation. Figure 5 Capacity curves and capacity spectra of the building studied by considering the mechanical properties of the materials as random variables 17 3.2 Calculation of the performance points by considering uncertainties From a capacity spectrum of a structure and a response spectrum, there are several simplified methods to estimate the structural seismic response, HAZUS 1999, ATC-40 1996, FEMA 440 2005. These simplified methods generally look for obtaining the performance point. This latter is a measure of the expected demand of a structure. Thus, in order to calculate the performance point of the analyzed building, due to the same seismic hazard used in the IDA calculations, the response spectra of the earthquake records presented in Table 3 should be calculated. These response spectra, together with the capacity spectra depicted in Figure 5, are displayed in Figure 6. Obviously, the response spectra should be scaled to different intensity levels in the same way as in the IDA method. So, to facilitate the comparison of the results, the intensity levels will be the same ones that were used in the IDA calculations; that is, PGAs ranging from 0.05g to 1g at intervals of 0.05g. Figure 6 also shows an example of calculation of the performance points, for a PGA of 1g. In this figure the inelastic spectra, reduced due to the ductility of the building, are also shown. Note that the spectral displacement of the performance point is also referred to as target displacement in the international literature. The methodology applied for obtaining the performance point is further explained in FEMA 440. 18 Figure 6 Performance point for a given demand and capacity spectra. The method applied is exposed in Chapter 6 of FEMA 440. After computing all the performance points (20000 in this case), 1000 curves relating PGAs and the spectral displacements of the performance points are obtained. Figure 7 displays these curves, which will be useful to easily combine the uncertainties related to the mechanical properties of the materials and those related to the seismic hazard. Figure 7 Relation between PGA to spectral displacement To summarize the simulation scheme for calculating the curves of Figure 7, a flow chart is depicted in Figure 8. The main steps of the proposed approach are as follows: step 1: a capacity curve is computed for a randomly generated structural model (counter i, i=1:100); step 2: take the response spectrum of the record of Table 3 corresponding to the iteration number (counter j, j=1:10); step 3: increase the seismic intensity and scale the spectrum to this PGA value (counter k, k=1:20); step 4: compute the performance point and store the spectral displacement of the performance point, SdPP, in the matrix MRSd-PGA(10*(i-1)+j, k)=SdPP; check for intensity PGA; if the PGA is lower than the maximum PGA considered, go to step 3; else, check for record number; if the record number is lower than the maximum number of records considered, then go 19 to step 2; else check for the SdPP statistical distribution; the convergence of the two first statistical moments of the distribution of SdPP is checked. If this convergence is unsatisfactory, then go to step 1; else end. It is worth noting that a number of 100 structural models together with 10 seismic actions warranties this convergence; thus, this last checking is avoided. Figure 8 Flow chart of the simulation scheme for calculating capacity curves and the performance point ‘PP’ by considering the response spectra for the earthquakes records shown in Table 3 20 3.3 Calculation of the new simplified damage index As said above, the Park and Ang damage index is the sum of the contributions to damage of the maximum ductility and of the energy dissipated by the structural elements. Pujades et al. 2015 proposed an analogue equation for estimating a seismic damage index based on the PA: 𝐷𝐷𝐷𝐷𝑃𝑃𝑃𝑃(𝑆𝑆𝑆𝑆) = 𝛼𝛼𝐷𝐷𝐷𝐷𝜇𝜇(𝑆𝑆𝑆𝑆) + 𝛽𝛽𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑆𝑆𝑆𝑆) (4) in this equation, DIPA(Sd) is the pushover-analysis-based global damage index of the structure; 𝛼𝛼 is a coefficient that weights the contribution to the global damage of the first term of DIPA(Sd) related to the ductility; 𝐷𝐷𝐷𝐷𝜇𝜇(𝑆𝑆𝑆𝑆) is the damage term related to the ductility; 𝛽𝛽 is a coefficient that weights the contribution to the global damage