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

Bayesian assessment of the compressive strength of structural masonry Nagel, Joseph B.; Mojsilovic, Nebojsa; Sudret, Bruno Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Bayesian Assessment of the Compressive Strength of StructuralMasonryJoseph B. NagelPhD Candidate, Institute of Structural Engineering, ETH Zürich, SwitzerlandNebojsa MojsilovicSenior Scientist, Institute of Structural Engineering, ETH Zürich, SwitzerlandBruno SudretProfessor, Institute of Structural Engineering, ETH Zürich, SwitzerlandABSTRACT: The application of hierarchical models for assessing the compressive strength of structuralmasonry is investigated. Based on current codified models the distribution of compressive strengthswithin an ensemble of masonry wall specimens is related to the statistical properties of the populations ofbrick units and mortar used. The parameters of this relation are calibrated with test data acquired at ETHZürich. This approach allows for heterogeneous material modeling, consistent uncertainty managementand optimal information processing. Costly compression tests of full-size masonry and inexpensive testsof brick and mortar samples are jointly utilized for learning about the masonry wall characteristics.1. INTRODUCTIONStructural masonry is a composite material thatconsists of brick units and mortar. The mechani-cal key characteristic of masonry is the compressivestrength perpendicular to the bed joints. Estimatingor predicting this material property are thus issuesof central importance to assessing the reliabilityof masonry structures. These problems are there-fore addressed in current standards (EN 1996-1-1, 2005; JCSS, 2001) and numerous enhancements(Dymiotis and Gutlederer, 2002; Glowienka andGraubner, 2006; Schueremans and Van Gemert,2006; Mojsilovic and Faber, 2009; Sýkora andHolický, 2010; Garzón-Roca et al., 2013; Sykoraand Holicky, 2014; Sykora et al., 2014).The motivation of this research study is twofold.Firstly, we observe a systematic discrepancy be-tween measured data and predictions of the ma-sonry compressive strength according to EN 1996-1-1 (2005). This suggests a recalibration of themodel code parameters. Secondly, it is noticed thatcurrent approaches either suffer from their semi-probabilistic character or their unsatisfactory treat-ment of the emerging uncertainties. Thus the goalof this paper to develop a fully probabilistic exten-sion of current codified models for assessing thecompressive strength of unreinforced masonry. Wewill rely on hierarchical models (Nagel and Sudret,2015, 2014) and Bayesian networks (Sankararamanet al., 2012; Urbina et al., 2012). This approachwill allow for heterogeneous modeling of structuralmasonry, quantification of various types of uncer-tainty and acquisition of information from diversesources.More specifically it is aimed at analyzing thecompressive strength of structural masonry withsystem-level data, i.e. measurements that are takenfrom full-scale masonry specimens, component-level data, i.e. results from testing brick units andmortar samples individually, and prior or expertknowledge. Compression tests of masonry speci-mens are rather costly, whereas data associated tocomponent-specific material characteristics are rel-atively inexpensive to acquire. Hierarchical models112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015enable the joint processing of information from dif-ferent levels of the overall system. This way theinformation is optimally utilized. Moreover a pre-dictive relationship is established that connects themasonry compressive strength with the component-level compressive strengths.The remainder of this document is organized asfollows. Previous approaches of assessing the com-pressive strength of structural