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Period lengthening, exhibited by structures when subjected to strong ground motions, constitutes an implicit proxy of structural inelasticity and associated damage. However, the reliable prediction of the inelastic period is tedious and multi-parametric task that is related to both epistemic and aleatory uncertainty. Along these lines, the objective of this paper is to investigate and quantify the elongated fundamental period of reinforced concrete structures using inelastic response spectra defined on the basis of the period shift ratio (Tin/Tel). Nonlinear oscillators of varying yield strength (expressed by the force reduction factor, Ry), post-yield stiffness (ay) and hysteretic laws are examined for a large number of strong motions. Constant-strength, inelastic spectra in terms of Tin/Tel are calculated to assess the extent of period elongation for various levels of structural inelasticity. Moreover, the influence that structural characteristics (Ry, ay and degrading level) and strong-motion parameters (epicentral distance, frequency content and duration) exert on period lengthening are studied. Determined by regression analyses of the data obtained, simplified equations are proposed for period lengthening as a function of Ry and Tel. These equations may be used in the framework of the earthquake record selection and scaling.

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... Only in extreme cases of severely degrading buildings a ratio of 2.0 was estimated. Katsanos and Sextos [10] proposed an empirical function for period elongation based on the structural period and the force-reduction factor. Moreover, they showed that Peak Ground Acceleration (PGA) has a low correlation with the predominant inelastic period in contrast to spectral acceleration. ...

... Using the total drifts in Table 3 7 shows the ratio of the degraded stiffness of the prismatic structural elements of the models for the C1M low-code building in damage state for ΔΤ1 equal to 20% and 40%. It is noted that short-period structures (< 0.50 s) may demonstrate higher period elongation than long-period structures with equal force reduction factor [10]. Therefore, the same amount of period elongation may correspond to different damage levels for different structures. ...

... It is also observed that the probability of damage increases even for aftershocks with a PGA less than half the predicted PGA for the main shock, although according to other studies the effect of such aftershocks could be ignored [54]. Moreover, the assumption that period elongation of 60 % corresponds to a state near collapse in the case of the mid-rise models is in agreement with the empirical estimations by Katsanos and Sextos [10] of the period elongation of structures with an elastic period of 0.60 s and a force-reduction factor of 3. ...

Safety assessment of structures and/or critical infrastructures is a key factor in post-seismic decision-making. In this context we present a performance-based framework for modeling time-variant vulnerability of reinforced concrete buildings during aftershock sequences. Structural damage is associated with first eigenperiod elongation, a performance metric whose measurement can complement visual inspection and assessment of structural health as a post-seismic operative tool. The proposed framework is applied for a series of reinforced concrete building models and two aftershock sequences. Damage states are defined using thresholds of period elongation. Numerical models of the buildings in each damage state are considered and their fragility curves are computed. The time-variant vulnerability is modeled with Markov chain as a function of the characteristics of the aftershocks sequence. Finally, the probabilities of the damage states are computed as a function of time during two real aftershock sequences.

... The increase in the fundamental period is highly dependent on the earthquake characteristics and the level of nonlinearities in the system. The period elongation ratio has been observed to be proportional to the stiffness and the force reduction 24 factor (ratio of the maximum seismic force to yield force), while the effect of soil conditions has been estimated to be of minor significance for buildings with large elastic periods (Katsanos and Sextos 2015). Thus, period elongation in the structures is generally associated to the nonlinear behavior of the structure regardless of the dynamic behavior of the soils. ...

... ASCE 7-16 (2017b) and Baker (2011) recommend to take into account the amount of first-mode period elongation caused by inelastic response effects when creating a scenario spectrum to fully capture the structure's response. Katsanos and Sextos (2015) evaluated nonlinear systems under strong motions to assess the period elongation due to the inelasticity of the structural materials. The authors found elongation of the fundamental period larger than 100%. ...

... However, this elongation tends to be larger for buildings with elastic natural period less than 1.0 s. 25 Figure 10. Period shift ratio ( / ) based on the hysteretic degradation of the structural system with reduction factors of 3.5 and 5 (Katsanos and Sextos 2015). Hans et al. (2005) performed in situ experiments and seismic analysis including SSI effects to find period elongation; however, period lengthening effects due to the flexibility of the support conditions could not be identified in the analyses. ...

Soil-structure interaction (SSI) effects are relevant for the seismic analysis of tall buildings on shallow foundations since the dynamic behavior of structures is highly affected by the interaction between the superstructure and supporting soils. As part of earthquake-resistant designs of buildings, considering SSI effects in the analysis provides more realistic estimates of its performance during a seismic event, particularly when both the structure and soil undergo large demands that can compromise serviceability. Oversimplifications of structural or soil modeling in the analysis introduces variability and biases in the computed seismic response. The main goal of this dissertation is to investigate the interaction between archetype tall buildings and its supporting soils using numerical simulations. This dissertation develops the following objectives: i) to estimate the differences in the seismic performance of archetype tall building under different SSI approaches and compared to idealized fixed-base conditions; ii) to evaluate the seismic performance of tall building models by estimating intensity measures and engineering demand parameters (EDPs); iii) to assess the influence of SSI in the earthquake-induced losses of the structures; and iv) to evaluate the interaction of soil-structure systems using nonlinear constitutive models. To achieve these goals, numerical models of linear-elastic and nonlinear-inelastic tall buildings supported on mat foundation, combined with either fixed-base conditions at ground level or an explicit soil domain, are subjected to different earthquake time histories. The influence of SSI is quantified using structural and soil demands. It is concluded that the seismic response of tall buildings is largely affected by the inclusion of SSI effects when compared to conventional fixed-base structure models. SSI changed the computed seismic demands of the tall buildings in terms of inter-story drifts, peak horizontal accelerations, seismic-induced settlements, and losses compared to idealized buildings with fixed-base conditions. Nonlinear analyses show a significant decrease of EDPs when compared to those demands obtained with linear models. Energy distribution among both supporting soils and structure vary significantly as EDPs induce stresses and strains in the building beyond the onset of structural yielding. SSI impacts the structural and soil behavior and has practical implications in seismic resistant designs.

... Other researchers have evaluated the lengthening of the vibration period based on the measured response of instrumented buildings and bridges during ground motions [14,[25][26][27] and by experimentally investigating full scale RC structures [28][29][30][31]. Recently, Katsanos and Sextos [32] investigated the effects of structural and ground motions parameters on the elongated fundamental period of RC structures modeled as nonlinear SDOF systems with deteriorating properties. They also proposed an empirical equation to estimate the period elongation as a function of elastic vibration period and strength reduction factor. ...

... In previous studies, e.g. [32], it has been shown that the post-yield stiffness ratio does not have a significant effect on the period elongation. Hence, the α parameter is a constant value and is set equal to zero for all the analyzed cases. ...

... The values of shown by the circles in Fig. 2 are T in . This technique also has been used widely in previous studies [13,32,33] to identify the elongation of the first mode period. The flowchart shown in Fig. 3 is used for the calculation of constant ductility spectra based on the period shift ratio. ...

The elongation of the fundamental period of structure is a typical indicator of structural inelasticity during seismic events and of consequent damage accumulation. This study investigates the lengthening of the fundamental periods of reinforced concrete (RC) structures subjected to mainshock-aftershock sequences. RC structures are schematized herein as nonlinear SDOF (Single Degree Of Freedom) systems with deteriorating properties. Inelas-tic response spectra are developed considering both ground motion characteristics (e.g. site condition, epicentral distance, PGAs ratio of aftershock to mainshock, and duration) and structural properties, such as ductility, softening , pinching, accumulated damage and unloading stiffness. Furthermore, analytical equations estimating the elongated fundamental period are proposed as a function of the elastic (undamaged) vibration period and the significant structural parameters. The proposed equations can be reliably used for the design of structures against mainshock-aftershock sequences in future generations of seismic design codes.

... Only in extreme cases of severely degrading buildings a ratio of 2.0 was estimated. Katsanos and Sextos [10] also proposed an empirical function for period elongation based on the structural period and the force-reduction factor. Moreover, they showed that PGA has a low correlation with the predominant inelastic period in contrast to spectral acceleration. ...

... According to these relationships the period elongation corresponding to EMS-98 [12] damage grade 1, 2 and 3-4 is equal to 20 %, 43 % and 65 % and is independent of the number of storeys. The period elongation ratio has been observed to be in an analogy to the stiffness and the force-reduction factor (ratio of the maximum seismic force to yield force) of SDOF oscillators [10] while the effect of earthquake magnitude, distance and soil conditions has been estimated to be of minor significance. ...

... The probability of damage state exceedance in the numerical results for the case studies of the aftershock sequences of the Thessaloniki (1978) and the Northridge (1994) follows a similar trend to the failure probability during the aftershock sequence period as estimated by Iervolino et al. [5]. Moreover, the assumption that period elongation of ΔΤ = 60 % corresponds to a state near collapse is in agreement with the empirical estimations by Katsanos and Sextos [10]. ...

... An elastic period of T 1 = 1 s is selected, which is typically used as representative fundamental period of mid-rise buildings. Note that the aforementioned period range is chosen based on (ASCE 2010) provisions and recent literature recommendations (Katsanos and Sextos 2015). Establishing a match in the period range below 1 s will not have any impact on the results because higher modes do not contribute to the SDOF system response, however, the match was established in this range for code consistency. ...

... Note that for large values of R, the nonlinear structural response is sensitive to spectral ordinates at periods much larger than the fundamental one Fig. 11 Dispersion (expressed through coefficient of variation) of hysteretic energy for EPH system with α = 3% for "constant-R" approach (e.g., due to period elongation stemming from the strong nonlinear behavior); the chosen period range for spectral compatibility (i.e., 0.2T 1 -1.5T 1 ) may not be conservative in those cases (Katsanos and Sextos 2015), yielding the observed large variability for larger values of R. ...

An important consideration for the adoption of stochastic ground motion models in performance-based earthquake engineering applications is that the probability distribution of target intensity measures from the developed suites of time-histories is compatible with the prescribed hazard at the site and structure of interest. The authors have recently developed a computationally efficient framework to modify existing stochastic ground motion models to facilitate such a compatibility. This paper extends this effort through a validation study by comparing the seismic demand of recorded ground motions to the demand of stochastic ground motion models established through the proposed modification. Suites of recorded and stochastic ground motions, whose spectral acceleration statistics match the mean and variance of target spectra within a period range of interest, are utilized as input to perform response history analysis of inelastic single-degree-of-freedom (SDoF) case-study systems. SDoF systems with peak-oriented hysteretic behavior, strain hardening, and (potentially) degrading characteristics, experiencing different degree of inelastic response, are considered. Response is evaluated using the peak inelastic displacement and the hysteretic energy given by the work of the SDoF restoring force as engineering demand parameters (EDPs). The resultant EDP distributions are compared to assess the effect of (and validate) the proposed modification. It is shown that the proposed modification of stochastic ground motion models can provide results that are similar to these from recorded ground motion suites, improving any (in some cases large) discrepancies that exist for the initial, unmodified stochastic ground motion model.

