Inelastic Single-degree-of-freedom Systems Research Articles

A simplified method for the dynamic analysis of structures under column loss

The progressive collapse of buildings as a dynamic phenomenon can be fully understood only within a transient dynamic analysis context considering for damping and inertia effects. The response of a typical beam-column substructure is usually dominated by a single mode of deformation, thus the single-degree-of-freedom (SDOF) modelling assumption provides an accurate representation of the problem. In this study, the dynamic response of an inelastic SDOF system with damping was investigated analytically. Closed form analytical equations of the model were derived that describe the complete dynamic response in the time domain. Moreover, simplified formulae to estimate peak responses and Dynamic Increase Factors (DIFs) were provided. Numerical dynamic analyses in OpenSees were performed to verify the accuracy of the analytical model. The results indicate that the analytical model can trace the full dynamic response accurately while peak displacements can be accurately estimated with the proposed formulas. The estimated DIFs were in agreement with available from the literature results and demonstrate the significant effect of damping and hardening. Application of the analytical methodology on the progressive collapse analysis of a 10-storey steel frame demonstrated the effectiveness and potential of the proposed SDOF system.

Effectiveness evaluation and optimal design of nonlinear viscous dampers for inelastic structures under pulse‐type ground motions

SummaryThis paper addresses the effectiveness and optimal design of nonlinear viscous dampers for inelastic structures. First, a nonlinear damping index is derived by using the dimensional analysis to estimate the damping induced by supplemental nonlinear dampers on inelastic single degree of freedom (SDOF) structures. Subsequently, the effects of the added nonlinear damping on the seismic responses of inelastic SDOF systems are analyzed in dimensionless forms when subject to various near‐fault ground motions. The structure‐to‐motion frequency ratio, the motion characteristics, and the structural nonlinearity are the main factors that will affect the damping effectiveness. Especially, it is shown that adding nonlinear viscous dampers will decrease displacement demands yet sometimes lead to amplified acceleration responses. Furthermore, an equivalency procedure is developed to match the inelastic multi‐degree of freedom (MDOF) structure that is equipped with multiple nonlinear viscous dampers to its corresponding SDOF system. Such equivalency justifies that the analysis results for the viscous damping efficiency on SDOF systems can be congruously applied to realistic multi‐story structures. Finally, the optimal designs of nonlinear dampers for MDOF inelastic structures are identified by implementing a hybrid genetic optimization framework along with a robust performance index.

Duration effect of near-fault pulse-like ground motions and identification of most suitable duration measure

The duration effect of near-fault pulse-like ground motions on structural demands and the identification of most suitable duration measure are systematically addressed in this paper. Firstly, the duration effect of the Mavroeidis & Papageorgiou (M&P) pulse on the normalized maximum displacement and normalized hysteretic energy of elastic-perfectly-plastic, bilinear and rigid-plastic (i.e., Newmark sliding block) single-degree-of-freedom (SDOF) systems is analyzed by using dimensional analysis. Particularly, an intrinsic length scale is proposed to present well the normalized responses. It is shown that the duration effect of M&P pulses on the normalized responses is affected by the frequency ratio and normalized yield displacement. Generally, the duration effect is significant on the normalized hysteretic energy, whereas it is not remarkable on the normalized maximum displacement of ordinary inelastic SDOF systems, except for the normalized sliding displacement of Newmark sliding blocks. Then, a set of 65 spectrally equivalent strongest near-fault ground motions is input to the aforementioned SDOF systems, and the correlation analysis between seismic responses and duration measures is conducted. It is shown that the associated results are consistent with those obtained by dimensional analysis. Moreover, the uniform duration generally presents the best correlation with seismic demands, and thus is identified as the most suitable duration measure. Finally, the correlation statements are further validated on three 5-, 10-, and 20-story shear frame buildings with strength and stiffness degradation under spectrally equivalent ground motions.

Ground motion scaling for seismic response analysis by considering inelastic response and contribution of the higher modes

This paper proposes a new scaling procedure to consider the inelastic response of structure along with the effect of higher modes in scaling the selected ground motions for seismic response analysis. This is done by obtaining the corresponding inelastic Single-Degree-of-Freedom (SDOF) system of the structure through performing a single-run modal pushover analysis with a load pattern consistent with the combined-modal-story-shear profile obtained from response spectrum analysis. Therefore, the effect of the higher modes and interaction between them in the nonlinear phase are reflected in the inelastic SDOF. The scaling process is performed such that the peak displacement of the inelastic SDOF system under the scaled record matches the inelastic spectral displacement (target displacement). The proposed procedure is evaluated through three regular and one irregular tall building. The results demonstrate the superiority of the proposed procedure in estimating the engineering demand parameters of structures with/without important higher modes effects.

