Issue 50

M. Godio et alii, Frattura ed Integrità Strutturale, 50 (2019) 194-208; DOI: 10.3221/IGF-ESIS.50.17 195 I NTRODUCTION nreinforced masonry (URM) walls are very vulnerable to the out-of-plane actions, even more so at the higher levels of a building where the accelerations are large and the walls relatively slender [1]. The supports of out-of-plane loaded masonry walls in a building are subjected to a motion that is filtered and amplified by the building structure and, in some cases, can be significantly different from the ground motion. Moreover, because these walls span one or several storeys, their top and bottom supports are subjected to motions that differ in phase and amplitude. Relative support motion is reportedly due to two factors affecting the path of the seismic action from the ground to the walls located at the upper storeys of a building [2]: first, the diaphragm response, notably the in-plane or membrane flexibility of the diaphragm, see e.g. [3,4], and, second, the filtering and amplifying effect due to the building structure and, in particular, the shear walls. The performance of unreinforced masonry walls subjected to out-of-plane support motion including these two factors has been the topic of only few numerical studies so far. Landi et al. [5] investigated the response of pinned-clamped walls undergoing vertical bending. The walls were modelled as assemblies of two axially-loaded rigid macro-blocks fully separated by a crack at a certain wall height and laterally restrained at the top by a spring. The wall motion was described by a Lagrangian system of equations based on two degrees of freedom and integrated on a series of gaussian impulses and natural records. Penner et al. [6] performed a parametric study on pinned-clamped vertically-spanning walls by using the software Working Model 2D [7]. The study extended the one by Sharif et al. [8], who analysed two rigid macro-blocks subjected to equal input support motion, to two rigid macro-blocks subjected to relative support motion. This was achieved by connecting two lateral springs to the macro-block ends. The study included variations of the spring stiffnesses, wall thickness and slenderness and axial load magnitude and eccentricity. Code-based records were used as input to the numerical model. Derakhshan et al. [9] investigated the response of walls with pinned-pinned elastically-restrained supports. The wall response was captured by means of a tri-linear force-displacement model built upon a two-rigid-macro-block description. With respect to the two above-mentioned works [5,6], in this work the filtering action played by the structure was considered in the implementation of the input ground motions. As a result of the adopted filtering process, the input code-compatible records applied to the wall supports had a dominant period close to that of the building [9]. The out-of-plane response of URM walls was also studied by Tondelli et al [10], who included as input to the out-of-plane wall support motion not only the effect of filtering due to the mixed unreinforced masonry – reinforced concrete wall structure but also the effect of rocking and therefore elongation of adjacent in-plane loaded URM walls on the top restraint of the out-of-plane loaded wall. Both effects were experimentally observed [11] and simulated by means of the software UDEC 6.0 [12], where each brick row was modelled as a discrete rigid block. The above-mentioned numerical studies all show that URM walls appear to be more vulnerable when they are subjected to relative out-of-plane support motions [5,6,9,10]. Providing some degree of flexibility to the wall supports by means of springs, which allows mimicking the situation in which the supports do not move simultaneously, leads in fact to a displacement demand that is higher than that of a wall with ‘fixed’ or highly stiff supports. While a wall with fixed supports collapses because of the excessive mid-height displacement [13–15], a wall with flexible supports can also collapse due to the excessive support displacement [9,16]. Furthermore, the more the supports are flexible and have a different degree of flexibility, the higher can be the difference between the top and the bottom motions and the more vulnerable becomes the wall [6,16]. In the case of a very flexible top support, the upper macro-block behaves similarly to a parapet wall elastically restrained at the top and rocking on the lower macro-block, which, in turn, rocks on the fixed bottom support. In this situation, the support motion can give rise to a deformation pattern that is characterized by both macro-blocks rocking in the same direction, see e.g. [17]. This pattern is unfavourable when compared to the one experienced by a wall spanning vertically on fixed supports, since it results in a reduced wall force capacity [18] and displacement capacity [9]. The wall acceleration capacity consequently reduces. Depending on the wall configurations, flexible supports were found to reduce the wall capacity by up to 1.3 times [6], but a systematic study quantifying the vulnerability of out-of-plane walls subjected to the relative support motion based on the input motion characteristics is missing. In the above-mentioned studies, the URM walls are modelled as vertical strips undergoing vertical one-way bending. Such wall layout is simple yet representative of many wall configurations that are vulnerable to out-of-plane loading. During earthquakes, the effectiveness of wall side restraints, which would cause two-way bending, could in fact decrease, e.g. due to the weakening of the wall corners, which finally comes to trigger vertically-spanning overturning mechanisms. Such wall configurations have also been tested on shake tables. Among these tests, [15,16,19,20] include observations on the relative motion of the wall supports. When subjected to relative support motions the walls exhibit an acceleration profile varying piecewise linearly along the wall height [15]. Furthermore, the motion of the top and bottom connections may not necessarily be in-phase [16,19,20]. These are certainly two sources of wall collapse that are, nonetheless, difficult to predict and quantify U