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A Multi-Mode Method for Estimation of Floor Response Spectra
Abstract
A new simplified procedure for estimation of floor response spectra (FRS) is proposed. This methodology enriches the most common procedures using nonlinear response-history analysis to predict FRS by including a direct multi-mode technique to estimate FRS. A novel feature of the procedure is that the coupling effect is considered to establish equivalent modal systems and the FRS are developed by incorporating capacity spectrum method in conjunction with ductility-based FRS for each modal system. Both the proposed method and the traditional method are applied to three steel moment frame structures, and a reasonable accuracy is demonstrated.
... In reality, higher-order vibration modes significantly influence the seismic responses of multistory buildings (Chopra 2007). Although modified ESDOF systems based on modal pushover analysis (Han and Chopra 2006;Han et al. 2010) can incorporate the effects of higher-order vibration modes, challenges persist due to the coupling effects among different vibration modes when structures reach a highly nonlinear state (Pan et al. 2017(Pan et al. , 2018. Consequently, the aforementioned ESDOF-based approaches may introduce substantial errors, either underestimating or overestimating the seismic responses of the corresponding structures (Kuramoto et al. 2000;Aschheim and Browning 2008). ...
... the Gauss-Lobato integration scheme (Taucer et al. 1991;Spacone et al. 1996a, b), similar to the approach for modeling RC frames. The validity of the numerical simulation for the beam-column model of this 3-story steel frame was established by Pan et al. (2018). ...
... numerical simulation for the beam-column model of this 9-story steel frame was confirmed by Pan et al. (2018). ...
A novel and efficient method based on shear models considering hysteretic characteristics is proposed for predicting structural seismic responses. This method simplifies an actual building by representing it as a lumped mass shear model, with a set of tunable parameters allocated to the interstory restoring force model of each floor. The shear model is calibrated by matching the cyclic interstory pushover curves between the equivalent inelastic spring of each floor and the refined beam–column element model using a metaheuristic optimization algorithm. The novelty of the proposed method lies in its consideration of both cyclic envelopes and hysteretic characteristics (stiffness and strength deterioration and pinching behavior) and its automatic parameter calibration. Validation of the parameter calibration procedure is performed by comparing it with empirical methods via the application on three lateral load tests of reinforced concrete (RC) columns that exhibit varying degrees of hysteretic degradation. The efficiency and accuracy of the proposed method are confirmed through four illustrative examples, including the seismic response predictions of a bare RC frame, two steel frames, and an infilled wall RC frame. Despite the relatively large errors in the acceleration response predictions, the results demonstrate that the proposed method can accurately and efficiently predict the displacement and velocity responses.
... A Rayleigh damping ratio of 5% is applied to the MDOF system. The accuracy of the numerical simulation for the steel frame has been verified in Pan et al. [65]. ...
Assessing the structural collapse capacity efficiently and accurately is a key issue in earthquake engineering. Here, an efficient framework for structural seismic collapse capacity assessment based on an accurate equivalent single-degree-of-freedom (ESDOF) system is proposed. The corresponding ESDOF system is calibrated by matching the cyclic static pushover (SPO) curves of a more complex multi-degree-of-freedom (MDOF) system (e.g., high-fidelity finite element models) through a meta-heuristic optimization method. In this way, both the backbone curve and hysteretic characteristics (the stiffness and strength deterioration, and pinching behavior) of the complex MDOF system are considered in the corresponding ESDOF system. Then, incremental dynamic analysis (IDA) is carried out to assess the structural seismic collapse capacity using the ESDOF system instead of the corresponding MDOF system to improve the computational efficiency. The efficiency and accuracy of the proposed framework have been validated through three case studies, including a bare reinforced concrete (RC) frame, a steel frame, and an infilled wall RC frame. These results affirm that the proposed framework can accurately and efficiently assess the collapse capacity of low-rise bare and infilled RC frame structures, as well as steel frame structures.
... Qin et al. [26,27] established the floor response spectrum method of a main structure based on the random vibration method, developed a seismic action calculation program for the nonstructural components on the basis of multistorey buildings, and proposed design suggestions for seismic action calculations of the nonstructural components in the relevant Chinese code. Xiaolan Pan [28] proposed a multimode method for the estimation of floor response spectra on the basis of the seismic response analysis of multistorey steel frame buildings in Los Angeles. This method takes the coupling effect into consideration, establishes equivalent modal systems, and develops floor acceleration response spectra by incorporating the capacity spectrum method in conjunction with ductility-based floor response spectra for each modal system. ...
In recent years, earthquake disasters have seriously damaged nonstructural components, so it is necessary to study their seismic performance. However, the existing scholarly research mainly concentrates on multistorey and high-rise buildings, and there are still deficiencies in the analysis of the seismic performance of the nonstructural components in large-span structures under seismic action. In this paper, the acceleration responses of a single-layer spherical reticulated shell structure are compared with those described in the current seismic design codes of the nonstructural components, and it is found that the current codes are not fully applicable to the seismic design of the nonstructural components in reticulated shell structures. The calculation formulas of the acceleration response spectra of single-layer spherical shell nodes are theoretically derived, and the shell node acceleration response spectra are affected by higher-order modes, orthogonal horizontal seismic input directions, and the membrane stiffness of the shell nodes. The variations in the acceleration responses of the shell nodes with node position and rise-to-span ratio are analysed, and a design method for the equivalent seismic action of the nonstructural components in a single-layer spherical reticulated shell with a roofing system is proposed.
... Seismic analysis, assessment and design of acceleration-sensitive non-structural components (NSCs) have gained prominence over the past several years, primarily due to the large contribution which they can have on building repair costs and injuries. Consequently, a significant number of relevant journal papers was recently published (e.g., Petrone et al. 2015Petrone et al. , 2016Vukobratović and Fajfar 2015, 2017Lucchini et al. 2017;Pan et al. 2018;Surana et al. 2018a, b;Filiatrault et al. 2018;Anajafi and Medina 2019a;Merino et al. 2020;Asgarian and McClure 2020a, b;Anajafi et al. 2020;Kazantzi et al. 2020a), which represent an important addition to the studies performed and published in the past. However, there is always a need for further investigations using data obtained in instrumented and experimentally examined buildings to evaluate the numerous previous findings, assumptions and approximations. ...
