# NONLINEAR ELASTIC RESPONSE TO MONITOR STRUCTURAL DAMAGE IN BUILDINGS OF DIFFERENT TYPOLOGIES

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## Building response and engineering demand parameters

The parameters related to building response (Table 1.3) are PTA, PTV, PTD and ∆.
In the NDE1.0 flat-file, the total drift ratio ∆ reported in Table 1.3 corresponds to the maximum value computed in four ways:
– Maximum relative displacement between the top and bottom
– Relative displacement computed between the maximum values at the top (PTD) and bottom (PGD)
– Relative displacement at the time of the PTD.
– Relative displacement between top and bottom at the time corresponding to the co-seismic frequency (i.e. fmin).
The NDE1.0 flat-file provides the pre-seismic fi and co-seismic fmin fundamental frequency values required to compute the spectral IM and to analyze nonlinear elastic processes in the following chapters. Values of fi and fmin were computed as explained in the section 1.3.

Intensity of ground motion

The NDE1.0 flat-file includes several ground motion intensity parameters, classified as ordinary and spectral intensity measures (Tables 1.4 and 1.5, respectively). In addition to PGA, PGV and PGD, Arias Intensity (i.e. Ia, Arias, 1970), Destructive Potential (DP, Araya and Saragoni, 1984), and Cumulative Absolute Velocity (i.e. CAV, EPRI, 1988) are computed from the acceleration time histories a(t). Ia is an energy-based parameter that includes both amplitude and duration of the seismic shaking. It is often linked to the cumulative damage experienced by a structure, where damage is considered to be proportional to the energy dissipated per unit weight during the overall duration of the motion. Ia is defined as:
where g is the acceleration due to gravity and tf is the total duration of the recording. DP is a modification of Ia, where the frequency content of the earthquake is considered, as follows:
where v02 is the number of zero crossings per unit of time. CAV is assumed to reflect the damaging potential of seismic loading. CAV is given by:
In Table 1.5, spectral value-based IMs are provided for a 5% damping ratio. The frequency information suggests that spectral values should be more closely related to damage potential than peak values. We use the algorithms given by Papazafeiropoulos (2015), based on Newmark and Hall (1982) to generate response or pseudo-response spectra values, considering both frequencies: pre-seismic (Sa1, Sv1 and Sd1) and co-seismic (Sa2, Sv2 and Sd2) frequencies for acceleration, velocity and displacement, respectively. In order to take into consideration the frequency shift during the seismic loading, the NDE1.0 flat-file also includes the mean spectral values computed between fmin and fi (Avg_Sa, Avg_Sv and Avg_Sd). This approach has been used by Bommer et al. (2004) and Perrault and Guéguen (2015) to take into account the co-seismic nonlinear response of buildings, also reducing the uncertainties in the prediction of EDP (Perrault and Guéguen, 2015).

### Strong motion duration

Ground motion duration is often considered as a key parameter determining structural damage (i.e. Bommer and Martínez-Pereira, 1999; Araya and Saragoni, 1980). However, most recent analyses related to duration as a damage predictor are based on numerical simulations rather than experimental data (i.e. Chandramohan et al. 2016; Barbosa et al. 2017). The seminal definition of strong motion duration was proposed by Trifunac and Brady (1975), who consider that the significant duration is achieved at 95% of Ia. In this study, we compute different strong motion durations, summarized in Table 1.6, and defined as follows:
– Bracketed duration, DB (Fig. 6a): total time between the first and the last exceedance of a specific acceleration threshold (i.e. a0). Four acceleration thresholds are defined: 0.05g (i.e. DB1), 0.1g (i.e. DB2), 0.15g (DB3), and 0.20g (DB4).
– Effective duration, DE (Fig. 6b): defined by DE = tf – t0, where t0 corresponds to the time at which 0.01m/s of cumulative energy is reached in the Husid diagram (i.e. energy build-up plot, Ia) and tf corresponds to the time at which Ia=0.125 m/s.
– Uniform duration, DU: sum of the time intervals during which acceleration exceeds a specific acceleration threshold. Four acceleration thresholds are considered in the flat-file: 0.05g (i.e. DU1), 0.1g (i.e. DU2), 0.15g (i.e. DU3), and 0.20g (i.e. DU4).
– Significant duration, DS (Fig. 6b): defined as the time interval over which a specific percentage of total energy is accumulated on the Husid diagram. Intervals corresponding to (5-75)% and (5-95)% of total energy are considered, indicated as DSa1 and DSa2, respectively. Durations based on cumulative energy computed from velocity (i.e. DSv1 and DSv2) and displacement signals (i.e. DSd1 and DSd2) are also determined, as suggested by Trifunac and Brady (1975).
Figure 1.6 Schematic view of several duration definitions given for two different acceleration time histories. a) Bracketed duration. The horizontal dashed line corresponds to the threshold level of acceleration. b) Significant strong motion duration computed for (5-75) % and (5-95) % of total energy based on the Husid diagram.

