Issue 50

M. Godio et alii, Frattura ed Integrità Strutturale, 50 (2019) 194-208; DOI: 10.3221/IGF-ESIS.50.17 199 Earthquake Year Station Label* Mag. Duration** (s) PGA (g) PGV (cm/s) PGD (cm) Kern County 1952 Taft Lincoln School TAF111 7.36 20 0.18 19 9 San Fernando 1971 Pacoima Dam (upp. left abut) PUL164 6.61 12 1.22 114 39 Friuli 1976 Tolmezzo TMZ000 6.5 10 0.36 23 5 Imperial Valley 1979 Bonds Corner BCR230 6.53 15 0.78 45 15 Imperial Valley 1979 El Centro (Array #7) E07230 6.53 15 0.47 113 47 Nahanni 1985 Site 1 S1280 6.76 10 1.2 41 10 Loma Prieta 1989 Los Gatos (Lexington Dam) LEX000 6.93 10 0.44 86 17 Northridge 1994 Rinaldi (Receiving Stat.) RRS228 6.69 10 0.87 148 42 Northridge 1994 Sylmar (Olive View Med FF) SYL360 6.69 10 0.84 129 32 Kobe 1995 Takatori TAK000 6.9 15 0.62 121 40 *The records were downloaded from the PEER-NGA database [38] **The original records were shortened by cutting the parts preceding and succeeding the main shock Table 1 : Natural records used in the parametric study. transform operates on the input velocities, which are expressed in the following form:             T 1 β 1 β u t u t u t 2 2                     B 1 β 1 β u t u t u t 2 2         (8) with u(t) the original input ground motion considered for the study and β a factor enabling to control the phase shift between the support motions. Fig. 3 shows for one of the input ground motions used in this study the displacement histories and the Lissajous diagrams of the top and bottom support displacements resulting from Eqn. (8). In the figure, u max is calculated as max{u(t)}. Lissajous is the name used in signal processing to designate diagrams where two signals are plotted on the two axes. This representation is very handy for comparing the two signals. If the signals have equal amplitude and frequency, as in the case of two sinusoidal signals, which are represented by dashed lines in the Lissajous diagrams of Fig. 3, the diagrams will take the shape of an ellipse. The shape and orientation of the ellipse will change according to the phase difference, or phase-shift, between the two signals. As apparent from these diagrams, by using Eqn. (8) for β=0, the top and bottom support motions are identical and therefore of equal amplitude and in-phase; for non-zero values of β, the support motions have same amplitude but are out-of-phase with a phase shift that increases as β increases: in particular, for β=0.5 the phase shift amounts to 45°, whereas for β=1 the support motions are in quadrature, i.e. with a phase shift of 90°. For all values of β, the overall support velocity, i.e. the average velocity of the top and bottom supports, due to phase shift is (Eqn. (1))       β u t u t u (t ) 2       (9) and the relative velocity between the top and bottom supports is (Eqn. (2))         β ˆu (t) β u t u t       (10) Generation of support motions with relative amplitude A relative amplitude is introduced between the top and bottom wall support input motions as follows [10]: (11)       T u t 1 α u t           B u t 1 α u t    