Multi-degree-of-freedom Structures Research Articles

A practical optimisation method for friction tuned mass dampers in multi-storey buildings subjected to earthquake excitations

To reduce lateral drifts, the parameters of friction tuned mass dampers (FTMDs) need to be “tuned up” during the design process, which can be a challenging task for multi-storey buildings subjected to real ground motions. To address this issue, this article proposes a practical particle swarm optimisation (PSO) algorithm to optimise simultaneously the parameters (frequency ratio and friction ratio) of FTMDs fitted on multi-degree of freedom (MDOF) systems and frames subjected to real seismic records. First, elastic MDOF systems with FTMDs were modelled in OpenSees software considering different number of storeys (5, 10 or 20), periods and stiffness distributions. Optimised frequency ratios and friction coefficients of FTMDs and the main structure response were then derived using an objective function that minimised the structures’ maximum average roof displacement. FTMDs with optimised parameters proved very effective at reducing average roof displacements of the examined MDOF structures by up to 39 %, but the magnitude of the reduction depended on the period of the structure. Subsequently, the effectiveness of FTMDs at controlling the response of inelastic steel moment-resisting frames (MRFs) was examined. For the examined structures and seismic records, the use of optimised TMDs led to marginally smaller (2 %) roof displacements when compared to FTMDs. Therefore, FTMDs can be as effective as TMDs devices but with added advantages of less susceptibility to aging and temperature variations. It is also shown that higher mode effects experienced by real MRFs can be captured accurately by equivalent MDOF systems (difference in responses <4 %), which can therefore simplify the design.

Surrogate models for seismic response analysis of flexible rocking structures

AbstractSeismic design tools are based on surrogate models of the designed structure and the responses of those surrogates to earthquake ground motions. To design symmetric flexible rocking structures, a surrogate model that includes rocking and flexure is needed. In this paper, we derive the equation of motion of a flexible multi‐degree‐of‐freedom (MDOF) structure rocking on its base in modal coordinates. Then, we introduce a set of two‐degree‐of‐freedom (2DOF) surrogate models that accounts only for the first elastic vibration mode of the multimass structure and its rotation about the base pivot points. We investigate the surrogates' ability to represent the dynamics of an elastic MDOF structure that uplifts and rocks and the interaction between rocking and flexure. Therein, we detail the simplifications for the equations of motion of the 2DOF surrogate models and the adopted rocking impact model, and develop and check the sliding initiation condition. We show that the simplified 2DOF surrogate model responses compare well to experimental results. Then we assess the 2DOF surrogate model accuracy in representing the earthquake response of the MDOF model using Cloud Analysis and the coefficient of determination approach. We find that simplified 2DOF surrogate of the MDOF model is quite accurate () in estimating its maximum relative top displacement and acceptably accurate () in estimating its maximum base rotation based on a thousand randomly generated flexible rocking structure earthquake response analyses. Lastly, we discuss using the simplified 2DOF surrogate model of symmetric flexible rocking structures in preliminary seismic design, and give examples featuring a continuous elastic hollow semi‐conical chimney and an inelastic flexible MDOF structure, both with a base that may uplift.

On the optimal design and placement of Tuned-Mass-Damper-Inerter for Multi-Degree-Of-Freedom structures