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#### Empirical prediction of building response P(EDP|IM) and associated uncertainties

One key step in the PBEE framework is the prediction of EDP for a given IM (i.e. P(EDP|IM)). These relationships are based on statistical regressions between IM and EDP. EDP follows a lognormal distribution for a uniform distribution of ground motion parameters (Perrault and Guéguen, 2015) and therefore a log-linear functional form is considered to estimate the value of EDP | IM. The functional form proposed corresponds to a first-degree polynomial written as:
log(∆) = a + b log(IM) + ε (1.10)
where a and b are coefficients obtained by linear regression, and e is the standard error used to determine the efficiency of the IM in predicting ∆ (Shome and Cornell, 1999; Luco, 2002).
In Figure 1.7, the functional form is applied to estimate ∆ variability of the buildings as a function of PGA and PGV for the Japanese (1.7a) and US (1.7b) data. We observe that it is PGV and not PGA that gives the smallest EDP variability ( .346 and = 0.437 for the Japanese and US data, respectively; compared with PGA values=of0 = 0.518 and = 0.518 for Japanese and US data, respectively). This has also been reported by several previous studies in different contexts (i.e. Wald et al. 1999, Bommer and Alarcon, 2008; Akkar and Bommer, 2007; Lesueur et al., 2013; Perrault and Guéguen, 2015). This observation does not depend on building design.

1. NDE1.0 – A NEW DATABASE OF EARTHQUAKE DATA RECORDINGS FROM BUILDINGS FOR ENGINEERING APPLICATIONS
1.1. INTRODUCTION
1.2. DATA DESCRIPTION
1.3. DATA PROCESSING: THE WIGNER-VILLE DISTRIBUTION
1.4. STRUCTURE OF THE NDE1.0 FLAT-FILE
1.4.1. Building and earthquake characteristics
1.4.2. Building response and engineering demand parameters
1.4.3. Intensity of ground motion
1.4.4. Strong motion duration
1.5. EMPIRICAL PREDICTION OF BUILDING RESPONSE P(EDP|IM) AND ASSOCIATED UNCERTAINTIES
1.6. UNCERTAINTIES RELATED TO BUILDING-SPECIFIC DAMAGE PREDICTION
1.7. SEISMIC VULNERABILITY ASSESSMENT AND PERFORMANCE-BASED ANALYSIS
1.8. CONCLUSIONS
2. NONLINEAR ELASTICITY OBSERVED IN BUILDINGS: THE CASE OF THE ANX BUILDING (JAPAN)
2.1. INTRODUCTION
2.2. DESCRIPTION OF THE ANX BUILDING AND DATA
2.3. DATA PROCESSING
2.4. VARIATION OF THE RESONANCE FREQUENCY
2.5. DISCUSSION OF THE ORIGIN OF THE NONLINEARITIES
2.6. OBSERVATIONS IN ANOTHER BUILDING: THE CASE OF THE THU BUILDING
2.7. CONCLUSIONS
3. SLOW DYNAMICS (RECOVERY) USED AS A PROXY FOR SEISMIC STRUCTURAL HEALTH MONITORING
3.1. INTRODUCTION
3.2. METHODOLOGY
3.3. RELAXATION MODELS
3.3.1. Relaxation function
3.3.2. Relaxation time spectrum
3.4. EVOLUTION OF RELAXATION PARAMETERS OVER TIME
3.5. RELAXATION PARAMETERS AND STRUCTURAL DAMAGE
3.6. THE MULTI-SCALE FEATURE OF FREQUENCY RECOVERY IN BUILDINGS
3.6.1. Backbone recovery curve and hysteresis during aftershocks
3.6.2. Relaxation models applied to long-term structural recovery
3.7. CONCLUSIONS
4. NONLINEAR ELASTIC RESPONSE TO MONITOR STRUCTURAL DAMAGE IN BUILDINGS OF DIFFERENT TYPOLOGIES
4.1. INTRODUCTION
4.2. FREQUENCY VARIATIONS OVER TIME
4.3. EVOLUTION OF RELAXATION PARAMETERS