Analytical derivation of seismic fragility curves for historical masonry structures based on stochastic analysis of uncertain material parameters

Abstract – This work presents a probabilistic method to assess the seismic vulnerability of historical masonry structures, such as churches or cathedrals, including uncertainty analysis of the material parameters. The proposed approach considers the pushover analysis using the finite element method for the structural evaluation of the seismic behaviour of masonry structures. A stochastic analysis based on Monte Carlo simulation investigates the effect of the uncertainty of the structural members’ mechanical parameters on the evaluation of the seismic fragility. The method is applied to the seismic assessment of the bay structure of Santa Maria del Mar church in Barcelona, Spain. This case study is characterised by complex geometry and materials heterogeneity, and shows to be sensitive enough to the uncertainty of the material properties to experience two possible collapse mechanisms in case of an earthquake. The study presents how to derive analytical seismic fragility curves by considering the uncertainties regarding the material properties and the different types of collapse mechanism.


Introduction
The evaluation of the seismic vulnerability of historical masonry buildings is a task of paramount importance nowadays. Such structures have shown to be very susceptible to damage and even to collapse in case of earthquakes, as demonstrated by the recent events in Europe, like the 2016-2017 Amatrice-Visso-Norcia seismic sequence in the central Italy.
Reliable methodologies able to deal with this problem are necessary especially for built cultural heritage masonry structures in order to ensure their conservation. This category of buildings usually includes complex structural typologies like churches, cathedrals, palaces and monuments. For these types of structures, seismic analysis is a very demanding task involving a high amount of uncertainty in the characterization of the geometry and complex morphology of the structural members, as well as the mechanical parameters of the constituent materials. This is true especially for historical masonry buildings, which present a significant variety of construction typologies and composite materials. The quality of historical masonry is related not only to the material constituents, but also to the constructive features such as the dimension of the blocks, bond, interlocking, and transversal connections. Much of this information, which would be essential for a reliable seismic analysis, is usually very difficult to acquire. The necessary inspection and experiments are often severely limited by their high cost and the restrictions posed by the cultural value of the buildings.
The evaluation of the seismic vulnerability of historical masonry structures still requires proper methodologies able to account for the variability of the structural and material parameters affecting the seismic response of the structure. The limitations induced by the uncertainty of the materials' properties on conventional deterministic and semi-probabilistic approaches for seismic assessment are well known (Franchin, Pinto and Rajeev, 2010;Tondelli et al., 2012). Another important source of uncertainty is due to the modelling hypotheses and adopted strategies for the analysis of the structure (Atamturktur, Hemez and Laman, 2012;Bracchi et al., 2015) Several European research projects have dedicated great attention to the problem of including the inherent probabilistic nature into the seismic risk assessment approach, such as RISK-UE, SYNER-G and PERPETUATE (Mouroux and Le Brun, 2006;Cattari, 2014b, 2014a). All these projects have developed different methodologies to derive seismic fragility functions for ordinary and cultural heritage masonry buildings, expressing the probability of occurrence of certain damage grades for different levels of seismic intensity.
The recent Italian guidelines CNR-DT 212/2013 have presented an organic methodology for the probabilistic vulnerability assessment of existing masonry buildings. The document suggests the derivation of analytical fragility curves as a function of the dispersion of the considered uncertain parameters.
Even though some authors have explored the effect of the uncertainty of material parameters in the probabilistic seismic assessment of masonry buildings (Rota, Penna andMagenes, 2010, 2014;Pagnini et al., 2011;Snoj and Dolšek, 2011;Parisi and Augenti, 2012;Bosiljkov, D'Ayala and Novelli, 2015;Bracchi et al., 2016), limited studies are available about the case of more complex structural typologies in historical constructions (Petromichelakis, Saloustros and Pelà, 2014;Bartoli et al., 2017;Saloustros et al., 2019). This paper presents a probabilistic methodology for the analytical derivation of seismic fragility functions of complex masonry historical structures including the uncertainty regarding the materials' mechanical parameters. The approach makes use of the Finite Elements Analysis (FEA) for the evaluation of the seismic structural capacity through nonlinear static (pushover) analysis. A stochastic nonlinear analysis based on Monte Carlo simulation allows the investigation of the effect of the material parameters uncertainty on the evaluation of the seismic fragility.