Significant attention has been recently paid to the determination of floor acceleration demands used for the design of acceleration-sensitive non-structural components in buildings. This study makes a contribution through the use of experimental shake table data from one 6-storey and two 3-storey reinforced concrete buildings tested at the E-Defense testing facility. All buildings were exposed to multiple excitations, and floor acceleration demands were studied through peak floor accelerations and floor response (acceleration) spectra. The obtained ratios of peak floor to peak ground accelerations confirmed findings from previous studies. The ratio distributions along the height provided in Eurocode 8, ASCE 7-16 and NZS 1170.5 were found to be conservative. Besides the known properties of floor response spectra, the results for two of the buildings revealed that in some cases, spectra can have two prominent peaks corresponding to the fundamental mode period of different damage states or degree of nonlinearity during a single shaking event. A prerequisite for this occurrence, which in this paper is termed the “Elongated Fundamental Mode Effect”, is that due to the characteristics of the input motion, a significant part of the energy is involved both before and after the change of structural response nature, from (mostly) linear elastic to nonlinear. Among others, such change is perceptible through an extension of the fundamental period. The Elongated Fundamental Mode Effect was investigated by using scalograms generated through the Continuous Wavelet Transform, which was found to be an efficient tool for the visualization of energy localization in the time–frequency domain.
... where i T , i , and i are the modal period, damping ration, and participation factor of mode i, ij is the mode shape for floor j of mode i, PFAij is the peak floor acceleration on floor j for mode i, i AMP is the empirical amplification factor for the considered mode i, as defined in Vukobratović and Fajfar (2017). Pan et al. (2017bPan et al. ( , 2018 proposed a method using equivalent SDOF (ESDOF) systems to estimate the FRS of an MDOF system based on modal pushover analysis (MPA). The FRS of each mode are obtained through the nonlinear dynamic analysis of each ESDOF system, and the total FRS is calculated for the considered modes using the SRSS combination rule. ...
Nonstructural components (NSCs) are parts, elements, and subsystems that are not part of the primary load-bearing system of building structures but are subject to seismic loading. Damage to NSCs may disrupt the functionality of buildings and result in significant economic losses, injuries, and casualties. In past decades, extensive studies have been conducted on the seismic performance and seismic design methods of NSCs. As the input for the seismic design of NSCs, floor response spectra (FRS) have attracted the attention of researchers worldwide. This paper presents a state-of-the-art review of FRS. Different methods for generating FRS are summarized and compared with those in current seismic design codes. A detailed review of the parameters influencing the FRS is presented. These parameters include the characteristics of ground motion excitation, supporting building and NSCs. The floor acceleration response and the FRS obtained from experimental studies and field observations during earthquakes are also discussed. Three RC frames are used in a case study to compare the peak floor acceleration (PFA) and FRS calculated from time history analyses (THA) with that generated using current seismic design codes and different methods in the literature. Major knowledge gaps are identified, including uncertainties associated with developing FRS, FRS generation methods for different types of buildings, the need for comprehensive studies on absolute acceleration, relative velocity, and relative displacement FRS, and the calibration of FRS by field observations during earthquakes.
... To this day, a significant number of attempts have been made to study the floor acceleration demands in RC bare frame buildings [1][2][3][4][5][6][7][8]. The crucial parameters affecting floor accelerations have already been identified including the frequency content of the ground motion [9][10][11][12], the dynamic characteristics of the supporting structure (the "building structure" throughout this article is referred as the "supporting structure"), the level of nonlinearity (inelasticity) of the supporting structure [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], and both the period and damping of the NSC [2][3][4][5][6]9,12,21]. Amplification of Peak Floor Acceleration (PFA) along the height of the building has been identified to be governed by the dynamic characteristics (i.e. frequency and mode shape) of the supporting structure [4,5,9,11,18]. ...
The effect of unreinforced masonry infills on the floor acceleration response of inelastically responding mid-rise RC frame buildings is studied using incremental dynamic analyses. Two structural building configurations, uniformly infilled, and with open ground storey are analyzed by using the FEMA P695 far-field ground-motion suite. It is observed that the effect of structural nonlinearity is much more pronounced in case of infilled RC frame buildings than bare RC frame buildings. Sequential failure of infills results in a significant elongation of period of vibration resulting in the shifting of peaks in floor spectrum towards longer periods. Modified floor spectral amplification
functions are presented for both uniformly infilled and open ground storey RC frame buildings. The proposed spectral amplification functions are validated using nonlinear analysis of typical buildings for recorded as well as spectrum-compatible ground-motions and can be used to estimate the floor response spectra directly from the code-based or site-specific design spectra.
... However, the code models significantly underestimate the component amplification (a p ), for periods close to higher modes of vibration of the supporting structure, as these models completely ignore the peaks in FRS corresponding to higher modes. This observation is in good agreement with previous studies on regular multistorey buildings (eg, Weiser et al, Lucchini et al, and Surana et al,[36][37][38] Rodriguez et al,49 and Vukobratović and Fajfar and Pan et al[52][53][54] ). Further, the component amplification (a p ) at the FE of a lower floor (Z/H = 0.28) is approximately 50% higher in comparison with the corresponding value at the CR (and also at the SE) under tuned conditions with the second and the kth modes of vibration. ...