... The estimation of the apparent period of a non-linear system has been studied extensively from methodologies oriented to design or from systems identification techniques. Following the proposal by Katsanos and Sextos (2015) [1], it is possible to analyze spectrally the expected elongation of the apparent period based on the variables that define the hysteretic response of the system and the seismic demand. In this article a critical revision of this approach is made, using instead of identification in space-frequency, tools of identification and equivalent linearization that grant an effective representativeness in terms of response adjustment of relative accelerations. ...

... A pesar de que las estructuras en general desarrollan propiedades no-lineales, desde finales de años sesenta hasta entonces se han llevado a cabo varios estudios de tipo experimental de monitoreo de propiedades dinámicas en base a técnicas de identificación de espacio-frecuencia que asumen una pseudo-linealización del sistema estructural [1][2][3][4][6][7][8][9], permitiendo la estimación de un periodo fundamental aparente (o inelástico) del sistema. El ejemplo más directo es el uso de espectros de respuesta en frecuencia y funciones de transferencia empíricas, aplicadas a la respuesta de estructuras reales ante movimientos sísmicos. ...

Resumen La estimación del período aparente de un sistema no-lineal ha sido estudiada extensamente a partir de metodologías orientadas al diseño o a partir de técnicas de identificación de sistemas. Siguiendo lo propuesto por Katsanos y Sextos en el 2015 [2], es posible reflejar de manera espectral la elongación esperada del período aparente en función de las variables que definen la respuesta histerética del sistema y la demanda sísmica. En el presente artículo se hace una revisión crítica de este enfoque, utilizando en lugar de identificación en espacio-frecuencia, herramientas de identificación y linealización equivalente que otorgan una efectiva representatividad en términos de ajuste de respuesta de aceleraciones relativas. Para llevar a cabo este análisis, se toma en consideración una base de datos de 300 registros sísmicos y los modelos histeréticos de un grado de libertad estudiados por Katsanos y Sextos. Se utiliza OpenSEES para la obtención de la respuesta de los sistemas no-lineales tratados y la construcción de espectros de respuesta. Finalmente, se comparan los resultados obtenidos a partir de metodologías de identificación en espacio-frecuencia y linealización equivalente. Se observan diversos problemas en la identificación en espacio-frecuencia aplicada a sistemas de respuesta no-lineal de grandes amplitudes. Se propone una mejora para la obtención de espectros de elongación del período aparente mediante el uso de métodos de identificación basados en análisis bayesiano para el ajuste de sistemas lineales-equivalentes. Palabras-Clave: Elongación del periodo, identificación de sistemas, linealización equivalente, análisis bayesiano, respuesta de sistemas no lineales.

... The well-known Idarc program [34] is used in the modelling of the frames to obtain the story capacity curves and structural responses in the later sections with the beamcolumn model. Idarc has been widely used by many researchers in inelastic structural damage analysis [35][36][37]. The OpenSees program is used to build the idealized shearbuilding models and obtain the structural responses. ...

... The actual story capacity curves of structure can be obtained by two methods: (1) incremental dynamic analysis (IDA) method, through which the story capacity curves can be obtained by increasing the PGA of ground motion records as input to the structures, the maximum story shear and maximum story ratio are obtained during the time history analysis, thus producing story shear force versus inter-story drift ratio (IDR) curves; and (2) the pushover method, through which the story shear versus IDR curves can be obtained by monotonically The well-known Idarc program [34] is used in the modelling of the frames to obtain the story capacity curves and structural responses in the later sections with the beamcolumn model. Idarc has been widely used by many researchers in inelastic structural damage analysis [35][36][37]. The OpenSees program is used to build the idealized shearbuilding models and obtain the structural responses. ...

The paper aims to investigate the accuracies of idealization methods of the well-known shear-building models. Five idealization methods are adopted to idealize the structural story capacity curve within the range from zero to the deformation corresponding to the peak shear point. After the peak shear point, a skew branch followed by a constant branch are used to approximate the capacity curve. The five idealization methods are verified by using four reinforcement concrete (RC) frames with 3, 8, 12, and 18 stories. Results reveal that all the five idealization methods may cause remarkable errors in prediction of the period, displacements and accelerations of the actual buildings. The errors of the structural period by the five idealization methods are almost above 10–40%. The errors of the structural displacements and accelerations by the five idealization methods are almost above 30–90%. For all the five idealization methods, the prediction accuracy on displacement and acceleration will be dramatically increased if the comparison is only focused on the maximum value within all story rather than the maximum values of each story. The initial stiffness method provides the best predictions on periods of the actual buildings. The farthest point method provides better prediction than the other four idealization methods.

... Monitoring the elongation of the fundamental period of buildings can help assess earthquake damage in buildings whose stiffness gradually degrades before failure [11]. The fundamental period (or frequency) is assumed to be a proxy for the apparent structural stiffness and structural health [12][13][14][15][16]. For instance, the residual stiffness of masonry buildings from period measurements has been used to study the effect of seismic damage accumulation on a macro-seismic intensity assessment [12]. ...

... Katsanos et al. [15] showed that the transient period elongation during a seismic response did not exceed 1.2 and 1.7 for a designed earthquake and twice the designed earthquake, respectively. Katsanos and Sextos [16] also showed the sufficiency (i.e., independence from magnitude and distance) of the structural period elongation in damage prediction using single-degree-of-freedom oscillators. Reuland et al. [11] predicted fragility curves for subsequent earthquakes based on a measured postearthquake structural frequency and a visual inspection. ...

Rapid post-earthquake damage assessment is critical to short-term earthquake crisis management. Reinforced concrete buildings may accumulate damage during an aftershock sequence, and short-term damage forecasts after the mainshock can aid in decision-making (in particular, on whether to allow immediate occupancy) before further damage actually occurs. This paper presents an operative damage forecasting and building tagging procedure for reinforced concrete buildings during synthetic aftershock sequences near Thessaloniki, Greece, for two hypothetical earthquake scenarios. The synthetic aftershock sequences are simulated, and the time-variant seismic vulnerability is modeled based on fragility curves for the damage state thresholds in terms of period elongation. Period elongation is chosen as a damage proxy because it is available for rapid damage assessment in buildings with permanent monitoring systems or for city-scale post-earthquake surveys. Time-variable damage state probabilities owing to aftershocks are estimated, and a building tagging scheme is proposed based on a traffic-light concept (red-orange-green) to assist in seismic crisis management during aftershock sequences.

... It has been pointed out that the stiffer the structure, the larger the period lengthening. Accordingly, for structures with short vibration periods, we adopt T N = 2.0T 1 , which agrees with recommendations made by Bianchini et al. (2009), Katsanos and Sextos (2015), and Tsantaki et al. (2017), for relatively stiff structures, and assuming a ductility demand between 2 and 3. ...

... At short to moderate vibration periods, the structural period lengthening diminishes somewhat linearly until it reaches a semi-constant behavior (which is independent of the level of nonlinearity developed by the structure) (Katsanos and Sextos, 2015). In this regard, Di Sarno and Amiri (2019) quantified the fundamental period lengthening of structures by the ratio of response spectra corresponding to the lengthened and the elastic structural vibration period (T in /T el ). ...

For earthquake-resistant design, structural degradation is considered using traditional strength modification factors, which are obtained via the ratio of the nonlinear seismic response of degrading and non-degrading structural single-degree-of-freedom (SDOF) systems. In this paper, with the aim to avoid the nonlinear seismic response to compute strength modification factors, a methodology based on probabilistic seismic hazard analyses (PSHAs), is proposed in order to obtain strength modification factors of design spectra which consider structural degradation through the spectral-shape intensity measure INp. PSHAs using INp to account for structural degradation and Sa(T1), which represents the spectral acceleration associated with the fundamental period and does not consider such degradation, are performed. The ratio of the uniform hazard spectra in terms of INp and Sa(T1), which represent the response of degrading and non-degrading systems, provides new strength modification factors without the need to develop nonlinear time history analysis. A mathematical expression is fitted to the ratios that correspond to systems located in different soil types. The expression is validated by comparing the results with those derived from nonlinear time history analyses of structural systems.

... e degenerate trilinear model is determined using parameters such as fracture load P c , yield load P y , elastic stiffness, fracture stiffness, and postyield stiffness. e stiffness of the model during unloading remains unchanged, which is always the tangential stiffness between the yield point and the displacement origin [16]. e degenerate trilinear model is more reasonable and accurate in simulating the variation in internal forces and displacements in reinforced concrete structural components in an elastoplastic state during large earthquakes. ...

This study aimed to analyze the formation and application of the time-domain elastoplastic response spectrum. The elastoplastic response spectrum in the time domain was computed according to the trilinear force-restoring model. The time-domain elastoplastic response spectrum corresponded to a specific yield strength coefficient, fracture stiffness, and yield stiffness. However, the force-restoring models corresponding to different structural systems and the states of the structural systems at different moments were not the same. Therefore, the dynamic characteristics of a particular periodic point corresponding to a particular structure were meaningful for the elastoplastic response spectrum. In addition, the curve in the time-domain dimension along the periodic point truly reflected the real-time response of the structure when the structure encountered a seismic load.

... Note that for large values of R, the nonlinear structural response is sensitive to spectral ordinates at periods much larger than the fundamental one (e.g., due to period elongation stemming from the strong nonlinear behavior); the chosen period range for spectral compatibility (i.e., 0.2T1-1.5T1) may not be conservative in those cases (Katsanos and Sextos 2015), yielding the observed large variability for larger values of R. ...

An important consideration for the adoption of stochastic ground motion models in performance-based earthquake engineering applications is that the probability distribution of target intensity measures from the developed suites of time-histories is compatible with the prescribed hazard at the site and structure of interest. The authors have recently developed a computationally efficient framework to modify existing stochastic ground motion models to facilitate such a compatibility. For a given seismicity scenario, the framework identifies the modified stochastic ground motion model that can sufficiently match the prescribed hazard while maintaining similarity to regional physical ground motion model characteristics. This paper extends this effort through a validation study. Suites of recorded and stochastic ground motions, whose spectral acceleration statistics match the mean and variance of target spectra within a period range of interest, are utilized as input to perform response history analysis of inelastic single-degree-of-freedom case-study systems. The resultant engineering demand parameters distributions are then compared to assess the effect of the proposed modification.

... Such a normalization scheme, suggested by Miranda [20] and widely used elsewhere (e.g., [21,22]), is expected to provide an accurate characterization of the inelastic deformation demands for simplified structures of varying vibration periods when they are founded on soft soil conditions. Based on Fig. 1 (left), the residual displacement ratios were found to be affected by the lateral strength ratio and lower mean C r values were calculated for the stronger systems, i.e., R y < 4, compared to the weaker ones, the latter being representative of frame-resisting multistory buildings, systems with limited seismic capacity or irregular structures [23]. Such a trend, already observed by relevant studies being, though, associated with firm soil sites [4,8], reveals the higher susceptibility of the aforementioned structural configurations to increased inelastic response and, hence, extensive earthquake-induced damages [7,12]. ...