The method is applied to the study of the church of Santa Maria del Mar in Barcelona, Spain. The paper evaluates the seismic vulnerability of the representative macro-element of the bay structure against transversal horizontal loading considering the uncertainties in the definition of the materials' mechanical properties. The uncertain material properties of the different structural members of the macro-element are considered as random variables with associated probability distributions and ranges of variation. The effect of these uncertainties is investigated by analysing 200 random samples of possible material combinations for the same structure by means of a Monte Carlo Simulation. The force-displacement capacity of the different models has been calculated by FEA on 200 pushover analyses. The application of the N2 method permits the evaluation of the seismic demand for the city of Barcelona corresponding to four damage limit states (slight, moderate, extensive and complete) for all the investigated cases. The derivation of analytical fragility curves allows the definition of the probability of occurrence of each damage grade for different seismic hazard scenarios. Due to the prediction of two different collapse mechanisms, fragility curves are derived either by separating the 200 analyses in two groups depending on the predicted collapse mechanism or by using all the analyses without making any such differentiation. The effect of these alternatives on the estimation of the seismic fragility of the structure is discussed.

Proposed methodology for probabilistic seismic assessment including material uncertainty
This work proposes a methodology to evaluate analytical fragility curves in which the seismic vulnerability of complex historical masonry building is evaluated taking into account the uncertainty of the materials' mechanical properties.
The first step of the procedure is to identify and characterize the uncertain data. This activity is the result of the analysis of the diverse material typologies existing in the different structural members composing the complex building. For this reason, it is important the level of knowledge of the investigated building, as derived from inspection, survey and, if possible, experimental testing. Different mathematical models are available to assess the characteristics of uncertain parameters within a computational framework (Graf, Götz and Kaliske, 2015). The best compromise in terms of simplicity, reliability and computational efficiency lays in the concept of random variables, as derived from probability theory. The definition of a suitable probability density function for each random variable can model its uncertainty within reasonable ranges of variation.
The effect of the uncertainty of parameters on the seismic response of the investigated structure is evaluated by means of stochastic analysis. Among the available stochastic methods, this paper considers a Monte Carlo Simulation due to its effectiveness and adequacy to structural problems, as suggested by previous works (Rota, Penna and Magenes, 2010;Pagnini et al., 2011;Petromichelakis, Saloustros and Pelà, 2014) and relevant standards (CNR-DT 212/2013.
The Monte Carlo Simulation generates a specified number of N independent and identically distributed random samples (population of structural instances/cases to be analysed) within the input space ⊆ ℝ , where n is the number of the assumed input random variables, i.e. the uncertain material parameters. The sampling process takes into account the probability distributions associated to each of the random variables. The selection of suitable probability density functions for the uncertain material parameters is based on recent studies for masonry structures (Sykora and Holicky, 2010;CNR-DT 212/2013 and the recommendations of technical codes (MIT, 2009;CNR-DT 212/2013. Then, a mapping model maps each of the N samples to the result space ⊆ ℝ , where m is the number of the result variables to be evaluated. The mapping model proposed in this research includes the development of N pushover seismic analyses by FEA, the evaluation of the seismic performance by the N2 method (Fajfar, 1999) and the identification of the seismic demands associated to conveniently defined damage limit states. Finally, analytical seismic fragility curves are plotted as a function of the output variables. Figure   1 shows a flowchart summarizing the different stages of the proposed methodology.  -10 -A detailed survey and inspection campaign performed on site (Vendrell et al., 2007) provided valuable information regarding the dimension and inner morphology of the structural members. Walls and buttresses are composed of three-leaves whose external layers are made of ashlar masonry with lime mortar joints, while the inner core consists of irregular rubble masonry. All the stone materials come from the quarries of Montjuïc hill in Barcelona. Sonic tomography carried out on the columns revealed the inner morphology of the octagonal piers, with rows consisting of four external hexagonal stones surrounding a square central one (González et al., 2008). Each row is rotated 45º with respect to the inferior ones to provide interlocking between stone blocs. Compared to the rest of vertical load-bearing members, the piers can benefit from a higher strength and stiffness due to the large dimensions of the stones and the thinness of the mortar joints. The church of Santa Maria del Mar experienced the effects of several earthquakes during its history (González et al., 2008). According to available historical documents, an earthquake in 1373 caused the failure of the upper gallery of one of the bell towers , and another one in 1428 caused about twenty casualties due to the collapse of the rose window at the end of religious ceremony (Fontserè, 1971).