Collapse Capacity of Inelastic Single-Degree-of-Freedom Systems Subjected to Mainshock-Aftershock Earthquake Sequences

ABSTRACTThis study investigated the collapse capacity of inelastic single-degree-of-freedom (SDOF) systems subjected to mainshock-aftershock earthquake sequences. A hysteretic model incorporating strength and stiffness deterioration was used to describe the inelastic behaviors of SDOF systems under seismic excitations. An ATC63 far-field record set was adopted as seed mainshock records to synthesize earthquake sequences. An extended IDA method was employed to calculate the collapse capacity of inelastic SDOF systems subjected to mainshock-aftershock earthquake sequences by scaling the earthquake sequence as a whole. An extensive parametric study was conducted for examining the effects of hysteretic model parameters and artificial earthquake sequences synthesized by different methods and with near-field and far-field mainshock earthquakes on system collapse capacity. Through comparing the collapse capacity for mainshock-aftershock earthquake sequences and that for mainshock earthquakes only, it is found that aftershocks can yield a significant reduction on the median collapse capacity for all systems and lead to an important increase on the dispersion of collapse capacity for the systems at most periods. Based on the parametric study results, empirical expressions were proposed to predict the aftershock-induced reduction and increase on the median and dispersion of collapse capacity, respectively.

Damage-based strength reduction factor for nonlinear structures subjected to sequence-type ground motions

This paper investigates the strength reduction factor of single-degree-of-freedom (SDOF) system subjected to the mainshock–aftershock sequence-type ground motions. Both displacement ductility and cumulative damage are considered in the reduction factor. Records of mainshock-aftershock earthquakes were collected and classified according to site properties. The aftershock ground motions in sequence are scaled to five relative intensity levels. Based on the nonlinear time-history analysis of inelastic SDOF systems, the effects of natural period, ductility factor, damage index and aftershock have been studied statistically. The results indicate that the aftershock ground motion has significant influences on strength reduction factors, and the damage-based strength reduction factor is about 0.6–0.9 times of the ductility-based strength reduction factor. Finally, an empirical expression for strength reduction factor was established by regression analysis.

Equivalent Linear Model for Existing Soil-Structure Systems

The concept of equivalent linearization is extended for the soil-structure systems, in which the strength ratio (defined as the ratio of the yielding strength to the elastic strength demand) is known rather than the ductility ratio. The nonlinear soil-structure system is replaced by a linear single-degree-of-freedom (SDOF) system, which can capture the response of the actual system with sufficient accuracy. The dynamic characteristics of the equivalent linear SDOF system are determined through a statistical approach. The super-structure is modeled by an inelastic SDOF system with bilinear behavior, and the homogeneous half space beneath the structure by a discrete model, following the Cone Model. To cover a wide range of soil-structure systems, a comprehensive parametric study is conducted using a set of nondimensional parameters for the soil-structure system. The accuracy of the equivalent linear parameters is then assessed. The results confirm that the proposed equivalent linear model can capture the simultaneous effects of soil-structure interaction (SSI) and nonlinearity in the super-structure concerning the maximum inelastic response of the soil-structure system.

Comparison of Peak Ductility Demand of Inelastic SDOF Systems in Maximum Elastic Response and Major Principal Directions

This study investigates the orientation dependency of the peak ductility demand of inelastic single-degree-of-freedom (SDOF) systems of known strengths by taking the record-to-record variability of seismic effects into account. The probabilistic characteristics of the peak ductility demand in recording directions on the horizontal plane are compared with those in the maximum elastic response direction and the major principal direction. The analysis results indicate that the peak ductility demand depends on how input ground motions are defined in terms of record orientation. The observed orientation dependency can be explained by noting the response spectral shapes of scaled ground motion records in specific directions. The results have important implications for current earthquake engineering practice, suggesting that separate peak ductility demand models should be used for different ground motion directions to evaluate the seismic performance of structures consistently and without bias.

