International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Pushover-based loss estimation of masonry buildings with consideration of uncertainties Snoj, Jure; Dolšek, Matjaž 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1Pushover-Based Loss Estimation of Masonry Buildings with Consideration of Uncertainties Jure Snoj Postdoctoral Researcher, Faculty of Civil and Geodetic Engineering, University of Ljubljana, Ljubljana, Slovenia Matjaž Dolšek Associate Professor, Faculty of Civil and Geodetic Engineering, University of Ljubljana, Ljubljana, Slovenia ABSTRACT: A seismic loss estimation methodology for masonry buildings is introduced. It enables the estimation of losses based on the simulated damage at the building’s level with consideration of epistemic and aleatoric uncertainties. The modelling uncertainties are incorporated through a set of structural models, which are defined by utilizing the Latin Hypercube Sampling technique. The damage of the building is simulated by the pushover analyses, whereas the relationship between the engineering demand parameter at the level of the building and the seismic intensity is estimated by incremental dynamic analysis, which is performed for the single-degree-of-freedom model taking into account the ground motion randomness and the modelling uncertainties. The loss estimation methodology requires the use of the fragility functions, which were in the case of masonry walls established from the experimental database, whereas for the other components, they were adopted according to FEMA P-58-1. The proposed method is demonstrated by means of an example of a three-storey masonry building. It is shown that consideration of the effect of modelling uncertainty increased the probability of collapse in 50 years and the expected annual loss, respectively, for factors of 1.5 and 1.25.   1. INTRODUCTION Several approaches for loss estimation were developed in the last decade (Aslani, 2005; Bradley, 2009; Haukaas, 2013). Among others, Pacific Earthquake Engineering Research Center (PEER) developed a probabilistic framework (e.g. ATC, 2012 or FEMA P-58-1), which enables the loss estimation and at the same time propagation of uncertainties through four independent parts of the methodology: hazard analysis, structural analysis, damage analysis and loss analysis. Monte Carlo simulations are used to compute losses given the statistical model of seismic response of a structure. The motivation of this study was to simulate losses directly for each seismic response analysis and to investigate the impact of uncertainties on the loss estimation in the case of masonry buildings. The PEER methodology was used as the basis of this study. However it was applied in a different manner than proposed in FEMA P-58. Thus the proposed approach enables explicit consideration of record-to-record variability and the modelling uncertainty and the estimation of losses based on actual demand and damage obtained from structural analysis. The complexity of the structural analysis is, according to the opinion of the authors, one of the main reasons that the loss estimation methodology is rarely applied to masonry buildings (Bothara et al., 2007; Borzi et al., 2008), which represent majority of building stock in Europe. However, it was shown elsewhere (Dolšek, 2012; Graziotti et al., 2014), that the pushover-based methods can provide sufficiently accurate results in the case of masonry buildings.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2In this paper, firstly the methodology for loss estimation is briefly presented and its application is demonstrated by means of an example of a three-storey masonry building. 2. LOSS ESTIMATION METHODOLOGY In this study the PEER methodology for loss estimation was used in a different manner than it was suggested by Aslani (2005), Bradley (2009) or FEMA P-58 (ATC, 2012). In the modified methodology (Snoj, 2014), the damage and losses are assessed for each simulation from the results of the seismic response analysis. Such approach does not require any assumptions regarding the correlation between the damage of structural components at various levels of ground motion intensity. In Figure 1 the proposed methodology is presented in five steps. In the first step it is necessary to assemble information about the building: the location of the building, its geometry, material and modelling characteristics, the classification of structural and non-structural components into fragility and performance groups and their corresponding fragility and loss functions and the estimation of the replacement cost of the building.  