Finite Element model
The proposed probabilistic approach is applied in this research for the evaluation of the seismic response along the transversal (east-west) direction of the representative bay structure of Santa Maria del Mar, including the main and lateral naves. The geometry of the macro-element has been extracted from a Laser Scanner survey of the entire building  The selected macro-element is represented numerically as a two-dimensional finite element model. Figure 4 presents the finite element mesh, composed of 37780 triangular constant strain plane stress finite elements and 17749 nodes. The mesh is refined in the parts of the structure where high stress gradients commonly occur, such as the vaults and the buttresses. Numerical analysis are performed by using the finite element code DI-ANA-FEA (TNO, 2017) and the mechanical response of masonry is simulated adopting the total strain rotating crack model. The nonlinear post-peak compressive behaviour is characterized by a parabolic stress-strain relationship, whereas the tensile one by exponential softening. The stress-strain relationships are regularized according to the crackbandwidth approach (Bažant and Oh, 1983), ensuring mesh-size independent results.
The numerical model has been calibrated following a two-step procedure. The first step includes its comparison with a detailed 3D finite element model of the same macro-element (Murcia, 2008;Roca et al., 2009), such that the two models present an equivalent response in terms of stiffness and capacity under both gravitational and horizontal inplane loading proportional to the mass distribution. The second-step considers the comparison of the numerical vibration characteristics with the results of the experimental dynamic identification reported in (Vendrell et al., 2007). The fundamental vibration mode predicted by the numerical analysis in the transversal direction of the church has an eigenfrequency of 1.39 Hz and a participating mass of 70.6% ( Figure 5), while the second numerical eigen-frequency in the same direction is 7.02 Hz, with a participating mass of 9.2%. The first numerical eigen-frequency is 5% lower than the experimental one, measured equal to 1.45 Hz.
-13 -The adopted calibration methodology of the numerical model, which has been used for similar historical churches (Roca et al., 2013;Petromichelakis, Saloustros and Pelà, 2014), has been detailed for the selected macro-element of Santa Maria del Mar church in Contrafatto (2017). The choice of a simplified two-dimensional model allows the execution of the large number of numerical simulations required by the probabilistic approach at an affordable computational cost.  solved though a regular Newton Raphson method with an arc-length strategy. Convergence is checked based on force and displacement norm ratios below 1%.

Random variables
Based on available studies on the construction materials of the church (Vendrell et al., 2007), the structural elements are classified into four categories of materials with similar -15 -mechanical properties. These are the vaults and single-leaf walls (group I), the three-leaf walls with the heavy infill (group II), the columns (group III) and the light vault infill (group IV), shown in Figure 6. The probabilistic analysis considers the vaults and the single-leaf walls (group I) as the "reference material", and six material parameters as random variables. Three of them are the mechanical properties of the reference material, namely the compressive strength fc, the tensile strength ft, and the Young's modulus E. These material properties are characterized by aleatoric and epistemic uncertainty. The aleatoric uncertainty is due to the natural variation of the mechanical properties of masonry. The epistemic uncertainty is attributed to the impossibility to achieve a complete knowledge of the variation of the mechanical properties of masonry within existing structures. The values of these three parameters are defined to vary according to a lognormal probability density function, in agreement with references (Park et al., 2009;CNR-DT 212/2013. The statistical description of these parameters has been defined through the following procedure. Each of the four categories of materials shown in Figure 6 is associated to a masonry typology described in Table C8A.2.1 of the Instructions to Italian standards (MIT, 2009). In particular, making reference to the Table C8A.2.1 of (MIT, 2009), the properties of group I and group III are defined according to the 5 th masonry typology ("squared stone masonry"). Despite being associated to the same masonry typology, group III presents higher mechanical properties compared to group I, due to the thin mortar joints and the good interlocking between the stone block in the columns, which was confirmed by sonic tomography tests (Vendrell et al., 2007). For this reason, a correction factor of 1.2 has been applied to the mechanical properties of group III following the suggestions of Table   C8A.2.2 in (MIT, 2009). Regarding the material properties of the three-leaf walls and the vaults' infill (Group II), these are assumed to fall within the limits of the 2 nd masonry typology ("roughly cut stone masonry, having wythes of limited thickness and inner core") in Table C8A.2.2 in (MIT, 2009). In this case, two correction coefficients have been applied following the suggestions of Table C8A Table 1.  Figure 7 shows that for the per- MPa. The Young's modulus is related to the compressive strength as E/fc=300÷500.
