PSI - Issue 78

Lorenzo Audisio et al. / Procedia Structural Integrity 78 (2026) 1277–1284

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3. Theoretical background In order to evaluate the FRS, an amplification factor R is introduced, derived from the analysis of the response of a damped oscillator subjected to a harmonic excitation. As it is well known, the general solution of the motion equation consists of the sum of the homogeneous and the particular solution. By defining the ratio between the forcing frequency and the system’s natural frequency as = ⁄ , and introducing the static displacement u st , the steady-state displacement response can be expressed as (Anil k. Chopra, 1995): ( ) = ∙ 1 √(1− 2 ) 2 + (2 ) 2 ∙ ( − ) (4) where the term: = 1 √(1− 2 ) 2 + (2 ) 2 (5) is called deformation (or displacement) response factor , measuring the ratio of the response amplitude u 0 of the dynamic deformation to the static deformation u st . The Eq. 5 forms the theoretical basis for defining the element amplification factor R (Eq. 1) used in the formulation adopted within the instructions for the application of the Italian Standards NTC 2018 (MIT - Italian Building Standard 2018). In this formulation it is assumed that the NSE is considered as a SDOF uncoupled from the principal structure, and subjected to external harmonic forces represented by the modal accelerations (Eq. 2). The NSE total acceleration (Eq. 3) is obtained as combination of the spectral ordinates, each of which evaluated on the FRS (Eq. 1).

Fig. 3. Sensitivity of Amplification Factor R to Coefficients

The generalized expression assumed for the element amplification factor of the NSE is the following: = [( ) +(1−( ) ) ] −

(6)