Even if Tuned-Mass-Damper-Inerter (TMDI) design for Multi-Degree-Of-Freedom (MDOF) structures is a topic largely addressed in the current literature, some aspects still deserving insight emerge from a literature review. The state of the art presented in the paper highlights that Inerter-Vibration-Absorber (IVA) design is typically conducted based on simplified models (e.g. generalized 2-Degree-Of-Freedom (2-DOF) models) or sophisticated ones but a comparison between the two approaches is rarely made. Moreover, the device placement along the structure is not often addressed in the design formulation; nevertheless, a proper location greatly influences the absorber performance and its physical dimensions. The paper aims to be a comprehensive study that investigates in a unique formulation the optimal design together with the optimal placement of a TMDI in a MDOF structure and aims to make a comparison between the use of a 2-DOF model and a complete MDOF model for its optimal design and placement. The overall design methodology and performance evaluation is formulated considering a stationary white noise input. An exemplificative literature case study of a 10-DOF structure is utilized to apply the design procedure, evaluate the performance and make the comparisons. It is demonstrated that a reduced order model is useful for a first estimate of the TMDI design parameters but some, especially the damping coefficient, need improvement adopting more refined models. Structural performance is well cached with reduced order models, especially for those quantities that mainly depend on the first natural mode. An optimal placement, tradeoff between structural performances and absorber physical dimensions, is proposed. TMDI effectiveness is assessed also considering non-stationary input with natural earthquakes and comparisons with conventional and non-conventional Tuned-Mass-Damper (TMD) are conducted.

Data set from shaking table tests of a slender MDOF structure with base-rocking mechanisms

Numerical models of seismic-resistant structures are typically validated by comparing numerically predicted key seismic response parameters to those obtained from benchmark shake table experiments. However, studies have shown that for structures rocking on rigid foundations, the seismic response can be unrepeatable in experiments and deterministically unpredictable by numerical models. Therefore, researchers have proposed to validate rigid rocking models using statistical approaches, yet no experimental study has examined whether the same observations apply to slender multi-degree-of-freedom (MDOF) free-rocking structures or controlled rocking structures with positive post-rocking stiffness and dedicated energy dissipation. This data paper describes the now publicly available data obtained from a series of 254 shaking table tests of a slender MDOF tower structure with two base-rocking mechanisms, which can be used to examine the predictability of the seismic response of an MDOF rocking structure and to statistically validate different approaches aiming to model tall and slender structures that utilize base-rocking mechanisms to mitigate the dynamic response due to higher-mode effects.

Design of Optimal Passive Tuned Mass Damper with Static Output Feedback and Updating Iterative Procedure

In this study, the optimal design issue of a passive tuned mass damper (TMD) was transformed into the nonsparse control gain matrix optimization problem, and a general passive TMD optimization design method to minimize structural mean square responses or impulse response is therefore proposed. The proposed optimization procedure combines the static output feedback (also known as direct output feedback) algorithm and the updating iterative procedure. The proposed method can be applied to variant design scenario, whether the main structure is single-degree-of-freedom (SDOF) or multidegree-of-freedom (MDOF) structures, undamped or damped structures, subjected to wind disturbances or earthquake excitations. In addition, the proposed method is capable to consider the excitation shaping filter, so the design results are more suitable for practical application. The design procedure of the proposed method is presented, and all the required weighting matrices are introduced and derived in detail. Firstly, the SDOF structures are used as the main structure to demonstrate the correctness of the proposed method. The numerical simulation results verify that the obtained optimal design parameters of TMD were found identical to some cases which contain the analytic solution. The accuracy and feasibility of the proposed design method are confirmed. Finally, a passive TMD is optimally designed for a 5-story MDOF structure subjected to Kanai–Tajimi spectrum comparable earthquakes and a 60-story high-rise MDOF structure subjected to Davenport spectrum comparable wind loads for demonstration.

Seismic response of slender MDOF structures with self‐centering base shear and moment mechanisms