Fractal Characteristic Analysis of Seismic Responses of SDOF Systems Subjected to Near-Fault Ground Motions

From the perspective of fractal geometry, the irregularity and complexity of near-fault ground motions and seismic responses of single degree of freedom (SDOF) are analyzed. Based on the box-counting method, the fractal dimensions of 30 acceleration time series of near-fault ground motions from Taiwan Chi-Chi earthquake and California Northridge earthquake are calculated. It is indicated that the acceleration time series of ground motions present the statistical fractal property, and the influence of characteristics of near-fault ground motions on their box dimensions is remarkable. Moreover, the box dimensions of ground motions reflect their frequency property to a large extent, and can be regarded as alternative index to represent their frequency content. Finally, for the time histories of seismic dynamic responses of elastic and inelastic SDOF systems subjected to near-fault ground motions, their box dimensions are computed to examine the fractal property.

Non‐iterative equivalent linearization of inelastic SDOF systems for earthquakes in Japan and California

AbstractThe seismic performance of existing structures can be assessed based on nonlinear static procedures, such as the Capacity Spectrum Method. This method essentially approximates peak responses of an inelastic single‐degree‐of‐freedom (SDOF) system using peak responses of an equivalent linear SDOF model. In this study, the equivalent linear models of inelastic SDOF systems are developed based on the constant strength approach, which does not require iteration for assessing the seismic performance of existing structures. To investigate the effects of earthquake type and seismic region on the equivalent linear models, four ground‐motion data sets—Japanese crustal/interface/inslab records and California crustal records—are compiled and used for nonlinear dynamic analysis. The analysis results indicate that: (1) the optimal equivalent linear model parameters (i.e. equivalent vibration period ratio and damping ratio) decrease with the natural vibration period, whereas they increase with the strength reduction factor; (2) the impacts of earthquake type and seismic region on the equivalent linear model parameters are not significant except for short vibration periods; and (3) the degradation and pinching effects affect the equivalent linear model parameters. We develop prediction equations for the optimal equivalent linear model parameters based on nonlinear least‐squares fitting, which improve and extend the current nonlinear static procedure for existing structures with degradation and pinching behavior. Copyright © 2010 John Wiley & Sons, Ltd.

Seismic Demand Estimation of Inelastic SDOF Systems for Earthquakes in Japan

Abstract An accurate estimation of the maximum inelastic displacement of a structure under seismic excitations is essential to quantitative seismic risk assessment. The seismic performance of existing structures can be evaluated by utilizing inelastic single-degree-of-freedom (SDOF) systems and carrying out nonlinear dynamic analysis. This article develops a probabilistic model of the peak ductility demand of inelastic SDOF systems with various hysteretic characteristics using comprehensive sets of strong ground-motion records observed in Japan. The use of a large database facilitates the systematic investigation of the effects of earthquake type, record selection criteria, seismological parameters, and seismic region on the inelastic seismic demand. Nonlinear dynamic analysis of inelastic SDOF systems is carried out for statistical analysis and probabilistic modeling of the peak inelastic seismic demand. Analysis results indicate that the inelastic seismic demand depends on earthquake type, selection criteria, and seismological parameters to some degree. The most notable differences in inelastic seismic demands are observed for interface records at short vibration periods in comparison with crustal and inslab records; the differences can be explained by different response spectral shapes of the datasets. The inelastic seismic demand for the California crustal records is greater than that for the Japanese crustal records at short vibration periods, whereas the demands are comparable at long vibration periods. The peak ductility demand can be modeled as a Frechet variate, and empirical equations for calculating its statistics are developed, which achieve simplicity and sufficiency in probabilistic seismic risk analysis.

Approximate modal analysis of multistory symmetrical buildings with restricted inelasticity

AbstractSimplified capacity curves are presented for modal decomposition of multistory inelastic cantilever bents. These structural systems are uniform over the height and plasticity is assumed to be restricted into the beams, since significant rotation ductility factors may be attained in these members without loss of strength. The approximate method of modal decomposition of multistory inelastic bents is based on the concept that the total response may be obtained from the contributions of equivalent nonlinear single‐degree‐of‐freedom (SDOF) modal systems, in combination with the technique of modal superposition. In particular, the contribution of the first mode equivalent inelastic SDOF system is examined, since its modal contribution is of higher importance. The response of such SDOF systems basically depends on their capacity curve, which may be formulated by using the dynamic properties (frequency, effective modal mass and mode shape) of the initially elastic bent, as well as the corresponding properties of the bent when the entire set of coupling beams is assumed to be yielded. The procedure is presented for plane cantilever bents, but it can be easily extended to symmetrical structural systems composed of different types of inelastic bents. Its application is illustrated by means of a 10‐story inelastic coupled‐wall bent subject to a strong earthquake motion, equal to 1·5× El Centro N–S ground excitation, and the results compare well with those obtained from a step‐by‐step nonlinear time history analysis of the discrete member model. Copyright © 2007 John Wiley & Sons, Ltd.

