PSI - Issue 78

Parvane Rezaei Ranjbar et al. / Procedia Structural Integrity 78 (2026) 615–622

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where  is the cumulative distribution function of a standard normal variable and has a mean of 0 and a standard deviation of 1. Using the above equation, once the two parameters, m x (varying with IM ) and σ x , of variable X are obtained, the fragility curves can be constructed for various damage states or capacities. 2.3. Incremental Dynamic Analysis (IDA) and Limit States At this stage, Incremental Dynamic Analysis (IDA) is carried out to assess the structural performance under seismic loading. IDA is a nonlinear dynamic procedure widely regarded as an accurate method for structural seismic analysis. It relies on identifying an intensity measure ( IM ) and a corresponding damage measure ( DM ). The process involves subjecting a structural model to one or more ground motion records, each scaled to multiple intensity levels. This ultimately generates structural response curves that relate the seismic intensity to the structural response (Vamvatsikos and Cornell, 2002). Four drift-based limit states, defined according to HAZUS (2024), are considered, based on the specific building typology (Concrete Frame Buildings with Unreinforced Masonry Infill Walls (C3)), as shown in Table 1. Table 2 outlines the four damage states corresponding to specific drift ratio thresholds. The maximum interstory drift ratios are obtained for each record, corresponding to the first-mode 5%-damped spectral acceleration, S a ( T 1 ,5%), corresponding to the fundamental period of the structure ( T 1 ) in the considered direction. These data are then used to perform regression analyses and calibrate the fragility functions described in Section 2.2. As a result, fragility curves corresponding to the various HAZUS-defined damage states are derived and plotted. Table 1. Specific Building Types (Hazus, 2024). Height Range Typical Label Description Name Stories Stories Feet C3M Concrete Frame with Unreinforced Masonry Infill Walls Mid-Rise 4-7 5 50 3. Case study: RC building retrofitted with rocking steel-braced dual frames. The case study focuses on a school building constructed in 1964 (Fig. 1). It was designed without any seismic regulations, relying solely on gravity loads. The foundation consists of isolated footings set at a depth of -3.10 meters. The structure includes a semi-basement and two above-ground floors, for a total of three levels, with an L-shaped floor plan. The heights, measured from ground level, are +0.60 meters, +3.80 meters, and +7.40 meters, respectively. Reinforcement consisted of FeB38k ribbed bars, while the concrete exhibited a characteristic cube strength ( R ck ) of 20 MPa. More details about this benchmark case study can be found in Ferraioli et al. (2025c). For the analyses conducted in this study, the building was assumed to be located in Syracuse, and the corresponding seismic actions were defined according to the Italian Seismic Code (NTC-Guidelines, 2018) based on a reference life of 75 years, sub- soil class type “C”, and slope category T 1 (i.e., flat surface). The peak ground accelerations are 0.073g, 0.0996g, 0.345g, and 0.434g for the IO, DL, LS, and CP limit states, respectively. The seismic retrofit was carried out using an exoskeleton composed of self-centering dual-frame rocking systems (Fig. 2). Rocking-based retrofitting solutions have gained attention for their ability to reduce residual deformations, improve energy dissipation, and promote self centering behavior. Several configurations have been proposed, including self-centering rocking frames, pivoting core systems, segmented rocking elements, and energy-dissipating column bases. Self-centering is typically achieved through gravity effects or post-tensioned elements, or shape memory alloys, enabling the structure to return to its original position after seismic loading. Additional components - such as friction devices, yielding fuses, or base plates - are integrated to dissipate energy without compromising structural integrity. In this paper, rocking steel-braced dual Table 2. Damage levels and drift ratios for structural components - Special buildings with pre-code seismic design level. Building Properties Interstory Drift at Threshold of Damage States Type C3M Height (m) Slight 0.0016 Moderate Extensive Complete 10.5 0.0032 0.0080 0.0187