AbstractAs the amplification of seismic force demands due to higher‐mode effects on tall buildings is increasingly recognized as a design challenge, a number of high‐performance systems have been proposed to limit such effects through combined base shear‐limiting and moment‐limiting dual mechanisms. To better understand the influence of these dual base systems on the overall seismic behavior of tall buildings, this paper presents a study on the seismic response of Multi‐Degree‐of‐Freedom (MDOF) systems incorporating combined base shear‐limiting and moment‐limiting mechanisms. For an MDOF system with a given initial period and base strength level, the base shear‐limiting mechanism is defined by a shear strength factor, an inelastic stiffness parameter, and an energy‐dissipation parameter, while the base moment‐limiting mechanism is defined by a moment strength factor. To determine the influence of these parameters on the overall seismic responses of MDOF structures, a comprehensive parametric study was conducted, and the results were presented and discussed in terms of base displacement and rotation demands, seismic force demand amplification at the base and along the height, peak floor acceleration, peak roof drift, and absorbed energy. The numerical modeling methodology used in the parametric study was validated against 200 small‐scale shaking table tests of a scaled MDOF specimen with a base shear and moment dual‐mechanism system and was used to model MDOF structures that are representative of tall buildings with initial periods ranging from 1.0 to 10 s and having various base‐mechanism properties. The parametric study was then conducted using an ensemble of 20 Ground Motions (GM) scaled to three code‐specified seismic hazard levels for Los Angeles, California. The results of this study can be used to facilitate the design of base shear and moment dual‐mechanism systems for mitigating higher‐mode effects and enhancing the seismic resilience of tall buildings.

Conditions for minimum stiffness of proportionally loaded structures

During proportional loading of structures a significant re-arrangement of stresses may occur. It may result e.g. in the transition from a softening to a stiffening structure. This may even be the case for constant material stiffness. Since the terms stiffening structure and softening structure are lacking universally accepted definitions, it is not astonishing that the search for conditions for minimum stiffness of proportionally loaded multi degree of freedom structures has so far attracted very little attention. Presentation of such a condition, in the framework of the Finite Element Method (FEM), is the task of this publication. It is based on a linear eigenvalue problem that renders complex eigenvalues and eigenvectors possible. The point of inflection in the real part of the complex section of a specific eigenvalue function is shown to mark the load-level of minimum stiffness of a proportionally loaded structure. Complex eigenvalues require the indefiniteness of both real symmetric coefficient matrices of the underlying linear eigenvalue problem. In the given case, the first one depends on a dimensionless load parameter, whereas the second one is obtained by specialization of the first one for the onset of loading. These matrices are established with the help of a special hybrid finite element, available in a commercial FE program. The proposed condition for minimum stiffness of proportionally loaded structures is verified numerically. The results obtained in a parameter study of the investigated structure are found to agree well with the ones following from an alternative condition for the transition from a softening to a stiffening structure. This condition is based on a mechanically objective arc length in the form of a representative displacement of the structure.

An experimental study of tuned liquid column damper controlled multi-degree of freedom structure subject to harmonic and seismic excitations.

A tuned liquid column damper (TLCD) is a passive vibration control device that not only mitigates unwanted structural vibrations but also acts as a water storage facility in a building. These aspects of TLCD make its application specifically suited for building structures. Previously, many experimental works on TLCDs have been conducted considering a single degree of freedom (SDOF) structure. However, the performance of TLCDs to control the response of multi-degree of freedom (MDOF) structure has rarely been studied experimentally. Therefore, this study has investigated the performance of a tuned liquid column damper (TLCD) on a multi-degree of freedom (MDOF) structure using shake table testing. A four-storey steel frame structure equipped with TLCD at the top of the fourth storey has been studied. Experimental normalized frequency response curves for MDOF structure equipped with TLCD have been determined. For this purpose, a series of harmonic loadings including frequencies 0.65 Hz, 1.17 Hz, 1.30 Hz, 1.43 Hz and 1.95 Hz have been applied in addition to historic earthquake loading. Peak and root-mean-square (RMS) accelerations have been discussed in detail for all the applied loadings at each storey level of the structure. For comparison purposes, the percentage reductions in peak and RMS accelerations have been calculated and compared. Also, RMS displacements and inter-storey drifts have been presented for resonant and seismic excitations. Both in time and frequency domains, responses of controlled MDOF structure have been analyzed and compared with uncontrolled structure. Results confirmed that TLCD has improved the MDOF structure responses at harmonic loadings frequencies near resonance and historic earthquake excitations. Furthermore, the improvement in the responses of MDOF structure with TLCD is more prominent at harmonic loadings compared to historic earthquake loading.