In the second step the results of hazard analysis at the location of the building has to be extracted from Probabilistic Seismic Hazard Analysis (PSHA). In addition to the seismic hazard curve, an adequate set of ground motions has to be selected. In order to consider the aleatoric uncertainty, na ground motions were selected from the PEER Ground Motion Database (PEER, 2013) according to the procedure proposed by Jayaram et al. (2011).  In the third step, the structural analysis is performed for the set of models, which are generated in order to incorporate the effect of epistemic uncertainties. Only selected modelling and material parameters are considered as random variables, which are later sampled by using the Latin Hypercube Sampling (LHS) technique (Vořechovský and Novák, 2002; Dolšek, 2012). Each combination of the input parameters represents one structural model m for which nonlinear static analysis (pushover analysis) is performed. The SDOF models are then defined for each of nm structural models. The pushover curves are idealized by a simple trilinear force-displacement relationship, followed by a simple transformation from an multi-degree-of-freedom (MDOF) model to a SDOF model (Fajfar, 2000). Finally, the incremental dynamic analysis is performed on the equivalent SDOF models (Dolšek, 2012). The results are na·nm SDOF-IDA curves where the engineering demand parameters (edp) for multiple intensity levels (im) are obtained until the seismic intensity imC, which causes dynamic instability of the building. By combining the seismic demand from incremental dynamic analysis and the damage analysis based on the results of pushover analysis, it is possible to estimate the conditional probability of building’s collapse P(C|IM), the collapse fragility.    Figure 1: Overview of the methodology for seismic risk assessment (Snoj, 2014).  NO 1) Building's input data Simulation s m = m+1;  a = a+1; im = im + Δim; 3) Structural analysis (LHS, pushover, SDOF-IDA) 4) Damage analysis 5) Loss analysis if im <  imCE(LT,NC(s)|IM); if im ≥ imC  E(LT,C)Is m < nm   a < na im < imC ? Expected total loss given intensity - E(LT|IM) YES 2) Hazard analysis Performance assessment - EAL, P(LT > lt), E(LT)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3In addition, based on the values of the EDPs for all the structural and non-structural components in each simulation (steps 4 and 5) and by knowing the relationship between damage and engineering demand parameters (fragility functions) and the relationship between loss and damage (loss functions), the expected loss in each component j given EDP – E(Lj|EDPj (im)) can be calculated:           | | |jj j j j j jall DSE L EDP im E L DS p DS EDP im   (1) where the expected loss in component j for each damage state E(Lj|DSj) is weighed by the probability of its occurrence p(DSj|EDPj(im)) and summed over all possible local damage states. The total loss in the building for simulation s is simply the sum of expected losses over all components:      , | |T NC j jall jE L s IM im E L EDP im    (2) Note, that in the proposed methodology value of edpj given im is obtained directly from structural analysis for each component. This sample is approximation of random variables EDPj(im) (Eq. (1)) and its size is na·nm. Index NC stands for the non-collapse case, which occurs in each simulation for the intensity levels im<imC. However, for intensities higher than imC, the expected total loss of the building is equal to its replacement cost including the cost of demolition E(LT,C). For each intensity level im, the size of the sample values of E(LT (s)|IM) is also equal to na·nm. A mean value of E(LT (s)|IM) represents the expected total loss given intensity E(LT|IM), which is often termed the vulnerability curve. The expected annual loss (EAL) can be obtained by convolving the mean annual frequency of exceeding the ground motion intensity λIM and the expected total loss given intensity:   im( )| IMTalld imEAL E L IM dIMdIM   (3) This performance measure is very important for the investors, owners and other stakeholders, since they can compare the expected annual loss to the insurance premiums or annual revenues for financial planning. By analogy to Eq. (3), the mean annual frequency of exceeding a certain total loss λ(LT > lt) (loss hazard curve) can be computed by integrating the conditional probability of exceeding a certain loss given intensity P(LT > lt|IM) over all possible levels of ground motion:    im( )| IMT t T talld imL l P L l IM dIMdIM       (4) where P(LT>lt|IM) is estimated from simulations.  