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Capacity, fragility and damage in reinforced concrete buildings: a probabilistic approach

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Abstract

The main goals of this article are to analyze the use of simplified deterministic nonlinear static procedures for assessing the seismic response of buildings and to evaluate the influence that the uncertainties in the mechanical properties of the materials and in the features of the seismic actions have in the uncertainties of the structural response. A reinforced concrete building is used as a guiding case study. In the calculation of the expected spectral displacement, deterministic nonlinear static methods are simple and straightforward. For not severe earthquakes these approaches lead to somewhat conservative but adequate results when compared to more sophisticated procedures involving nonlinear dynamic analyses. Concerning the probabilistic assessment, the strength properties of the materials, concrete and steel, and the seismic action are considered as random variables. The Monte Carlo method is then used to analyze the structural response of the building. The obtained results show that significant uncertainties are expected; uncertainties in the structural response increase with the severity of the seismic actions. The major influence in the randomness of the structural response comes from the randomness of the seismic action. A useful example for selected earthquake scenarios is used to illustrate the applicability of the probabilistic approach for assessing expected damage and risk. An important conclusion of this work is the need of broaching the fragility of the buildings and expected damage assessment issues from a probabilistic perspective.

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Acknowledgments

The thoughtful comments and suggestions of Julian J. Bommer helped us to address the probabilistic treatment of the seismic actions and to improve the manuscript. The thorough review of two anonymous referees is also acknowledged. Special thanks are given to Emilio Carreño and Juan M. Alcalde of the Instituto Geográfico Nacional (IGN) who provided us the Spanish accelerogram database. M. Mar Obrador has carefully revised the English. This work has been partially funded by the Spanish government and the European Commission with FEDER funds through research projects CGL2008-00869/BTE, CGL2011-23621, CGL2011-29063INTERREG: POCTEFA 2007-2013/ 73/08 and MOVE-FT7-ENV-2007-1-211590. Yeudy F. Vargas has a scholarship from an IGC-UPC joint agreement.

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Correspondence to Luis G. Pujades.

Appendix: Accelerograms used in NLDA

Appendix: Accelerograms used in NLDA

Nonlinear Dynamic Analysis requires accelerograms. Stochastic analyses require a significant number of accelerograms, all of them well-matched with a target spectrum. So, given a target response spectrum and a specific database of acceleration records, the procedure used in this article for selecting accelerograms is the following. Step 1: normalize the target spectrum at the zero period. Step 2: compute the corresponding normalized spectrum for each accelerogram. Step 3: compute a measure of the misfit between the computed and target spectra; in this case, this measure is the error computed according to the following equation:

$$\begin{aligned} \varepsilon _j =\frac{1}{n-1}\sqrt{\sum _{i=1}^{n } {\left( {y_{ji} -Y_i } \right) ^{2}} }\, j=1\ldots N,\quad i=1\ldots n \end{aligned}$$
(8)

where \(\varepsilon _j \) is a least square measure of the misfit between the spectrum of the accelerogram \(j\) and the target spectrum; \(y_{ji} \) is the spectral ordinate \(i\) of the spectrum of the accelerogram \(j, \) and \(Y_i \) is the corresponding \(i\) ordinate of the target spectrum; \(n\) is the number of spectral ordinates of the accelerogram, assumed to be the same for each accelerogram \( j,\) and \(N\) is the number of accelerograms. Step 4: organize the spectra according increasing errors. Step 5: let \(Sa_{ik} i=1\ldots n,\;k=1\ldots N\), be the \(i\)-th spectral ordinate corresponding to the spectrum of the accelerogram \(k\), once the accelerogram series has been arranged in such a way that \(\varepsilon _k \le \varepsilon _{(k+1)}\,\, k=1\ldots N-1\). Compute the following new spectra:

$$\begin{aligned} b_{im} =\frac{1}{m}\sum _{k=1}^m {Sa_{ik} }\qquad i=1\ldots n,\quad m=1\ldots N; \end{aligned}$$
(9)

\(b_{im} \) are now the spectral ordinates of the mean of the first \(m\) spectra, once they have been arranged. Step 6: compute the following new error function \((Er_m )\), which is similar to that given in Eq. (8).

$$\begin{aligned} Er_m =\frac{1}{n-1}\sqrt{\sum _{i=1}^{n} {\left( {b_{im} -Y_i } \right) ^{2}} }\, m=1\ldots N,\quad i=1\ldots n \end{aligned}$$
(10)

The value of \(m \) that minimizes the value of \(Er_m \) is considered as the optimum number of accelerograms that are compatible with the given target spectrum and that can be used of the given database. Of course, the value of \(Er_1 \) is also crucial for knowing the goodness of the fit. For databases with a large number of accelerograms, \(Er_1 \) is really small while we found \(m\) values of several tens. Some additional basic assumptions can be taken in order to reduce the size of the database. Information about the magnitude and focal mechanism of the earthquakes and about distance and soil type of the accelerometric stations can be used to significantly reduce the number of acceleration records to be tested. This procedure has been applied to the European (Ambraseys et al. 2002, 2004) and Spanish strong motion databases. As the European database is larger, most of the selected accelerograms are taken from this database but for spectra type 2, some accelerograms of the Spanish database were selected too. For each spectrum, 1A, 1D, 2A and 2D, more than 1,000 records of acceleration were tested. Table 10 shows the statistics of the distributions of the magnitudes of the events corresponding to the accelerograms selected compatibles with the 1A, 1D, 2A and 2D spectra. Figure 24 shows the error function,\(Er_m \) in Eq. (10), for the EC8 1D spectrum. The value of \(m\) minimizing this function has been found to be 20.

Table 10 Statistics: mean values, standard deviations and coefficients of variation of the magnitudes of the earthquakes corresponding to the selected accelerograms and spectra
Fig. 24
figure 24

Function \(Er_m \) used to optimize the number of compatible accelerograms. This example corresponds to the Eurocode EC8, 1D spectrum and \(m=20\). The main parameters of the selected records are shown in Table 11

Table 11 shows the main parameters of these accelerograms. The mean values of the magnitude, distance and depth are respectively 6.5, 67 km and 16.7 km and the corresponding standard deviations are 0.7, 53.6 km and 17.6 km. Figure 25a shows the target EC8 1D spectrum, the mean spectrum and the spectrum defined by the median values plus one standard deviation. The fundamental period of the building is also plotted in this figure. Figure 25b shows an example of compatible accelerogram.

Table 11 Main parameters of the accelerograms compatible with the EC81D spectrum
Fig. 25
figure 25

a Normalized EC8 1D spectrum. The mean and the mean plus one standard deviation spectra are also shown. The fundamental period of the structure is plotted too. b Example of one accelerogram matching this spectrum

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Vargas, Y.F., Pujades, L.G., Barbat, A.H. et al. Capacity, fragility and damage in reinforced concrete buildings: a probabilistic approach. Bull Earthquake Eng 11, 2007–2032 (2013). https://doi.org/10.1007/s10518-013-9468-x

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