NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July -, Anchorage, Alaska INELASTIC DEFORMATION RATIOS FOR NONSTRUCTURAL COMPONENTS SUBJECTED TO FLOOR ACCELERATIONS J. C. Obando and D. Lopez-Garcia ABSTRACT Nonstructural components (NSCs) located at the upper floors of multistory building structures are not subjected to the ground accelerations but to the accelerations at the floor to which they are attached. Since the frequency content of floor accelerations can be very different from that of ground accelerations, the well-known properties of the inelastic deformation ratio (IDR) of ground accelerations might not be applicable to floor accelerations. Although the inelastic response of NSCs in multistory building structures has been analyzed in a few investigations, the IDR of SDOF systems subjected to floor accelerations has not been comprehensively characterized. The objective of this study is to present the preliminary findings of a research effort aimed at investigating in detail the properties of the IDR of SDOF systems subjected to floor accelerations. It was found that: (a) the spectral shape of floor accelerations is not made up of acceleration-, velocity- and displacement-sensitive spectral regions; (b) when the natural period of the SDOF system is long, the IDR is still essentially equal to unity (i.e., the equal displacement rule still applies); (c) when the natural period of the SDOF system is short, the IDR does not increase monotonically as the natural period of the SDOF decreases, but exhibits local minima at the modal periods of the building; and (d) the IDR remains constant when the damping ratio varies between % and %. Finally, an existing analytical expression of IDRs of ground accelerations was examined, and its application to floor accelerations was evaluated. An improved equation, valid only for accelerations at the roof level, is then proposed. Ph.D. Candidate, Dept. of Structural & Geotechnical Eng., Pontificia Universidad Catolica de Chile, Chile Associate Professor, Dept. of Structural & Geotechnical Eng., Pontificia Universidad Catolica de Chile, Chile Obando JC, Lopez-Garcia D. Inelastic deformation ratios for nonstructural components subjected to floor accelerations. Proceedings of the th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK,.
NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July -, Anchorage, Alaska Inelastic Deformation Ratios for Nonstructural Components subjected to Floor Accelerations J. C. Obando and D. Lopez-Garcia ABSTRACT This study reports the preliminary findings of a research effort aimed at investigating in detail the properties of the inelastic deformation ratio of SDOF systems subjected to floor accelerations. It was found that: (a) the spectral shape of floor accelerations is not made up of acceleration-, velocity- and displacement-sensitive spectral regions; (b) when the natural period of the SDOF system is long, the IDR is still essentially equal to unity (i.e., the equal displacement rule still applies); (c) when the natural period of the SDOF system is short, the IDR does not increase monotonically as the natural period of the SDOF decreases, but exhibits local minima at the modal periods of the building; and (d) the IDR remains constant when the damping ratio varies between % and %. Finally, an existing analytical expression of IDRs of ground accelerations was examined, and its application to floor accelerations was evaluated. An improved equation, valid only for accelerations at the roof level, is then proposed. Introduction One of the main lessons learned from major recent earthquakes worldwide is that proper construction of building structures designed according to modern, rational seismic codes is, although certainly necessary, not quite sufficient to ensure: (a) building functionality after the earthquake; and (b) significant reductions of economic losses. Such objectives require that nonstructural components (from now on referred to simply as NSCs) be seismically designed as well. A particularly representative example is the recent Chile earthquake, which caused significant economic losses and where a large number of buildings (some of them of critical importance, such as hospitals and the Santiago International Airport) lost their functionality despite overall satisfactory structural performance []. The observation made in the former paragraph is the motivation of a sizable amount of research related to the seismic analysis and design of NSCs. Although significant progress has been made in recent years, some issues have not been thoroughly investigated. For example, in most of previous studies, NSCs were modeled as linear SDOF systems, but in the current seismic design approach, NSCs are expected to behave inelastically when subjected to the design earthquake loads []. Since former studies on the inelastic response of NSCs are very limited [, Ph.D. Candidate, Dept. of Structural & Geotechnical Eng., Pontificia Universidad Catolica de Chile, Chile Associate Professor, Dept. of Structural & Geotechnical Eng., Pontificia Universidad Catolica de Chile, Chile Obando JC, Lopez-Garcia D. Inelastic deformation ratios for nonstructural components subjected to floor accelerations. Proceedings of the th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK,.
