Equivalent SDOF Research Articles

Structural impedance based damage diagnosis by piezo‐transducers

AbstractAlthough structural mechanical impedance is a direct representation of the structural parameters, its measurement is difficult at high frequencies owing to practical considerations. This paper presents a new method of damage diagnosis by means of changes in the structural mechanical impedance at high frequencies. The mechanical impedance is extracted from the electro‐mechanical admittance signatures of piezoelectric‐ceramic (PZT) patches surface bonded to the structure using the electro‐mechanical impedance (EMI) technique. The main feature of the newly developed approach is that both the real as well as the imaginary component of the admittance signature is used in damage quantification. A complex damage metric is proposed to quantify damage parametrically based on the extracted structural parameters, i.e. the equivalent single degree of freedom (SDOF) stiffness, the mass, and the damping associated with the drive point of the PZT patch. The proposed scheme eliminates the need for any a priori information about the phenomenological nature of the structure or any ‘model’ of the structural system. As proof of concept, the paper reports a damage diagnosis study conducted on a model reinforced concrete (RC) frame subjected to base vibrations on a shaking table. The proposed methodology was found to perform better than the existing damage quantification approaches, i.e. the low‐frequency vibration methods as well as the traditional raw‐signature based damage quantification in the EMI technique. Copyright © 2003 John Wiley & Sons, Ltd.

등가 1 자유도계에 의한 철근콘크리트 모멘트 골조구조의 비선형 지진응답 평가법의 검토

건축구조물의 내진성능을 효율적으로 평가하기 위해서는 다자유도계 구조물을 등가 1자유도계로 표현하는 것이 필요하다. 본 연구는 다층 철근콘크리트 골조구조를 등가 1자유도계로 치환하는 방법을 제시하였다. 다층 철근콘크리트 모멘트 골조를 등가 1자유도계로 치환하여 모델화에 따른 비선형 시간이력해석을 통하여 다자유도계 구조물의 축약 방법의 타당성을 검토하는 것이 목적이다. 다층 골조구조의 정적 비선형 해석을 수행하여, 등가 1자유도계의 복원력 모델을 설정하였다. 다층 골조 구조의 비선형 시간이력해석 응답치와 등가 1자유도계의 시간이력해석 응답치를 비교하여, 등가 1자유도계 모델의 타당성을 검증하였다. 다층 철근콘크리트 골조구조를 등가 1자유도계로 치환하여 비선형 지진응답을 구하는 방법은 충분히 그 타당성이 있다는 것이 확인되었다. 다층 철근콘크리트 골조구조의 대표변위로서 1차모드참여 yector<TEX>$_1$</TEX>,<TEX>$\beta$</TEX>{<TEX>$_1$</TEX><TEX>$\mu$</TEX>}= 1인 높이가 등가 1자유도계의 응답을 가장 근사하게 나타내는 것을 알 수 있었다. 등가 1자유도계의 지진응답해석에 사용하는 복원력 모델의 이력곡선에 따라 응답파형에 차이가 생기므로, 실제 frame 구조의 복원력 특성을 반영한 이력모델을 선정하는 것이 중요하다. To evaluate the seismic performance of multistory building structures use an equivalent SDOF model to represent the resistance of the structure to deformation as it respond in its predominant mode. This paper presents a method of converting a MDOF system into an equivalent SDOF model. The principal objective of this investigation is to evaluate appropriateness of converting method through perform nonlinear time history analysis of a multistory building structures and an equivalent SDOF model. The hysteresis rules to be used an equivalent SDOF model is obtained from the pushover analysis. Comparing the peak inelastic response of a moment resisting reinforced concrete frames and an equivalent SDOF model, the adequacy and the validity of the converting method is verified. The conclusion of this study is following; A method of converting a MDOF system into an equivalent SDOF model through the nonlinear time history response analysis is valid. The representative lateral displacement of a moment resisting reinforced concrete frames is close to the height of the first modal participation vector \ulcorner<TEX>$_1{\beta}$</TEX><TEX>${_1{\mu}}=1$</TEX>. It can be found that the hysteresis rule of an equivalent SDOF model have influence on the time history response. Therefore, it necessary for selecting hysteresis rules to consider hysteresis characteristics of a moment resisting reinforced concrete frames.