3. CASE STUDY: THREE STOREY MASONRY BUILDING 3.1. Building’s input data The methodology is demonstrated by means of the three-storey unreinforced masonry building, which is built from hollow clay bricks. The plan and the elevation of the building are presented in Figure 2. The building is symmetric around the Y axis. It has 5.6 % and 5.3 % of shear walls in the X and Y direction, respectively. The wall thickness is 0.3 m and the storey height of all storeys is 3.2 m. Concrete slabs with thickness of 0.18 m are considered as rigid diaphragms. It is assumed, that the building is located in Ljubljana (Slovenia) on the soil type B (Eurocode’s terminology). The equivalent frame model of the building was made by using the research version of the program Tremuri (Penna et al., 2014), which is specialized for seismic analysis and performance assessment of masonry structures. The nonlinear model consisted of planar frames, which are connected at the corners and intersections of walls. Each wall of the building was divided into piers and spandrels, where the non-linear response is simulated in plastic hinges. The main advantage of such macroelement model is the capability of representing the shear sliding and flexural failure mechanisms with toe crushing and their evolution, controlling the strength and stiffness deterioration. Note, that the global behaviour was governed only by in-plane capacity of the walls, 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4since out-of-plane collapse was not considered. Another important aspect of the mathematical model is the definition of ultimate drifts. The macroelement’s lateral stiffness and strength were set to zero, if the drifts of the structural components exceeded the ultimate drifts.  The impact of epistemic uncertainty was estimated by 30 structural models, which were generated by using the LHS technique. The number of structural models is more than two times the number of random variables used to describe the modelling uncertainties, which is sufficiently accurate according to previous studies. For comparison, the deterministic model was also defined, where the median values of the 13 uncertain input parameters were used. The corresponding coefficients of variation and the assumed type of distribution function (normal, lognormal and truncated normal or lognormal) are presented in the Table 1. The information regarding the uncertain modelling parameters was mainly adopted from the literature. The only exceptions were the ultimate drifts in shear and flexure, which were determined based on the database of the experimental results.  The dead load of the first two storeys amounted to 6.2 kN/m2, and to 6.3 kN/m2 on the flat walkable roof. Since the building was assumed to be an office building, the live loads for the floors, balconies and staircases were 3 kN/m2, 2.5 kN/m2 and 2 kN/m2, respectively. Additionally, the non-structural components were also considered in the analysis. They were categorized into fragility groups (the same fragility function) and performance groups (a logical group of components with similar performance). The description and the quantities for each of these groups are shown in the Table 2 together with the corresponding parameters of fragility functions (median value and CoV of the EDP) and loss functions (expected cost of repair compared to the cost of new component per unit) for each damage state. Finally, the cost for building's replacement including demolition was estimated to be 590000 € based on the Slovenian cost databases.  a) GROUND FLOOR, 1. AND 2. FLOOR200 290 170 220 250 220 170 290 120 20050010010050030301230450290450280 120 540 540 120 280130 90 130 160 390 250 390 160 130 90 130W1 W2W3W4W5W3W4W6W6W7W8W9 W10W12W11W13W13290Office 1Office 2Office 3 Office 4KitchenWCServerO (160x140) O (160x140)O (90x140)O (120x140)2 x O (100x140)2 x O (100x140)P (130x300)P (250x300)P (450x300)P (180x300)P (150x300)P (220x300) P (135x300)2 x P(160x300)P (230x300) P (220x300)P (150x300)P (180x300)100120A AD (1500)Z (100)P (130x300)O(100x140) SS (210) - ceiling929292100100W - walls P (l x h) - partitionsO (l x h) - windowsD (h) - chimneyZ (h) - parapet wall SS (A) - ceilingLEGENDb) SECTION A-A300 700 250 700 300480XYO (90x140)O (170x140) O (170x140) O (120x140)D (1500)P (130x300)P (250x300)Office 6Office 5Z (100)P (130x300)O(100x140)P (130x300)O(100x140)SS (210) - ceilingSS (210) - ceilingP (130x300)P (130x300)P (130x300)O(100x140)O(100x140)O(100x140)Figure 2: (a) The typical plan and (b) the elevation of the building. Presented are the structural and non-structural components.  