of the second term of DIPA(Sd); and DIDE(Sd) is the damage term related to the dissipated energy. To obtain two functions from the capacity spectrum, to represent adequately the terms of equation 4, it is necessary to analyze the tangent and secant stiffness functions as well as the dissipated energy curve; as it will be seen further on, these curves are obtained easily from the capacity spectrum. As shown above, the first term of the equation of Park and Ang (see eq. 2) is related to the maximum ductility reached by a structural element and it can be related to the stiffness degradation of the building. In the case of the capacity spectrum method, the first derivative of the capacity spectrum is directly related to the stiffness degradation. However, two kind of stiffness degradation can be calculated: the first one based on the tangent stiffness and the second one on the secant stiffness. Figure 9a shows the tangent and secant stiffness of the capacity spectra depicted in Figure 5. These stiffness functions can be transformed into a damage index based on the tangent stiffness or into a damage index based on the secant stiffness by means of the following equations (see Figure 9b): 𝐷𝐷𝐷𝐷𝑇𝑇𝑆𝑆(𝑆𝑆𝑆𝑆) = 𝑘𝑘𝑒𝑒−𝑘𝑘𝑡𝑡(𝑆𝑆𝑆𝑆) 𝑚𝑚𝑚𝑚𝑚𝑚�𝑘𝑘𝑒𝑒−𝑘𝑘𝑡𝑡(𝑆𝑆𝑆𝑆)� (5) 21 𝐷𝐷𝐷𝐷𝑆𝑆𝑆𝑆(𝑆𝑆𝑆𝑆) = 𝑘𝑘𝑒𝑒−𝑘𝑘𝑠𝑠(𝑆𝑆𝑆𝑆) 𝑚𝑚𝑚𝑚𝑚𝑚�𝑘𝑘𝑒𝑒−𝑘𝑘𝑠𝑠(𝑆𝑆𝑆𝑆)� (6) In these equations, DITS(Sd) is the damage index based on the tangent stiffness; ke is the elastic stiffness; kt(Sd) is the tangent stiffness; DISS(Sd) is the damage index based on the secant stiffness; and ks(Sd) is the secant stiffness. Observe that both, DITS(Sd) and DISS(Sd), have been normalized by 𝑚𝑚𝑚𝑚𝑚𝑚(𝑘𝑘𝑒𝑒−𝑘𝑘𝑡𝑡(𝑆𝑆𝑆𝑆)) and 𝑚𝑚𝑚𝑚𝑚𝑚(𝑘𝑘𝑒𝑒−𝑘𝑘𝑠𝑠(𝑆𝑆𝑆𝑆)), respectively, in such a way that their highest value are 1. Figure 9 Tangent stiffness and secant stiffness of the capacity spectrum The next step consists in selecting which of the calculated stiffness, tangent or secant, is a better candidate to represent the first term of Equation 4. If the first term of the Park and Ang equation is analyzed in detail, one can see that the maximum ductility reached during the time history response is normalized by the ultimate ductility. This fact indicates that the rate of evolution of the ductility index is closer related to the secant stiffness than to the tangent stiffness. Besides, the curves of Figure 5 do not show strong and sudden losses of stiffness. The later behavior can be likely in rigid structures, for which it can be expected that a combination of the tangent and the secant stiffness should be considered. For instance, if the infill walls were modelled nonlinearly would certainly lead to sudden losses of stiffness and strength and, probably, a linear combination between the secant and the tangent stiffness will be a better option for the first term of equation 4. Further research on this issue will allow generalizing equation 4, for new structural typologies. 22 However, as pointed out above, for the building types considered in this article, it is expected that the major contribution to damage be due to secant stiffness degradation. Thus, the damage index based on the secant stiffness has been selected as the first term of Equation 4, i.e., 𝐷𝐷𝐷𝐷𝜇𝜇(𝑆𝑆𝑆𝑆)=𝐷𝐷𝐷𝐷𝑆𝑆𝑆𝑆(𝑆𝑆𝑆𝑆). The second term of the proposed damage index in equation 4 stands for the dissipated energy as a function of the spectral displacement. This term is obtained as the cumulative integral of the nonlinear part of the capacity curve as defined in Pujades et al. 2015. This cumulative integral also can be easily obtained by integrating the function defined as the subtraction of the capacity curve, Sa(Sd), from a projected line of the elastic part of the capacity spectrum. Then, this energy-based index is normalized so that its value at the ultimate capacity point is one. Thereby, this index holds for the relative energy dissipated as referred to the energy loss at the ultimate or collapse point. The following equation is used to compute this energy term and Figure 10 illustrates the meaning of the involved functions and areas. 