], further research is still needed. For instance, the inelastic deformation ratio, defined as the ratio of the peak inelastic displacement response to the peak displacement response of the corresponding linear system [], has not been comprehensively characterized for NSCs located at upper floors of multistory buildings. In contrast, the inelastic deformation ratio (from now on, IDR) of SDOF systems subjected to ground accelerations has been thoroughly investigated, and is still a topic of current research []. In the particular case of lightly damped elasto-plastic systems, all the former studies indicate, in a qualitatively way, the same characterization of the IDR. When the natural period T n of the SDOF system is long, the value of the IDR is essentially equal to unity regardless of the value of the response modification factor R (i.e., the well-known equal displacement rule ). When the natural period T n of the SDOF system is short, on the other hand, the value of the IDR increases monotonically as T n decreases and as R increases. Quantitatively, these studies differ on the quantification of the influence of R and T n on the IDR when T n is short, and differ on the value of the period T c up to which T n is short and beyond which T n is long. In the context of spectral regions [], however, most studies agree that T c is the period separating the acceleration- and velocity-sensitive regions. Since, as it will be shown later, the frequency content of floor accelerations is typically quite different from that of ground accelerations, the characterization of the IDR described in the former paragraph might not be valid for NSCs located at upper floors of multistory buildings. This is the motivation of an ongoing research program, and the objective of this paper is to present some preliminary (yet meaningful) findings. Description of the ground acceleration Modeling and Analysis Procedure The ground acceleration was modeled as a nonstationary random process defined by a modified Kanai-Tajimi stationary power spectral density function [] and a modulating time function [, ]. The values of the parameters of the modified Kanai- Tajimi function were set in such a way that the frequency content of the resulting process is similar to typical frequency contents of actual far-field seismic accelerations recorded on firm soil conditions. Realizations () of the process were generated following standard simulation techniques []. Median pseudoacceleration response spectra can be seen in Fig. (a). Description of the building models A total of building structures were considered. The first three structures are the -, - and - story steel building models developed for the SAC Phase II Steel Project considering the seismic hazard corresponding to Los Angeles, US []. Aditionally, three RC wall structures having, and stories were also considered. The characteristics of these structures are typical of traditional RC wall structures built in Chile []. Finally, two RC dual wall-frame structures having and stories were also considered. The characteristics of these structures are representative of more recent (and current) RC structures built in Chile [].
In the models of the steel structures, only the north-south perimeter frames were considered. They are steel moment-resisting frames, and are the sole lateral-load resisting system along the north-south direction. In the models of the RC structures, all the lateral-load resisting planes along one of the principal directions were considered, effective (rather than gross crosssection) member stiffness was considered based on recommendations found in the literature [], and diaphragm constraints were incorporated at each floor level. All buildings were modelled as D frame structures made up of -DOFs frame elements. Rigid offsets were considered at all joints. The lateral floor displacements were considered as the only DOFs, and the corresponding condensed stiffness matrices were obtained through static condensation techniques. Mass matrices were then set equal to diagonal matrices whose elements are equal to the seismic masses associated to each floor level. Damping matrices were obtained following the Clough- Penzien procedure []. Modal damping ratios of the steel structures were set equal to %, and those of the RC structures were set equal to %. In all models, buildings were assumed to behave in a linearly elastic manner. Description of the analysis procedure The floor acceleration at a given floor was assumed equal to the absolute acceleration response at the same floor. At each floor of each building model, a floor acceleration history was then obtained for each realization of the ground acceleration through time history analysis. The displacement response of a perfectly elasto-plastic SDOF system (defined by T n, R and the damping ratio ) and that of the corresponding linear system (same values of T n and ) were then obtained for each floor acceleration history through time history analysis. A sample value of the IDR was obtained as the ratio of the peak inelastic response to the peak elastic response. Finally, the IDR was then set equal to the median of the sample values. In other words, values of the IDR were obtained through Monte Carlo simulation. IDRs were calculated for a wide range of values of T n and for =.. Some results for =. (more typical of NSCs) are shown in the last part of the paper. The procedure described in the former paragraph is sometimes called the cascade approach, and is accurate provided that the mass of the NSC is much less than the mass of the structure []. This observation means that the IDRs described in this paper are valid only for light NSCs. Floor Accelerations vs. Ground Accelerations An example of the differences between the frequency content of floor accelerations and that of ground accelerations is illustrated in Fig., which shows median pseudo-acceleration spectra. It can be seen that, unlike the ground spectrum, floor spectra are characterized by local maxima at the modal periods of the building. Further, the values of these maxima differ from floor to floor, and the modal period at which the spectrum reaches the global maximum is in general different at each floor as well. For instance, the global maximum of the first floor spectrum is the one associated with the second mode, and the third-mode local maximum is slightly greater than that of the first mode. The global maximum of the third floor spectrum, on the other hand, is the one associated with the first mode, and the third-mode local maximum is hardly noticeable.
