PSI - Issue 78

Salvatore Mottola et al. / Procedia Structural Integrity 78 (2026) 623–630

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The chosen capacity curve is idealized using the Takeda trilinear model (Takeda et al. , 1970), as outlined in Section 2.1. To represent the system as a single-degree-of-freedom (SDOF) model, base shear V (S) and roof displacement d (S) are normalized by the modal participation factor Γ . The structure and exoskeleton are modeled as parallel springs with distinct nonlinear behaviors. NLTHA of the idealized SDOF system yields P-spectra, from which the target displacement and reduction factors are determined. By intersecting the corresponding line on the spectra (i.e., Points P1 and P2 in Fig. 4b), suitable values of both stiffness ratio and exoskeleton ductility ( α , µ ( E ) ), corresponding to an acceleration reduction factor R a , are identified. The ideal design minimizes floor acceleration ( R a ) while remaining within the exoskeleton’s ductilit limits . Once the target values for stiffness ratio α and exoskeleton ductilit µ ( E ) are established, the corresponding exoskeleton stiffness K 0 (E) and strength V y (E) are calculated. The design base shear V y (E) is then distributed vertically along the building height according to the first mode shape in the selected direction. This results in story-specific strength V yi and stiffness K i , which are used to size the exoskeleton based on its spatial configuration (2D or 3D), placement (parallel or orthogonal to the façade), and structural type, such as concentric or eccentric braced frames, reinforced concrete frames, or shear walls. 3. Validation 3.1. Seismic retrofit of a benchmark building using parallel dissipative EBFs based on SSDs The reference case involves a school facility located in Vibo Valentia, within the Calabria region of Italy, previously subjected to seismic vulnerability analyses and seismic retrofit using endoskeletons made of dissipative steel braces. More details about seismic verification and nonlinear static (pushover) analysis of the existing building modelled in SeismoStruct (Seismosoft, 2023) can be found in Ferraioli and Lavino (2018). Fig. 5 shows the pushover curves corresponding to the modal load pattern and its trilinear idealization. The strengthening approach adopted for the reference building involves a parallel exoskeletal system composed of eccentric braced frames (EBFs) integrated with steel strip dampers (SSDs), as illustrated in Fig. 6. Within the various SSD configurations examined in prior research, dumbbell-shaped dampers are particularly notable for their superior energy dissipation and cumulative ductility. According to Ferraioli et al. (2025), when lateral-torsional buckling is prevented, these dumbbell-shaped elements show stable hysteresis loops and significant energy dissipation. Their findings indicate that the onset of buckling is governed by parameters such as slenderness, aspect ratio, and initial imperfections. In their study, the damper configuration was validated to maintain performance even in the presence of an initial imperfection of e 0 = 1/250. This validation enables the use of a simplified kinematic elastic-plastic model to represent the damper, comprising multiple strips (Fig. 6d). The design procedure of Section 2 is applied. The target design displacement of the combined system (SE) is defined as follows: in the X-direction from pounding with an adjacent building ( d p =50 mm), while in the Y-direction a total drift ratio of 1.5% is considered (i.e., d p =103 mm). Nonlinear time-history analyses (NLTHA) of the idealized SDOF model were performed using a set of seven ground motions, each with two horizontal components, with a total of 14 records. These records were selected and scaled according to the NTC Guidelines (2018) and Eurocode 8 (2005). REXEL software (Iervolino et al. , 2010) was used for the identification of suitable spectrum-compatible records (Fig. 7). The NLTHA of the equivalent SDOF system of the combined system (ES) is used to generate the P-spectra (Fig. 8). Entering the P-spectra with the vertical line corresponding to the target displacement reduction factor R d,t gives the stiffness ratio α and the ductility ratio µ ( E ) , which are the starting point for sizing the exoskeleton.

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(b) Fig. 5. Pushover curve and corresponding trilinear idealization: (a) X-direction; (b) Y-direction. Roof displacement [mm] Roof displacement [mm]

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