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑆𝑆𝑆𝑆) = ∫(𝑘𝑘𝑒𝑒 ∗ ξ −𝑆𝑆𝑚𝑚( ξ )) ξ =𝑆𝑆𝑆𝑆 ξ =0 𝑆𝑆 ξ 𝑚𝑚𝑚𝑚𝑚𝑚�∫(𝑘𝑘𝑒𝑒 ∗ ξ −𝑆𝑆𝑚𝑚( ξ )) ξ =𝑆𝑆𝑆𝑆 ξ =0 𝑆𝑆 ξ � (7) Figure 10 Graph depicting the functions and areas involved in the definition of the Energy term in equation (4) as described above and formulated in equation (7) 23 Figure 11 depicts both terms of Equation 4, 𝐷𝐷𝐷𝐷𝜇𝜇(𝑆𝑆𝑆𝑆) and 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑆𝑆𝑆𝑆). Figure 11 Stiffness and energy functions for calculating the new damage index. According to Equation 4, the coefficient α and β are weights of the damage terms related to the ductility and energy functions, respectively. However, it is important to recall that, in this paper, the values of the Park and Ang damage index, when larger than 1 were set to 1. Pujades et.al 2015 proposed the following assumption to avoid that the values of the new damage index be larger than 1: 𝛼𝛼= 1 −𝛽𝛽 (8) Therefore, equation 8 allows rewriting Equation 4 as follows: 𝐷𝐷𝐷𝐷𝑃𝑃𝑃𝑃(𝑆𝑆𝑆𝑆) = 𝛼𝛼𝐷𝐷𝐷𝐷𝜇𝜇(𝑆𝑆𝑆𝑆) + (1−𝛼𝛼)𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑆𝑆𝑆𝑆) (9) If equation 9 is applied for 𝛼𝛼= 0.5, the curves shown in Figure 12 are obtained. 24 Figure 12 New damage index for 𝛼𝛼= 0.5. However, these curves do not take into account the seismic action yet. Thus, to take account of it, a change of variable is performed by transforming the spectral displacement into a parameter explicitly linked to the seismic hazard as, for instance, PGA. It is important to recall that 1000 curves, connecting PGA and spectral displacement for the building and the seismic actions, were shown in Figure 7. These functions allow making the appropriate change of variable, as each of these curves were obtained by calculating the performance points in an incremental way. Then, the new damage index can be expressed as a function of the peak ground acceleration, PGA. It is important to note that these performance point computations have been made by using the same PGA values that were used in the nonlinear dynamic analysis. In Figure 13, the PDC calculated by using IDA are compared with the probabilistic damage curves, calculated by using PA, for an 𝛼𝛼= 0.7. 25 Figure 13 Comparison between damage indices by using IDA and PA Even if this comparison seems to exhibit a relatively good fit among the Park and Ang damage index and the proposed one, important differences appear when likening the percentiles of each group of PDC. In order to obtain a coefficient 𝛼𝛼, which provides the best fit between the curves obtained via IDA and PA, the Mean Squared Errors (MSE) for percentiles 5, 50 and 95 are calculated for a suite of 𝛼𝛼 values. Figure 14 shows the evolution of the MSE with 𝛼𝛼, for each percentile. 𝛼𝛼 values go from 0 to 1, with increments of 0.01. Figure 15 shows the comparison between the curves obtained by using IDA and PA approaches for percentiles 5, 50 and 95. The best fit is obtained by using 𝛼𝛼=0.65, 𝛼𝛼=0.78 and 𝛼𝛼=0.67 for percentiles 5, 50 and 95, respectively. It is important to observe the good adjust exhibited by the curves calculated with both methods. This fact demonstrates the validity of the simplified proposed approach based on PA. 32 scale strong ground motions to perform the comparison between the results of the nonlinear static and dynamic analyses. However, it is recognized that more realistic target spectra and more sophisticated scaling methods, should be investigated in order to analyse and discuss the results in specific sites where PSHA results are available. Finally, several advantages of the new method are summarized as follows: 1) the computational effort is much lower than the required by IDA. To be precise, at least a reduction to one sixth of the computational effort was observed when using the new procedure; 2) if the seismic hazard changes, the dynamic analysis requires to perform new complete nonlinear dynamic structural analyses, while the new procedure only requires to compute the new performance points as the capacity curve simply depends on the structure; 3) the new method also allows to take into account the uncertainties of the structural behavior and the ones of the seismic actions. 6 Acknowledgements This research has been partially funded by the Ministry of Economy and Competitiveness (MINECO) of the Spanish Government and by the European Regional Development Fund (FEDER) of the European Union (UE) through projects referenced as: CGL2011-23621 and CGL2015-65913 -P (MINECO / FEDER, UE). References Ambraseys, N., Smit, P., Sigbjornsson, R., Suhadolc, P. and Margaris B. 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