Considered intensity measures

An IM is a single ground motion parameter (scalar IM) or set of ground motion parameters (vector IM), which are representative of the earthquake damage po- tential with respect to a specific class of engineered systems. Typical engineering applications (e.g., performance-based assessment and design) require the choice of an IM which is suitable to predict the response of the system with the smallest scatter (“efficiency”) and providing a significant amount of information, down- grading the effect of other seismological parameters (“sufficiency”) to predict the response quantities involved in the performance objectives. In addition, many researchers have investigated other IM selection criteria, related for example to “hazard computability”, “proficiency”, and “practicality” (Padgett et al., 2008). Conventional IMs, including PGA, PGV, PGD, and (pseudo-) spectral accelera- tion at the initial fundamental period (for a damping ratio of 5%), Sa(T1), are

the most commonly used IMs. In general, PGA and Sa(T1) poorly predict the

structural response of mid-rise to high-rise moment resisting frames, although the latter IM sufficiently captures the elastic behaviour of first-mode dominated MDoF systems, especially in the case of low to moderate fundamental periods (Shome et al., 1998). However, the behaviour of highly nonlinear structures (sensitive to periods greater than T1 due to period lengthening) or structures dominated

by higher-mode periods (less than T1) are not very well represented by utilising

Therefore, it has become essential implementing advanced IMs that account for the elongated periods and/or consider nonlinear demand-dependent structural pa- rameters. Kazantzi and Vamvatsikos (2015) and Kohrangi et al. (2017) among several others have investigated the adequacy of numerous advanced scalar IMs that take into consideration the aforementioned parameters.

For the illustrative application presented here, we then use the advanced scalar IM proposed by Boj´orquez and Iervolino (2011). This IM, denoted as INp, is based

on Sa(T1) and the parameter Np, and is defined as

INp = Sa(T1)N α

p (3.4)

where the parameter α is taken as α = 0.4 based on the tests conducted by the authors and Np is defined as

Np = Sa,avg(T1, ..., TN) Sa(T1) = [ QN i Sa(Ti)] 1/N Sa(T1) (3.5)

TN corresponds to the maximum period of interest and lies within a range of 2

and 2.5T1, as suggested by the authors. In this study, INp is computed for four

different fundamental periods T1: 0.5, 1, 2, and 4 s. For the Np computation,

3 periods are considered: T1, 1.5T1 and 2T1. Figure 3.1 shows an example of

scatter plot for the structural demand in terms of inelastic displacement versus INp for an inelastic SDoF with T1 = 1s, a strength reduction factor (Rµ) equal

to 8 (typical of severely inelastic structures), and a non-degrading elastic-plastic with positive strain-hardening, α = 3%, model (EPH). 121, 2-component, ground motion records from the Northridge earthquake have been used; seeGalasso et al.

(2012) for details. For the considered case, INp outperforms all the conventional

and advanced scalar IMs in terms of all the criteria for optimal IMs.

Integral (i.e., duration-related) IMs, such as the Arias intensity or significant ground motion duration, are commonly used, but they are considered to be related

0 0.5 1 1.5 I Np(T1=1s) [g] 0 0.1 0.2 0.3 0.4 0.5 In el as ti c D is p la ce m ent [m ]

Figure 3.1: Example of scatter plot of the inelastic displacement versus INp for an inelastic SDoF with T1 = 1s, Rµ = 8, and EPH model with, α = 3%

(Northridge earthquake). SeeGalasso et al.(2012) for details.

more to the cyclic energy dissipation rather than to the peak structural response. In fact, some studies (e.g., Iervolino et al., 2006) investigated how ground motion duration-related parameters affect nonlinear structural response and particularly structural collapse (e.g.,Raghunandan and Liel,2013;Chandramohan et al.,2016). It is widely acknowledged that, generally, spectral ordinates are sufficient (i.e., du- ration does not add much information) if one is interested in the ductility demand, while duration-related measures do play a role only if the hysteretic structural re- sponse is to be assessed; i.e., in those cases in which cyclic deterioration and cumulative damage potential of the earthquake are of concern. Chandramohan et al. (2016) highlight the need to consider ground motion duration, in addition to intensity and response spectral-shape, in regions where significant hazard due to long duration shaking exists, such as locations susceptible to large magnitude, subduction zone earthquakes. Finally, integral IMs are also important for several other engineering applications, for example, in geotechnical engineering, such as landslide and liquefaction risk assessment. Therefore, the engineering validation of simulated ground motions in terms of duration-related parameters is also of significant importance.

The term duration is typically used to identify only the portion of a record in which the ground motion amplitude can potentially cause damage to engineering

and geotechnical structures. Several definitions are proposed to this aim; the most commonly used one is the significant duration, introduced byTrifunac and Brady

(1975), defined as the time interval over which the integral of the square of the ground acceleration (Husid plot, Husid, 1969) is within a given range of its total value. Usually, this range is between 5% and 95% (as in this study), denoted as D5−95, or between 5% and 75%.

Finally, Cosenza and Manfredi (1997) introduced the dimensionless ID-factor de-

fined as ID = RtE 0 a 2_{(t) dt} P GAP GV (3.6)

which has proven to be a good proxy for cyclic structural response (Manfredi,

2001). Here, a(t) is the acceleration time-history and tE is the complete duration

of the ground motion (length of the record). Figure 3.2 shows an example of scat- ter plot for the structural demand in terms of equivalent number of cycles (Ne –

i.e., the cumulative hysteretic energy normalised with respect to the largest cycle) versus ID and D5−95for an inelastic SDoF with T1= 1s, a strength reduction factor

(Rµ) equal to 2 (typical of mildly inelastic structures), and a degrading/evolution-

ary model (ESD) comprising a negative strain-hardening (i.e., a softening branch), −α = 10%, and a residual strength equal to 10% of the maximum strength. The simple peak-oriented model is considered to account for the cyclic stiffness degra- dation. Also in this case, 121, 2-component, ground motion records from the Northridge earthquake have been used; see Galasso et al. (2012) for details. For the considered case, D5−95 outperforms other integral IMs (including ID) in terms

of all the criteria for optimal IMs. However, the authors found that this result is dependent on the considered level of nonlinearity, with ID outperforming the other

integral IMs in the case of severely inelastic structures (i.e., Rµ ≥ 4). Therefore,

both metrics are kept in our validation exercise.

0 5 10 15 20 25 30 I_D 0 3 6 9 12 15 N e 0 5 10 15 20 25 30 35 40 D_5-95 [s] 0 3 6 9 12 15 N e

Figure 3.2: Example of scatter plot of the equivalent number of cycles versus ID (left) and D5−95 (right) for an inelastic SDoF with T1 = 1s, Rµ = 2, and ESD model with, α = 10% (Northridge earthquake). See Galasso et al. (2012)

for details.

inDreger et al. (2015) was to validate elastic spectral response by using the BBP v14.3. The parameters proposed in our study - as well as those introduced in

Burks and Baker(2014)- are intended as a supplement, not a replacement, to that validation. It is understood that many other metrics would be necessary to fully assess the simulation methods’ ability to produce reasonable ground motions as a whole. An important property of the proposed validation parameters is that they are hazard computable, i.e., empirical models or GMPEs exist (e.g., for ID

and D5−95 see Iervolino, Giorgio, Galasso and Manfredi, 2010) or may be easily

derived (e.g., for INp Boj´orquez and Iervolino, 2011) combining existing tools and

can be used as a baseline comparison against simulations for a very broad range of conditions, including future earthquake scenarios.