PSI - Issue 78

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

446

ξ [%]

0 10 20 30 0.0 1.0 2.0 3.0 4.0 5.0 μ M ξ [%] (c) 0 10 20 30 0.00 0.01 0.02 0.03 θ max [rad] ξ [%] c (f)

0 10 20 30 0.0 0.2 0.4 0.6 0.8 T 1 / T sec ξ [%] (b) 0 10 20 30 0.00 0.01 0.02 0.03 θ max [rad] ξ [%] b (e)

T sec = 1 T sec = 3 T sec = 5

0 10 20 30

(a)

T 1 [s]

0.0 1.0 2.0 3.0

ξ [%]

0 10 20 30

(d)

0.0 0.5 1.0 1.5

λ M

Fig. 2. Experimental values of the equivalent viscous damping ratio of the SDOF system versus: (a) elastic period of vibration T 1 , (b) ratio of elastic to secant period of vibration T 1 / T sec , (c) displacement ductility of the system μ , (d) ratio λ M , (e) first achieved chord rotation capacity of beams and (f) first achieved chord rotation capacity of columns. The overdamped response spectra are then scaled to this value of the PGA (later referred to as a g u ), and the optimal value of the viscous damping ratio exp eq  of the elastic SDOF system is identified as the one that virtually nullifies the difference between the spectral displacement corresponding to the secant period of vibration S d ( T sec ) a gu

and the displacement capacity u SDOF . 4. Results of the parametric analysis In Figure 2 the values of the equivalent viscous damping ratio exp

eq  obtained through the parametric analysis of the SDOF systems are plotted against the elastic period of vibration of the system T 1 , the ratio of T 1 to the secant period of vibration T sec , the displacement ductility μ of the system, the ratio  M (evaluated at the end of the design of the system and thus considering the actual longitudinal reinforcements) and the chord rotation demands in beams and columns ( b max  and c max  ) at the first achievement of the chord rotation capacity in the system. Note that the ductility demand has been estimated considering the ratio of the lateral displacement u m corresponding to the first achievement of the chord rotation capacity, to the lateral displacement corresponding to the chord rotation at yield. This latter quantity has been evaluated according to Biskinis and Fardis (2010). As could be seen in Figure 2a, the value of exp eq  is inversely proportional to the period of vibration T 1 . This trend is more evident if exp eq  is plotted against the ratio T 1 / T sec as shown in Figure 2b. Also, in this plot, the results obtained for the three values of T sec (i.e., 1.0 s, 3.0 s and 5.0 s) gather along a single curve. As expected, a clear correlation exists between exp eq  and the displacement ductility μ †‡ƒ† ƒ• Š‹‰ŠŽ‹‰Š–‡† ‹ Figure 2c. The relationship between exp eq  and the ratio  M is less apparent, as shown in Figure 2d. However, even though a great dispersion exists, the ratio  M effectively separates the cases in which the target limit state is achieved in the beams or in the columns. At a first glance at the plots shown in Figures 2e, and 2f, the correlation between exp eq  and the values of the chord rotation demands in beams and columns ( b max  and c max  ) appears doubtful. However, on a closer examination of the data, the authors note that in the plots of exp eq  versus b max  and the cases on the right side are the ones in which the beams have achieved first the chord rotation capacity. Similarly, in the plot of exp eq  versus c max  the cases on the right side are the ones in which the columns have achieved first the chord rotation capacity. Conversely, data on the left side of the plots refers to the cases in which the chord rotation capacity has been achieved in a member that is different from the one referred to on the horizontal axis of the plot. This means that when the chord rotation capacity is achieved in one type of member, i.e. beam or column, the chord rotation demand in the other type of element is generally low. In this perspective, the existence of a correlation between exp eq  and b max  and c max  is apparent.