A suite of reinforced-concrete frame buildings located on hill sides, with two different structural configurations, viz. step-back and split-foundation, are analyzed to study their floor response. Both step-back and split-foundation structural configurations lead to torsional effects in the direction across the slope due to the presence of shorter columns on the uphill side. Peak floor acceleration and floor response spectra are obtained at each storey’s centre of rigidity and at both its stiff and flexible edges. As reported in previous studies as well, it is observed that the floor response spectra are better correlated with the ground response spectrum. Therefore, the floor spectral amplification functions are obtained as the ratio of spectral ordinates at different floor levels to the one at the ground level. Peaks are observed in the spectral amplification functions corresponding to the first two modes in the upper portion of the hill-side buildings, whereas a single peak corresponding to a specific kth mode of vibration is observed on the floors below the uppermost foundation level. Based on the numerical study for the step-back and split-foundation hill-side buildings, simple floor spectral amplification functions are proposed and validated. The proposed spectral amplification functions take into account both the buildings’ plan and elevation irregularities and can be used for seismic design of acceleration-sensitive non-structural components, given that the supporting structure’s dynamic characteristics, torsional rotation, ground-motion response spectrum and location of the non-structural components within the supporting structure are known, since current code models are actually not applicable to hill-side buildings.
In the present study, an effective analysis framework is developed for generating the floor response spectra (FRS) of nuclear power plants based on the explicit time-domain method (ETDM), an efficient approach recently proposed for nonstationary random vibration analysis of large-scale complex structures. The FRS obtained by the present approach are fully consistent with the code-specified ground response spectra (GRS), and the problems encountered in the traditional time-history method and direct spectra-to-spectra method can be completely avoided. The FRS of a real nuclear power plant are generated to demonstrate the feasibility of the proposed method to engineering practice.
In this study, the effects of inherent building torsion on the seismic response of acceleration-sensitive non-structural components are investigated. To achieve this objective, a group of elastic torsionally irregular step-back reinforced concrete moment-resisting frame buildings are analyzed under bi-directional earthquake excitations. The floor torsional amplification factors, defined as the ratio between the floor spectral ordinate at the flexible/stiff edge to the floor spectral ordinate at the center of rigidity, are obtained as a function of the tuning ratio for varying damping ratios of non-structural components. The correlations of the peak torsional amplification factors at different floors for the rigid and flexible non-structural components are studied with their floor displacement-based torsional irregularity indices, recommended in the national building codes of the United States and India. It is observed that the torsional amplification factors are building characteristics and tuning ratio-dependent. These torsional amplification factors are further observed to be well-correlated with the corresponding floor displacement-based torsional irregularity indices for both the rigid and flexible non-structural components. Contrarily, the torsional amplification factors for the very flexible non-structural components tend to unity and are thereby observed to be independent of the characteristics of both the building and the non-structural components. Simplified and numerically validated, floor displacement-based models are proposed to compute the torsional amplification factors, which can be used in aggregation with the existing codes to design acceleration-sensitive non-structural components in torsionally irregular buildings.
In most current building codes, seismic design of Non-Structural Components (NSCs) is addressed through empirical equations that do not capture NSC response amplification due to tuning effects with higher and torsional modes of buildings and that neglect NSC damping. This work addresses these shortcomings and proposes a practical approach to generate acceleration NSC Floor Design Spectra (FDS) in buildings directly from their corresponding Uniform Hazard Spectra (UHS). The study is based on the linear seismic analysis of 27 reinforced concrete buildings located in Montréal, Canada, for which ambient vibration measurements (AVM) are used to determine their in situ three-dimensional dynamic characteristics. Pseudo Acceleration Floor Response Spectra (PA-FRS) are derived at every building floor for four different NSCs damping ratios. The calculated roof FRS are compared with the 5% damped UHS and a formulation is proposed to generate roof FDS for NSCs with 5% damping directly from the UHS.
A set of mid-rise bare and uniformly infilled reinforced-concrete frame buildings are analyzed for two different
seismic intensities of ground-motions (i.e., „Design Basis Earthquake‟ and „Maximum Considered Earthquake‟) to study their
floor response. The crucial parameters affecting seismic design force for acceleration-sensitive non-structural components are
studied and compared with the guidelines of the European and the United States standards, and also with the recently developed
NIST provisions. It is observed that the provisions of both the European and the United States standards do not account for the
effects of the period of vibration of the supporting structure and seismic intensity of ground-motions and thereby provides
conservative estimates of the in-structure amplification. In case of bare frames, the herein derived component amplification
factors for both the design basis earthquake and the maximum considered earthquake exceeds with their recommended values in
the European and the United States standards for non-structural components having periods in vicinity of the higher modes of
vibration, whereas, in case of infilled frames, component amplification factors exceeds with their recommended value in the
European standard for non-structural components having periods in vicinity of the fundamental mode of vibration, and only for
the design basis earthquake. As a consequence of these observations, as well as capping on the design force (in case of United
states standard and NIST provisions), in case of the design basis earthquake, the combined amplification factor is underestimated
for non-structural components having periods in vicinity of the higher modes of vibration of bare frames, and also for nonstructural components having periods in vicinity of the fundamental mode of vibration of infilled frames. At the maximum
considered earthquake demand, excepting non-structural components having periods in vicinity of the higher modes of vibration
of bare frames, all provisions generally provide conservative estimates of the design floor accelerations.
Severe damage to acceleration sensitive nonstructural components in recent earthquakes has resulted in unprecedented losses. Recent research has been aimed at increasing the understanding of acceleration demands on nonstructural components in buildings. This investigation subjects a set of four special moment resisting frame (SMRF) building models to a suite of 21 far-field ground motions using the incremental dynamic analysis procedure. Full three-dimensional models including floor slabs are used to extract both the horizontal and vertical responses. Floor acceleration response spectra are generated to assess the acceleration demands on elastic nonstructural components. Changes to the current code provisions that include the influence of structural period are proposed. An alternative design approach that directly amplifies the ground acceleration spectrum to achieve the desired floor acceleration spectrum is presented.