Stochastic dynamic analysis of nonlinear MDOF systems under combined Gaussian and Poisson noise excitation based on DPIM

Random vibration analysis of structures subjected to combined Gaussian and Poisson white noise excitation is a challenging issue. In this paper, a novel direct probability integral method (DPIM) is suggested to address stochastic responses and dynamic reliability of nonlinear multi-degree-of-freedom (MDOF) systems under combined excitation. Firstly, probability density integral equation (PDIE) of MDOF system under combined excitation is derived based on the principle of probability conservation. Then, DPIM is proposed to achieve the probability density function of stochastic response by solving deterministic motion equation of MDOF system and PDIE in sequence. To solve PDIE, two techniques, i.e., partition of input probability space and smoothing of Dirac delta function, are introduced. From the perspective of probability conservation, furthermore, the equivalent relationship between the PDIE and the corresponding probability density differential equation of a Markov system under combined Gaussian and Poisson noise is established. Since Dirac delta function is analytically integrated as Heaviside function, the first-passage dynamic reliability of MDOF system under combined excitation is readily evaluated by introducing the extreme value mapping of stochastic response. Finally, two nonlinear MDOF systems, including multiple-span bridge under combined vehicle load and non-stationary seismic excitation, are solved. Results demonstrate that the proposed DPIM is effective for random vibration and dynamic reliability analyses of MDOF structures excited by combined Gaussian and Poisson noise, and considering the randomness of vehicle load and seismic excitation simultaneously in the bridge design benefits the safe operation of bridge.

Various damper forces and dynamic excitation nonparametric identification with a double Chebyshev polynomial using limited fused measurements

By introducting a double Chebyshev polynomial as a general nonparametric model for various nonlinear restorying forces (NRFs) and an updated observation equation, and with the help of an Updated Extended Kalman filter with unknown input (UEKF-UI) algorithm and a data fusion technology, a nonparametric identification approach for both NRFs at multiple locations and excitation for multi-degree-of-freedom (MDOF) lumped mass structures is presented. In order to investigate the generality of the proposed identification method, both numerical and experimental studies on different MDOF structure models equipped with a shape memory alloy (SMA) damper and/or a magnetorheological (MR) damper mimicking different types of nonlinearities that widely exist in practical engineering structures under unknown excitation are carried out. Identification results considering different initial estimation errors and measurement noise show that the NRF provided by different types of dampers at multiple locations, excitation and unmeasured dynamic responses can be identified nonparametrically with acceptable accuracy.

Design of a pair of isolated tuned liquid dampers (ITLDs) and application in multi-degree-of-freedom structures

The tuned liquid damper (TLD) has been proven as an effective vibration mitigation device. However, it is less efficient and robust under seismic excitations. Addressing this issue, this study proposes a dual-mode-based design methodology utilizing a pair of isolated TLDs (ITLDs) for the multi-performance vibration mitigation of multi-degree-of-freedom (MDOF) structures. Oriented by the dual-mode control target, a series of 2DOF structures were considered, and a mechanical model of the considered ITLD was established. The modal analysis was performed for the primary structure, on the basis of which feasible installation methods for the ITLDs are proposed. Correspondingly, an extensive parametric analysis and a frequency response analysis were conducted, which revealed the control advantages of a pair of ITLDs, and we explored the potential optimal design for the liquid mass and added isolation layer. Inspired by the parametric analysis results, a mode-based design procedure is proposed for a pair of ITLDs by incorporating the modal characteristics of the primary structure. Finally, the ITLDs were applied in a multi-story building structure to illustrate the effectiveness of the devices and proposed design method. The obtained analysis results show that the dual-mode mitigation effect can be satisfactorily achieved using a pair of ITLDs with the proposed design approach, as well as synthetic functionality of the pair of ITLDs. The employment of a pair of ITLDs provided effective and robust mitigation of seismic responses for MDOF structures, yielding an improved multi-response reduction compared with the TLDs with the same parameters. In particular, the seismic-excited structural acceleration responses can be suppressed for different floors, which cannot be achieved by conventional TLDs.

