PSI - Issue 44

Stefano Bracchi et al. / Procedia Structural Integrity 44 (2023) 394–401

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Stefano Bracchi et al. / Structural Integrity Procedia 00 (2022) 000 – 000

study, i.e. weak (KL2-A) or good quality mortar (KL2-B), leading to the definition of different intervals of mechanical properties, due to the application (or not) of the correction coefficients proposed by EC8 for good quality mortar. In case of KL3, it is considered that the engineer can perform in-situ tests. Three sub-cases are then considered with reference to the ranges of values of mechanical properties suggested by EC8, i.e. weak (KL3-A-LOW, KL3-B LOW), average (KL3-A-CEN, KL3-B-CEN) and good (KL3-A-UPP, KL3-B-UPP) quality masonry. To this aim, the intervals of mechanical properties identified at KL2 are divided into three parts of equal width. It was also decided to consider the possibility of performing more than one test. For each KL3, two additional cases are hence defined: a first one where the engineer performs two tests, i.e. a diagonal compression test and a double flat-jack test (KL3-A LOW-1, KL3-A-CEN-1, KL3-A-UPP-1, KL3-B-LOW-1, KL3-B-CEN-1, KL3-B-UPP-1) and a second one, in which four tests are performed, i.e. a diagonal compression test and three double flat-jack tests (KL3-A-LOW-2, KL3-A CEN-2, KL3-A-UPP-2, KL3-B-LOW-2, KL3-B-CEN-2, KL3-B-UPP-2). Nonlinear static analyses are performed by means of the TREMURI software, based on an equivalent-frame modelling, and adopting the multi-linear element of Cattari and Lagomarsino (2013). To characterize the epistemic uncertainties related to mechanical properties, the following material properties are generated by means of statistical sampling: Young’s modulus E , compressive strength of masonry f m , shear strength of masonry τ 0 (to be used in the shear strength criterion of Turnsek and Sheppard, 1980), compressive strength of the unit f b , shear and flexural drift limits. The procedure proposed by Franchin et al. (2018) is adopted, considering also element-to-element variability. For the other properties, an intra-element correlation matrix is defined. At KL1, the engineer is able of defining the typology of masonry (masonry made of clay bricks and lime mortar) based on visual survey, without information on mortar quality and presence of transversal connections; uncertainties related to these factors have to be considered in the reference interval, applying the correction coefficients to the upper bound of the interval. EC8 is not giving values of shear strength τ 0 , but only of the initial shear strength f v0 ; for this reason, it was decided to refer to the Italian code (NTC18, 2018) for the definition of τ 0 . The interval of compressive strength of units, f b , is not defined and it was derived from the experimental results of Binda et al. (1996). Regarding shear and flexural drift limits, EC8 is giving only a mean value, whereas the standard deviation was assumed based on experimental results collected by Morandi et al. (2018) or based on expert judgment. Finally, the lognormal distributions of the material properties were defined to perform statistical sampling. At KL2, it was assumed that non-destructive tests are performed, with the aim of identifying the mortar quality (weak or good). In the latter case, the engineer can apply the correction coefficient for good quality mortar; uncertainty related to transversals connections was considered in both the cases. At KL3, uncertainties on mechanical properties can be reduced performing in-situ tests; in this work only diagonal compression tests (to measure τ 0 ) and double flat-jack tests (to measure E ) were considered. Three possibilities were considered, corresponding to a test on an element with bad, average and good masonry quality. For the measured properties only, the reference interval for KL3 (derived from KL2) was divided into three parts of equal width, corresponding to the three cases of masonry quality. EC8 accounts for test results by means of a Bayesian updating of the prior distribution of the measured property (Bracchi et al. 2016), defined from the interval considered at KL2 and leading to the definition of a posterior distribution narrower than the prior. Prior distribution (with mean µ’ and standard deviation σ ’ ) is updated based on test results and number of tests, obtaining a posterior distribution with mean µ’’ and standard deviation σ ’’ calculated as: ′′ = ̅+ ′ + , ′′ = ′ √ + , = ′ + ′′ , ′′ = 2 ′2 (1) where n is the number of performed tests, ̅ is the average of the experimental results, k’ depends on the typology of test and s is the standard deviation of the experimental tests. For each case at KL3, the distributions from which the results of the tests were sampled were defined starting from the corresponding interval of KL2. With the posterior distribution, it was finally possible to sample the mechanical parameters and generate the set of models used in the analysis. In this way, the intervals of E and τ 0 are updated, whereas the ones of f m , f b and the drift limits are the same as KL2, since these properties are not directly measured.