Stability and accuracy analysis of outer loop dynamics in real‐time pseudodynamic testing of SDOF systems

AbstractReal‐time pseudodynamic (PSD) testing is an experimental technique for evaluating the dynamic behaviour of a complex structure. During the test, when the targeted command displacements are not achieved by the test structure, or a delay in the measured restoring forces from the test structure exists, the reliability of the testing method is impaired. The stability and accuracy of real‐time PSD testing in the presence of amplitude error and a time delay in the restoring force is presented. Systems consisting of an elastic single degree of freedom (SDOF) structure with load‐rate independent and dependent restoring forces are considered. Bode plots are used to assess the effects of amplitude error and a time delay on the steady‐state accuracy of the system. A method called the pseudodelay technique is used to derive the exact solution to the delay differential equation for the critical time delay that causes instability of the system. The solution is expressed in terms of the test structure parameters (mass, damping, stiffness). An error in the restoring force amplitude is shown to degrade the accuracy of a real‐time PSD test but not destabilize the system, while a time delay can lead to instability. Example calculations are performed for determining the critical time delay, and numerical simulations with both a constant delay and variable delay in the restoring force are shown to agree well with the stability limit for the system based on the critical time delay solution. The simulation models are also used to investigate the effects of a time delay in the PSD test of an inelastic SDOF system. The effect of energy dissipation in an inelastic structure increases the limit for the critical time delay, due to the energy removed from the system by the energy dissipation. Copyright © 2007 John Wiley & Sons, Ltd.

Role of Damage Functions in Evaluation of Response Modification Factors

From the study of the nonlinear behavior of single degree of freedom (SDOF) oscillators, response modification factor spectra corresponding to failure conditions of structures are assessed by means of damage indices. Two different damage indices are used: one that neglects and one that takes into account the way in which the hysteretic energy is dissipated. Ground motion records obtained from several earthquakes and grouped as a function of magnitude of the event, source-to-site conditions and site-soil conditions are used as input for nonlinear dynamic analysis of inelastic SDOF systems in order to evaluate constant damage spectra of the strength reduction factors. Simplified relations depending on a limited number of significant parameters are then proposed to approximate the computed mean reduction factors, in order to derive inelastic response spectra directly from the elastic ones.

Accumulation of wind induced damage on bilinear SDOF systems

The evaluation of the accumulation of permanent set for inelastic structures due to wind action is important in establishing a criterion to select a reduced design wind load and in incorporating the beneficial ductile behaviour in wind engineering. A parametric study of the accumulation of the permanent set as well as the ductility demand for bilinear single-degree-of-freedom (SDOF) systems is presented in the present study. The dynamic analysis of the inelastic SDOF system is carried out using the method of Newmark for artificially generated time history of wind speed. Simulation results indicate that the mean of the normalized damage rate is highly dependent on the natural frequency of vibration. This mean value is relatively insensitive to the damping ratio if the damping ratio is larger than 5%. Thernscatter associated with the accumulation of the permanent set is very significant. The consideration of the postyield stiffness can significantly reduce the accumulation of the permanent set if the ratio of the yield strength to the expected peak response is small. The results also show that the ductility demand due to the wind action over a period of one hour for flexible structures can be much less than that for rigid structures or structures with larger damping ratio if the SDOF systems are designed with a reduced peak response caused by the fluctuating wind.

Estimation of Inelastic Deformation Demands of SDOF Systems

Two recently proposed and closely related approximate methods to estimate the maximum lateral inelastic deformations demands of single-degree-of-freedom (SDOF) systems subjected to earthquake ground motions are investigated. In both methods, maximum inelastic displacements are estimated from maximum elastic displacements through the use of parameters that relate the response of elastic and inelastic SDOF systems. In the first method, the maximum inelastic displacement is estimated from the maximum elastic displacement through mean strength reduction factors derived from statistical studies. In the second method, the maximum inelastic displacement is estimated from the maximum elastic displacement through inelastic displacement ratios also derived from statistical studies. It is shown that the first method is actually a first-order approximation of the second method and that only in the absence of variability the two methods provide the same result. Furthermore, it is shown that the systematic error introduced by the first method leads to underestimations of maximum inelastic displacement demands. The error is evaluated and examples using results from statistical results of the seismic response of SDOF systems are presented.