PSI - Issue 78

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

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3. Analysis of the SDOF system For each SDOF system a pushover analysis (PO) is first performed to obtain the lateral displacement u SDOF and the base shear V base ( u SDOF ) corresponding to the first achievement of a unit value of the ratio of the chord rotation demand ϑ † to the chord rotation capacity ϑ — of the members. This point of the capacity curve is later referred to as the target point and the condition ϑ † Ȁ ϑ — =1 is later referred to as target limit state. The lateral secant stiffness K sec corresponding to the target point is evaluated as the ratio of the base shear to the lateral displacement of the system, i.e. K sec = V base ( u SDOF )/ u SDOF . Then, nonlinear dynamic analyses are carried out by means of 15 artificial accelerograms to identify the value of the PGA (later indicated as a gu ) corresponding to the achievement of the lateral displacement u SDOF . Finally, a set of overdamped response spectra is constructed and the value of the equivalent viscous damping ratio of the SDOF system is identified as the one that virtually nullifies the difference between the spectral displacement corresponding to the secant period of vibration T sec of the inelastic SDOF system and the lateral displacement u SDOF . 3.1. Cyclic pushover analysis Gravity loads are first applied in ten steps through a load control integrator. Then, a cyclic pushover analysis is carried out by applying a displacement loading protocol to node 5 along the horizontal direction through a displacement control integrator. The considered displacement protocol has been proposed by Pampanin et al (2007) and consists of eight series of symmetric displacement cycles with increasing amplitude, each consisting of three cycles with constant maximum displacement. The maximum displacement of the eight series of cycles is equal to 0.2, 0.6, 0.8, 1.0, 1.5, 2.0, 3.0, 3.5 % of the height of the column elements (i.e., H /2). At each step of the analysis, the response of the system is used to calculate the chord rotation demand ϑ d of the members. Specifically, the chord rotation demand ϑ d at one end of a member is evaluated as the ratio of the transverse displacement u infl of the inflection point (i.e., the null point of the bending moment diagram) to the shear span of the member (i.e., the distance between the considered end and the inflection point) calculated with reference to the considered end, i.e. ϑ d = u infl / L v . Due to its dependency on the shear length L v and the normalized axial force ν of the member, the chord rotation capacity ϑ u is also evaluated at each step of the pushover analysis and at both ends of the elements. The value of ϑ u is computed according to the expression proposed by Biskinis and Fardis (2010) as reported in Eurocode 8 (EN 1998-1 2004) for the near collapse (NC) limit state. At the end of the analysis, the target point [ u SDOF , V base ( u SDOF )] of the capacity curve is identified as the one corresponding to the first achievement of the chord rotation capacity ϑ u in one of the members (i.e., ϑ d / ϑ u =1). The secant stiffness K sec of the system is evaluated at the target point as the ratio of the base shear V base ( u SDOF ) to the lateral displacement u SDOF . 3.2. Dynamic analysis The seismic input consists of 15 artificial accelerograms generated by some of the authors in a previous study (Amara et al 2014) using the SIMQKE computer program (Gasparini and Vanmarcke 1976). The artificial accelerograms were generated with a total duration of 30.5s considering a “compound” intensity function with rise time set equal to 4.0s and parameters IPOW and ALFA0 set equal to 2 and 0.25 respectively. The duration of the stationary part of the compound function was calibrated so that the generated signals had dynamic properties as close as possible to those of a set of reference natural accelerograms. More details may be found in Amara et al. (2014). The same set of artificial accelerograms is also used to generate a set of average elastic displacement response spectra for values of the viscous damping ratio ranging from 5% to 50% in steps of 2.5%. For each SDOF system, the dynamic analyses are performed assuming a trial value of the PGA to obtain the median value of the maximum lateral top displacements of the system u m . The value of the PGA is then adjusted following a trial-and-error process until the lateral displacement u m matches the lateral displacement capacity of the SDOF system u SDOF previously determined through the cyclic pushover analysis.