These limit values are also consistent with the range of variation reported for the selected masonry typologies in Table C8A.2.1 of (MIT, 2009), despite the variability found in the literature (Vanin et al., 2017). In fact, the suggested ratio E/fc is 400 for the 5 th masonry typology ("squared stone masonry") in Table C8A.2.1 of (MIT, 2009). The tensile strength is also related to the compressive strength with limits assumed in the range ft /fc=0.02÷0.05 according to previous studies by the authors on similar stone masonry typologies (Roca et al., 2013;Saloustros et al., 2014;Pelà et al., 2016). Similarly to the compressive strength, the values of the standard deviation of the lognormal distributions of Young's modulus and tensile strength (Table 2) have been defined such that the majority of the values falls within the defined lower and upper bounds, see Figure 7.    (Lourenço, 2009). Due to the lack of experimental data for the investigated structure, the correlation between the mechanical properties of a material is based on technical codes (MIT, 2009;CEB-FIP, 2013) and expert judgement. In the presence of experimental data, a comprehensive statistical analysis could be followed for an accurate characterization of the correlation structure between the chosen random variables as in (Franchin et al., 2018). Table 4 presents the mean values for the different materials. These values, which represent a plausible choice in the absence of experimental data, define the properties of the "reference model", i.e. the combination of values for the mechanical parameters that might be considered in a conventional deterministic or semi-probabilistic approach to the problem. The outcome from the reference model will be compared with the results from the probabilistic analysis proposed in this work.

Uncertainty analysis of material parameters
The showing that the chosen one of N=200 seems to be appropriate for the selected case study.
The number of random variables in this study is n = 6 due to the assumptions made in Section 3.3. Figure 7 illustrates the distribution of the three first random variables for the reference material, i.e. the compressive strength, the tensile strength and the Young's modulus. These plots give an additional indication that the selected sample size is sufficient to give an input that converges to the lognormal distributions with the statistical parameters of Table 2.
-21 - The number of results variables is m = 4, corresponding to the seismic demand values, expressed in terms of Peak Ground Acceleration (PGA), associated to each of the four limit states that will be defined in the next Section 3.5. the tensile strength f t , (middle) and the elastic modulus E (right) and corresponding lognormal probability density functions (in dashed line) for the reference material.

Damage limit states and seismic demand
The seismic vulnerability of the selected macro-element is related to the probability of the structure to reach different limit states. Each of these limit states, denoted as , represents distinct damaged states of the structure corresponding to different seismic demands, expressed in terms of spectral displacements of the equivalent Single Degree Of Freedom System (SDOF). This works defines the limit states by adopting the definitions proposed by Lagomarsino & Giovinazzi (2006) for masonry buildings. According to these authors, four limit states are considered as a function of the yield displacement dy and the ultimate displacement du of the idealized capacity curve corresponding to the equivalent SDOF. The latter curve is constructed according to the Italian Ministry of Infrastructure and Transport (2009) by considering the ultimate displacement as the one corresponding to a decrease of 20% of the maximum load capacity of the structure. Table   5 presents the definition of each limit state and their association with the damage state of the structure. It is noted that the definition of the limit states for complex masonry structural typologies, as the one studied in this work, is still an open issue. Different approaches relating the limit states with the cracking affecting the investigated structure or the formation of the collapse mechanism have been presented in previous studies (Lagomarsino and Resemini, 2009;Petromichelakis, Saloustros and Pelà, 2014;Ortega et al., 2018). The Peak Ground Acceleration (PGA) corresponding to each limit state is identified through the N2 method (Fajfar, 1999) (Vendrell et al., 2007), carried out in the surrounding of the church, revealed the existence of a layer of poor quality rubble material of anthropogenic origin. Due to this, a Soil Type D is used for the definition of the elastic spectrum according to the Eurocode 8. Figure 8 illustrates the application of the N2 method for a capacity curve obtained for one of the studied cases. As soon as the limit states have been identified for each curve (shown with dots in Figure 8), the N2 method allows the identification of the corresponding seismic demand for each of them in terms of Peak Ground Acceleration. Figure 8. Illustration of the application of the N2 method for obtaining the seismic demand in terms of PGA for the four limit states.