120 earthquake ground motions recorded on soft soil sites were employed to assess, through response history analysis of simplified systems, the residual displacement demand, Cr, defined as the ratio of the residual displacement to the maximum elastic displacement. Single degree of freedom systems were considered and the lateral strength ratio was parametrized to account for varying structural inelasticity. Four hysteretic laws were chosen to represent degrading and non-degrading performance while variation in the post-yield stiffness was considered. The analysis scheme enabled assessing the relationship of the residual displacement with the aforementioned structural characteristics. The residual displacement demand was found to be sensitive in the post-yield stiffness ratio. An equation was finally introduced to accommodate the reliable estimation of residual displacement ratio, the latter being beneficial for evaluating the seismic performance of existing structures built on soft soil sites.

... This observation is likely due to the inappropriate selection of lengthened period, which is simply chosen as the twice of the fundamental period of structure. If a procedure like the one proposed by Katsanos and Sextos [51] is utilized to predict this extended period, the better results in terms of small dispersion in the structural responses is expected. Fig. 8 explains that SI is the most efficient IM for predicting the structural responses in the low-and medium-rise buildings with the record-to-record variability less than 0.27. ...

Although several research studies have examined some effects of ground motion (GM) duration on the structural responses, many questions in this field remain unexplored and unaddressed. One area that remains a topic of debate in this field is that no outcomes can be found with regard to the effect of GM duration on structural responses when considering different choices of conditioning intensity measures (IMs). This study examines the role of the conditioning IM in the degree that GM duration influences the structural responses. To this end, the seismic demand in three different structural systems from low- to high-rise buildings are estimated using multiple stripe analyses subjected to different sets of GMs from shallow crustal seismic zone. It is found that duration of GMs from shallow events affects the structural response but not as much as that reported for GMs from subductions events. The results also reveal that the importance of GM duration mainly depends on the considered conditioning IM. Specifically, GM duration does not substantially affect the structural responses in terms of probability of collapse if peak ground acceleration, peak ground velocity, spectrum intensity, spectral acceleration at higher modes are implemented as the conditioning IMs. On the other hand, in the case of cumulative absolute velocity and spectral acceleration at the fundamental and lengthened periods, the structural responses are considerably affected by GM duration.

... During high-intensity seismic events, it is known that the cross-sections in plastic hinge areas of a building can be severely cracked or even present steel yielding, resulting in structural stiffness degradation. Therefore, an increase in the building's flexibility, and as such its fundamental period, is expected [60]. The DI DC is based on the above-mentioned increase in the fundamental period and is calculated according to Equation (4). ...

Advanced machine learning algorithms have the potential to be successfully applied to many areas of system modelling. In the present study, the capability of ten machine learning algorithms to predict the structural damage of an 8-storey reinforced concrete frame building subjected to single and successive ground motions is examined. From this point of view, the initial damage state of the structural system, as well as 16 well-known ground motion intensity measures, are adopted as the features of the machine-learning algorithms that aim to predict the structural damage after each seismic event. The structural analyses are performed considering both real and artificial ground motion sequences, while the structural damage is expressed in terms of two overall damage indices. The comparative study results in the most efficient damage index, as well as the most promising machine learning algorithm in predicting the structural response of a reinforced concrete building under single or multiple seismic events. Finally, the configured methodology is deployed in a user-friendly web application.

... Before deriving the standard values of the seismic response in the time domain, the multiple-mass-points system's basic vibration modes must first be determined. On this basis, the natural vibration periods of different vibration modes and the displacement of each mass point for each vibration mode can be determined [12][13][14][15]. During the basic vibration modes calculation, the mass points undergo purely translational motion with no rotation. ...

The response to earthquake ground motion is composed of three basic elements, namely, amplitude, frequency, and duration. The seismic response of a structure is controlled by the particular combination of these three elements. The seismic response spectra reflect the earthquake ground motion’s frequency-domain features and provide the maximum response amplitude of a single-degree-of-freedom system to a given earthquake ground motion but do not consider the duration factor. However, the analysis of post-earthquake damage shows that the seismic response duration has a strong impact on the damage to structures. Therefore, it is necessary to develop a simple and practical analytical method to account for the seismic response duration. The present study was conducted based on the response spectra theory. We introduce an analytical method of elastic seismic response, which considers its duration by adding the time-domain dimension of earthquakes. The time-domain spectral matrix is used to solve the time-dependent seismic response through the vibration mode decomposition method. The time-domain vibration mode decomposition reaction spectrum not only takes into account the maximum seismic reaction of each vibration mode but also considers the seismic reaction of different vibration modes occurring at the same time, at each moment. The dynamic time duration of the structure’s seismic reaction is quantified by the time-domain seismic reaction spectrum to obtain a more accurate analysis method for the seismic reaction of the structure.

... Several researchers introduced damage models based on period elongation. For instance, Katsanos and Sextos (2015) propose an inelastic spectrum to predict period elongation of structures under earthquake loading. Nevertheless, the above models do not allow an easy association of the numerical results to the observable damage. ...

When strong earthquakes occur, buildings are usually affected by multiple, consecutive seismic shakings causing a progression of the damage. This paper presents a comprehensive discussion on how the cumulative damage caused by multiple shocks could influence the assignment of macroseismic intensity. The potential consequences in case of single or multiple shocks are investigated through a large set of simulations, applying a simplified non-linear structural model on a stock of masonry buildings taken from real cases. The results allow an estimation of the changes in vulnerability, with reference to the European Macroseismic Scale classification, for the buildings that suffered a specific damage in a prior shock. Simulations reckon a significant difference in the macroseismic intensity assignments, especially when the same damage is associated to single or multiple seismic shocks. We highlight the potential bias due to the cumulative damage on the correlation between macroseismic intensity and ground motion parameters, and we formulate a proposal for testing on the field, in the case of future earthquakes, what has been here investigated through a simplified simulation.

... To estimate the structural peak responses under earthquakes in a simple way, the response spectra method has been widely used in design practices. The studies on inelastic response spectra are booming over these years under the framework of performance-based earthquake e n g i n e e r i n g ( R i d d e l l e t a l . 2 0 0 2 , C h o p r a a n d Chintanapakdee 2004, Thermou et al. 2012, Dimakopoulou et al. 2013, Han et al. 2014, Katsanos and Sextos 2015, Esfahanian and Aghakouchak 2015, Rahgozar et al. 2016. The existing inelastic response spectra generally belong to constant-strength spectra or constant-ductility spectra, whose specific yielding strength levels or ductility levels of the SDOF system are predefined, respectively. ...

Evaluation on the sidesway seismic collapse capacity of the widely used low- and medium-height structures is meaningful. These structures with such type of collapse are recognized that behave as inelastic deteriorating single-degree-of-freedom (SDOF) systems. To incorporate the deteriorating effects, the hysteretic loop of the nonlinear SDOF structural model is represented by a tri-linear force-displacement relationship. The concept of collapse capacity spectra are adopted, where the incremental dynamic analysis is performed to check the collapse point and a normalized ground motion intensity measure corresponding to the collapse point is used to define the collapse capacity. With a large amount of earthquake ground motions, a systematic parameter study, i.e., the influences of various ground motion parameters (site condition, magnitude, distance to rupture, and near-fault effect) as well as various structural parameters (damping, ductility, degrading stiffness, pinching behavior, accumulated damage, unloading stiffness, and P-delta effect) on the structural collapse capacity has been performed. The analytical formulas for the collapse capacity spectra considering above influences have been presented so as to quickly predict the structural collapse capacities.

... For small values of R µ , there is a strong correlation of the results to the R µ =1 case for ∆ in and therefore to the results reported in Figures 6.3 and 6.4 or the reported F 1 values in Figure 6.1. Note that for large values of R µ , the nonlinear structural response is sensitive to spectral ordinates at periods much larger than the fundamental one (e.g., due to period elongation stemming from the strong nonlinear behaviour); the chosen period range for spectral compatibility (i.e., 0.2T 1 -1.5T 1 ) may not be conservative in those cases (Katsanos and Sextos, 2015), yielding the observed large variability for larger values of R µ . Figure 6.8: Median hysteretic energy for EPH system with α=3% for "constant-R µ " approach. ...

The recent advances in computational efficiency and the scarcity/absence of recorded ground motions for specific seismicity scenarios have led to an increasing interest in the use of ground motion simulations for seismic hazard analysis, structural demand assessment through response-history analysis, and ultimately seismic risk assessment. Two categories of ground motion simulations, physics-based and stochastic site-based are considered in this study. Physics-based ground motion simulations are generated using algorithms that solve the fault rupture and wave propagation problems and can be used for simulating past and future scenarios. Before being used with confidence, they need to be validated against records from past earthquakes. The first part of the study focuses on the development of rating/testing methodologies based on statistical and information theory measures for the validation of ground motion simulations obtained through an online platform for past earthquake events. The testing methodology is applied in a case-study utilising spectral-shape and duration-related intensity measures (IMs) as proxies for the nonlinear peak and cyclic structural response. Stochastic site-based ground motion simulations model the time-history at a site by fitting a statistical process to ground motion records with known earthquake and site characteristics. To be used in practice, it is important that the output IMs from the developed time-histories are consistent with these prescribed at the site of interest, something that is not necessarily guaranteed by the current models. The second part of the study presents a computationally efficient framework that addresses the modification of stochastic ground motion models for given seismicity scenarios with a dual goal of matching target IMs for specific structures, while preserving desired trends in the physical characteristics of the resultant time-histories. The modification framework is extended to achieve a match to the full probability model of the target IMs. Finally, the proposed modification is validated by comparison to seismic demand of hazard-compatible recorded ground motions. This study shows that ground motion simulation is a promising tool that can be used for many engineering applications.

... Based on 300,000 nonlinear seismic response time history analysis data, Katsanos and Sextos (2015) used the theory of elastoplastic response spectrum to study the calculation method of the period elongation of the building structure under the seismic damage state. Research results shows that the periodic elongation rate of damaged structures is signi cantly affected by the period of structural elasticity and the rate of structural stiffness degradation. ...

The seismic damage state of building structure can be evaluated by observing the fundamental period change of structure. Firstly, the fundamental period calculation formula that adapts to the deformation pattern and distribution mode of horizontal seismic action for reinforced concrete frame structure is derived. Secondly, the seismic damage assessment standard of building structure considering period variation is established. Then, the seismic damage assessment method of building structure is constructed. Finally, the seismic damage example is used to verify the established evaluation method. The results show that the established research method has high accuracy and good engineering practicability.

