Issue 50

M. Godio et alii, Frattura ed Integrità Strutturale, 50 (2019) 194-208; DOI: 10.3221/IGF-ESIS.50.17 198 Figure 2 : Discrete element model of a vertically-spanning URM wall with different top and bottom support motions used in the study. on the wall response of the elastic modulus of masonry, the wall height-to-thickness ratio, effective thickness and the vertical stress applied at the wall top. The parameters E m , t w , r b and σ v are therefore changed during the study, as explained later in the paper - see Tab. 2. The other parameters are defined as follows [32]: ν=0, f t =0 MPa, c=2 MPa, φ=35° and ψ=0°. A stiffness-proportional Rayleigh damping model is used in the numerical simulations [12]. Following the strategy validated in [32], the damping centred on the first flexural frequency of the wall, which is also reported in Tab. 2, is obtained considering the wall as a double-clamped Euler’s beam: 2 m w I w w w 1 4.730 E I f 2π H ρ t L        (5) with ρ the wall density and I w =1/12L w t w 3 the moment of inertia of a generic cross-section of the wall. A damping ratio ζ of 20% operates at the above frequency [32]. Time-history analyses are carried out through the numerical model by applying velocity histories in the out-of-plane direction of the wall. The velocity histories that are applied to the top and the bottom wall supports are different and are implemented following the strategies described in the next section. R ELATIVE SUPPORT MOTION GENERATION ab. 1 contains the suite of records used in this study. The records are the same as those used by Godio and Beyer [32] and introduced by Sorrentino et al. [37]. The set of selected records covers a wide window of peak ground acceleration (PGA), velocity (PGV) and displacement (PGD), which is thought to diminish the record-to-record variability of the results obtained in the study. Starting from these records, synthetic relative support motions are generated by introducing a phase shift and a relative amplitude. Generation of support motions with phase shift The Hilbert transform is a convenient method for generating signals that have the same amplitude but are phase-shifted, or asynchronous. The Hilbert transform is the integral transform [39]: (6) in which s(t) is the original signal and PV is the Cauchy principal value of the integral [39]. The Hilbert transform operates on the signal as a filter and allows the analytical signal z(t) [40] to be generated starting from the original one: (7) In this study, the Hilbert transform is used to introduce a phase shift between the input motions of the top and bottom wall supports. Since the numerical model used in this study requires as input the velocity histories at the wall supports, the Hilbert O t w h b r b t w,eff σ v u T ( t ) u B ( t ) H w       1 s t τ s t PV dτ π τ               z t s t i s t    T