The objective of this article is to study the effects of structural nonlinear behavior on Floor Response Spectra (FRS) of existing reinforced concrete frames. This study examines how the FRS vary with the level of post-elastic behavior in buildings of different number of stories and masonry infill wall configurations. The effect of damping modeling assumptions is also investigated. Differences and similarities with findings from the literature are discussed. On the basis of the obtained results, a commentary on the adequacy of basic assumptions used in predictive equations proposed by different seismic codes is offered.
The modal pushover analysis (MPA) procedure, which includes the contributions of all significant modes of vibration, estimates seismic demands much more accurately than current pushover procedures used in structural engineering practice. Outlined in this paper is a modified MPA (MMPA) procedure wherein the response contributions of higher vibration modes are computed by assuming the building to be linearly elastic, thus reducing the computational effort. After outlining such a modified procedure, its accuracy is evaluated for a variety of frame buildings and ground motion ensembles. Although it is not necessarily more accurate than the MPA procedure, the MMPA procedure is an attractive alternative for practical application because it leads to a larger estimate of seismic demands, improving the accuracy of the MPA results in some cases (relative to nonlinear response history analysis) and increasing their conservatism in others. However, such conservatism is unacceptably large for lightly damped systems, with damping significantly less than 5%. Thus the MMPA procedure is not recommended for such systems.
It has been observed during previous earthquakes that the damage to operational and functional components of buildings often result in more injuries, fatalities and property damage than those inflicted by structural damage. Operational and functional components of a building include architectural components, mechanical and electrical equipment, as well as building contents. A rational approach to designing these elements against seismic excitations involves the use of floor design spectra. The development of such design spectra for buildings in Canada constitutes the objective of the paper. This objective was achieved by conducting comprehensive analyses of selected reinforced concrete buildings, with different lateral force resisting systems and building heights, under code compatible earthquake records for an eastern and a western Canadian city. It was observed that the floor response was significantly amplified, especially for buildings with short periods. Generally, the higher floors showed higher amplifications with differences in spectra between the floors being more pronounced in low-rise buildings and shear wall buildings with short fundamental periods. The results provided a large volume of data to generate floor response spectra for the design of operational and functional components of buildings in Canada. The details of the approach and the design spectra are presented in the paper.
A statistical analysis of the peak acceleration demands for nonstructural components (NSCs) supported on a variety of stiff and flexible inelastic regular moment-resisting frame structures with periods from 0.3 to 3.0 s exposed to 40 far-field ground motions is presented. Peak component acceleration (PCA) demands were quantified based on the floor response spectrum (FRS) method without considering dynamic interaction effects. This study evaluated the main factors that influence the amplification or decrease of FRS values caused by inelasticity in the primary structure in three distinct spectral regions namely long-period, fundamental-period, and short-period region. The amplification or decrease of peak elastic acceleration demands depends on the location of the NSC in the supporting structure, periods of the component and building, damping ratio of the component, and level of inelasticity of the supporting structure. While FRS values at the initial modal periods of the supporting structure are reduced due to inelastic action in the primary structure, the region between the modal periods experiences an increase in PCA demands. A parameter denoted as acceleration response modification factor (Racc) was proposed to quantify this reduction/increase in PCA demands. Copyright © 2007 John Wiley & Sons, Ltd.
This paper investigates the response of nonstructural components in the presence of nonlinear behavior of the primary structure using floor response spectra method (FRS). The effect of several parameters such as initial natural frequency of the primary structure, natural frequency of the nonstructural components (subsystem), strength reduction factor and hysteretic model have been studied. A database of 164 registered ground acceleration time histories from the European Strong-Motion Database is used. Results are presented in terms of amplification factor and resonance factor. Amplification factor quantifies the effect of inelastic deformations of the primary structure on subsystem response. Resonance factor quantifies the variation of the subsystem response considering the primary structure acceleration. Obtained results differed from precedent studies, particularly for higher primary structure periods. Values of amplification factor are improved. Obtained results of resonance factor highlight an underestimation of peak values according to current design codes such as Eurocode 8. Therefore a new formulation is proposed.
KeywordsNonlinear seismic behavior-Nonstructural components-Floor response spectra-Resonance factor-Amplification factor-Nonlinear time history analysis-Hysteretic model-Recorded earthquake ground motions
The peak acceleration demands for acceleration-sensitive nonstructural components supported on elastic and inelastic regular moment-resisting frame structures are statistically analyzed. The responses of a variety of stiff and flexible frame structures (with 3, 6, 9, and 18 stories) subjected to a set of 40 ground motions are evaluated. The nonstructural components under consideration are those that can be represented by single-degree-of-freedom systems with masses that are small compared to the total mass of the supporting structure. This study evaluates and quantifies the dependence of peak component accelerations on the location of the nonstructural component in the structure, the damping ratio of the component, and the properties of the supporting structure such as its modal periods, height, stiffness distribution, and strength. The results show that current seismic code provisions will not always provide an adequate characterization of peak component accelerations. Recommendations are provided to estimate peak acceleration demands for the design of nonstructural components mounted on inelastic frame structures.
The inability of current seismic design codes to reliably predict building performance during earthquakes, and to satisfy secondary goals of controlling property damage and maintenance of building function in moderate and frequent earthquakes, prompted the earthquake engineering community to set its future target at performance-based seismic design. Of various methods proposed for the development of design methodologies for future design codes, two reliability-based methods are considered in this study. The first part of this research focuses on a design approach based on the optimization of life cycle cost, which is the sum of the initial building cost and the estimated damage cost due to seismic events during the lifetime of the building. Two moment resisting steel frame buildings are considered for this study. Seismic damage costs are computed for a large set of simulated ground motion records using the concepts of HAZUS RTM 99 loss estimation methodology. Expected seismic damage costs for selected design lives of the buildings are obtained through Monte Carlo simulation. Optimum design strength of each building is selected by comparing life cycle costs of alternative designs. The results show that current codes should be made more stringent in order to achieve life cycle cost-optimum designs. Results of the optimization procedure are found to be sensitive to the selected design life and to two key parameters: cost sensitivity factor to weight change and discount rate. The second part of this research proposes a design checking methodology based on a probabilistic hysteretic energy demand criterion. Hysteretic energy demand is considered to be the best means for quantifying structural damage. Probabilistic seismic hazard analysis is performed by constructing uniform hazard spectra for hysteretic energy demand at a specific site using simulated ground motions. An equivalent single degree of freedom system-based methodology is adopted for using the uniform hazard spectra information in demand assessment of multi-degree of freedom structures. Finally, a deterministic equation is developed for checking the target probabilistic performance criterion for the building. The proposed procedure allows a designer to check a design for a target probabilistic criterion without performing any nonlinear time-history analysis or any uncertainty analysis.