... values of period lower than !" , which are associated with higher modes, are not considered herein, since this study focuses on SDoF systems. Katsanos & Sextos [51] demonstrated that the period elongation of SDoF systems ranges from 120% to 250% and strongly depends on the ratio between the yielding displacement with respect to the elastic displacement demand at the first period. Therefore, two versions of are considered, setting equal to 1.5 and 2 (respectively indicated as *.P , 8 hereafter). ...

This paper investigates a number of computational issues related to the use of nonlinear static procedures in fragility analysis of structures. Such approaches can be used to complement nonlinear dynamic procedures, reducing the computational and modelling effort. Specifically, this study assesses the performance of the Capacity Spectrum Method (CSM) with real (i.e. recorded) ground motions (as opposed to code-based conventional spectra) to explicitly account for record-to-record variability in fragility analysis. The study focuses on single-degree-of-freedom systems, providing a basis for future multi-degree-of-freedom system applications. A case-study database of 2160 inelastic oscillators is defined through parametric backbones with different elastic periods, (yield) base shear coefficients, values of the ductility capacity, hardening ratios, residual strength values and hysteresis rules. These case studies are analysed using 100 real ground motions. An efficient algorithm to perform the CSM with real spectra is proposed, combined with a cloud-based approach (Cloud-CSM) to derive fragility relationships. Simple criteria to solve the issue of multiple CSM solutions (i.e. two or more points on the backbone satisfying the CSM procedure) are proposed and tested. It is demonstrated that the performance point selection can be carried out based on a particularly efficient intensity measure detected via optimal intensity measure analysis. The effectiveness of the proposed Cloud-CSM in fragility analysis is discussed through extensive comparisons with nonlinear time-history analyses, the code-based N2 method, and a simple method involving an intensity measure as a direct proxy for the performance displacement. The Cloud-CSM provides errors lower than ±20% in predicting the median of the fragility curves in most of the analysed cases and outperforms the other considered methodologies in calculating the fragility dispersion.

... Empirical relations can be employed for predicting the lengthening of modal periods due to inelastic response. 30 These relations however are derived for the mean values of inelastic (effective)-to-elastic periods of single degree of freedom systems under large number of ground motions. Since IDS is performed on single ground motions, lengthening of modal periods improve the scale factors for some ground motions whereas it worsens for some others. ...

A novel amplitude scaling procedure is proposed in this study where the ground motion scaling factors are defined as the ratio of interstory drift distributions under target spectrum versus under the associated ground motion spectrum. The advantage of employing interstory drift ratio in ground motion scaling, compared to employing spectral intensity directly, is that it provides a strong theoretical link between the target spectrum intensity and the fundamental dynamic characteristics of the structure. Hence, scaling is conditioned on structural response, which is in turn a function of seismic intensity. The interstory drift‐based scaling procedure (IDS) is presented herein for planar frames for brevity. Accuracy and efficiency of the IDS procedure is assessed under a set of near fault strong motions from large magnitude events. The results revealed that the proposed procedure is accurate since the resulting bias in estimating linear elastic interstory drifts is negligibly small. Further, it is noticeably more effective as compared to the conventional procedures suggested in recent seismic codes, yet it is simpler.

... In the context of vibration-based seismic damage quantification, the elongation of fundamental period is a widely adopted damage indicator to estimate the overall inelastic structural performance after earthquakes [3]. In reinforced concrete (RC) structures, the period elongation is a function of the ground motion parameters [4] as well as the geometry of the structure and the presence of infill walls [5]. Some studies experimentally assessed the changes in dynamic properties on a few structures affected by seismic damage using OMA data, and proposed period elongation ranges as a function of the level of damage experienced by such structures during a specific earthquake [5][6][7]. ...

One of the main issues in civil engineering is to estimate the level of damage for existing buildings subjected both to decay due to natural ageing and to additional loads that could affect their serviceability during lifetime. This paper looks at the process of Structural Health Monitoring (SHM) in the context of seismic damage detection and quantification. The aim of the paper is to define ranges of variation for parameters commonly monitored as structural damage indicators, like fundamental periods, residual drift ratios and lateral stiffness, in relation with seismic damage levels. For this purpose, a methodology is proposed for simulating the seismic damage of existing reinforced concrete (RC) columns and assessing the correlation between the variation of such parameters with increasing damage levels. Seismic damage levels are defined based on the Park and Ang damage index, and preliminary ranges of variation of damage indicators are derived for the seismic damage quantification in existing RC columns.

... To avoid any such bias, the 2D frame model was subjected to 300 strong ground motions obtained from the PEER NGA Database [32], with a wealth of different characteristics in terms of seismological parameters (earthquake magnitude, sourceto-site distance and rupture mechanism), amplitude and frequency content as well as soil conditions, in which the motions were recorded. A comprehensive description of the selected earthquake records can be found elsewhere [33]. ...

The seismic design of an eight-story reinforced concrete space frame building is undertaken using a yield frequency spectra (YFS) performance-based approach. YFS offer a visual representation of the entire range of a system's performance in terms of the mean annual frequency (MAF) of exceeding arbitrary global ductility or displacement levels versus the base shear strength. As such, the YFS framework can establish the required base shear and corresponding first-mode period to satisfy arbitrary performance objectives for any structure that may be approximated by a single-degree-of-freedom system with given yield displacement and capacity curve shape. For the eight-story case study building, deformation checking is the governing limit state. A conventional code-based design was performed using seismic intensities tied to the desired MAF for safety checking. Then, the YFS-based approach was employed to redesign the resulting structure working backwards from the desired MAF of response (rather than intensity) to estimate an appropriate value of seismic intensity for use within a typical engineering design process. For this high-seismicity and high-importance midrise building, a stiffer system with higher base shear strength was thus derived. Moreover, performance assessment via incremental dynamic analysis showed that while the code-design did not meet the required performance objective, the YFS-based redesign needed only pushover analysis results to offer a near-optimal design outcome. The rapid convergence of the method in a single design/analysis iteration emphasized its efficiency and practicability as a design aid for practical application. Copyright

... Based on the above line of thought, the selection methodology presented herein makes use of the findings of a recent study to bound the upper spectral matching period range by the first-mode elongated (inelastic) period T 1,in (Eq. 4), the latter being defined as a function of the corresponding elastic period, T 1 , and the force reduction factor, R y (or behavior factor, q, in EN1998-Part 1 2004), for which the building or bridge has been designed (Katsanos and Sextos 2015): ...

A decision support process is presented to accommodate selecting and scaling of earthquake motions as required for the time domain analysis of structures. Code-compatible suites of seismic motions are provided being, at the same time, prequalified through a multi-criterion approach to induce response parameters with reduced variability. The latter is imperative to increase the reliability of the average response values, normally required for the code-prescribed design verification of structures. Structural attributes like the dynamic characteristics as well as criteria related to variability of seismic motions and their compliance with a target spectrum are quantified through a newly introduced index, δsv–sc, which aims to prioritize motions suites for response history analysis. To demonstrate the applicability of the procedure presented, the structural model of a multi-story building was subjected to numerous suites of motions that were highly ranked according to both the proposed approach (δsv–sc) and the conventional one (δconv), that is commonly used for earthquake records selection and scaling. The findings from numerous linear response history analyses reveal the superiority of the proposed multi-criterion approach, as it extensively reduces the intra-suite structural response variability and consequently, increases the reliability of the design values. The relation between the target reliability in assessing structural response and the size of the suite of motions selected was also investigated, further demonstrating the efficiency of the proposed selection procedure to achieve higher response reliability levels with smaller samples of ground motion.

Earthquake damage in structures is related to the co-seismic elongation of the structural period that is used as a global damage proxy. The study introduces a novel and straightforward technique to estimate the predominant elastic and inelastic period of buildings during earthquake shaking. The proposed technique introduces the Decoupled Fourier Amplitude Spectrum (DFAS), which aims to decouple the spectrum of the response from the seismic excitation and allows the identification of the co-seismic vibration period during damaging earthquakes. The DFAS technique is applied to numerically-calculated linear and nonlinear responses of earthquake-excited SDOF systems in order to highlight the usefulness of the DFAS-based period identification. Further corroboration of the DFAS technique is provided by calculating the predominant inelastic frequency using structural responses recorded from three existing buildings during their excitation by severe earthquake events that led to structural damages. The DFAS-identified predominant inelastic frequency of the systems was found in good agreement with results based on well-established time-frequency transformations. The proposed technique is expected to facilitate the damage assessment and the associated quantification of limit states on the basis of either monitored responses of existing structures or numerical predictions.

The influence of bond-slip behavior, shown by pinched hysteretic curves in the inelastic response of degrading structural systems, is investigated in this paper. Inelastic response spectra are calculated and compared for deteriorating structures with and without manifestation of bond-slip. The response spectra are generated by using a smooth hysteretic model with distinct degrading parameters that are calibrated to the generic inelastic envelopes. In addition to the ductility demands, degraded residual strengths and stiffness of structural systems after an earthquake are considered as the relevant response measures. These measures can serve as design parameters for structures that should remain functional after an extreme event. Based on the response spectra analyses, it is found that nonlinear responses of degrading structural systems with bond-slip develop larger ductility demands and greater loss of strength and stiffness than the responses without bond-slip. Numerical results show that, ductility demand is increased up to 50% to 150% with consideration of slip in the model, as well as residual strength and stiffness decreased up to 10% to 40%.
Keywords response spectra, strength and stiffness deteriorations, pinching and bond-slip 2

Forward-directivity effects cause most of the seismic energy from the rupture to arrive at the beginning of the motion as a large pulse. This characteristic of ‘early-arriving’ pulses is often adopted to identify pulse-like motions caused specifically by directivity effects. However, not all early-arriving pulses are caused by the forward-directivity effects. Also, current criteria to select near-source directivity pulses mainly include the presence of large pulses in the velocity time series, whether the velocity pulse is early-arriving or located within 20 or 30 km closest distance to the fault. This study is intended to provide important insights into the differences in the response attributes of reinforced concrete (RC) frame structures subjected to early-arriving pulse-like ground motions caused by different physical processes, i.e., forward-directivity versus non-directivity effects. Nonlinear time-history analyses of three generic RC frame structures to two extensive suites of unscaled and scaled ground motions were performed to examine the distinct effects of these two types of ground motions. Results indicate that for the short period (2-story) frame, the mean drift demands by directivity motions are equal to or smaller than that caused by non-directivity motions. For the medium (6-story) and long period (20-story) frames, the directivity-induced motions result in larger demands in the lower half of the structures compared to non-directivity motions, especially for high-intensity ground motions. Spectral analyses of the ground motions provide key information as to why the non-directivity records cause larger demands in the 2-story frame. Simulations using pulse models that represent pulse-like ground motions are carried out to gain additional understanding on the primary findings for the 6-story and 20-story structures.