地震時繰り返し挙動を考慮したエネルギー釣合手法に基づく層崩壊型RC構造物の最大応答変形予測

The observed damage patterns of R/C structure in 1995 Hyogo-ken Nanbu Earthquake include first-story collapse and intermediate-story collapse. It is necessary to estimate the maximum deformation for R/C structure before earthquake occurs. Generally, maximum deformation of R/C structure is computed by dynamic response analysis for Multi-story Frame. On the other hand, there are many, simple methods to estimate the maximum deformation for single degree of freedom (SDOF) system without dynamic response analysis. In order to evaluate rationally and simply the seismic performance of existing buildings, it is thought these methods on SDOF are applicable to the estimation of maximum deformation for Multi-story Frame with Soft story (MF-S). This paper deals with two points for estimation method on maximum deformation of R/C MF-S. One is the method to transform R/C MF-S to equivalent SDOF which has the yielding period of R/C structure and the shear strength coefficient of soft story. The other is the each accuracy of two simple estimation methods using equivalent SDOF, which one is to estimate directly the maximum deformation by dynamic response analysis for equivalent SDOF and the other is to estimate the maximum deformation by solving energy balance based on energy input velocity concept in soft story. The result indicates that the accuracy of both methods are generally good.

RC造ピロティ形式建物の制振補強手法に関する研究

The first story collapse of R/C building with soft first story is one of the characteristic damage by the Hyogoken-Nanbu Earthquake. A large deformation occurs in the soft first story which almost absorbs input energy, and to control the large deformation is necessary. In this paper, the seismic response control method for R/C building with soft first story is proposed, where it is thought that the presentation of seismic response control criteria for SDOF, which has a general characteristics, is rational and simple. This paper deals with the validity about conversion of the R/C building with soft first story into the equivalent SDOF and the design method to apply equivalent SDOF to decision of requisite damper strength to control a large deformation of first story. As a result, it is shown that maximum deformation of R/C buildings and the equivalent SDOF approximately has a good relation by using equivalent SDOF which has same yielding period and yielding base shear coefficient for R/C building with soft first story. The estimation method of requisite damper strength to control a large deformation of first story is applicable to R/C building with soft first story.

The effects of repeated earthquake ground motions on the non‐linear response of SDOF systems

AbstractIn many parts of the world, the repetition of medium–strong intensity earthquake ground motions at brief intervals of time has been observed. The new design philosophies for buildings in seismic areas are based on multi‐level design approaches, which take into account more than a single damageability limit state. According to these approaches, a sequence of seismic actions may produce important consequences on the structural safety. In this paper, the effects of repeated earthquake ground motions on the response of single‐degree‐of‐freedom systems (SDOF) with non‐linear behaviour are analysed. A comparison is performed with the effect of a single seismic event on the originally non‐damaged system for different hysteretic models in terms of pseudo‐acceleration response spectra, behaviour factor q and damage parameters. The elastic–perfect plastic system is the most vulnerable one under repeated earthquake ground motions and is characterized by a strong reduction of the q‐factor. A moment resisting steel frame is analysed as well, showing a reduction of the q‐factor under repeated earthquake ground motions even larger than that of an equivalent SDOF system. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Ductility and ductility reduction factor for MDOF systems

Ductility capacity is comprehensively studied for steel moment-resisting frames. Local, story and global ductility are being considered. An appropriate measure of global ductility is suggested. A time domain nonlinear seismic response algorithm is used to evaluate several definitions of ductility. It is observed that for one-story structures, resembling a single degree of freedom (SDOF) system, all definitions of global ductility seem to give reasonable values. However, for complex structures it may give unreasonable values. It indicates that using SDOF systems to estimate the ductility capacity may be a very crude approximation. For multi degree of freedom (MDOF) systems some definitions may not be appropriate, even though they are used in the profession. Results also indicate that the structural global ductility of 4, commonly used for moment-resisting steel frames, cannot be justified based on this study. The ductility of MDOF structural systems and the corresponding equivalent SDOF systems is studied. The global ductility values are very different for the two representations. The ductility reduction factor F-mu is also estimated. For a given frame, the values of the F-mu parameter significantly vary from one earthquake to another, even though the maximum deformation in terms of the interstory displacement is roughly the same for all earthquakes. This is because the F-mu values depend on the amount of dissipated energy, which in turn depends on the plastic mechanism, formed in the frames as well as on the loading, unloading and reloading process at plastic hinges. Based on the results of this study, the Newmark and Hall procedure to relate the ductility reduction factor and the ductility parameter cannot be justified. The reason for this is that SDOF systems were used to model real frames in these studies. Higher mode effects were neglected and energy dissipation was not explicitly considered. In addition, it is not possible to observe the formation of a collapse mechanism in the equivalent SDOF systems. Therefore, the ductility parameter and the force reduction factor should be estimated by using the MDOF representation.