masonry will be re-viewed in Section 2. Hierarchical models will beintroduced in Section 3. In Section 4 the acquireddata will be discussed and Section 5 will show theresults of Bayesian updating. Lastly we will sum-marize and conclude in Section 6.2. CURRENT MODELSIn EN 1996-1-1 (2005) it is tried to relate the com-pressive strength of masonry to the resistances ofits brick and mortar components. The relationshipis realized as a power functionfw = k′ f α′b fβ ′m . (1)On the one hand, the compressive strength of ma-sonry is summarized by the characteristic value fw,i.e. a 5%-quantile. On the other hand, fb denotesthe normalized mean compressive strength of theunits and fm denotes the mean compressive strengthof the mortar. Estimates of the constants (k′,α ′,β ′)are given for different types of masonry. In JCSS(2001) the empirical relation Eq. (1) is interpretedsimilarly. Here fw, fb and fm represent the meanvalues of the corresponding distributions. Differ-ent prior estimates of the coefficients (k′,α ′,β ′) areprovided. The coefficients are often set so that they(approximately) satisfy α ′+ β ′ = 1. This choicecan be justified for reasons of the physical dimen-sion in Eq. (1).Ensuing from these semi-probabilistic models, avariety of extensions have been proposed in the lit-erature. There are probabilistic reinterpretations ofEq. (1) based on lognormal distributions (Schuere-mans and Van Gemert, 2006; Sýkora and Holický,2010; Sykora and Holicky, 2014; Sykora et al.,2014). In other studies the model uncertainty ofEq. (1) is quantified (Dymiotis and Gutlederer,2002; Glowienka and Graubner, 2006). A conju-gate Bayesian updating approach based on Gaus-sian distributions is presented in Mojsilovic andFaber (2009). Another idea is to establish a con-nection between the compressive strengths of ma-sonry and its components via artificial neural net-works (Garzón-Roca et al., 2013).These previous approaches suffer from the factthat they either do not clearly distinguish betweenepistemic and aleatory shares of uncertainty or theyneglect material heterogeneity. Fitting the parame-ters of a probabilistic extension of Eq. (1) is a prob-lem that has hardly been satisfactorily solved as yet.3. HIERARCHICAL MODELSIn the following hierarchical Bayesian modeling isintroduced as a tool for distinguishing and handlinguncertainty in codified models of the form Eq. (1).The aim of this section is to establish a Bayesianmodel and updating strategy for the following ex-perimental situation. The compressive strength ismeasured for a number of clay block masonry spec-imens. Specimens can be grouped according tothe ensembles of brick units and mortar that wereused for their construction. Here ensembles of claybricks are characterized by the same ingredientsused and the same manufacturing procedure. Simi-larly in every ensemble of mortar samples identicalconstituents were used for mixing. In this modelingapproach material heterogeneity is accounted for bydistinguishing between brick and mortar samplesused in constructing the masonry wall systems. Thefinal goal is the assessment and prediction of thecompressive capacity of structural masonry by uti-lizing system- and component level information.3.1. Aleatory ModelWithin an ensemble of masonry wall specimens, thecompressive strength of the masonry wall is repre-sented as a random variableFw = kFαb Fβm . (2)This is a probabilistic extension of the codifiedmodel in Eq. (1). We remark that the coefficients(k,α,β ) of the relation Eq. (2) are not immediatelyidentified with the ones of Eq. (1).212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The compressive strengths of the bricks and themortar are modeled as lognormal random variablesFb∼LN ( fb|µb,σ2b ) and Fm∼LN ( fm|µm,σ2m).Their distributions are determined by