In many researches and engineering projects, in order to control the uncertainty of ground motions, a large number of Time History Analyses (THAs) are used with a time-consuming process. To reduce the computation time, this study proposed four truncation methods that truncates the trailing records based on the principle of structural maximum displacement response equivalence. By investigating the changes of ground motion parameters and structural response after truncating, an optimal method is confirmed and then compared with commonly used truncation methods. The results show that the optimal truncation method leads to a significant time-saving procedure and reliable computation results.

Soil-structure interaction (SSI) effects are of interest for the seismic analysis and the design of tall buildings on shallow foundations, particularly when both the structure and soil undergo inelastic demands. The objective of this paper is to evaluate the interaction of nonlinear soil-structure systems using the direct fully-coupled approach for modeling SSI. Numerical simulations of linear and nonlinear tall buildings, combined with either fixed-base conditions at the ground level or an explicit soil domain, are performed. The soil domain was modeled assuming either linear elastic isotropic or multiple yield surface plane strain continuum constitutive models. An archetype 30-story building supported on a mat foundation was modeled using nonlinear link elements to control geometry, stiffness, and strength. Structural stiffness and mass profiles were algorithmically generated to satisfy prescribed modal characteristics of tall buildings and achieve a more realistic response. The influence of nonlinear material responses for both structure and supporting soils subjected to selected earthquake time histories is quantified using drifts, accelerations, displacements, hysteretic energy, and transfer functions. Nonlinear analyses considering SSI largely influenced the computed seismic structural response presented in this paper, showing a significant decrease of the seismic demands when compared to those demands obtained with linear SSI models, which impacts the structural behavior and has practical implications in seismic-resistant designs.

This volume gathers the latest advances and innovations in the field of structural health monitoring, as presented at the 8th Civil Structural Health Monitoring Workshop (CSHM-8), held on March 31–April 2, 2021. It discusses emerging challenges in civil SHM and more broadly in the fields of smart materials and intelligent systems for civil engineering applications. The contributions cover a diverse range of topics, including applications of SHM to civil structures and infrastructures, innovative sensing solutions for SHM, data-driven damage detection techniques, nonlinear systems and analysis techniques, influence of environmental and operational conditions, aging structures and infrastructures in hazardous environments, and SHM in earthquake prone regions. Selected by means of a rigorous peer-review process, they will spur novel research directions and foster future multidisciplinary collaborations.

Abstract. For earthquake resistant design, structural degradation is considered using traditional strength modification factors, which are obtained via the ratio of the nonlinear seismic response of degrading and non-degrading structural single degree of freedom (SDOF) systems. In this paper, with the aim to avoid the nonlinear seismic response to compute strength modification factors, a methodology based on probabilistic seismic hazard analyses (PSHA) is proposed in order to obtain strength modification factors of design spectra which consider structural degradation through the spectral-shape intensity measure I<sub>Np</sub> . PSHA using I<sub>Np</sub> to account for structural degradation, and Sa ( T <sub>1</sub>) which represents the spectral acceleration associated with the fundamental period and does not consider such degradation, are performed. The ratio of the uniform hazard spectra in terms of I<sub>Np</sub> and Sa ( T <sub>1</sub>), that represent the response of degrading and non-degrading systems, provide new strength modification factors without the need to develop nonlinear time history analysis. A mathematical expression is fitted to the ratios that correspond to systems located in different soil types. The expression is validated by comparing the results with those derived from nonlinear time-history analyses of structural systems.

This report summarizes the modeling of inelastic structures and enhancements to
the program series IDARC developed for analysis, design and support of experimental
studies. It includes a synthesis of all the material presented in previous reports NCEER-
87-0008, NCEER-92-0022, and NCEER-96-0010 (and in other related reports). The
report also presents new developments regarding modeling of inelastic elements and
structures with supplemental damping devices, infill panels, etc.
The analytical models described herein include frame structures with rigid or semirigid
connections made of beams, columns, shear walls, connecting beams, edge elements,
infill masonry panels, inelastic discrete springs (connectors), and damping braces
(viscoelastic, fluid viscous, friction, hysteretic). The formulations are based on
macromodels in which most structural members are represented by a singlecomprehensive
element with nonlinear characteristics.
The nonlinear characteristics of the basic macromodels are based on a flexibility
formulation and a distributed plasticity with yield penetration. Properties of members are
calculated by fiber models or by formulations based on mechanics. The solutions are
obtained using step-by-step integration of equations of motion using the Newmark beta
method. One-step correction and iterative computations are performed to satisfy
equilibrium. The nonlinear dampers are treated as time dependent Maxwell models,
Kelvin Models or hysteretic models. Their solution is obtained by simultaneously solving
their individual equations using a semi-implicit Runge-Kutta solution.

The estimation of MDOF nonlinear structural response given an earth-quake of magnitude M at distance R is studied with respect to issues such as the benefits and harms of (1) first scaling the records, (2) selecting records from the "wrong" magnitude, (3) alternative choices for how to scale the records, and (4) scaling records to a significantly higher or lower intensity, etc. We find that properly chosen scaling can reduce the necessity of the number of nonlinear analyses by a factor of about four, and that proper scaling does not introduce any bias. Several global and local nonlinear damage measures are considered. A five-DOF model of a steel structure is used; other cases are under study. The paper finishes with a demonstration of the use of such results in the estimation of the annual probability of exceeding a specified interstory ductility (drift) or other damage measures.

Ground motion intensity parameters are used to express the relationship between expected structural damage and the seismic forces imposed. The graphical representation of damage probability as a function of ground motion intensity leads to fragility curves that are generally used in loss estimation studies. The most typical parameters used to represent the ground motion intensity are peak ground acceleration, peak ground velocity, spectral acceleration, and spectral displacement. Other parameters obtained from the ground motion trace and response spectra have been recommended in literature, but no consensus on which intensity parameter to use exists because of the various drawbacks of these ground motion intensities. A new spectrum ground motion intensity parameter that relies on the expected elongated period of the structure under seismic forces has been developed. This intensity measure takes into account the approximate yield capacity of the structure and the area between the fundamental and elongated period of the structure under the elastic response spectrum of the given ground motion. The correlation of this intensity measure with the calculated demand parameter, maximum interstory drift in our case, is investigated for a set of 100 ground motion records in order to verify its accuracy. This intensity measure is primarily proposed for the selection of ground motions to be used for the analyses of individual structures that are desired to respond at various levels of nonlinearity.

Field investigations after the recent Tohoku and Christchurch earthquakes reported failure of structural systems due to multiple earthquakes. In most failure cases the reported damage was mainly due to dramatic loss of stiffness and strength of structural elements as a result of material deterioration due to repeated earthquake loading. This study aims to investigate the degrading behavior of reinforced concrete frame systems subjected to Tohoku and Christchurch earthquake sequences. Numerical models of RC frames that incorporate damage features are established and inelastic response history analyses are conducted. The results presented in this study indicate that multiple earthquake effects are significant.

The recent concerns regarding the seismic safety of the existing building stock have placed the review of current seismic assessment procedures on the top of the agenda. Alongside with the development of more advanced commercial software tools and computational capacities, non-linear dynamic analysis is becoming, more and more, a common and preferable procedure in the seismic assessment of existing buildings. Besides the complexity associated with the formulation of the mathematical model, major issues arise related with the definition of the seismic action, which can lead to different levels of uncertainty in terms of local and global building response. Aiming to address this issue, a comparative study of different code-based record selection methods proposed by EC8-3, ASC41-06 and NZSEE is presented herein. The various methods are employed in the seismic assessment of four steel buildings, designed according to different criteria, and the obtained results are compared and discussed. Special attention is devoted to the impact of the number of real ground motion records selected (three and seven) in the response of the buildings and the inclusion of additional selection criteria based on the control of the spectral mismatch of each individual record with respect to the reference response spectrum adopted in the seismic assessment. The results obtained indicate that the Eurocode selection procedure slightly underestimates both the local and global response of the buildings in comparison with the American and New Zealand procedures, which conversely, lead to similar results. It is also concluded that record sets incorporating the additional selection criteria significantly reduce the level of uncertainty in the response.

This paper investigates the elongation of the fundamental period of reinforced concrete buildings that occurs during earthquake loading and its correlation with various intensity measures and engineering demand parameters. For this purpose, five buildings designed according to modern seismic codes are studied through equivalent single degree of freedom nonlinear systems with hysteretic laws that represent various levels of stiffness degradation, strength deterioration and pinching. By means of an extensive parametric analysis using a large set of earthquake ground motions and a rigorous validation procedure, the period elongation is quantitatively assessed as a function of building configuration and design (structural system and ductility class), ground motion characteristics (peak ground acceleration, spectral acceleration, frequency content) and demand parameters (displacement ductility). The results indicate that structures, designed according to modern seismic codes, are expected to exhibit low-to-moderate period elongation even for twice the intensity of the design earthquake. Given that the fundamental period of buildings is a key parameter in most seismic code procedures for ground motion selection, design and assessment, the implications of the predicted period lengthening are also discussed. The results are of interest to designers and analysts, as well as code-development committees.

Assessment of imposed damage to a structure after earthquakes of different intensities is very important. Since a large number of buildings experience seismic loads in their lifetime, determining the usability of a building or a structure and its resistance to future earthquakes is crucial. This paper considers period elongation and calculates damage by Park-Ang damage index and studies the relation between these two parameters. In this paper, a mathematical relation between damage index and the period elongation is proposed. The advantage of this method is the consideration of period elongation for calculation of the damage. The period varies with changes in structure’s stiffness which reflects all the elements stiffness. This helps to understand the correlation between the seismic behavior of a structure with distribution and order of plastic hinges and cracks.

A large mainshock triggers numerous aftershocks, exposing evacuees and residents to significant risk and hampering building reoccupation and restoration activities in a post-disaster situation. It is thus important to take into account the seismic effects of mainshock-aftershock (MSAS) sequences, not just those of mainshocks. To assess the nonlinear damage potential caused by aftershocks, this study investigates the effects of aftershocks on peak ductility demand of inelastic single-degree-of-freedom systems using real as well as artificial MSAS sequences. The real sequences are constructed from the K-NET and KiK-net databases for Japanese earthquakes. Comparison of peak ductility demand due to real mainshock events alone and real MSAS sequences renders empirical assessment of the aftershock impact on peak ductility demand. Moreover, time-history data of artificial MSAS sequences are generated based on the generalized Omori's law and suitable aftershock record selection procedure that takes into account key characteristics of aftershock records (magnitude, distance, and site classification). The validity of artificially generated MSAS sequences is evaluated by comparing probabilistic characteristics of peak ductility demand caused by artificial sequences with those caused by real sequences. The results indicate that peak ductility demands from real and artificial sequences are similar; thus, artificial sequences can be substituted for real sequences. Such calibration is particularly useful when an extensive data set of real MSAS sequences is not available.