Seismic Fragility
The seismic fragility of the analysed macro-element of Santa Maria del Mar church is expressed in terms of analytical fragility curves, representing the probability that the structure will exceed each considered damage limit state as a function of the PGA.
This work considers the fragility function generally accepted in available standards (ATC-58, 2009;Federal Emergency Management Agency, 2010), i.e. the log-normal cumulative distribution function. According to the latter, the conditional probability of being in, or exceeding, a particular limit state , given a value of PGA, is where is the median value of the PGA at which the analysed macro-element reaches the limit state , is the standard deviation of the natural logarithm of the PGA for limit state and is the standard normal cumulative distribution function. The above values of and are computed through the following functions where N is the selected number of structural samples analysed in the Monte Carlo stochastic simulation and ( ) is the Peak Ground Acceleration for the case = 1, . Figure 9 presents the results of the N=200 pushover analyses in terms of the horizontal acceleration against the horizontal displacement at the key of the vault in the main nave.

Seismic capacity curves
It is easy to identify two groups of capacity curves with important differences in load and displacement capacities. The first group, composed by NG = 163 cases, is characterized by load capacities ranging between 0.075 g and 0.11 g, and a more ductile post-peak response. The second group, composed of NL = 37 cases, presents much lower load capacity levels, ranging between 0.03 g and 0.07 g, and a brittle post-peak response. The presence of the two groups of the capacity curves in Figure 9 is due to the possibility of obtaining two different collapse mechanisms in the finite element analysis of the studied macro-element after varying the mechanical properties of the materials. The analysed cases falling within the group with higher capacity present a global collapse mechanism of the macro-element, shown in Figure 10a. For these cases, cracking at the main and lateral naves and at the two lateral buttresses provokes the global failure of the structure.
Contrariwise, the second group with lower capacity is characterized by a local collapse mechanism of the analysed macro-element, as illustrated in Figure 10b, with the collapse of the right buttress due to shear cracking.
The three black lines in Figure 9 are the 16%, 50% and 84% percentile curves, representing for each displacement the horizontal acceleration that is not exceeded by the 16%, -27 -50% and 84% of the analysed cases, respectively. Using the side percentile curves it is easy to identify that the large majority of the capacity curves falling above the 16% percentile curve, correspond to cases predicting a global collapse mechanism of the studied macro-element. Hence, the visual presentation of the side percentile curves in Figure 9 demonstrates that the structure is most likely (roughly 84% of probability) to present a global collapse mechanism for the investigated combination of material parameters. It is noted that these percentile curves are conventionally adopted also in FEMA (Federal Emergency Management Agency 2010), and have been derived from all the samples. Figure 10. Contour of the crack widths for the two collapse mechanisms predicted by the pushover analyses after considering the variation of material parameters: (a) global mechanism involving the buttresses and the main and lateral naves, (b) local collapse of the right buttress. Figure 11 presents the mean, the median and the pushover curve of the reference case.
The latter corresponds to an analysis adopting as mechanical properties the mean values for each material category presented in Table 4 and obtained following the procedure described in Section 3.4. The mean curve represents the average acceleration of all the analysed cases for each displacement level, while the median corresponds to the aforementioned 50% percentile curve. The higher position of the median curve compared to the mean one implies that for the adopted distributions of the uncertain parameters, the distribution of the horizontal acceleration for a given horizontal displacement is unsymmetrical and shifted to the higher values. In other words, for each value of the horizontal displacement the capacity curves below the median are located slightly farther from the median than the capacity curves above it. For the analysed structure, this happens due to the quite distinct response of the NL cases predicting the local collapse mechanism, which increases the dispersion of the capacity curves, as can be seen in Figure 9. The above result suggests that samples with one or more input parameters below the mean, push the capacity curve downwards more than samples above the mean push it upwards. Consequently, the use of a single numerical model with deterministic mechanical properties, adopted following the suggested values from the literature, may predict a very different structural capacity compared to the average of the analysed cases. Due to this, the reference model predicts a higher capacity than that given by the mean and median curves, as shown in Figure 11. This implies that the use of a single numerical model with deterministic/semi-probabilistic mechanical properties from the literature would overestimate the structural capacity for the analysed case. This result underlines the importance of considering the uncertainties of the mechanical properties in the seismic assessment of complex masonry structures.  Table 4. Figure 12a and Figure 12b present the mean and median capacity curves computed by considering only the results of the NG and NL cases predicting the global and the local collapse mechanisms of the selected macro-element, respectively. Considering only the cases leading to the same collapse mechanism minimizes the differences between the mean and the median capacity curves. This means that for both cases there is a very similar distribution of the horizontal accelerations around the mean value for each displacement level. Once again, the difference between the reference case (grey lines in Figure   12) and the average response of the analysed cases (dashed black line in Figure 12) demonstrates the potential erroneous estimation of the capacity of the structure by using a deterministic/semi-probabilistic approach. This difference becomes particularly important for the NL cases, as illustrated in Figure 12b.