Sa (g) Sa (g) Sa (g) Sa (g).. Ground acceleration (a).. -story steel building st floor (b)........................ -story steel building nd floor (c).. -story steel building rd floor (d)...................... Figure. Median pseudo-acceleration spectra ( =.) Another important difference is revealed when spectra are shown in tripartite plots. As expected, the ground spectrum can be characterized in terms of acceleration-, velocity- and displacement-sensitive regions (Fig., left). Such characterization, however, does not apply to typical floor spectra (Fig., right). When the value of T n is neither too short or too long, none of the response quantities (i.e., pseudo-acceleration, pseudo-velocity or displacement) remains essentially constant at any range of values of T n, which is clearly a consequence of the complex spectral shape due to the local maxima. Qualitative Description Inelastic Displacement Ratios of Floor Accelerations Typical examples of IDRs of floor accelerations can be seen in Fig.. In accordance with common notation, the IDR is denoted C d []. It was found that the value of C d is still essentially equal to unity when T n is greater than both T c (it will be shown later that the value of T c of the ground acceleration considered in this study is. s) and the fundamental period of the
Pseudo-velocity (m/s) Pseudo-velocity (m/s) building. As T n decreases, however, the value of C d does not increase monotonically, as in the case of ground accelerations. Rather, there are local minima at the modal periods of the building. Further analysis revealed that such local minima are the consequence of the shape of displacement spectra of floor accelerations. The elastic spectra exhibit sharp local maxima, particularly at the fundamental period of the building, whereas such maxima are absent in the inelastic spectra. A representative example is shown in Fig. (left). In contrast, the elastic spectrum of the ground acceleration does not exhibit local maxima (Fig., right)..... m E- m Spectrum Peak ground response g. m g. m. m. g Ground acceleration T c =.s. g. g. g.. m... (b). m E- m Spectrum Peak floor response g. m g. m. m. g. g. g. g.. m -story steel building th floor Fig.. Tripartite plots of median spectra ( =.) -story RC wall building th floor - R =........... -story steel building th floor - R =........... Fig.. IDRs of floor accelerations ( =.). Vertical dashed lines indicate the first modal periods of the building. Quantitative Characterization Although the qualitative characterization described above was found to be of general validity,
Displacement (m) Displacement (m) significant quantitative differences were found not only among different buildings but also among different floors of a given building, and the development of an analytical expression of C d was found very challenging. Preliminarily, however, an effort was made to evaluate if existing expressions of C d valid for ground accelerations could also be used, at least in an approximate manner, to predict values of the C d factor of floor accelerations.. -story RC wall building th floor. Ground acceleration... Elastic Inelastic.............. Fig.. Median elastic and inelastic (R = ) displacement response spectra ( =.). Vertical dashed lines indicate the first modal periods of the building. Of the many expressions of C d presented in the literature, the one proposed in [] was selected because it was shown to be valid for inelastic SDOF bilinear systems subjected to virtually any kind of ground acceleration, provided that the post-yielding stiffness ratio is small and period T c is correctly characterized. For the particular case of %-damped, perfectly elastoplastic SDOF systems and a constant value of R, factor C d is given by []: C d R. Tn. T c. () According to [], it is necessary to characterize the spectral regions of the seismic acceleration in order to correctly evaluate the value of T c. However, it has also been shown that period T c coincide with the period associated with the main frequency of the seismic acceleration, being the main frequency the one at which the power spectral density function reaches its maximum value []. Based on this criterion, the value of T c of the ground acceleration considered in this study is. s (Fig. ). This value is also shown in Fig. (left), where it can be seen that, indeed, it essentially coincides with the period separating the acceleration- and velocity-sensitive spectral regions. Comparisons between values of factor C d of the ground acceleration considered in this study and values of C d indicated by Eq. with T c =. s can be seen in Fig., where the good agreement is evident. In passing, it must be mentioned that such agreement indicates that the ground acceleration considered in this study is indeed realistic, at least in a median sense.