Inelastic displacement ratios have been generally investigated for fixed-base systems without taking into consideration soil structure interaction. In this study, inelastic displacement ratios are investigated for SDOF systems with period range of 0.1–3.0 s with elastoplastic behavior considering soil structure interaction for 64 different earthquake motions recorded on different site conditions such as rock, stiff soil, soft soil and very soft soil. Soil structure interacting systems are modeled with effective period, effective damping and effective ductility values differing from fixed-base case. For inelastic time history analyses, Newmark method for step by step time integration was adapted in an in-house computer program. Results are compared with those calculated for fixed-base case. A new equation is proposed for inelastic displacement ratio of interacting system as a function of structural period of interacting system
$(\tilde T)$
, ductility ratio (μ) and period lengthening ratio
$(\tilde T/T)$
. The fitness of the regressed function of the inelastic displacement factor is shown in figures. The regressed equation for
$\tilde C_\mu$
should be useful in estimating the inelastic deformation of structure where the global ductility capacity can be estimated.

SUMMARY This paper presents the case for the significant elongation of the period of vibration of reinforced concrete (RC) buildings during strong ground shaking due to earthquakes. This viewpoint is substantiated by the results of experimental tests on RC structures and the strong ground-motion measurements obtained from damaged RC buildings during earthquakes, wherein a large increase in the period of vibration is observed during ground shaking. The increase in the fundamental period is obviously dependent on the level of shaking and the associated extent of non-linearity that is attained within the structure and/or foundation; this behaviour has been more frequently observed in experimental tests than in the field due to the lack of instrumented buildings that have been subject to large strong ground shaking. Analytical models which replicate the results of the experimental tests are introduced and additional studies on the elongation of the period during seismic action are presented.

The important role played by the duration of ground shaking in the response of saturated soil deposits is universally acknowledged, but no such consensus exists regarding the degree of influence that duration exerts on structural damage. There are several hundred papers in the literature that link structural damage to parameters related either directly or indirectly to the duration of strong ground motion. The conclusions of these studies differ widely with regard to the influence of strong-motion duration on structural demand. This paper provides a summary and critical review of the literature on this subject. It is found that studies employing damage measures related to cumulative energy usually find a positive correlation between strong-motion duration and structural damage, while studies employing damage measures using maximum response generally do not find strong correlations between duration and damage.

A key component of the NGA research project was the development of a strong-motion database with improved quality and content that could be used for ground-motion research as well as for engineering practice. Development of the NGA database was executed through the Lifelines program of the PEER Center with contributions from several research organizations and many individuals in the engineering and seismological communities. Currently, the data set consists of 3551 publicly available multi-component records from 173 shallow crustal earthquakes, ranging in magnitude from 4.2 to 7.9. Each acceleration time series has been corrected and filtered, and pseudo absolute spectral acceleration at multiple damping levels has been computed for each of the 3 components of the acceleration time series. The lowest limit of usable spectral frequency was determined based on the type of filter and the filter corner frequency. For NGA model development, the two horizontal acceleration components were further rotated to form the orientation-independent measure of horizontal ground motion (GMRotI50). In addition to the ground-motion parameters, a large and comprehensive list of metadata characterizing the recording conditions of each record was also developed. NGA data have been systematically checked and reviewed by experts and NGA developers.

The relationship between the peak deformations of inelastic and corresponding linear single-degree-of-freedom (SDF) systems is investigated. Presented are the median of the inelastic deformation ratio for 214 ground motions organized into 11 ensembles of ground motions, representing large or small earthquake magnitude and distance, and National Earthquake Hazards Reduction Program (NEHRP) site classes B, C, and D; near-fault ground motions are also included. Two sets of results are presented for bilinear nondegrading systems over the complete range of elastic vibration period, Tn:Cμ for systems with known ductility factor, μ, and CR for systems with known yield-strength reduction factor, Ry. The influence of postyield stiffness on the inelastic deformation ratios Cμ and CR is investigated comprehensively. All data are interpreted in the context of acceleration-sensitive, velocity-sensitive, and displacement-sensitive regions of the spectrum for broad applications. The median Cμ versus Tn and CR versus Tn plots are demonstrated to be essentially independent of the earthquake magnitude and distance (over their ranges considered), and of site class. In the acceleration-sensitive spectral region, the median inelastic deformation ratio for near-fault ground motions is systematically different when plotted against Tn; however, when plotted against normalized period Tn/Tc (where Tc is the period separating the acceleration- and velocity-sensitive regions) they become very similar in all spectral regions. Determined by regression analysis of the data, two equations—one for Cμ and the other for CR—have been developed as a function of Tn/Tc, and μ or Ry, respectively, and are valid for all ground motion ensembles considered. These equations for Cμ and CR should be useful in estimating the inelastic deformation of new or rehabilitated structures—where the global ductility capacity can be estimated—and existing structures with known lateral strength.

In this, the second of a two-part paper, the analysis of the apparent frequency of a seven-story reinforced-concrete hotel building in Van Nuys, Calif., is extended to consider its time-dependent changes, both short and long term. The instantaneous apparent frequency is measured by two methods: windowed Fourier analysis and zero-crossings analysis. The results show that it changes from earthquake to earthquake and during a particular earthquake. The results also suggest "self healing" believed to result from settlement of the soil with time and dynamic compaction from aftershock shaking. Implications of such high variability of the system frequency on structural health monitoring, control of response, as well as on the design codes are discussed. Nonlinear response of the foundation soil acts as a sink of the incident seismic wave energy. It is suggested that it could be exploited in future designs to serve as a powerful and inexpensive energy-dissipation mechanism.

2 SUMMARY The Building Research Institute (BRI) of Japan is a national institute that is carrying out research and development on building engineering, architecture and urban planning. The BRI is operating a strong motion observation network for buildings throughout Japan as one of its research activities. The BRI annex building is one of the stations of the BRI strong motion network and is densely instrumented with twenty-two accelerometers. In this paper, the variation of the dynamic characteristics of the annex building is discussed through the analysis using strong motion records. The natural frequencies and damping ratios of the annex building were analysed based on a single- degree-of-freedom system for 158 strong motion records. The fall of the identified first natural frequencies with elapse of time was clearly recognised. Sixteen strong motion records with relatively large response were selected in order to examine the cause of the fall in detail. The Evolution Strategies (ES) algorithm was applied to the analysis using a swaying-rocking multi-degree-of-freedom system. The ES algorithm is a powerful problem-solving tool based on natural evolution. The building stiffness, rocking stiffness, swaying stiffness and first natural modal damping ratio were identified for sixteen strong motion records by using the ES algorithm. The rocking stiffness, swaying stiffness and first natural modal damping ratio of the building showed stable values independent of the elapse of time. Consequently, it was inferred that the fall of the natural frequencies was caused by the softening of the building stiffness.

SUMMARY Ambient vibration analysis is proposed as an alternative way to inspect buildings before or after an earthquake. This fast and low-cost method is well-adapted to large-scale studies for which a large amount of buildings has to be checked. One of the most common critics usually done on the use of ambient vibrations in structures is the very low level of vibrations. Because of the low amplitude range of the ambient vibration (PGA0.1g). The objective of this paper is to present a comparison of the structural dynamic characteristics deduced from strong, moderate and weak motion recordings for a set of twelve Californian buildings and four European buildings. The present study differs from numerous previous investigations in two aspects: (a) the number of compared building is larger; and (b) for two buildings, more than 10 earthquakes records are available, which allows to investigate the dependence of dynamic characteristics with shaking intensity.

During the 2002 seismic sequence in Molise (Italy), the town of Bonefro suffered moderate damage (IMCS VII) except for two reinforced concrete (RC) buildings. These buildings are located on soft sediments, close to each other and very similar in design and construction. The main difference is the height: the most damaged one (European Macroseismic Scale damage 4) has four stories, whereas the less damaged (EMS damage 2) has three stories. The M 5.4 shock on 31 October damaged both of them. The second shock on 1 November (M 5.3) increased the damage on the four-story building substantially, just while a 5-min. seismic recording was taken. We analyzed the recorded data by four different techniques: short-time fourier transform (STFT), wavelet transform (WT), horizontal-to vertical spectral ratio (HVSR), and horizontal-to-vertical moving window ratio (HVMWR). All the results agree upon the estimate of the main building frequency before the second shock and upon the shift of frequency due to damage. All the fundamental frequencies (pre-, during, and postdamage) are in the range 2.5-1.25 Hz. The fundamental fre- quency of the less damaged building was estimated at about 4 Hz. To test if the soil-building resonance effect could have increased the damage, we also evaluated the soil fundamental frequency by three different techniques: noise HVSR, strong motion HVSR of seven aftershocks, and 1D modeling based on a velocity profile derived from noise analysis of surface waves (NASW) measurements. The results are again in good agreement, showing that resonance frequencies of the soil and of the more damaged building are very close.

The more recent earthquake engineering research has highlighted the mismatch between experimental and
theoretical period-height relationships not only for undamaged buildings but also for damaged ones. For the first
time in Italy, after the 2009 L’Aquila earthquake, we have estimated the fundamental periods of 48 RC buildings
after a strong seismic sequence. Performing ambient vibration measurements we investigated the fundamental
translational frequencies of 48 buildings with different built typology, structural characteristics, age and heights
affected by different damage levels (four out of the 5 damage levels defined by EMS 98). The distribution of
fundamental period versus buildings height and damage level has been evaluated. The fundamental period of RC
damaged buildings, for low damage level, is close to undamaged buildings ones. When damage levels gets
higher, the fundamental periods show as expected a general increase, but reaching values lower than those
provided some recent codes.

A method to evaluate the seismic collapse performance of frame structures is presented, considering uncertainties in both the ground motion hazard and inelastic structural response to extreme input ground motions. The procedure includes a new seismic-intensity scaling index that accounts for period softening and thereby reduces the large record-to-record variability typically observed in inelastic time-history analyses. Equations are developed to combine results from inelastic time history analyses and a site-specific hazard curve to calculate the mean annual probability of a structure exceeding its collapse limit state.

In the framework of the research activity of the ELSA Laboratory of the Joint Research Centre, pseudo-dynamic testing of a real-size plan-wise irregular 3-storey frame structure is being carried out as the core of the research project SPEAR (Seismic PErformance Assessment and Rehabilitation of existing buildings). The project, funded by the European Commission, sees the participation of many European and overseas Partners and is specifically aimed at throwing light onto the behaviour of existing old RC frame buildings lacking seismic provisions. The main goal of the SPEAR project is contributing to the improvement of current design, assessment and retrofitting techniques and the development of new simplified approaches for the assessment and rehabilitation of existing building structures. This goal is pursued by means of a balanced combination of experimental and numerical activities. In the paper the pre-test numerical work on the specimen is presented; the PsD test set-up is then described. Moreover, the results from the first PsD test are presented and discussed in detail in relation to the open issues of research in the field of torsionally unbalanced buildings.