-30 -  Figures 13 and 14 show the fragility curves as a function of the Peak Ground Acceleration (PGA) constructed by considering a different sample number of N in Equations (2) and (3). In the fragility curves of Figure 13 all the analysed cases have been considered, which means that N is equal to 200 in Equations (2)-(3). These graphs represent the probability that the structure will reach or exceed a considered limit state for different levels of PGA.

Seismic fragility curves
The vertical solid line corresponds to the seismic demand of the municipality of Barcelona according to the Spanish seismic standard for a 500-year return period (Comisión Permanente de Normas Sismo resistentes, 2002), i.e. 0.04 g. For this seismic demand, and considering all the analysed cases, there is a 100% probability that the structure will reach the limit state LS1, 50.9% probability for LS2, 12.4% probability for LS3 and 5.2% probability for LS4.  Figure 14a and Figure 14b show the fragility curves considering only the group of analyses predicting a global and a local collapse mechanism, respectively, while Table 6 summarizes the probabilities for the occurrence of each limit state. In particular, the fragility curves of Figure 14a are constructed considering only the NG = 163 cases predicting a global mechanism of the studied macro-element for the computation of the median and lognormal standard deviation (i.e. N=NG in Equations (2)- (3)). In the same way, the fragility curves of Figure 14b are constructed considering only the NL = 37 cases predicting a local mechanism of the studied macro-element for the computation of the median and lognormal standard deviation (i.e. N=NL in Equations (2)- (3)). This differentiation of the fragility curves according to the collapse mechanism shows that the cases predicting a global collapse mechanism are characterized by a much lower seismic vulnerability compared to those predicting the local collapse of the right buttress. Specifically, there is zero probability for the occurrence of limit states LS3 and LS4, whereas the respective probabilities for these two limit states for the local mechanism cases are 68.7% and 51.6%. The probability of limit state LS2 presents also an important difference of 39.4% between the two groups, being 50.7% for the global mechanism group and 90.1% for the local one.
On the contrary, there is a 100% probability for the occurrence of the first limit state for both groups. Another possibility for assessing the fragility of the structure is combining the fragility curves developed for each group by considering the number of cases giving each mechanism (i.e. NG and NL). This combined probability PC can be formally expressed as where and are the number of cases predicting a global and a local collapse mechanism, and [ ] and [ ] are the probabilities of reaching a limit state by considering only the and cases, respectively. The last column of Table 6 presents the combined probabilities [ ] derived from Equation (4) for each limit state. It is evident that Equation (4) gives higher predictions for the probability of all the limit states than those predicted using the total number of the analysed cases (i.e. P [LSi] in the 6 th column of Table 6). The only exception is given by limit state LS1 that is satisfied by all analysed cases. This difference implies that in complex historical structures presenting distinct collapse mechanisms, as the one studied in this research, the use of all the results for obtaining the fragility of the structure may underestimate the vulnerability of the analysed structure. The results of the two approaches used for the estimation of the seismic fragility of the structure can be appreciated in the histograms of Figure 15. Figures 16 and 17 illustrate the damage corresponding to the four limit states for a global and a local collapse mechanism case, respectively. In the following, the probabilities for each limit state correspond to column 3 of Table 6 for the local mechanism group and to column 5 of Table 6 for the global mechanism group. Considering the cases predicting a global collapse mechanism, the first limit state LS1 with a probability of 81.5%, corresponds to cracking at the lateral naves and at the bottom of the right buttress ( Figure 16a).
The second limit state LS2 has a probability of 41.3% and corresponds to further propagation of the damage in the lateral aisles and initiation of cracking at the central nave ( Figure 16b). The last two limit states LS3 and LS4 are characterized by the propagation of damage at all the naves and the two buttresses and have a 0% probability (Figure 16cd).
