Caracterización y análisis de los anillos de crecimiento

Incremental dynamic analysis is a procedure to semi-empirically estimate probabilistic seismic structural demand and capacity. This well-established procedure, typically entails a non-linear numerical model of the structure which is subjected to a suite of ground motion records, all scaled at a common IM level. This IM level is gradually increased by applying a common scale factor simultaneously to all the records, in order to reveal the entire range of post-yield response of the structure, conditional to several IM values, up to global dynamic instability and consequent collapse.

During IDA, structural response to a single record is usually represented by plotting two scalars against each other: a ground motion intensity measure characterizing the various scaled incarnations of the record and an engineering demand parameter (EDP) characterizing the amplitude of response. EDP is usually selected to be some measure of local or global structural deformation (e.g., maximum roof displacement or maximum interstory drift for a frame structure). The ground motion IM should be monotonically scalable and should ideally possess some further desirable properties, such as sufficiency, efficiency and scaling robustness (see Luco and Cornell, 2007). Commonly used IMs are PGA and 5% damped, first mode period spectral acceleration S T ,5% . _a





By plotting EDP responses to the various scaled versions of a single record on the abscissa and corresponding IM level on the vertical axis, one obtains a single record IDA curve. IDA curves start with a linear segment corresponding to elastic response and then evolve into, generally speaking, non-monotonic functions of ground motion IM. An IDA curve eventually culminates into a flat-line, a horizontal segment of continuously increasing EDP at constant IM level, signifying the onset of global dynamic instability.

Once a set of IDA curves has been collected, representing the entire suite of ground motions, it is an efficient practice to summarize the curves into sample fractile statistics. Typically sample medians, 16% and 84% fractiles are calculated; employing these particular statistics to obtain summary IDA curves has certain advantages:

PULSE-LIKE SEISMIC DEMAND FOR MULTI-LINEAR BACKBONE OSCILLATORS

75

 fractiles are invariant with respect to monotonic one-on-one transformations of the variables (see Benjamin and Cornell, 1970).

 these fractile values fit well with the common assumption that the conditional distribution of EDP given IM can be represented by a lognormal distribution, when 16% and 84% would be one standard deviation of the logs distant from the median.

 once a certain portion of the records begin to collapse the structure, other statistics such as sample mean, become impossible to calculate, while counted sample fractiles are still a valid option (see Shome and Cornell, 2000).

While single IDA curves may be non-monotonic and even discontinuous, summary fractile IDA curves are usually better-behaved, being monotonic and continuous more often than not. For more details on the intricacies of this method, the interested reader is referred to Vamvatsikos and Cornell (2002 and 2004). Given that the structural model should ideally be sufficiently complex so as to be able to represent the full repertoire of non-linear responses and eventual failure mechanisms and the suite of records large enough to account for the inherent variability of seismic loading, it is fair to say that IDA can be a computationally intensive procedure. This fact motivated Vamvatsikos and Cornell (2006) to develop a software tool which provides a shortcut, at the cost of introducing some approximation in the process. Having observed that summary IDA curves of SDOF systems with multi- linear backbone curves exhibit a consistent behavior in correspondence with each segment of the backbone (elastic, post-yield hardening, post-cap softening and residual strength segments, the first three represented in Figure 5.1), they used IDA to investigate the response of a large population of oscillators with varying backbone parameters.

Having thus mapped the behavior of many backbone shapes against a suite or ordinary ground motions, not affected by directivity, they proposed an intricate analytical model, aptly named the SPO2IDA tool, capable or reproducing the IDA curves of these SDOF systems without having to run any analysis. Taking into consideration the well-established methodologies that allow studying the inelastic response of first-mode dominated MDOF systems by means of a substitute SDOF approximation (which were discussed in some length in Chapter 3), it becomes clear that SPO2IDA is essentially nothing less than a complex R-μ-T relation. What sets SPO2IDA apart from the more traditional R-μ-T relations, its complexity

PULSE-LIKE SEISMIC DEMAND FOR MULTI-LINEAR BACKBONE OSCILLATORS

76

notwithstanding, is the fact that it also provides information on the dispersion of seismic response around the central value.

A sample application of SPO2IDA can be seen in Figure 5.2, which shows estimated fractile IDA curves for an oscillator with natural period of vibration T=1.0s, plotted over its backbone curve (SPO2IDA tool available online at the time of writing at

http://users.ntua.gr/divamva/software/spo2ida-allt.xls , last accessed on the ides of

March, 2015).

Figure 5.2. SPO2IDA estimates of the 16%, 50% and 84% fractile IDA curves for an oscillator with natural period of vibration T=1.0s, superimposed over the oscillator’s trilinear backbone curve. The backbone is defined by hardening post-yield slope 20% of the elastic, capping ductility at  _c 3.0

and a softening branch with descending slope of -200%.

The objective of the study presented in this chapter, is to follow in the footsteps of Vamvatsikos and Cornell (2006) and employ IDA on SDOF systems using the set of pulse-like records assembled in Chapter 4 (Table B.1 of Appendix B) in order to develop the equivalent of an R-μ-T T_p relation appropriate for NS FD ground motions, which also takes the shape parameters of a trilinear backbone curve into account.

PULSE-LIKE SEISMIC DEMAND FOR MULTI-LINEAR BACKBONE OSCILLATORS

77

5.3 MODELLING NEAR-SOURCE PULSE-LIKE SEISMIC DEMAND