PSI - Issue 44

Luca Umberto Argiento et al. / Procedia Structural Integrity 44 (2023) 1388–1395 Luca Umberto Argiento et al. / Structural Integrity Procedia 00 (2022) 000 – 000

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equivalent non-linear single degree of freedom (SDOF) system. Hence, such a conversion can be simply obtained by assuming a =  g and d = d CG being g the gravitational acceleration. Fig. 2 shows the capacity curves a ( d ) related to both the ‘free’ and ‘restrained’ rocking conditions for the 14 selected churches. It clearly emerges that the frictional resistances strongly increase the acceleration, a 0 , which activates the mechanism, and the trend of the capacity curves along the evolution of the mechanism is strictly related to the variation of the frictional resistances. In the case of ‘restrained’ condition, the first branch of the capacity curves is linearly decreasing as it corresponds to the constant contribution of the frictional resistances. Then, the curves have a stepwise linear decreasing trend due to the progressive loss of contact along the blocks starting from the top of the interlocked walls. When all the frictional resistances become null, the capacity curves for the restrained façades coincide with those corresponding to the free rocking condition.

3.5

SMC SRO SG SFS SRE SL SMP SAP SPB CA SOC SMA SFA SD

3

2.5

a [m/s 2 ]

2

1.5

1

0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

d [m]

Fig. 2. Capacity curves related to the free (dashed lines) and restrained rocking (continuous lines) conditions for the 14 selected facades.

3. Seismic assessment of the selected rocking façades 3.1 The seismic input: response spectra indicated by the Italian seismic code

According to the capacity spectrum method (CSM) indicated by the Italian seismic code (MIT 2018, MIT 2019), the seismic demand is represented in terms of over-damped elastic acceleration-displacement response spectra (ADRS) derived from analytical functions of the hazard parameters provided by the code. For the seismic parameters, a soil type B (deposits of very dense sand, gravel, or very stiff clay) is assumed, and the soil factor S is fixed equal to 1.2 accounting for topographic and stratigraphic conditions. Three limit states are considered for the seismic assessments, which are related to the following threshold values of the capacity parameters: the activation of the rocking mechanism (LS0), a moderate (LS1) and a severe (LS2) rocking motion. LS0 conventionally occurs when PGA is greater than or equal to the minimum acceleration that triggers the motion; more accurate approaches take into account an initial pseudo-elastic capacity that generally falls within the range of the maximum spectral accelerations (Griffith et al. 2003, Lagomarsino 2015). The second limit state, LS1, occurs when the displacement of the control point attains the value d LS1 = 0.4 d 0 , where d 0 corresponds to the loss of static equilibrium (e.g., when d 0 becomes equal to the half-thickness of the façade wall in the unconstrained condition). Lastly, LS2 is associated with d LS2 = 0.6 d 0 , that is representative of a severe rocking motion. While LS0 corresponds to a serviceability limit state, LS1 and LS2 are considered representative of life safety and collapse limit states, respectively. Once defined the reference displacement capacities for LS1 and LS2, the application of the CSM requires deriving the corresponding values of the equivalent periods T through the following relations: 1 = 1.68 √ 1 ( 1 ) 2 = 1.58 √ 2 ( 2 ) (3)