Equivalent SDOF systems for the estimation of seismic response of multistory frame structures

In this research an approximate investigation has been carried out on the seismic response of multistory frame structures. The simplifying hypothesis on which the adopted methodology is based is the assumption of an assigned time-independent global deflection shape for the lateral displacements of the frame. This hypothesis corresponds to consider the dynamic response of the frame limited to the first vibration mode only, both in elastic and in inelastic behavior. The above mentioned simplification firstly allows the reduction of the response of MDOF systems to the response of an equivalent simple system (ESDOF), defined by means of appropriate values for mass, stiffness and viscous damping; on the other hand, it allows an approximate extension of the calculation of parameters characterizing the seismic response, easily obtainable for SDOF systems, to multistory frame systems. The analyses have been concerned different aspects of seismic response: strength, displacement, input energy and dissipated energy demands have been evaluated. The study has been extended to some damage indexes: although these parameters still do not have an exact and complete definition. The simplifying choice of using an assigned modal shape also for the inelastic response, although affected by the approximations implicitly present in this formulation, offers the possibility to investigate the influence of the adopted global deflection shape on response parameters; the importance of this aspect is in the fact that the shapes can correspond to different stiffness and strength distribution along the height of the building.

Seismic Design Method for Thin-Walled Steel Frame Structures

A seismic design method for thin-walled steel frame structures is proposed in the present paper. The method employs a pushover analysis to investigate the structural nonlinear behavior. A failure criterion accounting for the characteristics of thin-walled steel members and an equivalent single–degree-of-freedom (ESDOF) system are used to estimate the capacity and demand of a structure. The proposed method is a displacement-based design procedure in which not only the global maximum drift but also the residual displacement are evaluated. Limitations of the method due to higher modal effects are also studied, and an applicable range of the ESDOF system is proposed. The method is applied to some examples, and the computed results are compared with those from multi–degree-of-freedom time-history analyses. It is found that the dynamic analysis of the ESDOF model with a simplified bilinear hysteretic model can reliably predict the global maximum drift but cannot provide satisfactory estimations for the residual...

Assessment of inelastic response of buildings using force- and displacement-based approaches

In recognition of the increasing importance of accurate seismic vulnerability assessment, this paper deals with procedures and the application of inelastic acceleration and displacement spectra in the seismic assessment of buildings. An identification procedure is outlined, whereby an equivalent single degree of freedom (SDOF) system is devised to represent the building. The SDOF system characteristics (stiffness, strength, post-peak force response and ductility) are readily evaluated from observation of the seismic response of buildings and simple mechanics. The characteristics are then tuned using measurements from instrumented buildings. Based on the earthquake scenario and structural response characteristics, appropriate inelastic acceleration and displacement spectra are selected and used to ‘predict’ the response. Comparison between the measured and predicted responses for the five buildings studied in the paper confirm the feasibility of the procedure and the realism of the results. Copyright © 2000 John Wiley & Sons, Ltd.

Instability of buildings during seismic response

The issue of gravity induced instability during response to severe seismic excitation is examined. It is contended that while static instability is fully determined by the existence of at least one negative eigenvalue in the second-order tangent stiffness, this condition is necessary, but not sufficient for instability during dynamic response. It is noted that the likelihood of collapse is strongly dependent on the shape of the mechanism that controls during the critical displacement cycle and that this shape can be reasonably identified using a pushover analysis with an appropriately selected lateral load distribution. A characterization of the instability limit state based on reduction of a multistorey building to an equivalent SDOF system is presented.

ENERGY INPUT AND DISSIPATION BEHAVIOUR OF STRUCTURES WITH HYSTERETIC DAMPERS

This paper presents qualitative investigations on the energy behaviour of structures into which hysteretic dampers are incorporated. Emphasis was given to the ratio of the structural stiffness after the yielding of hysteretic dampers to the initial elastic stiffness, with a premise that this ratio, termed α in this study, tends to be large for structures with hysteretic dampers. Structures concerned were represented by discrete spring-mass systems having bilinear restoring force behaviour, in which the second stiffness relative to the initial stiffness is α. It was found that with the increase of α the total input energy tends to increase, but the increase is confined to a narrow range of natural periods. Both the total input energy and hysteretic energy were found to become less sensitive to the yield strength with the increase of α. A simple formula was also proposed to estimate the maximum deformation given the knowledge of the hysteretic energy. Analysis of MDOF systems revealed that, even when α is large, the total input energy and hysteretic energy for MDOF systems are approximately the same as those of the equivalent SDOF system, and the hysteretic energy can be distributed uniformly over the stories if α is large.

Two methods for seismic analysis of non-classically damped systems

Two methods of seismic analysis of non-classically damped systems are compared. The first method is an extension of the Foss method (called the modified Foss method here) in which the response in the time domain is represented in terms of real modal vectors and relative displacements and velocities of the equivalent single-degree-of-freedom (SDOF) systems. In the second method, known as the canonical method, a different set of real modal vectors along with sine and cosine responses of the equivalent SDOF systems are used. It is shown that the canonical method can be derived along the same lines as the modified Foss method. The time domain responses from the two methods are identical and the corresponding response spectra responses comparable. The modified Foss method is preferred because of the physical significance of the relative displacements and velocities used in the method.

