PSI - Issue 78

Melina Bosco et al. / Procedia Structural Integrity 78 (2026) 441–448

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For each SDOF system, three values of T sec are considered, namely 1.0 s, 3.0 s or 5.0 s. Note that the mass of the system can be calculated only once the pushover analysis of the SDOF system has been performed and the value of the secant stiffness K sec has been determined. 2.2. Numerical model of the SDOF systems The numerical analyses are performed by means of the software OpenSees (Mazzoni et al 2006). Beams and columns of the SDOF system are modelled as force-based elements (ForceBeamColumn with HingeRadau integration method) with ending plastic hinges (Scott and Fenves 2006). The length of the plastic hinge normalized with respect to the height of the member’s cross -section L pl / h is assigned based on the following expression, proposed by the authors in a previous study (Barbagallo et al 2022) The response of steel and concrete is simulated by means of the uniaxial material models “ReinforcingSteel” and “Concrete04” respectively. Based on test data reported in a database assembled by Verderame et al (2011), the average values of the yield strength and the ultimate tensile strength of FeB44k steel are assumed equal to f ym =470 MPa and f um =700 MPa respectively. The value of the average ultimate strain at peak stress is assumed equal to ε um =0.10. Based on stress-strain diagrams of FeB44k steel bars tested at the laboratory of the University of Catania during the 1990s, the strain at incipient hardening ε sh and the normalized initial tangent of the hardening branch, i.e. E sh / E s , are assumed equal to 7 ε y (where  y is the yield strain) and 2.5 respectively. In order to account for the strength degradation of concrete due to ageing, the average compressive strength of C20/25 concrete is assumed equal to 20 MPa (corresponding to a characteristic value of 12 MPa). The mechanical properties of unconfined concrete, namely the ultimate strain ε cu , the strain at peak stress ε c , the elastic modulus E c , the peak tensile stress f t and the corresponding strain ε t , have been calculated according to analytical formulations available in the literature. The mechanical properties of the uniformly confined concrete (i.e., peak compressive stress f cc and ultimate strain ε cu ) have been obtained based on the confinement model proposed by Chang and Mander (1994). The effectiveness of confinement has been considered as reported in (Barbagallo et al 2022). The accuracy of these modelling assumptions has been previously validated against laboratory tests (Barbagallo et al 2022). The ends of the central beam of the model are forced to have equal lateral displacements to reproduce the rigid slab behavior. However, this causes the development of an unexpected axial force in the beam which may lead to an overestimation of the resisting bending moment. To nullify this axial force while maintaining the transfer of the shear force and bending moment, an axial buffer element has been added between node 5 and 7 of the model. For further details about this modelling strategy refer to Barbagallo et al (2020). A uniformly distributed vertical load p s is applied to the beams to represent the gravity loads acting at floor level in the seismic design situation. A vertical axial force P = A c f c – p s L is applied to nodes 5 and 6 to simulate the gravity loads transferred to the columns from the upper stories of the multi-story frame. Finally, an external bending moment b,mid 2 s Ed / 24 = M p L is applied to nodes 8 and 9 to simulate the actual shape of the bending moment diagram due to the gravity loads. The magnitude of p s is determined considering a combination coefficient s = 0.33 (D.M. 1996) and based on the assumption that the permanent and the variable components of the distributed gravity load p in the non-seismic design situation represent the 70% and 30% of the total respectively. Assuming that the internal forces in the beams are caused by the load p and that the design bending moment at the ends of the beam is equal to pL 2 /10, the value of the uniform load p is given as ( ) top 2 s s 10 [ 0.9 ] / =  −  p A h c L . The mass of the system is divided into two equal parts and applied to nodes 5 and 6. Second order effects are not included in the model; a linear geometric transformation has been preferred. This choice stems from the fact that it is not possible to properly include in a SDOF model an effect that depends on the gravity loads and lateral displacements of the floors of a MDOF frame. Referring to the dynamic analyses, the viscous damping coefficient of each SDOF system is assumed to be mass-proportional and defined so that the motion of the system be characterized by a viscous damping ratio ξ 0 equal to 5%. 0.35 pl w 0.19 − =  L h (3)