This paper deals with floor acceleration spectra, which are used for the seismic design and assessment of acceleration-sensitive equipment installed in buildings. In design codes and in practice, not enough attention has been paid to the seismic resistance of such equipment. An ‘accurate’ determination of floor spectra requires a complex and quite demanding dynamic response history analysis. The purpose of the study presented in this paper is the development of a direct method for the determination of floor acceleration spectra, which enables their generation directly from the design spectrum of the structure, by taking into account the structure's dynamic properties. The method is also applicable to inelastic structures, which can greatly improve the economic aspects of equipment design. A parametric study of floor acceleration spectra for elastic and inelastic single-degree-of-freedom (SDOF) and multiple-degree-of-freedom structures was conducted by using (non)linear response history analysis. The equipment was modelled as an elastic single-degree-of-freedom system. The proposed method was validated by comparing the results obtained with the more accurate results obtained in a parametric study. Due to its simplicity, the method is an appropriate tool for practice. In the case of inelastic structural behaviour, the method should be used in combination with the N2 method, or another appropriate method for simplified nonlinear structural analysis. Copyright
The earthquake response problems of structure and of equipment mounted therein are examined both by time history and frequency domain analysis method. Remaining totally in the frequency domain, so-called ground and floor response spectra are developed and compared to results obtained by the more usual time-domain analysis. The study emphasizes throughout the Fourier spectral aspects of earthquake response analysis and tends in particular to elucidate some basic issues involved in the development of floor response spectra.
Floor response spectra, which are used for the seismic design of equipment, are often based on the assumption that the behaviour of a structure and its equipment is linearly elastic. Significant reductions in the peak values of floor acceleration spectra can be achieved if inelastic behaviour of the structure is taken into account. This paper presents the most important results of an extensive parametric study of floor acceleration spectra, taking into account inelastic behaviour of the structure, and linear elastic behaviour of the equipment. The structure and the equipment were modelled as single-degree-of-freedom systems. The influences of the input ground motion, ductility, hysteretic behaviour and the natural period of the structure, as well as that of damping of the equipment, have been studied. A simple practice-oriented method for direct determination of floor acceleration spectra from an inelastic spectrum for the structure and an elastic spectrum for the equipment is proposed and validated. In this method, the floor response spectra in the resonance region, where the natural period of the equipment is close to the natural period of the structure, are based on the empirical values obtained in the parametric study, whereas the spectra in the pre- and post-resonance regions are based on the principles of dynamics of structures. The method is intended for a quick estimation of approximate floor acceleration spectra.
This paper summarizes the results of a comprehensive study of floor acceleration demands on nonstructural components. The study is based on analytical studies on buildings responding elastically and nonlinearly to earthquake ground motions. Buildings analyzed range from 3 to 18 stories. Parameters evaluated include: fundamental period of vibration, lateral resisting system, damping ratio, level of ground motion intensity relative to the intensity that triggers nonlinear behavior. The study included the following aspects: (1) parametric study of peak floor acceleration demands in buildings responding elastically; (2) parametric study of floor response spectral ordinates in buildings responding elastically; (3) development of response spectrum analysis for estimating peak floor acceleration demands; (4) dynamic interaction between primary and secondary systems; and (5) peak floor acceleration demands in buildings responding nonlinearly. Results indicate that the amplitude of acceleration demands in buildings and their variation along the height are strongly dependent on the period of vibration, lateral resisting system and damping ratio of the building and therefore, current US recommendations that compute acceleration demands that are independent of the period, damping and lateral resisting are inadequate. It is also shown that alternative code procedures based on linear response spectrum modal analysis are equally inadequate. The most important trends are summarized.
In this paper, a parametric study is conducted in order to evaluate the seismic demand on light acceleration‐sensitive nonstructural components caused by frequent earthquakes. The study is motivated by the inconsistent approach of current building codes to the design of nonstructural components; the extensive nonstructural damage recorded after recent low‐intensity earthquakes also encouraged such a study.
A set of reinforced concrete frame structures with different number of stories, that is, 1 to 10 stories, are selected and designed according to Eurocode 8. The structures are subjected to a set of frequent earthquakes, that is, 63% probability of exceedance in 50 years. Dynamic nonlinear analyses are performed on the reference structures in order to assess the accuracy of the equations to predict seismic forces acting on nonstructural components and systems in Eurocode. It is concluded that the Eurocode equations underestimate the acceleration demand on nonstructural components for a wide range of periods, especially in the vicinity of the higher mode periods of vibration of the reference structures; for periods sufficiently larger than the fundamental period of the structure, instead, the Eurocode formulation gives a good approximation of the floor spectra. Finally, a novel formulation is proposed for an easy implementation in future building codes based on the actual Eurocode provisions. The proposed formulation gives a good estimation of the floor spectral accelerations and is able to envelope the floor spectral peaks owing to the higher modes. Copyright © 2014 John Wiley & Sons, Ltd.