Considering the local collapse mechanism, the first limit state LS1 has 100% of probability and corresponds to damage at the buttresses above the vaults of the lateral naves and initiation of cracking at the two buttresses (Figure 17a). For the second limit state LS2 (16.7% probability) there is already important shear cracking at the right buttress ( Figure   17b). The cracks present in LS2 increase their propagation in the third limit state LS3 (probability 12.7%) as no important new damage occurs in the structure (Figure 17c).
Finally, the last limit state LS4 (probability 9.5 %) corresponds to the shear failure of the right buttress ( Figure 17d).  Considering the above results, as well as the fragility of the structure presented in Table   6, the most vulnerable parts of the building are the lateral naves and the top portions of the buttresses, since they are expected to experience damage for the expected seismic hazard in Barcelona for a 500-year return period (PGA = 0.04 g).

Effect of the sample size choice
This section investigates the effect of the sample size on the performed probabilistic analysis through a convergence study of the Monte Carlo simulation based on the approach presented in (Ballio and Guadagnini, 2004). The metrics used for the determination of the convergence are the median ( Equation (2)) and the standard deviation of the lognormal distribution ( Equation (3)) of the PGA for each limit state, since these two statistical measures are used in the definition of the fragility curves in Equation (1).  (2) and (3). The values of these two statistical measures converge faster for the first two limit states (LS1 & LS2) compared to the last two (LS3 & LS4). This is because the last two limit states are defined considering the ultimate displacement of the structure (see Table 6), which has been considered the one corresponding to 80% of the maximum capacity of the structure. Due to this, the computed PGAs for LS3 & LS4 depend heavily on the post-peak response of each analysed case, which as can be seen in Figure 9 can vary significantly. This variation is related not only to the different material properties of each analysis, but also to numerical aspects such as the convergence strategy, the used constitutive model and the finite element technology (Vlachakis et al., 2019). For this reason, the median and standard deviation of LS3 & LS4 present a slower convergence, characterized by an oscillating trend around a constant value for a sample size above 100. In particular, considering the case of LS4, there is a 9% difference between the maximum and the minimum reported values of the standard deviation for sample sizes above 100. For the median value of LS4, this difference is limited to 3%. Considering the above results, the two statistical measures used in the definition of the fragility curves start converging for a sample number N>100.
Therefore, the chosen sample size of N=200 seems to be appropriate for the selected case study.

Conclusions
This work has presented a methodology for the probabilistic seismic assessment of com- direction. An important outcome of the work is the proposal of seismic fragility curves for the representative bay structure of this valuable heritage monument.
The application of the proposed probabilistic methodology has shown its ability to identify different damage and collapse mechanisms that might be overlooked in conventional approaches based on a deterministic/semi-probabilistic evaluation of the material properties. The probabilistic seismic assessment procedure has demonstrated that the seismic response of complex masonry structures, such as historical churches and cathedrals, may show a significant sensitivity to the variation of the material properties and thus to the uncertainty linked to them. The use of the proposed method allows a more accurate evaluation of the seismic safety of the building compared to the conventional deterministic/semi-probabilistic approaches.
The adopted methodology predicts that two collapse mechanisms are possible for the analysed macro-element of the church of Santa Maria del Mar. The first one is a global collapse mechanism, predicted by 81.5% of the analysed cases, with cracking in the main and lateral naves and the lateral buttresses. The second collapse mechanism, predicted by 18.5% of the analysed cases, is characterized by the local failure of the lateral buttress due to shear cracking. The seismic fragility of the structure depends importantly on the final collapse mechanism. The cases resulting in the global collapse of the structure present low seismic fragility for the seismic demand of the city of Barcelona, and only slight and moderate damage states are probable. On the contrary, the cases with a local collapse mechanism present low capacity and a very brittle post-peak response resulting in a high seismic fragility, which exceeds 50% for all the investigated limit states.
Taking into account the mentioned distinct collapse mechanisms and structural responses, the seismic fragility of all the analysed cases has been investigated using two approaches. In the first one, the fragility curves have been determined using all the analysed cases without making any distinction between those predicting a local and a global collapse mechanism. In the second one, the probability of having a different collapse mechanism is considered by assigning specific weights to the cases predicting a local and a global collapse mechanism. The latter differentiation results in increased levels for the seismic fragility of the investigated structure for all the limit states.
The main aim of this work is to contribute to the discussion on the possibility to apply probabilistic approaches for the seismic assessment of complex historical masonry struc-