Using Component Mode Synthesis and Static Shapes for Tuning TMDs

The application of the component mode synthesis (CMS) method to calculate the optimum tuned mass‐damper parameters is extended by adding a number of static mode shapes to the formulation. It is shown that the addition of these modes can result in a better accuracy of the responses. A three‐dimensional floor model is used to illustrate this technique. In addition, the tuning parameters are compared to values found using the customary equivalent single degree of freedom (SDOF) model for optimization. It is shown that when several mass dampers are tuned to a single mode, the equivalent SDOF model for optimization cannot generate the optimum parameters for these tuned mass dampers. It is also found that using the CMS method with the addition of the static mode shape can result in significant reduction in the computational effort for the system response. It is also concluded that when n number of tuned mass dampers are used to reduce the level of vibration of a single mode, the optimality criterion is expected to produce n+1 number of equal peaks in the response curve near that natural frequency.

Tuned Mass Dampers for Balcony Vibration Control

Problems associated with floor vibrations caused by human movements are becoming more prevalent. In this paper tuned mass dampers are used to reduce the excessive vibrations of a long-span balcony due to spectator-induced resonance. A three-dimensional computer model of the system is developed and its main characteristics are established through matching the dynamic properties of the model with the results of a number of dynamic tests. Five pairs of tuned mass dampers are considered to act together to suppress the resonant response of the system. These mass dampers are tuned to the first two natural modes of the structure, which had the highest possibility of excitation. Optimum design parameters of these tuned mass dampers are developed using the equivalent single degree of freedom (SDOF) model. Test data obtained after the installation of tuned mass dampers is used to evaluate their effectiveness for the reduction of the vibration amplitude.

Tuned Mass Dampers to Control Floor Vibration from Humans

The vibration of floor systems caused by human movements can be reduced using tuned mass dampers. The component mode synthesis (CMS) method is used to compute the response of the floor‐tuned mass damper system using only a few natural modes of the floor. Models representing typical two‐dimensional floor‐system behavior are used to illustrate this technique. Tuning parameters obtained using the CMS method and an equivalent single degree of freedom (SDOF) model to represent the vibration characteristics of a multiple degree of freedom system are compared. It is found that using the CMS method can result in the optimum parameters of the tuned mass dampers regardless of the closeness of the natural frequencies of the system. This is not the case when the equivalent SDOF model is used. Therefore, it is concluded that use of CMS method to represent the model for optimization produces tuned mass damper parameters that provide global reduction in the amplitude of vibration for cases with and without closely space...

Normalizing Inelastic Seismic Response of Structures having Eccentricities in Plan

The potentially disastrous consequences of “plan eccentricities” on the seismic response of structures is well recognized, but the exact influence of various parameters on the inelastic response of such systems remains unclear. This paper presents the results of an extensive parametric study conducted to investigate the effect of the uncoupled translational frequency ωx, the ratio of uncoupled frequencies Ω, the normalized eccentricity (e/r), and the level of excitation, on the inelastic response of simple structures having eccentricities in plan. A procedure is developed to ensure a logical comparison between the response of the eccentric systems and equivalent single‐degree‐of‐freedom (SDOF) benchmark. This procedure hinges on the definition of an equivalent SDOF system, and on a relationship between the excitation levels of the eccentric and SDOF systems. The findings and observations noted from this parametric study, initially conducted on simple monosymmetric bilinear two‐element single‐story systems...

Effects of currents on the stochastic response to earthquakes of multiple degree of freedom models of guyed offshore towers

The dynamic response of multiple degree of freedom (MDOF) guyed offshore towers to stochastic earthquake loads and a steady uniform current is investigated. The nonlinear cable stiffness and the fluid-structure interaction were linearized by using the stochastic linearization method. To investigate the importance of higher modes on the stochastic forces and moments of the guyed tower, numerical results for several load cases are presented and discussed. The displacement statistics from the MDOF linear analysis were compared with the statistics of an equivalent single degree of freedom (SDOF) time simulation that incorporated the nonlinearities of the structure. The increased damping of the structure with increasing current was found to significantly reduce the stochastic forces and moments of the structure.

Earthquake response of irregular R/ C structures in the nonlinear range

The application of a simple model ( Q-Model) for calculation of nonlinear seismic response history of irregular planar structures has been demonstrated. The model represents the structure by an equivalent SDOF oscillator. Stiffness degradation effects are accounted for through a simple hysteresis model. The Q-Model is evaluated for small-scale test structures in addition to a full-scale hypothetical frame. It is shown that the model exhibits satisfactory performance. The application of the Q-Model, or an approach similar to that, in seismic response computation for three-dimensional irregular structures is briefly discussed.