Energy analysis of particle tuned mass damper systems with applications to MDOF structures under wind-induced excitation

In recent years, particle damper has been proved to be an effective vibration control strategy in various fields. In this study, an energy-evaluation model of particle tuned mass damper (PTMD) attached on a Multi Degree of Freedom (MDOF) structure (76-story high-rise building) under wind excitation is established based on an equivalent simplified method, and the simulation results are validated by a wind tunnel experiment. The energy related in the whole structural system, including the kinetic energy, elastic potential energy, energy dissipated by the structural damping and external input energy, has been analyzed in detail, and the energy transmission effect between the main structure and the attached PTMD has been investigated through the time history response of the input energy distribution. It is found that PTMD shows favorable energy dissipation performance, a large percentage of the input energy transfers from the main structure to PTMD in the first few seconds, and after that the particle-container wall collision consumes the main proportion of the input energy. Moreover, a series of systematic parametric analyses are conducted, including volume filling ratio, PTMD auxiliary mass ratio, particle damping coefficient and frequency ratio of PTMD. Furthermore, a full factorial study is conducted to investigate the interaction effect of different parameters. The results show that the combination of the parameters has a significant effect on the energy dissipation performance of PTMD, a parametric designing procedure has been given accordingly.

Theoretical and Experimental Study of the Pounding Response for Adjacent Inelastic MDOF Structures Based on Dimensional Analysis

Dimensional analysis is applied to study the pounding response of two inelastic multidegree of freedom (MDOF) structures under simplified earthquake excitation. The forces and deformations of the collision processes are simulated by adopting the improved Kelvin pounding model. The inelastic characteristics of MDOF structures are described by the bilinear interstory resistance model, and the representations of dimensionless impact force and the dimensionless motion equation in the pounding process are derived. On the basis of the above‐mentioned theoretical deduction, the superiority of the improved Kelvin model is verified by comparing the impact response of the improved Kelvin model and the Kelvin model. Finally, the validity of the proposed theoretical method is further proved by the comparison between results from the shaking table tests of adjacent four‐story and three‐story steel frame structures and the corresponding numerical results obtained by the MATLAB program.

Dimensional pounding response analysis for adjacent inelastic MDOF structures based on modified Kelvin model

The dimensional analysis method is used to study the pounding response of two inelastic MDOF (multi-degree-offreedom) structures under simplified earthquake excitation. The improved Kelvin pounding model is adopted to simulate the force and deformation of the collisions during the contact process. The bilinear interstory resistance model is used to describe the inelastic characteristics of the MDOF structures. The expression of dimensionless pounding force and the equation of dimensionless motion during the collision process are derived. Based on the above theoretical derivation, the accuracy of the improved Kelvin model is verified by comparing the pounding responses in the form of spectra between the improved Kelvin model and Kelvin model. The effects of the pounding on the response of the left structure (with a smaller mass and stiffness) are analyzed in different trend (amplification region, suppression region and unaffected region), and the self-similarity of the pounding response for the two inelastic MDOF structures is revealed. The effects of the story mass ratio, post-yield stiffness ratio, yield displacement and structure spacing on the pounding response are studied. The peak displacement response of the left side structure increases with the increasing of story mass ratio and decreases with the increasing of yield displacement and postyield stiffness ratio. With the increasing of structure spacing, the peak displacement decreases in the first spectrum region, and in the second spectrum region, the peak displacement increases. Moreover, the change of the parameters has little effect on the response of the right structure (with a larger mass and stiffness).