Normaliced PSD........s Frequency (rad/s) Fig.. Normalized power spectral density of the ground acceleration Simulations Eq. [] R =........... R =........... Fig.. Comparison between empirical and analytical values of factor C d of the ground acceleration ( =.). Possible application of Eq. to floor accelerations was then evaluated. As described before, floor spectra do not exhibit spectral regions similar to those of ground spectra, hence the value of T c could not be defined as the period separating the acceleration- and velocity-sensitive spectral regions. The main frequency of floor accelerations was then examined. It was found that the power spectral density function of floor accelerations exhibit sharp local maxima at the modal frequencies, and that the mode whose frequency is the main frequency (i.e., the frequency associated with the global maximum) is in general different at each floor of each building, and cannot be predicted without performing frequency domain analysis. Hence, period T c could not defined, at least not in a practice-oriented manner, in terms of the period associated with the main frequency of a floor acceleration. However, it was also found that the main frequency of the acceleration at the roof level is in general equal to the fundamental frequency of the building.
Therefore, the application of Eq. to floor accelerations, with T c equal to the fundamental period of the building, was evaluated only at the roof level of each building. Examples can be observed in Figure. The accuracy of Eq. was found to vary widely: slightly unconservative in some cases (Fig., top), slightly conservative in some other cases (Fig., middle) and very conservative in the remaining cases, especially when the fundamental period of the building is relatively long and the value of R is relatively high (Figure, bottom). Motivated by these findings, Eq. was then modified in order to get a more accurate expression. Based on regression analysis, the following expression is proposed:. T c T c n. T. sin. R. () It is recalled that T c in Eq. is equal to the fundamental period of the building, and that Eq. is valid only for floor accelerations at the roof level. Results show that the proposed equation is conservative, slightly so in most cases (Fig., top and middle). When it is definitely conservative, it is still less so than Eq (Fig., bottom). Influence of the damping ratio So far the damping ratio of NSCs has been set equal to %. It was necessary to do so in order to fairly examine the applicability of Eq., which is valid for =.. However, the damping ratio of actual NSCs is typically equal to a somewhat lesser value, usually assumed as %. In order to evaluate whether the observations described in this paper are still valid for NSCs having a more realistic damping ratio, values of factor C d were calculated considering =.. Representative results are shown in Fig., where it can be seen that they are essentially identical to those valid for =.. Since there no reasons to believe that intermediate (i.e.,..) damping ratios might lead to different values of factor C d, it can be safely assumed that all the observations made in this paper are in fact valid for... Summary Preliminary results of a study on inelastic deformation ratios (IDRs) of SDOF nonstructural components (NSCs) subjected to floor accelerations indicate that the relationship between the IDR and the natural period of the NSC is somewhat different from that typical of SDOF systems subjected to ground accelerations. Such differences were found to be significant in some cases. The accurate characterization of such differences is the objective of continuing research. Acknowledgements The research reported in this paper was supported by the Vicerrectoria de Investigacion of the Pontificia Universidad Catolica de Chile through a Instructor Becario scholarship, and by the Comision Nacional de Investigacion Cientifica y Tecnologica CONICYT and the Agencia de Cooperacion Internacional de Chile AGCI: CONITYT-AGCI/Becas Doctorado Nacional para Estudiantes Extranjeros/ Convocatoria.
-story RC wall building th floor - R = -story RC wall-frame building th floor, R = Simulations Eq. [] Eq. (proposed)................... -story steel building rd floor - R = -story RC wall building th floor - R =................ -story steel building th floor - R = -story steel building th floor - R =................ Fig.. Comparison between empirical and analytical values of factor C d of floor accelerations ( =., T c = fundamental period of the building).
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