Table 1: The median, coefficient of variation and assumed type of distribution for uncertain material and modelling parameters of modern hollow clay masonry buildings. Name X  COV Distribution Specific weight γ (kN/m3) 14 0.10 Logn. Comp. strength fm (MPa) 5 0.20 Logn. Shear strength fv0 (MPa) 0.20 0.30 Trunc. logn. (0.10 - 0.30) Elastic modulus E (MPa) 5000 0.25 Logn. Shear modulus G  (MPa) 500 0.25 Logn. Ultimate shear drift δs (%) 0.41 0.57 Trunc. logn. (0.11 - 0.80) Ultimate flexural drift δf  (%) 0.72 0.47 Trunc. logn. (0.50 - 1.80) Friction coefficient μ 0.40 0.19 Trunc. norm. (0.30 - 0.50) Damping coefficient ξ 0.05 0.40 Trunc. norm. (0.03 - 0.07) Acc. eccentricity ex (% LY) 0 3 Norm. Acc. eccentricity ey (% LX) 0 3 Norm. Deform. parameter Gc 7 0.21 Trunc. norm. (5 - 9) Softening parameter β 0.30 0.17 Norm.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5Table 2: The database of fragility and performance groups used in this study. The median and coefficient of variation of the corresponding EDP define the fragility functions and the expected cost of repair compared to the cost of new element define the loss functions. Fragility functions were assumed to be lognormally distributed. Fragility and performance groups Fragility functions Loss functionsComponents Unit Floors Quantity DS EDP X  CoV New Unit  Cost (€) E(L'|DS) X YStructural Masonry walls shear m2 1, 2, 3 166 159 DS1IDR (%) 0,11 0,26 101 0,21DS2 0,29 0,47 101 0,86DS3 0,41 0,57 101 1,21Masonry walls flexure DS1IDR (%) 0,05 0,50 101 0,21DS2 0,33 0,52 101 0,86DS3 0,72 0,47 101 1,21Non-structural Partition walls m2 1, 2, 3 26 66 DS1IDR (%) 0,21 0,60 37 0,30DS2 0,71 0,45 37 0,60DS3 1,20 0,45 37 1,20Windows # of windows (1,4m x 1,4m) 1, 2, 3 7.7 2.9 DS1IDR (%) 1,60 0,29 560 0,10DS2 3,20 0,29 560 0,60DS3 3,60 0,27 560 1,20Masonry parapet m2 3 25 45 DS1 PFA (g) 0,20 0,60 78 0,60DS2 0,40 0,60 78 1,20Masonry chimney m 1 30 DS1 PFA (g) 0,35 0,60 150 1,20DS2 0,50 0,60 150 1,20Suspended ceiling m2 1, 2, 3 210 DS1PFA (g) 0,27 0,40 22,5 0,12DS2 0,65 0,50 22,5 0,36DS3 1,28 0,55 22,5 1,20Server and computers / floor 1, 2, 3 12000 DS1 PFA (g) 1,00 0,50 1000 1,00Generic drift   sensitive components / floor 1, 2, 3 20000 DS1IDR (%) 0,40 0,50 1000 0,025DS2 0,80 0,50 1000 0,10DS3 2,50 0,50 1000 0,60DS4 5,00 0,50 1000 1,20Generic acceleration sensitive components / floor 1, 2, 3 20000 DS1PFA (g) 0,25 0,60 1000 0,02DS2 0,50 0,60 1000 0,12DS3 1,00 0,60 1000 0,36DS4 2,00 0,60 1000 1,203.2. Hazard and structural analysis The hazard curve (Figure 3a) was obtained from previous study (Brozovič and Dolšek, 2013).  The ground motions were selected to match the Eurocode’s spectrum (Figure 3b), which provides conservative estimates of the fragility functions at the level of the structure and losses.    Figure 3: (a) The hazard curve and (b) the elastic spectra of the 30 selected ground motions and target spectrum from Eurocode 8 for soil type B.   For the ground motion selection, the soil type B was assumed with consideration of the interval of the magnitudes (5.5 ≤ M ≤ 7.5), the source-to-site distances (5 km ≤ r ≤ 50 km) and scale factor (sf ≤ 3). The peak ground acceleration was selected for the intensity measure.  In Figure 4 the structural analysis of the deterministic model is summarized. Firstly, the pushover analysis is performed, and then the pushover curve is idealized with a simple tri-linear force-displacement relationship and transformed from MDOF to SDOF model (Figure 4b). Note the sudden strength deterioration in the pushover curve that occurred due to the formation of plastic mechanism in the first storey, where multiple walls failed at approximately the same displacement. In Figures 4c and 4d, the SDOF-IDA curves are shown, 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6respectively, for only one ground motion and for the set of 30 ground motions, which reflect the impact of ground-motion randomness. The seismic intensities causing collapse vary quite significantly although the structure had very low vibration period. For deterministic model, a sample of 30 intensities, which cause collapse of the building, was estimated. The median collapse capacity  pga50,DET,DS4 and the corresponding dispersion βDET,DS4 amounted to 0.58 g and 0.18, respectively.   The effect of the epistemic uncertainties on the seismic response is shown in the Figure 5. The pushover analyses were performed for 30 structural models (Figure 5a). The values of base shear were observed between 2500 kN and 4100 kN. Much greater variation was observed in the case of displacement capacity of the building for the near-collapse limit state (from 1.6 cm to 12.1 cm). It was shown that the most influential parameters which affected base shear and deformation capacity were: initial shear strength, friction coefficient, ultimate shear drift in the walls and the evolution of different global plastic mechanisms. In Figures 5c and 5d the SDOF-IDA curves for 900 simulations (30 ground motions · 30 models) are presented including the 16th, 50th and 84th percentile curves for both, stochastic and deterministic model. The slight decrease of the median SDOF-IDA curve and larger dispersion of global collapse capacities for the stochastic model compared to the deterministic model can be observed. The median collapse capacity corresponding to the stochastic model pga50,STOCH,DS4 and the corresponding dispersion βSTOCH,DS4 amounted to 0.55 g and 0.37, respectively. 