The concept of how two techniques, Best Estimate Method and Evaluation Method, may be applied to the traditional seismic analysis and design of a nuclear power plant is introduced. Only the four links of the seismic analysis and design methodology chain (SMC) - seismic input, soil-structure interaction, major structural response, and subsystem response - are considered. The objective is to evaluate the compounding of conservatisms in the seismic analysis and design of nuclear power plants, to provide guidance for judgments in the SMC, and to concentrate the evaluation on that part of the seismic analysis and design which is familiar to the engineering community. An example applies the effects of three-dimensional excitations on a model of a nuclear power plant structure. The example demonstrates how conservatisms accrue by coupling two links in the SMC and comparing those results to the effects of one link alone. The utility of employing the Best Estimate Method vs the Evaluation Method is also demonstrated.
A detailed characterization of potential structural damage is essential to performance-based seismic design. The Park–Ang damage index is selected in this work as the seismic damage measure, since it is one of the most realistic measures of structural damage. Response spectra constitute the most common tool used for characterizing the seismic hazard at a site, and these spectra represent the demand on single-degree oscillators. To use these spectra for estimating the Park–Ang damage index demand on an MDOF system, three equivalent single-degree system-based approximate schemes are proposed. These schemes are tested on three moment resisting frames under several ground motion scenarios. The effectiveness of an equivalent system scheme is measured by comparing with the estimates from a nonlinear response-history analysis of the MDOF model. These schemes are tested for both global and storey-level damage indices. Variation of the non-dimensional parameter β is also considered in these case studies. Overall, all the three schemes are found to be effective with varying degrees of accuracy. The proposed methods are recommended for damage-based seismic design and performance evaluation of structures because these schemes can use response spectra for demand estimation and reduce computation cost.
Acceleration floor spectra curves are commonly used as seismic design inputs for light equipment and other secondary systems. Earlier the writer had presented a method for generation of floor spectra directly from prescribed ground spectra. Further complementary developments in this method are made. These are: (1)The development of the method for generation of floor spectra at structural frequencies (an important resonance case); and (2)evaluation of nonstationarity of earthquake motions on floor spectra curves. Also, some approximations made in earlier work are further evaluated. As a result of these developments and evaluations, a comprehensive procedure based on stochastic principles is presented in the paper for definition of seismic design inputs for secondary systems. This procedure avoids the use of often used and recently criticized spectrum-consistent time history as seismic input for generation of floor spectra curves.
A simple method is presented for generation of floor response spectra directly from a given ground response spectrum. A common practice to generate floor spectra is to carry out time-history analysis for a ground spectrum-consistent accelerogram. Unless an ensemble of such accelerograms is used the response results cannot be relied upon, but the use of an ensemble of accelerograms is cumbersome and costly. A method which can directly use a prescribed spectrum is preferred. The proposed method is based on transfer characteristics of the structure for random excitation, and makes use of the structural frequencies, mode shapes and participation factors, and the prescribed spectrum to generate floor spectra. In the paper, the use of the proposed method is demonstrated by generating floor spectra of a building. The proposed method is simple and especially convenient for use on digital computers.
An extensive parametric study is conducted with eight code-designed steel moment-resisting frames and a series of linear and nonlinear single-degree-of-freedom nonstructural components to investigate to what degree, how often, and under what conditions the nonlinear behavior of a building may amplify the seismic response of a nonstructural component attached to it. The study comprises the time-history analysis of each of the considered frames with one of the considered nonstructural components connected to it. In each case, an ensemble of 25 recorded earthquake ground motions is used, alternatively scaling them to three different intensity levels. The study also comprises a comparison between the nonstructural component responses obtained when the supporting structure is modeled alternatively as a nonlinear and a linear system. The influence of the nonstructural component location, nonlinearity, and damping ratio and the relative values of the building and component masses and natural frequencies is investigated. It is found that, in general, the nonlinear behavior of the supporting structures reduces the seismic response of the nonstructural components in comparison with the linear counterparts. In a few cases, however, the nonstructural component response is amplified by a factor that can be as large as 5.2. These cases correspond to components located on the lower building floors, with a natural period equal to the second or third natural period of the building, and subjected to a narrow-band excitation with a dominant period close to the building's fundamental natural period.
This paper presents two simplified methods for calculating the seismic force coefficients for flexible nonstructural components in building structures. The methods utilize the dynamic characteristics, expressed in terms of the fundamental periods and damping ratios, of the component and the supporting structure to calculate the force coefficient. The cases with the building period and component period known or unknown are considered. The formulas with less information tend to provide more conservative estimates of the force to cover the worst case situations. The new formulas now include the effect of possible resonance with higher modes. The validity of the proposed formulas is verified by a comprehensive numerical study of several buildings of different fundamental periods.
The paper examines the seismic design force formulas that are currently prescribed in the 2003 NEHRP Provision for the design of nonstructural components in buildings, and presents new formulas to improve the these recommendations. The current code provisions are building independent and overly conservative, especially for the design of nonstructural components in tall buildings. This conservatism can be reduced by utilizing the information about the fundamental period of the building. New force formulas that depend on the building period are proposed. The proposed formulas are intended to avoid more involved analyses. They are validated by a comprehensive numerical study of several buildings of different periods analyzed for an ensemble of recorded earthquake acceleration time histories.
Preliminary analyses are performed to obtain insight into the seismic response of light, acceleration sensitive nonstructural subsystems supported on structures that yield during severe earthquake ground motions. The effects of the severity of the inelastic deformations, of different hysteretic characteristics of the structure and of the amount of viscous damping of the subsystem are. thoroughly investigated. Current design recommendations for subsystems accounting for yielding of the supporting structure are assessed and found to be unconservative. An amplification factor is defined to quantify the effects of inelastic deformations of the supporting structure on subsystem response. Design guidelines are formulated for predicting the amplification factor based on statistical evaluation of the results generated for ten earthquake ground motions. Using these values, design floor response spectrums can be obtained from conventional linear elastic floor response spectrums accounting for yielding of the supporting structure without having to perform inelastic analysis.