Building Research Institute (BRI) is conducting strong motion observation of a base isolated building in Kushiro City, Hokkaido. A number of valuable strong motion records including the record from the 2003 Off-Tokachi Earthquake have been obtained since the building was completed in 2000.
From the analysis of the strong motion records from the Off-Tokachi Earthquake, the non-linear behavior of the surface geology and the base isolation device was recognized. The performance of the base isolation device, which consists of natural rubber bearings, lead dampers and steel dampers, was verified through the seismic response analysis using a multi-degree-of-freedom system model.
The result of the analysis estimating the natural frequency and the damping ratio of the upper structure showed amplitude-dependency of the stiffness. The natural frequency decreased as the maximum displacement of the upper structure increased. In addition, the difference in the natural frequencies between the cases before and after the Off-Tokachi Earthquake was found out.
The estimation of the stiffness and the damping ratio were made using a model with the base isolated storey. The result showed good agreement with the equivalent stiffness and the equivalent damping ratio evaluated from the hysteresis characteristics of the base isolation devices in large displacement range.

This paper summarizes the results of a comprehensive statistical study aimed at evaluating peak lateral inelastic displacement demands of structures with known lateral strength and stiffness built on soft soil site conditions. For that purpose, empirical information on inelastic displacement ratios which are defined as the ratio of peak lateral inelastic displacement demands to peak elastic displacement demands are investigated. Inelastic displacement ratios were computed from the response of single-degree-of-freedom systems having 6 levels of relative lateral strength when subjected to 118 earthquake ground motions recorded on bay-mud sites of the San Francisco Bay Area and on soft soil sites located in the former lake-bed zone of Mexico City. Mean inelastic displacement ratios and their corresponding scatter are presented for both ground motion ensembles. The influence of period of vibration normalized by the predominant period of the ground motion, the level of lateral strength, earthquake magnitude, and distance to the source are evaluated and discussed. In addition, the effects of post-yield stiffness and of stiffness and strength degradation on inelastic displacement ratios are also investigated. It is concluded that magnitude and distance to the source have negligible effects on constant-strength inelastic displacement ratios. Results also indicate that weak and stiffness-degrading structures in the short spectral region could experience inelastic displacement demands larger than those corresponding to non-degrading structures. Finally, a simplified equation obtained using regression analyses aimed at estimating mean inelastic displacement ratios is proposed for assisting structural engineers in performance-based assessment of structures built on soft soil sites. Copyright © 2006 John Wiley & Sons, Ltd.

The Southern California Seismic Network (scsn) has recently installed seismic stations in two buildings on the Caltech campus (Millikan Library and the Broad Center). Continuous real-time accelerometer data from these structures are now freely available to the community. This dataset provides a new opportunity to observe, and better understand, the variances in the primary dynamic property of a building system, its natural frequencies. Historical data (triggered strong-motion records, ambient and forced vibration tests) from the well-studied Millikan Library show dramatic decreases in natural frequencies, attributed mainly to moderately large local earthquakes. The current forced vibration east–west fundamental frequency is 22% lower than that originally measured in 1968. Analysis of the new continuous data stream allows the examination of other previously unrecognized sources of measurable change in the fundamental frequencies, such as weather (wind, rain, and temperature), as well as nonlinear building vibrations from small local and moderate regional earthquakes. Understanding these nonlinear shifts is one of the long-term goals of real-time building instrumentation and is critical if these systems are to be used as a postearthquake damage assessment tool.

Minimum Design Loads for Buildings and Other Structures provides requirements for general structural design and the means for determining dead, live, soil, flood, wind, snow, rain, atmospheric ice, and earthquake loads, as well as their combinations, which are suitable for inclusion in building codes and other documents. This Standard, a complete revision of ASCE/SEI 7-02, includes revised and significantly reorganized provisions for seismic design of structures, as well as revisions in the provisions for determining live, flood, wind, snow, and atmospheric ice loads. Supplement No. 1, which is included with the Standard, ensures full and complete coordination between ASCE/SEI 7-05 and the 2006 International Building Code. The updates which comprise Supplement No. 1 are seamlessly integrated into this volume and are not available anywhere else.
ASCE/SEI 7-05 is an integral part of building codes in the United States. The earthquake load provisions in ASCE 7-05 are substantially adopted by reference in the 2006 International Building Code and the 2006 NFPA 5000 Building Construction and Safety Code. Many other provisions, including calculations for wind and snow loads, are also adopted by reference by both IBC and NFPA model building codes.
Structural engineers, architects, and those engaged in preparing and administering local building codes will find this Standard an essential reference in their practice.

Fundamentals of Earthquake Engineering combines aspects of engineering seismology, structural and geotechnical earthquake engineering to assemble the vital components required for a deep understanding of response of structures to earthquake ground motion, from the seismic source to the evaluation of actions and deformation required for design. The nature of earthquake risk assessment is inherently multi-disciplinary. Whereas Fundamentals of Earthquake Engineering addresses only structural safety assessment and design, the problem is cast in its appropriate context by relating structural damage states to societal consequences and expectations, through the fundamental response quantities of stiffness, strength and ductility. The book is designed to support graduate teaching and learning, introduce practicing structural and geotechnical engineers to earthquake analysis and design problems, as well as being a reference book for further studies. Fundamentals of Earthquake Engineering includes material on the nature of earthquake sources and mechanisms, various methods for the characterization of earthquake input motion, damage observed in reconnaissance missions, modeling of structures for the purposes of response simulation, definition of performance limit states, structural and architectural systems for optimal seismic response, and action and deformation quantities suitable for design. The accompanying website at www.wiley.com/go/elnashai contains a comprehensive set of slides illustrating the chapters and appendices. A set of problems with solutions and worked-through examples is available from the Wley Editorial team. The book, slides and problem set constitute a tried and tested system for a single-semester graduate course. The approach taken avoids tying the book to a specific regional seismic design code of practice and ensures its global appeal to graduate students and practicing engineers.

SUMMARY Aftershocks induced by a large mainshock can cause additional damage to structures and infrastructure, hampering building reoccupation and restoration activities in a post-disaster situation. To assess the nonlinear damage potential due to aftershocks, this study investigates the effects of aftershocks by using real as well as artificially generated mainshock–aftershock sequences. The real mainshock–aftershock sequences are constructed from the Pacific Earthquake Engineering Research Center—Next Generation Attenuation database for worldwide shallow crustal earthquakes; however, they are deemed to be incomplete because of missing records. To supplement incomplete real dataset, artificial sequences are generated on the basis of the generalized Omori's law, and a suitable aftershock record selection procedure is then devised to simulate time-series data for mainshock–aftershock sequences. The results from nonlinear dynamic analysis of inelastic single-degree-of-freedom systems using real and artificial sequences indicate that the incremental effects of aftershocks on peak ductility demand using the real sequences are relatively minor and that peak ductility demand estimates based on the generalized Omori's law are greater, particularly in the upper tail, than those for the real sequences. The results based on the generalized Omori's law also suggest that the aftershock effects based on the real sequences might underestimate the aftershock impact because of the incompleteness of the real dataset. Copyright © 2012 John Wiley & Sons, Ltd.

The frequency content of an earthquake ground motion is important because it affects the dynamic response of earth and structural systems. Four scalar parameters that characterize the frequency content of strong ground motions are (1) the mean period (T-m), (2) the average spectral period (T-avg), (3) the smoothed spectral predominant period (T-o), and (4) the predominant spectral period (T-p). T-m and T-avg distinguish the low frequency content of ground motions, while T-o is affected most by the high frequency content. T-p does not adequately describe the frequency content of a strong ground motion and is not recommended. Empirical relationships are developed that predict three parameters (T-m, T-avg, and T-o) as a function of earthquake magnitude, site-to-source distance, site conditions, and rupture directivity. The relationships are developed from a large strong-motion database that includes recorded motions from the recent earthquakes in Turkey and Taiwan. The new relationships update those previously developed by the authors and others. The results indicate that three site classes, which distinguish between rock, shallow soil, and deep soil, provide a better prediction of the frequency content parameters and smaller standard error terms than conventional "rock" and "soil" site classes. Forward directivity significantly increases the frequency content parameters, particularly T-m and T-o, at distances less than 20 kin. Each of the frequency content parameters can be predicted with reasonable accuracy, but T-m is the preferred because it best distinguishes the frequency content of strong ground motions.

This paper follows the evolution of the NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures from the development of ATC 3-06 to the 1997 edition of the document. The features of the 1997 NEHRP Provisions are described in detail. Complementary information about the NEHRP Provisions is found in several other papers in this volume of Earthquake Spectra. Subject areas covered elsewhere are specifically referenced in this paper.

This paper gives a review of seismic damage indices, with particular reference to their use in retrofit decision making. Damage indices aim to provide a means of quantifying numerically the damage in concrete structures sustained under earthquake loading. Indices may be defined locally, for an individual element, or globally, for a whole structure. Most local indices are cumulative in nature, reflecting the dependence of damage on both the amplitude and the number of cycles of loading. The main disadvantages of most local damage indices are the need for tuning of coefficients for a particular structural type and the lack of calibration against varying degrees of damage. Global damage indices may be calculated by taking a weighted average of the local indices throughout a structure, or by comparing the modal properties of the structure before and after (and sometimes during) the earthquake. The weighted‐average indices are prone to much the same problems as the local indices. The modal indices vary widely in their level of sophistication, those capable of detecting relatively minor damage requiring the accurate determination of a large number of modes of vibration. The development and application of damage indices has until now concentrated almost exclusively on flexural modes of failure; there is a clear need to investigate the ability of the indices to represent shear damage.

The results of a comprehensive statistical study of inelastic displacement ratios that permit the estimation of maximum lateral inelastic displacement demands on a structure from maximum lateral elastic displacement demands are presented. These ratios were computed for single-degree-of-freedom systems undergoing different levels of inelastic deformation when subjected to a relatively large number of recorded earthquake ground motions. The study is based on 264 acceleration time histories recorded on firm sites during various earthquakes in California. Three types of soil conditions with shear-wave velocities higher than 180 m/s rue considered. The influences of period of vibration, level of ductility demand, site conditions, earthquake magnitude, and epicentral distance are carefully evaluated and discussed. Inelastic displacement ratios associated with mean values are presented. Special emphasis is given to the disperson of the results. It is concluded that for sites with average shear-wave velocities higher than 180 m/s the influence of soil conditions is relatively small and can be neglected for design purposes. Finally, results from nonlinear regression analyses are presented that provide a simplified expression to be used in the design to approximate mean inelastic displacements ratios for structures on firm sites.

