Prediction of the number of equivalent cycles for earthquake motion

Abstract

This paper addresses new prediction models for computing the number of equivalent cycles in liquefaction analyses. Three models are presented for the Italian seismicity as a function of synthetic ground motion parameters that are often available from earthquake data web-sites immediately after events. In particular, it is observed that the number of earthquake cycles can be best estimated from the five following parameters: peak ground acceleration; epicentral distance; Arias Intensity; the mean period; and, the frequency of zero crossings. However, strong estimations can be obtained from the first three parameters only. We use statistical indicators to determine the goodness of the models and the usefulness of the selected independent variables, and we present a comparative analysis to validate our predictive equations. Moreover, this paper describes the existing correlation between magnitude and cycle numbers. The study is primarily based on Italian acceleration records, even if the database is also expanded to recorded European, Japanese and American events to amplify the magnitude values range. These simplified models are useful in addressing practical earthquake engineering problems which require the knowledge of number of equivalent cycles.

Appendices

Appendix A

This appendix illustrates an example of the computation the equivalent number of cycles following Seed et al.’s (1975) procedure.

Referring to the HNE horizontal acceleration recording for 1.5 to 5 s of the Atina ATQ station, at the epicentral distance of 17.42 km, for the Lazio-Abruzzo earthquake (Italy) of May 11, 1984, with a moment magnitude of 5.5 (Luzi et al. 2016), a simplified example of the complete procedure is shown in Fig. 14 and Table 10 applying the following steps:

Fig. 14

Example of the peak counting method

Table 10 Example of Seed et al.’s (1975) procedure

• 1. The recorded accelerogram is normalized on its peak value a_max;

• 2. Peaks of the signals are identified via the peak counting method, using a threshold value of 0.15 (red dots in Fig. 14 and accelerograms peaks columns in Table 10) and for each one the absolute value of the a/a_max ratio is extracted (|a/a_max| column in Table 10);

• 3. The liquefaction resistance curve is selected; in this specific example, Liu et al.’s (2001) curve is used. It is defined by the range between the direct simple shear curves DSS+ and DSS-. An arithmetic average is generated between the values of their parameters as reported by Biondi et al. (2004) and then R = 0.65 and m = 0.344 (Fig. 1). By adding the graph of the liquefaction resistance curve with each peak ratio a/a_max to the ordinate axis, it is possible to obtain the number of cycles from the horizontal one (Neq column of Table 10);

• 4. The total number of cycles is obtained by summing the values of each peak (the last value shown under the Neq column in Table 10).

Appendix B

This appendix illustrates an example of computation of the equivalent number of cycles, following the Rainflow Counting Method.

The Rainflow Counting Method is schematized in Fig. 15, where the cascade logical procedure for cycles counting is outlined following the ASTM Standards (1985). Several parameters introduced in Fig. 15 (peaks, valleys and range) are clarified in Fig. 16.

Fig. 15

Rainflow counting method (Hancock and Bommer 2004)

Fig. 16

Cycle parameters

From a physical point of view, it identifies hysteresis stress cycles of the material response (the soil response in this specific case) with the maximum cycle identified from maximum and minimum efforts of the stress and strain sequences (peaks A and D in stress and strain histories of Fig. 16, corresponding to points A and D of the hysteresis loop). Therefore, any temporal history can be thought to be composed of a main loop from the absolute minimum value to the absolute maximum value or vice versa (points A and D in Fig. 17), interrupted by several inversions forming secondary cycles (B–C, E–F and G–H in Fig. 17). The application of a time dependent acceleration load to a soil deposit leads to the development of stress and strain histories that are responsible of damage induced by the irregular load in the ground. Consequently, Rainflow Counting is particularly useful in assessing cycles counting for time history input data (Benasciutti 2004). The hysteresis loop describes energy dissipated in the soil for the entire duration of earthquake motion. Some quantities are sketched in the stress–strain plan shown in Fig. 18, where W is the maximum amount of energy stored in elastic material with the same shear modulus G as the visco-elastic one and where ΔW₁ is energy dissipated per unit of material volume in one loading cycle. In referring to dissipated energy, the number of equivalent cycles can be computed from the ratio between the energy dissipated per unit of volume during earthquake motion, ΔW, and ΔW₁ as indicated in Eq. 15. The ΔW₁ contribution can be easily expressed as 2πDτγ where D is the damping ratio, τ is shear stress and γ is shear strain. This equation explains the relationship between the number of equivalent cycles and the material hysteresis loops for energy-type equivalence approaches. Further issues related to this topic are elaborated in Green (2001).

$${\text{Neq}} = \frac{\Delta W}{{\Delta W_{1} }} = \frac{\Delta W}{2\pi D\tau \gamma }$$

(15)

Fig. 17

Representation of cycles in a stress–strain plane

Fig. 18

Hysteresis loop quantities (Green 2001)

As done for the previous approach (cf. “Appendix A”), an example of the Rainflow Counting Method is shown in Fig. 19, where the computed cycles and half cycles are identified for the same portion of the accelerometric recording (i.e., HNE horizontal acceleration recording between 1.5 and 5 s for the ATQ station for the Lazio-Abruzzo earthquake). Relative information on the extracted cycles is reported in Table 11 in terms of amplitudes (a_i), means, numbers (N_Rc), extraction start times and cycle periods. The example shows the final equivalent cycle number obtained from Hancock and Bommer’s (2004) procedure, summarized in the last column of the table for the i-th half cycle with the total number shown in the last row (NeqHB).

Fig. 19

Example of the rainflow counting method

Table 11 Parameters of Rainflow cycles extracted from the signal and example of Hancock and Bommer’s (2004) procedure

It is worth noting that in this specific example, the number of cycles derived from Seed et al.’s (1975) method (6.43) is slightly lower than the number obtained using Hancock and Bommer’s (2004) procedure (8.28).