hyperparam-eters θ b = (µb,σb) and θ m = (µm,σm) that arethe mean and standard deviation of log(Fb) andlog(Fm), respectively. Consequently the masonrywall compressive strength in Eq. (2) is a randomvariable that follows a lognormal distributionFw ∼LN(fw|µw,σ2w), (3a)with µw = αµb +βµm + logk, (3b)and σ2w = α2σ2b +β 2σ2m. (3c)The distribution Eq. (3a) represents the variability,i.e. the frequency distribution, of the masonry com-pressive strengths within the population of speci-mens. It is parametrized by hyperparameters θ w =(µw,σw) that are determined by the statistical prop-erties of component populations due to Eqs. (3b)and (3c).The mean value and the variance of the dis-tribution LN(fw|µw,σ2w)in Eq. (3) are simplygiven as E[Fw] = exp(µw + σ2w/2) and Var[Fw] =(exp(σ2w)− 1)exp(2µw + σ2w), respectively. The5%-quantile ofLN(fw|µw,σ2w), e.g. for compar-ison with Eq. (1), follows as Qw,5% = exp(µw −1.645σw).3.2. Epistemic ModelIf the coefficients (k,α,β ) of Eqs. (2) and (3)are not perfectly known, one can represent theirepistemic uncertainty as prior random variables(K,A,B) ∼ pi(k,α,β ). In the following we willconfine the analysis to the case β = 1−α . We con-sider mutually independent prior random variablesK ∼ pi(k), A∼ pi(α). (4)Their joint prior uncertainty pi(k,α) = pi(k)pi(α)can be reduced by Bayesian data analysis of exper-imental measurements. In the following two dif-ferent updating approaches are outlined for exper-imental situations where the assumption of knownhyperparameters, i.e. the distributional parametersof the ensembles of masonry wall components, iseither justified or rather unfounded.3.3. Known HyperparametersLet us consider experiments of the following type.In each batch of experiments i = 1, . . . ,n the ma-sonry compressive strength fw,i j is measured for anumber of different specimens j = 1, . . . ,Ji froman ensemble. We use 〈 fw,i j〉 = ( fw,11, . . . , fw,nJn)to denote the set of these measurements. Thehyperparameters θ b,i and θ m,i are measured forthe bricks and the mortar used in experiment i,too. This can be accomplished by a statisti-cal analysis of data 〈 fb,ik〉 = ( fb,11, . . . , fb,nKn) and〈 fm,il〉 = ( fm,11, . . . , fm,nKn) with k = 1, . . . ,Ki andl = 1, . . . ,Li. These data must be numerous and theymust be observed for the ensembles of brick unitsand mortar used. The Bayesian multilevel modelfor this scenario can be written as(Fw,i j |k,α)∼ pi( fw,i j |k,α), (5a)(K,A)∼ pi(k)pi(α). (5b)Here the conditional distributions Eq. (5a) are givenby Eq. (3), where batch-specific knowns θ b,i andθ b,i are plugged in. The epistemic prior uncertaintyof the coefficients (k,α) is encoded in Eq. (5b). Aslong as not indicated otherwise, all random vari-ables in Eq. (5) are assumed to be (conditionally)independent. A directed acyclic graph (DAG) asin Fig. 1 serves as an intuitive visualization of themodel Eq. (5).Figure 1: Known hyperparameters. Nodes symbolizeknown ( ) or unknown ( ) quantities. Arrows representdeterministic ( ) or probabilistic ( ) relations.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015As usual, Bayesian updating is accomplishedby conditioning the prior distribution pi(k,α) =pi(k)pi(α) on the acquired data 〈 fw,i j〉. One obtainspi(k,α |〈 fw,i j〉) ∝ pi(k)pi(α)n∏i=1Ji∏j=1pi( fw,i j |k,α).(6)Note that Eq. (6) is based on exact values the hy-perparameters θ b,i and θ m,i for every batch i.3.4. Unknown HyperparametersThe requirement of known hyperparameters θ b,iand θ m,i restricts the applicability model Eq. (5) tosituations that are rarely met in practice. Thereforewe consider the situation when only prior knowl-edge pi(θ b,i,θ m,i) = pi(θ b,i)pi(θ m,i) about the hy-perparameters is available. Additionally in eachbatch of experiments i a variable number of mea-surements fb,ik and fm,il for k = 1, . . . ,Ki and l =1, . . . ,Li are taken of the brick unit and the mor-tar compressive strength, respectively. The corre-sponding hierarchical Bayesian model reads(Fw,i j |k,α,θ b,i,θ m,i)∼ pi( fw,i j |k,α,θ b,i,θ m,i),(Fb,ik|θ b,i)∼ pi( fb,ik|θ b,i), (7a)(Fm,il |θ m,i)∼ pi( fm,il |θ m,i),(Θb,i,Θm,i)∼ pi(θ b,i)pi(θ m,i),(K,A)∼ pi(k)pi(α).