Derivation of yield force coefficient for RC frames considering energy balance and P-delta effects

Currently, prominent energy balance concept can efficiently be used to calculate the yield base shear force of structures. Energy balance equation is an additional constraint for the balance of seismic input energy by the energy absorption of the structure. This equation can be defined as the sum of elastic and plastic energies of structural systems using elastic and plastic load-deformation characteristics and the total energy dissipation capacity can finally be equated to the seismic energy demand. The objective of this paper is to present a procedure for the determination of energy-based yield force coefficient of reinforced concrete (RC) frame structures considering P-delta effects. The total of elastic and plastic energies are computed by means of elastic spectral velocity and energy modification factor, which is originally derived for structural systems with geometric nonlinearity. Total inelastic energy of single-degree-of-freedom (SDOF) system is transformed into the total inelastic energy of the multi-degree-of-freedom (MDOF) system using the conversion based on structural dynamics. Plastic energy of MDOF system is formulated after total seismic energy demand is obtained. It is accordingly equated to the total work done by inelastic MDOF structure with P-delta effects and consequently, a dimensionless yield force coefficient is derived. Nonlinear static analyses are performed for selected multi-story RC frames and yield force coefficients are compared with the results of the energy-based formulation. The results show that the proposed formulation taking P-delta effects into account can be effectively used to estimate the yield force coefficient of RC frame structures.

Cross-Layer Installed Cable-Bracing Inerter System for MDOF Structure Seismic Response Control

Inerter-based vibration control systems have been developed rapidly in recent years. However, previous studies mainly focus on the development of new devices and parameter optimization strategies, while ignoring the improvements in the utilization efficiency of the inerter system that the bracing system and novel installation methods may bring. In this paper, a cross-layer installed cable-bracing inerter system (CICBIS) is proposed to improve the utilization efficiency of the inerter system, which can cross more layers and is suitable for shear-type multi-degree-of-freedom (MDOF) structures. A demand-based cable-bracing inerter system (CBIS) design method is developed. The mass enhancement and utilization efficiency improvement of the inerter system caused by the cross-layer installation are quantified through calculating the effective inerter-mass ratio of the CBIS-equipped MDOF structure. A 10-story benchmark structure is used to verify the control performance of the CICBIS and the design method. The analysis results show that the proposed design method can exert the cable-bracing system’s adjustability and the damping enhancement of the inerter system. The CICBIS can reduce the total apparent mass and damping coefficient requirements of the inerter systems without increasing the control force. It means that the proposed design method is effective, and the CICBIS has a high efficiency.

A generalized 2-DOF model for optimal design of MDOF structures controlled by Tuned Mass Damper Inerter (TMDI)

The paper illustrates a methodology to perform the optimal design of a Tuned Mass Damper Inerter (TMDI) equipped on a multi degree of freedom (MDOF) structure through a generalized model. The TMDI is considered in both configurations grounded and ungrounded. A generalized 2-DOF model is obtained: the primary oscillator, representing the original MDOF structure and the secondary oscillator, representing the control system. A Gaussian zero mean white noise random process is adopted for the excitation. The TMDI design is carried out adopting an energy objective function, maximized over the space of the design parameters which are: the mass and inertance ratios, the frequency ratio, the damping factor and a parameter that accounts for the TMDI location on the MDOF structure. Synthetic performance-based design maps of the TMDI parameters are carried out. It is shown that is possible to have different design configurations to achieve a target performance level. The maps furnish, as limit cases, the design of a Tuned Inerter Damper (TID) and a Tuned Mass Damper (TMD), when the mass or the inertance ratios are respectively assumed null. It is shown straightforwardly the regions of the design parameters where the inerter enhances the performances achievable by classical TMD. Furthermore, an analytical expression which links the optimal design variables necessary to obtain a desired target performance is proposed. Frequency response functions and modal parameters of certain 2-DOF models with optimally designed control system are reported and considerations on how the design parameters influence the system dynamics and the damping capabilities are highlighted. Comparisons of the TMDI optimally designed with cases of no control and same mass TMD are evidenced throughout the paper. Finally, the main responses of the primary structure and the TMDI are estimated in the space of variation of the design parameters in order to assess the effectiveness of the control system.