The truncation of the high frequency modes, so commonly used in dynamic structural analysis, sometimes can cause significant errors in the calculated response; this is caused by the so-called missing mass effect. Such mode truncations can also affect the accuracy of floor spectra generated for stiff structural systems, and also for floors close to the base in even not so stiff structural systems, especially if the mode displacement methods are employed in the analysis. However, this problem due to mode truncation can be alleviated and virtually removed by employing the mode acceleration formulation in the analysis. Here, a direct response spectrum approach for generation of floor response spectra has been developed on the basis of the mode acceleration formulation and random vibration principles. As seismic input, this approach requires the relative acceleration spectra in lieu of the pseudo-acceleration spectra— so commonly used with the mode displacement approaches. Because the relative spectrum values become very small for frequencies higher than the highest frequency component in the design ground motion, such high structural frequencies need not be considered in the analysis. Numerical results demonstrating the effectiveness of the proposed new approach are presented.
This paper describes a method for the direct calculation of floor response spectra from ground response spectra. The procedure utilizes the Fourier transform of the ground movement. The mathematical derivations are given in detail, and the method is applied to the calculation of the floor response spectra for a structure which is a simplified model of the SUPER-PHENIX fast breeder reactor power plant.
An approximate method is proposed to estimate the seismic response of nonlinear nonstructural components attached to nonlinear building structures. The method is based on a previously developed procedure for the analysis of linear secondary systems mounted on a linear primary structure, the introduction of simplifying assumptions similar to those made in the derivation of the equivalent lateral force procedure for the seismic analysis of conventional buildings, and the use of strength reduction factors to account for the nonlinear behaviour of nonstructural components and supporting structure. Its application to any given nonstructural component only requires knowing the geometric characteristics, weights, and target ductilities of the nonstructural component and the structure to which it is connected, in addition to the fundamental natural period of the structure and the elastic response spectrum specified for the design of the structure. Presented also are a numerical example that illustrates the application of the method and the results of a comparative numerical study that is carried out to assess the method’s adequacy. Based on its simplicity and rationality and the results from the comparative study, it is concluded that the proposed method represents a simple but effective procedure for the seismic design of nonstructural components in buildings.
A new pushover analysis procedure derived through adaptive modal combinations AMC is proposed for evaluating the seismic performance of building structures. The methodology offers a direct multimode technique to estimate seismic demands and attempts to integrate concepts built into the capacity spectrum method recommended in ATC-40 1996, the adaptive method originally proposed by Gupta and Kunnath 2000 and the modal pushover analysis advocated by Chopra and Goel 2002. The AMC procedure accounts for higher mode effects by combining the response of individual modal pushover analyses and incorporates the effects of varying dynamic characteristics during the inelastic response via its adaptive feature. The applied lateral forces used in the progressive pushover analysis are based on instantaneous inertia force distributions across the height of the building for each mode. A novel feature of the procedure is that the target displacement is estimated and updated dynamically during the analysis by incorporating energy-based modal capacity curves in conjunction with constant-ductility capacity spectra. Hence it eliminates the need to approximate the target displace- ment prior to commencing the pushover analysis. The methodology is applied to two existing steel moment-frame buildings and it is demonstrated that the AMC procedure can reasonably estimate critical demand parameters such as roof displacement and interstory drift for both far-fault and near-fault records, and consequently provides a reliable tool for performance assessment of building structures.
In this paper a method to determine floor response sepctra is proposed which is based on the modal analysis of a support structure with interaction-free, one-degree-of-freedom system attached. The time-consuming methods using real or artificial soil accelerations are avoided as well as some of the arbitrarinesses in the approaches of Biggs or Kapur-Shao.
The ATC-40 and FEMA-274 documents contain simplified nonlinear analysis procedures to determine the displacement demand imposed on a building expected to deform inelastically. The Nonlinear Static Procedure in these documents, based on the capacity spectrum method, involves several approximations: The lateral force distribution for pushover analysis and conversion of these results to the capacity diagram are based only on the fundamental vibration mode of the elastic system. The earthquake-induced deformation of an inelastic SDF system is estimated by an iterative method requiring analysis of a sequence of equivalent linear systems, thus avoiding the dynamic analysis of the inelastic SDF system. This last approximation is first evaluated in this report, followed by the development of an improved simplified analysis procedure, based on capacity and demand diagrams, to estimate the peak deformation of inelastic SDF systems.
Several deficiencies in 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, the same as that determined by Procedure A if it converged.
The peak deformation of inelastic systems determined by ATC-40 procedures are shown to be inaccurate when compared against results of nonlinear response history analysis and inelastic design spectrum analysis. The approximate procedure underestimates 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 procedure is deficient relative to even the elastic design spectrum in the velocity-sensitive and displacement-sensitive regions of the spectrum. For periods 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.
Finally, an improved capacity-demand-diagram method that uses the well-known constant-ductility design spectrum for the demand diagram has been developed and illustrated by examples. This method gives the deformation value consistent with the selected inelastic design spectrum, while retaining the attraction of graphical implementation of the ATC-40 methods. One version of the improved method is graphically similar to ATC-40 Procedure A whereas a second version is graphically similar to ATC-40 Procedure B. However, the improved procedures differ from ATC-40 procedures in one important sense. The demand is determined by analyzing an inelastic system in the improved procedure instead of equivalent linear systems in ATC-40 procedures.
The improved method can be conveniently implemented numerically if its graphical features are not important to the user. Such a procedure, based on equations relating Ry and µ for different Tn ranges, has been presented, and illustrated by examples using three different Ry - µ - Tn relations.
Using representative numerical models of eight code-designed steel moment-resisting frame buildings and several ground motions, time-history analyses are performed and a critical evaluation of Peak Horizontal Floor Acceleration (PHFA) demands is conducted. The frames are modeled alternatively as linear and nonlinear systems to isolate the effect of building nonlinearity on PHFA. In most cases, PHFA is reduced when nonlinear behavior of a building is considered; however, in some cases, significant amplification of PHFA is observed. Results from the numerical study provide insight into the trend of modal response modification factors presented taking ground motion spectral shape into account.