(7b)While Eq. (7a) summarizes the aleatory uncertain-ties, Eq. (7b) contains the epistemic uncertainties.The model Eq. (7) is visualized as the DAG inFig. 2. We remark that the observations 〈 fb,ik〉 and〈 fm,il〉 inform about the statistical properties θ b,iand θ m,i of the component ensembles. This waythey give information about the unobservable prop-erties of the brick and mortar samples used for con-structing the masonry wall i.Bayesian analysis proceeds by updat-ing the joint prior pi(k,α,〈θ b,i〉,〈θ m,i〉) =pi(k)pi(α)∏ni=1pi(θ b,i)pi(θ m,i). Conditionedon the data (〈 fw,i j〉,〈 fb,ik〉,〈 fm,il〉) one obtains forFigure 2: Unknown hyperparameters. The batch-specific hyperparameters θ b,i and θ m,i are unknown.They can be inferred from the data 〈 fb,ik〉 and 〈 fm,il〉.the joint posteriorpi(k,α,〈θ b,i〉,〈θ m,i〉|〈 fw,i j〉,〈 fb,ik〉,〈 fm,il〉)∝ pi(k)pi(α)n∏i=1pi(θ b,i)pi(θ m,i)·Ji∏j=1pi( fw,i j |k,α,θ b,i,θ m,i)·Ki∏k=1pi( fb,ik|θ b,i)Li∏l=1pi( fm,il |θ m,i).(8)Notice that the posterior Eq. (8) gathers informationfrom both system- and component-level data.4. EXPERIMENTAL DATAIn the years 2009-2012 and 2014 the compres-sive strength of clay block masonry was measuredfor a variable number of specimens in a series ofcompression tests. In addition, the compressivestrengths of bricks and mortar were recorded for re-alizations from the same ensembles that were laterused for the construction of the masonry wall. Thetests were performed at the laboratories of the De-partment of Civil, Environmental and Geomatic En-gineering of ETH Zürich. In Table 1 the experimen-412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: Experimental data. Data are shown for tests of clay block masonry that were performed in the years2009-2012 and in 2014. Blocks from the same ensemble were used in 2009 and 2010. Thus the correspondingrows show duplicate data entries.2009 batch 1fw [MPa] 9.41 5.53 7.98 8.86 6.67 7.17 7.92fb [MPa] 33.58 34.55 37.1 39.21 39.63 36.1 35.46 37.61 35.6 36.26 35.2 32.7 36.6fm [MPa] 15.8 16.1 14.4 14.8 16.1 15.4 13.9 14.6 14.6 14.4 16.8 16.12010 batch 2fw [MPa] 6.67 6.12 5.91 8.3 6.44 5.32 6.7fb [MPa] 33.58 34.55 37.1 39.21 39.63 36.1 35.46 37.61 35.6 36.26 35.2 32.7 36.6fm [MPa] 12.5 12.94 12.43 13.33 12 12.322011 batch 3fw [MPa] 4.32 3.71 6.06 4.95 4.29 2.8 6.28 4.22 5.23fb [MPa] 23.8 26.8 25.7fm [MPa] 14.9 14.7 14.9 15.4 14.7 14.62012 batch 4fw [MPa] 8 7.87 8.1 7.53 8.14 6.99 7.82 9.13 5.87 7.71fb [MPa] 37 39.9 38fm [MPa] 26.9 28.1 17 16.2 18.7 21.12014 batch 5fw [MPa] 6.53 7.01 6.12 5.94 7.14 5.69 5.82 6.34 5.96fb [MPa] 28.15 27.74 28.05 27.20 26.25 23.15 26.69 28.04 27.69 26.73fm [MPa] 11.73 12.19 12.13 10.49 10.34 10.44tal data are summarized. Five batches of experi-ments were performed in total. At the system- andthe component level the available data is generallyscarce. Especially in the years 2011 and 2012 thenumber of component-level tests was very limited.Moreover, in the years 2009 and 2010 brick unitsfrom the same ensemble were used.We observe that the empirical relation Eq. (1)generally overpredicts the masonry wall compres-sive strength. In Fig. 3 the actually acquired data fori = 1, i.e. for the year 2009, is shown together withthe correspondingly predicted characteristic value.The values k′ = 0.45, α ′ = 0.7, β ′ = 0.3 providedin EN 1996-1-1 (2005) were used. Brick unit data〈 fb,ik〉 have been normalized according to their ge-ometry. Moreover a lognormal distribution of theform Eq. (2) is shown, where α = α ′ and β = β ′have been identified with the corresponding coeffi-cients from EN 1996-1-1 (2005). The remainingcoefficient k = k′ · exp(1.645√α2i σ2b,i +β 2i σ2m,i +αi[σ2b,i/2]+βi[σ2m,i/2]) has been set in order that the5%-quantile equals Eq. (1). Note that the above-mentioned identification/transformation of the co-efficients establishes another way of extendingEq. (1) and comparing it to Eq. (2). In this paperwe do not pursue this approach, though.compressive strength fw,1j [MPa]0 5 10 15 20probabilitydensity00. valueidentified distributionFigure 3: Data & predictions for 2009. The data, itsexpected 5%-quantile and a corresponding lognormaldistribution are shown. The data are overpredicted.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Of course, the unexpected code/measurementdiscrepancy raises important questions. Anticipat-ing our results it is said that we will not able to sat-isfactorily explain this discrepancy. Instead we willcalibrate the coefficients k and α in a way that leadsto better predictions. Those predictions are validfor the testing machine and the materials used inour laboratory. Using the predictions outside theirscope of applicability is questionable and shouldonly be done with utmost caution.5. BAYESIAN ANALYSISThe Bayesian framework discussed in Section 3is now applied to analyze the experimental datathat was presented in Section 4. More specificallywe use the first two batches of experiments thatwere conducted in 2009 and 2010 to calibrate theunknown coefficients of the model Eq. (5). Forthose batches the amount of component-level datais deemed sufficient to fit the hyperparameters andto treat them as knowns subsequently. Moreover thefirst four batches will be analyzed with the modelEq. (7) that allows to treat the hyperparameters asunknowns. Especially in the years 2011 and 2012the small amount of component-level data does notallow to proceed in another way. The fifth batch ofexperiments from 2014 will be used as an indepen-dent test set.Since the coefficients in Eqs. (2) and (3) cannotbe identified with those of Eq. (1), it is not pos-sible to elicit informative priors about the formerby exploiting expert knowledge or code informa-tion about the latter. Hence uninformative priorsare used. Specifically we assign uniform prior dis-tributions pi(k) = U (0,1) and pi(α) = U (0.5,1).Due to β = 1− α the latter assignment enforcesα ≥ β . This reflects the intuition that, regarding themasonry compressive strength, the brick units aremore influential than the mortar. For the Bayesianmodel in Eq. (7) priors pi(θ b,i) = pi(µb,i)pi(σb,i)and pi(θ m,i) = pi(µb,i)pi(σb,i) have to be elicitedfor the unknown hyperparameters. We use indepen-dent uniform hyperprior distributions with reason-able bounds for the means and standard deviations.The posteriors Eqs. (6) and (8) can be sam-pled by means of Markov chain Monte Carlo(MCMC) techniques (Brooks et al., 2011). InFigs. 4 and 5 the resulting posterior marginalsof k and α are depicted. It can be seen thatpi(k,α |〈 fw,i j〉,〈 fb,ik〉,〈 fm,il〉) contains a higher de-gree of posterior uncertainty than pi(k,α |〈 fw,i j〉).Since more data has entered the former posterior,at first sight this seems to be surprising. This factcan be attributed to the differences of the modelsEqs. (5) and (7) in treating the hyperparameters andtheir uncertainties, though.k0.15 0.2 0.25 0.3probabilitydensity0102030405060708090priorpi(k|〈fw,ij〉)pi(k|〈fw,ij〉, 〈fb,ik〉, 〈fm,il〉)Figure 4: Posterior of k. The posteriors pi(k|〈 fw,i j〉)and pi(k|〈 fw,i j〉,〈 fb,ik〉,〈 fm,il〉) are shown. It can beseen that the latter is broader than the former.