PSI - Issue 78

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

443

Table 1. Reference values for the longitudinal reinforcement ratio and volumetric transverse reinforcement ratio of the columns

Non-Seismic Design GL

Seismic Design (ASD)

Seismic Design (LSD)

ρ s [%] ρ w [%]

0.5-0.7 0.3-0.4

0.7-0.9 0.8-0.9

1.0-1.3 1.5-2.0

restrain both the horizontal and vertical translations of the nodes; the supports at nodes 3, 4, 8 and 9 consist of rollers that restrain only the vertical translation of the nodes. To simulate the constraint effects of the RC slab, nodes 5 and 6 are forced to have the same horizontal displacements. A pin supported rigid element connected to nodes 3 and 5 is used to promote a uniform deformation of the columns. An additional node 7, coincident with node 5 but not subject to constraints, is added to the scheme for modelling purposes, as discussed in the following subsections.

2.1. Parametrization of the SDOF system

The cross-sections of the columns are rectangular with dimensions ranging from 30×30 to 30×80 cm² and oriented so as to have their longest side either parallel or orthogonal to the plane of the frame. Reinforcement detailing of the cross section is defined to obtain pre-fixed values of the longitudinal reinforcement ratio ρ s and the volumetric transverse reinforcement ratio ρ w that are typical of non-seismic design and seismic design in the framework of both the Allowable Stress Design method or the Limit States Design method (see Table 1). The value of the axial force N to be applied at the top of the columns is calculated for two values of the normalized axial force ν = N / A c f c equal to either 0.20 or 0.40, where A c is the gross cross-sectional area of the column and f c is the average cylindrical compressive strength of concrete. The beams ’ cross-section is rectangular, with size equal to either 30×50 cm² or 30×60 cm² or 60×24 cm² (flat beams). The mechanical concrete cover is equal to 4 cm. In order to consider different yielding sequences of beams and columns, a parameter   is defined at the single beam-to-column node of the system as the ratio of the sum of the plastic bending moments of the beams to the sum of the plastic bending moments of the columns framing into the same node. The considered values of   are 0.5, 0.7, 1.0 and 1.3. The ratio   is calculated as where b pl,bot M is the plastic bending moment of the beam on one side of the node (bottom reinforcement assumed as tensile reinforcement), b pl,top M is the plastic bending moment of the beam on the other side of the node (top reinforcement assumed as tensile reinforcement) and c pl (N) M is the plastic bending moment of the single column reduced to consider the interaction with the axial force due to gravity loads in the seismic design situation. The SDOF systems are designed in the framework of the allowable stress design (ASD) method stipulated in (D.M. 1972, D.M. 1992). Longitudinal reinforcements are assumed to be made of FeB44k steel, which was used in Italy until the late XX century. This type of steel is characterized by a characteristic value of the yield strength equal to 430 MPa. The class of resistance of concrete is assumed to be C20/25. The allowable compressive stress c  of C20/25 concrete is equal to 8.50 MPa, whereas the allowable tensile stress s  of FeB44k steel is equal to 255 MPa. Note that the amount of top and bottom reinforcement of the beams is determined so as to satisfy Eq. (1) once that both   and c pl (N) M have been fixed. Owing to geometric constrains relative to the arrangement of the longitudinal rebars, some cases have been found to be impractical and thus discarded. In total, the design process has been completed for 1717 cases. The mass m of each SDOF system is calculated so as obtain an assigned value of the secant period of vibration T sec at the attainment of a pre-fixed limit state (see Section 3). b b pl,bot M M M c l p (N) 2 + pl,top M  = (1)

2 sec sec 2 4

 T k

m

=

( 2 )