3.3. Damage and loss analysis The collapse fragility curves, which are defined as the conditional probability of global collapse given intensity P(C|IM) for the stochastic and deterministic model are shown in the Figure 6a. Note that the probability of collapse in t years can be computed by integrating the P(C|IM) with the seismic hazard over all possible intensities. If the modelling uncertainties are not considered in the analysis (deterministic model), there is 1.2 % probability of collapse in 50 years for this building, however if the modelling uncertainties are taken into account, the risk of collapse increased to 1.8 %.     Figure 4: (a) The pushover curve, (b) the tri-linear force-displacement relationship for the SDOF model (c) SDOF-IDA curve for the accelerogram 1 and (d) 30 SDOF-IDA curves for deterministic model for all considered ground motions.     Figure 5: The comparison of the results for stochastic and deterministic model in terms of: (a) pushover curves, (b) force-displacement relationship of SDOF models, (c) SDOF-IDA curves including intensities causing collapse and (d) the corresponding 16th, 50th, and 84th percentiles.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7The expected loss at the component level and the total loss for the non-collapse cases given the intensity were estimated according to Eqs. (1) and (2), respectively. At each intensity level im the top displacement for the MDOF model was obtained by transformation of the corresponding displacement from the SDOF-IDA curve. The so-determined top displacement was used to estimate the engineering demand parameters from pushover analysis. For example, the expected losses for non-collapse cases given the intensity are presented for different fragility groups (Figure 6b). The contribution of the non-structural components is significant. They contribute more than 65 % to total loss for all the intensities. Note, that negligible losses were observed for windows. The main reason for this is the assumed fragility function (Table 2), because masonry buildings typically don’t experience very high interstorey drifts and the windows remain undamaged in the simulations. In the reality, this is of course not the case, hence the fragility functions for some of the non-structural components should be investigated thoroughly in the future research, or the correlations between damage of various components should be considered (i.e. windows are damaged if the adjacent masonry wall fails). Although the proposed methodology enables explicit consideration of such correlations, they were not considered in this study due to simplicity. In the Figure 6c the expected total loss given intensity E(LT|IM) is presented, considering also the simulations where the collapse of the building occurs. Note that the collapse cases occurred even at quite low level of pga = 0.25 g. Therefore the total loss due to collapse start to dominate quickly, because the ratio between replacement cost of the building E(LT,C) and the expected losses given no collapse E(LT,NC|IM) is very high in this case. The collapse fragility of the building and the ratio E(LT,C)/E(LT,NC|IM) were also the most influential parameters for the E(LT|IM) (Snoj, 2014).   Figure 6: The results of the damage and loss analysis for stochastic and deterministic model: (a) the building’s collapse fragility, (b) the contribution of the considered fragility groups to the E(LT,NC|IM), (c) the disaggregation of the expected losses given intensity and (d)  loss curves for multiple timeframes.  It is also interesting to investigate the impact of modelling uncertainties on the expected annual loss (EAL, Eq. 3). The EAL was estimated to 380 € (~0.06 % replacement cost or 50 € per 100 m2 of floor area) when the impact of modelling uncertainties was neglected and to 470 € (~0.08 % of the replacement cost or 62 € per 100 m2 of floor area) for the stochastic model. Finally, the results of the loss assessment are presented in terms of loss curves (Eq. (4), Figure 6d). The investors can get very interesting information from these curves. For example, the expected loss which is exceeded with 10 % probability in 50 years for this building was estimated to 17000 €. Another way to communicate to the stakeholders in terms of losses is to estimate the probability that the loss will exceed a certain value. For example, there is 0.44 % probability that the loss will exceed 50000 € in 10 years.  