α0.7 0.75 0.8 0.85 0.9 0.95 1probabilitydensity0102030405060priorpi(α|〈fw,ij〉)pi(α|〈fw,ij〉, 〈fb,ik〉, 〈fm,il〉)Figure 5: Posterior of α . Both the posterior marginalspi(α |〈 fw,i j〉) and pi(α |〈 fw,i j〉,〈 fb,ik〉,〈 fm,il〉) peak attheir upper boundary.Specifically the modes kˆ = 0.21 and αˆ =1 are found for the posterior pi(k,α |〈 fw,i j〉)that represents the situation that hyperparame-ters are assumed to be known. The posteriorpi(k,α |〈 fw,i j〉,〈 fb,ik〉,〈 fm,il〉), for the scenario that612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015hyperparameters are treated as unknowns, featuresthe modes kˆ = 0.22 and αˆ = 1.The fact that the posterior of α in Fig. 5 peaksat the upper bound of its prior is somewhat sur-prising. As a consequence of βˆ = 1− αˆ = 0, theinfluence of mortar occurs to be negligible. More-over, such a behavior may indicate that the inverseproblem is improperly solved, e.g. the true parame-ter value was accidentally excluded a priori. It wastherefore tried to relax the assumption β = 1−αby permitting arbitrary values α > 0 and β > 0. Tothat end independent priors pi(α) and pi(β ) were as-signed. We had to conclude that the limited amountof available data is not sufficiently informative inorder to calibrate this extended model.Plugging the point estimates kˆ and αˆ in Eq. (3)establishes a predictive relation of the frequencydistribution of structural masonry. For that purposeone has to specify the values or estimates of thehyperparameters θ b and θ m for the ensembles ofbricks and mortar used in the construction of themasonry wall. The predicted distributions, that areobtained this way for the actually analyzed batchesof experiments, describe the masonry wall resis-tances adequately well. Since the estimations of thecoefficients were informed by the very same data,this does not seem to be very surprising. Yet thissignifies that the representation Eq. (3) is adjustableenough to match the data. In turn this may indicatethat Eq. (3) is indeed a suitable representation of themasonry wall compressive strength.When applied to the fifth batch of experimentsthe procedure described above can serve as a vali-dation test, i.e. the data collected in 2014 are usedas an independent test set. In Fig. 6 the measuredmasonry wall compressive strengths are shown to-gether with the their predicted distribution. Theplot is supplemented with the corresponding 5%-quantile. Here the point estimates kˆ = 0.22 andαˆ = 1 that were obtained by analyzing the previ-ous four batches are used on one side. On the otherside component-level data 〈 fb,5 j〉 and 〈 fm,5 j〉 for thefifth batch are used to estimate θ b,5 and θ m,5. Thepredictive distribution captures the data fairly well.Obviously it is of higher quality than the poor code-forecast shown in Fig. 3.compressive strength fw,4j [MPa]0 2 4 6 8 10probabilitydensity00. valuepredicted distributionFigure 6: Data & predictions for 2014. The gathereddata, the predicted distribution and its 5%-quantile areshown. Predictions conform to data tolerably well.6. SUMMARY & CONCLUSIONIt was demonstrated how hierarchical Bayesianmodels can serve the purpose of assessing the com-pressive strength of structural masonry. This estab-lishes a fully probabilistic alternative to the exist-ing semi-probabilistic approaches. The hierarchi-cal framework offers versatile and powerful tools ofuncertainty quantification and information aggrega-tion at multiple system levels. Different types ofuncertainty, i.e. ignorance and variability, are thor-oughly managed, while heterogeneous types of in-formation, e.g. data and expert knowledge, are con-sistently utilized. This way the analysis of the ma-sonry wall resistance can be based on large-scalecompression tests as well as on inexpensive tests ofbrick unit and mortar samples.Our hope is that this possibility will encourageexperimenters in entirely publishing their collecteddata. In fact it seems to be commonplace to quotestatistical data summaries