Investigated in this paper is the approximation in the ATC-40 nonlinear static procedure (NSP) that the earthquake-induced deformation of an inelastic single-degree-of-freedom (SDF) system can be estimated by an iterative method requiring analysis of a sequence of equivalent linear systems. Several deficiencies in the ATC-40 Procedure A are demonstrated. This iterative procedure did not converge for some of the systems analyzed. It converged in many cases, but to a deformation much different than dynamic (nonlinear response history or inelastic design spectrum) analysis of the inelastic system. The ATC-40 Procedure B always gives a unique value of deformation, same as that determined by Procedure A if it converged. These approximate procedures underestimate significantly the deformation for a wide range of periods and ductility factors with errors approaching 50%, implying that the estimated deformation is about half the ‘‘exact’’ value. Surprisingly, the ATC-40 procedures are deficient relative to even the elastic design spectrum in the velocity-sensitive and displacement-sensitive regions of the spectrum. For systems with a period in these regions, the peak deformation of an inelastic system can be estimated from the elastic design spectrum using the well-known equal displacement rule. However, the approximate procedure requires analyses of several equivalent linear systems and still produces worse results.

Global damage indices based on equivalent modal parameters are defined using the vibrational parameters of an equivalent linear structure. In continuum mechanics-based damage models, effects of the growth and coalescence of microcracks are accounted for through the definition of an internal or local damage variable. Damage to engineering materials essentially results in a decrease of the free energy stored in the body with consequent degradation of the material stiffness. It is shown that parameter-based global damage indices can be related to local damage variables through operations of averaging over the body volume. The relationship obtained is applied to the case of numerical simulations of damage events as well as to the case of actual structures tested on shaking tables. Some results are also presented, relating parameter-based global damage indices to averages in space and time of local plastic strain.

A theory for an enhanced mathematical model of R/C frame members is presented and its accuracy is verified by simulating various laboratory experiments for which data were available in the literature. New member and global damage parameters are defined. These damage parameters are useful for subsequent reliability analysis of damaged concrete frames.

A simple definition of the duration of strong earthquake ground motion based on the mean-square integral of motion has been presented. It is closely related to that part of the strong motion which contributes significantly to the seismic energy as recorded at a point and to the related spectral amplitudes. Correlations have been established between the duration of strong-motion acceleration, velocity, and displacement and Modified Mercalli intensity, earthquake magnitude, the type of recording site geology, and epicentral distance. Simple relations have been presented that predict the average trend of the duration and other related param- eters as a function of Modified Mercalli intensity, earthquake magnitude, site geology and epicentral distance.

Accurate estimation of inelastic displacements is important for the evaluation of the seismic performance of structures with desired ductile response. In this paper, nonlinear dynamic analyses results from a companion numerical study investigating the response of ductile-designed bridge structures, were compared with a commonly applied inelastic displacement estimation approach and an alternative approach. The extended pile-shaft-supported bridge structures considered are susceptible to amplified response under long-period velocity pulses, and hence an evaluation of design methods for estimating inelastic displacement demands is warranted. In this case, force-reduction-displacement-ductility-period (R-AA-7-) relations and a mean spectral displacement approach are investigated. The alternative approach estimates inelastic displacement demand using the mean elastic spectral displacement between two spectral periods that are important for the structure's response. Results support the conceptual merits of using the mean spectral displacement method, indicating that the approach is capable of reducing the uncertainty in predicting inelastic displacement demands for the types of structures considered when subjected to near-fault ground motions.

This paper presents results of a study aimed at evaluating the effect of aftershocks in steel framed buildings. For that purpose, three frame models representing existing steel moment-resisting frames were subjected to a set of as-recorded mainshock–aftershock seismic sequences. For this purpose, 64 as-recorded seismic sequences registered as a consequence of the 1994 Northridge and 1980 Mammoth Lakes earthquakes were considered in this study. In particular, this investigation employed 14 seismic sequences recorded in 7 accelerographic stations in the near-fault region. An examination of the as-recorded seismic sequences shows that the frequency content of the mainshock and the main aftershock is weakly correlated. The response of the frame models was measured in terms of the peak and residual (permanent) drift demands at the end of the earthquake’s excitation. From the results of this investigation, unlike previous results based on artificial seismic sequences, it was found that as-recorded aftershocks do not significantly increase peak and residual drift demands since the predominant period of the aftershocks (i.e. frequency content) is very different from the period of vibration of the frame models. In addition, it was shown that artificial seismic sequences could significantly overestimate median peak and residual drift demands as well as the record-to-record variability.

The duration of strong ground shaking during earthquakes can play an important role in the response of foundation materials and structures, particularly when strength or stiffness degradation is encountered. A thorough seismic hazard assessment should therefore include an estimation of the expected duration of strong motion, which first requires criteria to define the part of an accelerogram considered to represent the duration of strong ground motion. Some 30 different definitions of strong motion duration are reviewed and classified into generic groups. Problems that arise with the use of these definitions for duration are highlighted. A new definition of duration is presented using a previously unexplored option which identifies the part of the record where the main energy is contained and constrains this strong shaking phase by absolute criteria. This new definition is shown to give consistently meaningful durations for strong earthquake accelerograms from an engineering viewpoint. The correlations between the new definition of duration and magnitude, soil conditions and distance are explored as a first step towards the development of predictive equations.

Many vital reinforced concrete (RC) buildings experience moderate or severe earthquakes in their lifetime because they are located in hazardous areas. However, their importance cause to be evaluated by different types of damage functions. In these procedures, structures are usually modelled. These models neither correctly display the effects of the cracks that emerge and plastic hinges nor precisely consider the effects of asymmetric configuration and infill panels. Furthermore, the actual nonlinear dynamic behaviour of existing buildings could be evaluated by assessing nonlinear dynamic characteristics such as the fundamental period. These dynamic characteristics, which are obtained by some field tests such as forced and/or ambient vibration methods, comprise the aforementioned effects. This paper offers a damage index (pattern) for seismic damage assessment of RC buildings based on the variation of the nonlinear fundamental period, which is obtained by field tests. Finally, the seismic situation of existing RC buildings that have experienced an earthquake is precisely and expeditiously assessed by this new damage index. Copyright © 2010 John Wiley & Sons, Ltd.

Modelling assumptions, boundary and loading conditions have a significant effect on analytical assessment of ductility supply and demand measures for RC bridges, a structural form which had suffered extensively in recent earthquakes. In recognition of the important role played by analysis in advancing seismic design of bridges, this paper is concerned with assessing the effect of model characteristics and earthquake strong-motion selection on analytical action and deformation seismic design parameters. This is of particular significance when viewed in the light of the large capital investment and problems with the satisfaction of dynamic similitude encountered in physical testing of piers and pier-deck assemblies. The models studied range between simple fixed-base cantilever and inclusion of both soil and deck effects, represented by assemblies of springs in translational and rotational degrees of freedom. Moreover, two sets of earthquake records are used in dynamic analysis, each comprising six records covering low, intermediate and high a/v, where a and v are the peak ground acceleration and velocity, respectively. The two sets differ in the scaling procedure employed to bring them to a common level of severity; the first set is obtained by direct acceleration scaling whilst the second utilizes the concept of velocity spectral intensity. The results from static and dynamic analysis, using advanced material characterization and solution procedures, are assessed and discussed. Subject to the limitations of the study, outlined in the paper, the results indicate that the inclusion of deck stiffness and/or soil representation is essential to avail of accurate seismic response parameters. However, the effect of variations in soil stiffness and/or deck torsional rigidity applied in the analysis is rather small, compared to the inclusion/exclusion of the model feature. Moreover, it is also observed that using acceleration scaling leads to much larger scatter in the results than when velocity spectral intensity scaling is used. Finally, the results from two particular earthquakes, Friuli and El Centro, highlight the peril of using a small number of records selected without due consideration to the relationship between their wave form, predominant periods and spectral shapes on the one hand and the response periods of the structure on the other.

This paper summarizes the results of a comprehensive statistical study of inelastic displacement ratios that allow the estimation of maximum lateral inelastic displacement demands from maximum elastic displacement demands for structures built on soft soil sites. These ratios were computed for single-degree-of-freedom systems undergoing six levels of inelastic deformation when subjected to 116
earthquake ground motions recorded on bay-mud sites of the San Francisco Bay Area and on sites in the former lake-bed zone of Mexico City. These soft soil deposits are characterized by low shear wave velocities, high water contents, and high plasticity indices. The influence of period of vibration normalized by the predominant period of the ground motion, the level of inelastic deformation, earthquake magnitude, and epicentral distance are evaluated and discussed. Mean inelastic displacement ratios and their corresponding dispersion are presented. The effect of stiffness degradation on inelastic displacement ratios is also considered. For this purpose, mean ratios of
maximum inelastic displacement demands of stiffness degrading systems to maximum inelastic displacement demands of nondegrading systems are presented. Finally, a simplified equation to estimate mean inelastic displacement ratios obtained through nonlinear regression analyses is provided to aid designers estimate inelastic displacement demands of structures built on soft soil sites.

Results of a detailed statistical study of constant relative strength inelastic displacement ratios to estimate maximum lateral inelastic displacement demands on existing structures from maximum lateral elastic displacement demands are presented. These ratios were computed for single-degree-of-freedom systems with different levels of lateral strength normalized to the strength required to remain elastic when subjected to a relatively large ensemble of recorded earthquake ground motions. Three groups of soil conditions with shear wave velocities higher than 180m/s are considered. The influence of period of vibration, level of lateral yielding strength, site conditions, earthquake magnitude, distance to the source, and strain-hardening ratio are evaluated and discussed. Mean inelastic displacement ratios and those associated with various percentiles are presented. A special emphasis is given to the dispersion of these ratios. It is concluded that distance to the source has a negligible influence on constant relative strength inelastic displacement ratios. However, for periods smaller than 1s earthquake magnitude and soil conditions have a moderate influence on these ratios. Strain hardening decreases maximum inelastic displacement at a fairly constant rate depending on the level of relative strength for periods of vibration longer than about 1.0s while it decreases maximum inelastic displacement non-linearly as the period of vibration shortens and as the relative-strength ratio increases for periods of vibration shorter than 1.0s. Finally, results from non-linear regression analyses are presented that provide a simplified expression to be used to approximate mean inelastic displacement ratios during the evaluation of existing structures built on firm sites. Copyright © 2003 John Wiley & Sons, Ltd.

The Southern California Seismic Network (SCSN) has recently installed seismic stations in two buildings on the Caltech campus (Millikan Library and the Broad Center). Continuous real-time accelerometer data from these structures are now freely available to the community. This dataset provides a new opportunity to observe, and better understand, the variances in the primary dynamic property of a building system, its natural frequencies. Historical data (triggered strong-motion re-cords, ambient and forced vibration tests) from the well-studied Millikan Library show dramatic decreases in natural frequencies, attributed mainly to moderately large local earthquakes. The current forced vibration east–west fundamental frequency is 22% lower than that originally measured in 1968. Analysis of the new continuous data stream allows the examination of other previously unrecognized sources of measurable change in the fundamental frequencies, such as weather (wind, rain, and temperature), as well as nonlinear building vibrations from small local and moderate regional earthquakes. Understanding these nonlinear shifts is one of the long-term goals of real-time building instrumentation and is critical if these systems are to be used as a postearthquake damage assessment tool.