PSI - Issue 44

Andrea Natale et al. / Procedia Structural Integrity 44 (2023) 1768–1775 Andrea Natale et al./ Structural Integrity Procedia 00 (2022) 000–000

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this process not convenient to implement in the design practice. In order to provide to designers an effective tool for a quick and reliable estimation of the PBT, a simplified framework is proposed and implemented in a MATALB code. Few data are required as input (i.e. geometry, site) to allow the practitioners to have first preliminary estimation of the PBT. The code implements the full PBEE procedure and allow to evaluate the EALs and the PBT considering different and alternative retrofit solutions. A case study building is used in this paper to show the potential and the results of the application of this procedure.

Nomenclature PBT

Pay-Back Time EALs Expected Annual Losses PBEE Performance Based Earthquake Engineering EDPs Engineering Demand Parameters NLTHs Non-Linear Time Histories

2. Methodology The methodology to quantify the PBT described in Natale (2020) and based on PBEE is considered to develop the simplified procedure at the base of this study. In particular, some basic assumptions are made to simplify the structural analysis needed to assess the EDPs, while the PBEE framework for loss-assessment is fully implemented in the Matlab code. The code is developed as a series of intercommunicating scripts that allow to conduct the structural analysis, the loss-assessment and the calculation of the PBT for 4 configurations of the reference building: As-built, retrofitted with FRP, Rebuilt (matching the modern requirement of NTC 2018 (MIT 2018) and retrofitted with base isolation (and considering possible local FRP strengthening of critical RC members). The estimation of the curve of losses combined with the cost of each retrofit solution allows to define the PBT. The calculation is completely automatic once the data in the “Input Script” are defined. In particular, the input data are: the total mass of each floor, including deadloads and live loads in the seismic combination; the number of floors and the height of each floor; the thickness and the length of the infills (clear distance between two columns); the mechanical properties of the base isolation system used as retrofit solution in terms of equivalent stiffness ( k_ieff1 ), damping ( smorz1 ), elastic stiffness ( k_iiso ), post-yielding stiffness ( k_p1 ), yielding strength ( F_y1 ) and yielding displacement ( d_y1 ), the type of bearing choosing between a sliders (Fenz and Constantinou 2006) or elastomeric bearings. The mechanical properties of the steel reinforcement in terms of yielding strength are needed, while the user may decide to conduct a sensitivity analysis varying the concrete compressive strength and reinforcement details in case, they are not known at the beginning of the design process. Furthermore, the selection of 14 records, is needed considering the 9 return periods (from 30 years to 2475 years). In case that concrete compressive strength and reinforcement details are not defined, a random extraction of concrete strength based on a gaussian distribution (Masi and Vona 2009) is performed and combined with 6 different transverse reinforcement ratio to account for the variability of internal reinforcement. Once all these data have been defined in the “Input Script”, the code composes the mass and the stiffness matrix. The structural model adopted in the code is an MDOF, where each degree of freedom is represented by the total mass of the floor for the fixed base configuration (As-built, FRP, Rebuilt), in addition a degree of freedom is considered for the base isolation configuration that represent the base isolation system. The degrees of freedom related to the floors assume a linear behavior while the base isolation is characterized by a bilinear behavior. The mass matrix is enabled considering the mass of each floor, where the mass accounts for all contribution from the slab to the column and beams and infills. The stiffness matrix is assembled considering the stiffness of each floor, calculated as the sum of the columns’ stiffness (assuming the RC frame as shear type) and the lateral stiffness of the infills calculated according to the Panagiotakos and Fardis (1996)’s model. As regard the base isolation, the stiffness is calculated as the sum of the stiffness of each device installed under the columns of the floor. Once that the MDOF structural model is defined a modal analysis, to assess the fundamental period of vibration and NLTHs, to assess the EDPs, are performed. furthermore, the damping matrix proportional to mass and stiffness matrix is defined (Adhikari 2006). NLTHs are