4. CONCLUSIONS In this study, a methodology was presented for seismic risk assessment in terms of probability of collapse and the expected losses due to earthquakes. Some issues of the PEER loss estimation methodology were revealed. For 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8example, the component fragility functions of a certain component are often considered independent of the other components, which may result in biased results as discussed in this paper. Thus the correlations between damage of some components should be taken into account in order to increase the reliability of loss estimation.  The combination of pushover analysis, which is performed for the model of entire structure, and incremental dynamic analysis for the SDOF model was applied to realistic structure in order to study the effect of both, epistemic and aleatoric uncertainties. For the stochastic model the probability of the collapse increased for 50 % compared to the deterministic model of the three-storey unreinforced masonry building. The expected annual loss for this building was estimated to 0.06 % and 0.08 % of the replacement cost in the case of deterministic and stochastic model, respectively.  Hence, the modelling uncertainties have a large impact on the intermediate results, including the collapse fragility of the building. The collapse fragility, in addition to hazard curve, fragility and loss functions and the ratio between the replacement cost of the building and the losses without consideration of building’s collapse also have a large impact on the loss estimation; therefore the modelling uncertainties also have a large impact on losses.  5. REFERENCES  Aslani, H. (2005). “Probabilistic earthquake loss estimation and loss disaggregation in buildings” Doctoral dissertation. Stanford, Stanford University, Department of Civil and Environmental Engineering. ATC. (2012). “FEMA P-58-1: Seismic performance assessment of buildings, Volume 1 - Methodology” Washington D.C., FEMA. Borzi, B., Crowley, H., and Pinho, R. (2008). “Simplified pushover-based earthquake loss assessment (SP-BELA) method for masonry buildings” International Journal of Architectural Heritage, 2(4), 353–376. doi:10.1080/15583050701828178 Bothara, J. K., Mander, J. B., Dhakal, R. P., Khare, R. K., and Maniyar, M. M. (2007). “Seismic performance and financial risk of masonry houses” Journal of Earthquake Technology, 44(4), 1–27.  Bradley, B. A. (2009). “Structure-specific probabilistic seismic risk assessment” Doctoral dissertation. Christchurch, University of Canterbury, Department of Civil and Natural Resources Engineering. Brozovič, M., and Dolšek, M. (2013). “Envelope-based pushover analysis procedure for assessing the collapse risk of buildings” In Proceedings of the 11th International Conference on Structural Safety and Reliability, New York, USA, June 16-20, 2013. Dolšek, M. (2012). “Simplified method for seismic risk assessment of buildings with consideration of  aleatory and epistemic uncertainty” Structure and Infrastructure Engineering, 8(10), 939–953.  Fajfar, P. (2000). “A nonlinear analysis method for performance based seismic design” Earthquake Spectra, 16(3), 573–592. Graziotti, F., Penna, A., Bossi, E., and Magenes, G. (2014). “Evaluation of displacement demand for unreinforced masonry buildings by equivalent SDOF systems” In Proceedings of 9th International Conference on Structural Dynamics, Porto, Portugal, June 30 - July 2, 2013.  Haukaas, T. (2013). “Probabilistic models, methods, and decisions in earthquake engineering” In Proceedings of the 11th International Conference on Structural Safety and Reliability, New York, USA, June 16-20, 2013. Jayaram, N., Lin, T., and Baker, J. W. (2011). “A computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance” Earthquake Spectra, 27(3), 797–815.  PEER. (2013). “NGA strong motion database” (Retrieved 7/18/2013)  Penna, A., Lagomarsino, S., and Galasco, A. (2014). “A nonlinear macroelement model for the seismic analysis of masonry buildings” Earthquake Engineering & Structural Dynamics, 43(2), 159–179.  Snoj, J. (2014). “Ocena potresnega tveganja zidanih stavb” Doctoral dissertation. Ljubljana, University of Ljubljana, Faculty of Civil and Geodetic Engineering.  Vořechovský, M., and Novák, D. (2002). “Correlated random variables in probabilistic simulation” In Proceedings of the 4th International Ph. D. Symposium in Civil Engineering, Munich, Germany, September 19-21, 2002.  


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