This paper examines the potential development of a probabilistic design methodology, considering hysteretic energy demand, within the framework of performance-based seismic design of buildings. This article does not propose specific energy-based criteria for design guidelines, but explores how such criteria can be treated from a probabilistic design perspective. Uniform hazard spectra for normalized hysteretic energy are constructed to characterize seismic demand at a specific site. These spectra, in combination with an equivalent systems methodology, are used to estimate hysteretic energy demand on real building structures. A design checking equation for a (hypothetical) probabilistic energy-based performance criterion is developed by accounting for the randomness of the earthquake phenomenon, the uncertainties associated with the equivalent system analysis technique, and with the site soil factor. The developed design checking equation itself is deterministic, and requires no probabilistic analysis for use. The application of the proposed equation is demonstrated by applying it to a trial design of a three-storey steel moment frame. The design checking equation represents a first step toward the development of a performance-based seismic design procedure based on energy criterion, and additional works needed to fully implement this are discussed in brief at the end of the paper. Copyright © 2006 John Wiley & Sons, Ltd.
Hysteretic energy dissipation in a structure during an earthquake is the key factor, besides maximum displacement, related to the amount of damage in it. This energy demand can be accurately computed only through a nonlinear time-history analysis of the structure subjected to a specific earthquake ground acceleration. However, for multi-story structures, which are usually modeled as multi-degree of freedom (MDOF) systems, this analysis becomes computation intensive and time consuming and is not suitable for adopting in seismic design guidelines. An alternative method of estimating hysteretic energy demand on MDOF systems is presented here. The proposed method uses multiple ‘generalized’ or ‘equivalent’ single degree of freedom (ESDOF) systems to estimate hysteretic energy demand on an MDOF system within the context of a ‘modal pushover analysis’. This is a modified version of a previous procedure using a single ESDOF system. Efficiency of the proposed procedure is tested by comparing energy demands based on this method with results from nonlinear dynamic analyses of MDOF systems, as well as estimates based on the previous method, for several ground motion scenarios. Three steel moment frame structures, of 3-, 9-, and 20-story configurations, are selected for this comparison. Bias statistics that show the effectiveness of the proposed method are presented. In addition to being less demanding on the computation time and complexity, the proposed method is also suitable for adopting in design guidelines, as it can use response spectra for hysteretic energy demand estimation. Copyright © 2008 John Wiley & Sons, Ltd.
An Erratum has been published for this article in Earthquake Engng. Struct. Dyn. 2004; 33:1429.
Based on structural dynamics theory, the modal pushover analysis (MPA) procedure retains the conceptual simplicity of current procedures with invariant force distribution, now common in structural engineering practice. The MPA procedure for estimating seismic demands is extended to unsymmetric-plan buildings. In the MPA procedure, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non-linear static analysis using the inertia force distribution for each mode, which for unsymmetric buildings includes two lateral forces and torque at each floor level. These ‘modal’ demands due to the first few terms of the modal expansion are then combined by the CQC rule to obtain an estimate of the total seismic demand for inelastic systems. When applied to elastic systems, the MPA procedure is equivalent to standard response spectrum analysis (RSA). The MPA estimates of seismic demand for torsionally-stiff and torsionally-flexible unsymmetric systems are shown to be similarly accurate as they are for the symmetric building; however, the results deteriorate for a torsionally-similarly-stiff unsymmetric-plan system and the ground motion considered because (a) elastic modes are strongly coupled, and (b) roof displacement is underestimated by the CQC modal combination rule (which would also limit accuracy of RSA for linearly elastic systems). Copyright
Developed herein is an improved pushover analysis procedure based on structural dynamics theory, which retains the conceptual simplicity and computational attractiveness of current procedures with invariant force distribution. In this modal pushover analysis (MPA), the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by a pushover analysis using the inertia force distribution for each mode. Combining these ‘modal’ demands due to the first two or three terms of the expansion provides an estimate of the total seismic demand on inelastic systems. When applied to elastic systems, the MPA procedure is shown to be equivalent to standard response spectrum analysis (RSA). When the peak inelastic response of a 9-storey steel building determined by the approximate MPA procedure is compared with rigorous non-linear response history analysis, it is demonstrated that MPA estimates the response of buildings responding well into the inelastic range to a similar degree of accuracy as RSA in estimating peak response of elastic systems. Thus, the MPA procedure is accurate enough for practical application in building evaluation and design. Copyright © 2001 John Wiley & Sons, Ltd.
Total structural acceleration regulation is a means of managing structural response energy and enhancing the performance of civil structures undergoing large seismic events. A quadratic output regulator that minimizes the total structural acceleration energy is developed and tested on a realistic non-linear, semi-active structural control case study. Suites of large scaled earthquakes are used to quantify statistically the impact of this type of control in terms of changes in the statistical distribution of controlled structural response. Structural responses are shown and statistically characterised for a three-story steel moment-resisting frame with realistic non-linear behavior. Total structural acceleration control is shown to be more effective than typical displacement focused optimal structural control methods, by providing equivalent or better performance in terms of displacement and hysteretic energy reductions, while also significantly reducing peak story accelerations and the associated damage and occupant injury. For earthquake engineers faced with the dilemma of balancing displacement and acceleration demands this control strategy essentially eliminates that concern, making it a very attractive solution to mitigate seismic hazards. These results are also presented in a unique graphical form showing the statistical distribution to highlight the impact of control for a suite of ground motions with a given probability of occurrence. Hence, by presenting structural response in terms of statistical distribution, rather than specific data, the results, as best seen graphically, are immediately applicable to a variety of standard hazard analysis methods, which lie at the core of the trend towards performance-based design in earthquake engineering.
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