only, e.g. sample meansor characteristic values. The proposed methodol-ogy, however, allows to process the acquired dataas a whole.A number of questions have arisen. It is queriedif Eq. (3) is an adequate representation of the distri-bution of masonry compressive strength in terms ofdistributional parameters of the components. Withregard to the complexity of structural masonry, itsfailure modes and their dependency on the qual-ity of workmanship, the relations Eqs. (1) and (2)are oversimplifying. They were inspired by the712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015structure of current models but lack a solid phys-ical foundation. For future studies this motivatesthe introduction of model uncertainty in additionto the emerging parameters uncertainties. Beyondthat future work will also involve the constructionand objective selection of better system-level mod-els of aleatory variability. A more fundamentalquestion concerns the general suitability of empiri-cal relations for any probabilistic extension what-soever. Another raised issue relates to the ob-served mismatch between measurements and code-predictions. We were not able to explain this dis-crepancy.7. REFERENCESS. Brooks, A. Gelman, G. L. Jones, and X.-L. Meng, eds.(2011). Handbook of Markov Chain Monte Carlo.Handbooks of Modern Statistical Methods. Chapman& Hall/CRC.Dymiotis, C. and Gutlederer, B. M. (2002). “Allowingfor uncertainties in the modelling of masonry com-pressive strength.” Constr. Build. Mater., 16(8), 443–452.EN 1996-1-1 (2005). “Eurocode 6: Design of masonrystructures. Part 1-1: General rules for reinforced andunreinforced masonry structures. European Commit-tee for Standardization (CEN), Brussels, Belgium.Garzón-Roca, J., Obrer Marco, C., and Adam, J. M.(2013). “Compressive strength of masonry made ofclay bricks and cement mortar: Estimation based onneural networks and fuzzy logic.” Eng. Struct., 48,21–27.Glowienka, S. and Graubner, C.-A. (2006). “Probabilis-tic modelling of the load carrying capacity of mod-ern masonry.” Proc. 4th Int. Probabilistic Symposium,Berlin, Germany (October 12-13).JCSS (2001). “Probabilistic model code. Joint Commit-tee on Structural Safety (JCSS), Zürich, Switzerland.Mojsilovic, N. and Faber, M. H. (2009). “Probabilisticassessment of masonry compressive strength.” Proc.10th Int. Conf. on Structural Safety and Reliability(ICOSSAR2009), Osaka, Japan (September 13-17).Nagel, J. B. and Sudret, B. (2014). “A unified frameworkfor multilevel uncertainty quantification in Bayesianinverse problems. Probab. Eng. Mech. (submitted forpublication).Nagel, J. B. and Sudret, B. (2015). “Bayesian multilevelmodel calibration for inverse problems with perfectdata in the presence of epistemic and aleatory uncer-tainty. J. Aerosp. Inf. Syst. (in press).Sankararaman, S., McLemore, K., and Mahadevan, S.(2012). “Bayesian methods for uncertainty quantifi-cation in multi-level systems.” Topics in Model Val-idation and Uncertainty Quantification, Volume 4,T. Simmermacher, S. Cogan, L. G. Horta, and R.Barthorpe, eds., Conference Proceedings of the So-ciety for Experimental Mechanics Series, Springer,New York, 67–74.Schueremans, L. and Van Gemert, D. (2006). “Probabil-ity density functions for masonry material parameters– a way to go ?.” Proc. 5th Int. Conf. on StructuralAnalysis of Historical Constructions: Possibilities ofNumerical and Experimental Techniques, New Delhi,India.Sykora, M., Cejka, T., Holicky, M., and Witzany, J.(2014). “Probabilistic model for compressive strengthof historic masonry.” Safety, Reliability and RiskAnalysis: Beyond the Horizon, R. D. J. M. Steen-bergen, P. H. A. J. M. van Gelder, S. Miraglia, andA. C. W. M. Ton Vrouwenvelder, eds., CRC Press,2645–2652.Sýkora, M. and Holický, M. 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