Identification of abrupt stiffness changes of structures with tuned mass dampers under sudden events

This paper presents a recursive system identification method for multi-degree-of-freedom (MDoF) structures with tuned mass dampers (TMDs) considering abrupt stiffness changes in case of sudden events, such as earthquakes. Due to supplementary non-classical damping of the TMDs, the system identification of MDoF+TMD systems disposes a challenge, in particular, in case of sudden events. This identification methods may be helpful for structural health monitoring of MDoF structures controlled by TMDs. A new adaptation formulation of the unscented Kalman filter allows the identification method to track abrupt stiffness changes. The paper, firstly, describes the theoretical background of the proposed system identification method and afterwards presents three parametric studies regarding the performance of the method. The first study shows the augmented state identification by the presented system identification method applied on a MDoF+TMD system. In this study, the abrupt stiffness changes of the system are successfully detected and localized under earthquake, impulse and white noise excitations. The second study investigates the effects of the state covariance and its relevance for the system identification of MDoF+TMD systems. The results of this study show the necessity of an adaptive definition of the state covariance as applied in the proposed method. The third study investigates the effects of modeling on the performance of the identification method. Mathematical models with discretization of different orders of convergence and system noise levels are studied. The results show that, in particular, MDoF+TMD systems require higher order mathematical models for an accurate identification of abrupt changes.

Computing input energy response of MDOF systems to actual ground motions based on modal contributions

The use of energy concepts in seismic analysis and design of structures requires the understanding of the input energy response of multi-degree-of-freedom (MDOF) systems subjected to strong ground motions. For design purposes and non-time consuming analysis, however, it would be beneficial to associate the input energy response of MDOF systems with those of single-degree-of-freedom (SDOF) systems. In this paper, the theoretical formulation of energy input to MDOF systems is developed on the basis that only a particular portion of the total mass distributed among floor levels is effective in the nth-mode response. The input energy response histories of several reinforced concrete frames subjected to a set of eleven horizontal acceleration histories selected from actual recorded events and scaled in time domain are obtained. The contribution of the fundamental mode to the total input energy response of MDOF frames is demonstrated both graphically and numerically. The input energy of the fundamental mode is found to be a good indicator of the total energy input to two-dimensional regular MDOF structures. The numerical results computed by the proposed formulation are verified with relative input energy time histories directly computed from linear time history analysis. Finally, the elastic input energies are compared with those computed from time history analysis of nonlinear MDOF systems.

Comparative investigation on optimal viscous damper placement for elastic-plastic MDOF structures: Transfer function amplitude or double impulse

The effectiveness of the optimal viscous damper placement for elastic-plastic multi-degree-of-freedom (MDOF) structures under the critical double impulse is demonstrated through the comparison with the optimal damper placement for elastic MDOF structures designed with respect to transfer function amplitudes at natural frequencies. The method for optimal viscous damper placement was proposed for elastic-plastic MDOF structures subjected to the critical double impulse as a representative of near-fault ground motions in the previous paper [1]. The double impulse is composed of two impulses with opposite directions and the critical interval is determined by using the criterion on the maximum input energy. The critical timing of the second impulse was found to be the timing which requires the vanishing of the sum of the restoring force and the damping force in the first story. Three models with different story stiffness distributions of main structures are treated to demonstrate the effectiveness of the optimal viscous damper placement under the critical double impulse. The double impulse pushover (DIP) procedure proposed in the previous paper for determining the input velocity level of the critical double impulse is examined further in this paper. It is demonstrated that the optimal viscous damper placement for elastic-plastic MDOF structures under the critical double impulse is more effective for pulse-type recorded earthquake ground motions than the optimal damper placement designed with respect to transfer function amplitudes at natural frequencies because several higher modes arising in the elastic-plastic response under pulse-type recorded earthquake ground motions can be well controlled by the design for the double impulse.