Issue 50

M. Godio et alii, Frattura ed Integrità Strutturale, 50 (2019) 194-208; DOI: 10.3221/IGF-ESIS.50.17 196 [15]. Moreover, test results [15] have shown a significant decrease in the initial lateral wall stiffness when cracking occurs. The stiffness also decreases through subsequent tests due to damage, that is, even if no new major cracks appear. The influence of the elastic modulus of masonry was, however, not included in previous numerical studies [5,6,9,10] and needs therefore to be investigated in a more systematic way. The objective of the here presented study is to quantify the vulnerability of walls subjected to the relative support motion by investigating the influence of the two fundamental input motion characteristics, which are (i) the relative amplitude and (ii) the phase shift between the top and bottom support motions. The numerical modelling of rocking masonry structures is a challenging task and many numerical modelling techniques based on e.g. finite element [21–23], discrete element [24– 28], applied element [29,30] and rigid-block models [31] have been put forward in the recent years. In this study, the URM walls are modelled by means of a discrete element model, which was already validated and employed in a previous work on walls with fixed support motions [32]. Unlike the models used in previous numerical studies analysing URM walls under relative support motions [4,5,8] – except [10] – the here presented study makes no assumption on the height at which the macroscopic crack forms dividing the one-way spanning wall into a lower and an upper rocking body. This more versatile modelling approach allows some essential features of the out-of-plane behaviour of URM walls to be captured, namely, the opening/closure of the interfaces following the impacts and the change in height of the macroscopic crack during the shake due to the activation of ‘modes’ of rocking other than the two-macro-block mechanisms usually investigated [15]. Relative support motions are generated from a suite of natural records. Starting from these records, support motions are derived by changing, on the one hand, the phase shift, and, on the other hand, the relative amplitude. This is done by applying appropriate signal processing techniques. The strategy followed in this paper to produce records with relative amplitude formalises and extends the one already adapted by Tondelli et al. [10]. In this paper, this strategy is applied for different wall configurations and, as a novelty, a strategy to produce out-of-phase motions is introduced. The resulting support motions are not necessary representative of the real floor motions, as these latter are usually amplified around the building frequency [2,9,33,34]. The reason of using synthetically-generated instead of experimentally-derived floor motions is that the former allows us examining a wide range of configurations controlling precisely the input motion characteristics. The vulnerability of the walls against relative support motions is quantified for different wall configurations through fragility curves. A parametric study investigates the acceleration capacity and the failure mechanisms of URM walls in which the masonry elastic modulus, wall height-to-thickness ratio, wall effective thickness and applied axial load (overburden) are varied. Godio and Beyer [32] carried out a parametric study on the same wall configurations and records, but for wall with identical support motions. R ESPONSE OF URM WALLS UNDER RELATIVE SUPPORT MOTION bservations from shake table test show that uncracked URM walls undergo a uniform or a linearly varying acceleration profile along their height, depending on whether the input motions at the top and bottom of the wall are the same, i.e. have equivalent amplitude and are in-phase, or not [15]. When the walls are cracked, the acceleration profile becomes piece-wise linear between the cracks [15], and, for the general case in which the supports undergo relative motion, consists of three components (Fig. 1): (a) a constant one, generated by the overall support motion; (b) a linear one, provided by the relative motion between the top and bottom supports, as a result, for instance, of the diaphragm flexibility; (c) a piece-wise linear one, engendered by the inertia of the macro-blocks separated by the cracked middle section undergoing rigid-body rocking motion. As illustrated by Meisl et al. [15], the components (b) and (c) may not be necessarily in phase with the overall support acceleration (a), which can be a source for wall collapse [15]. The acceleration profile given by the support motion only is given by the sum of the components (a) and (b). The sum of these two components gives a trapezoidal acceleration profile. Denoting with ü T (t) and ü B (t) respectively the top and bottom support accelerations, the overall support acceleration results in the average value: T B u (t ) u (t ) u(t ) 2      (1) The relative support acceleration, which is also a useful parameter in the present work, is defined as:     T B ˆu(t ) u t u t      (2) O