Bilinear Hysteretic Model Research Articles

Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy (SMA) composite structures, the force-displacement characteristics of the SMA layer are studied. The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material. Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process, the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode. The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement. Furthermore the energy dissipated by SMA layer during one period is obtained by integration, then its dependencies are discussed on the vibration amplitude and the SMA’s strain (Ms-Strain) value at the beginning of martensitic phase transformation. It is shown that SMA’s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain. The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing. The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude. The obtained results are helpful for designing beams laminated with shape memory alloys.

Fragility analysis of R/C frame buildings based on different types of hysteretic model

Estimation of damage probability of buildings under a future earthquake is an essential issue to ensure the seismic reliability. Fragility curves are useful tools for showing the probability of structural damage due to earthquakes as a function of ground motion indices. The purpose of this study is to compare the damage probability of R/C buildings with low and high level of strength and ductility through fragility analysis. Two different types of sample buildings have been considered which represent the building types mentioned above. The first one was designed according to TEC-2007 and the latter was designed according to TEC-1975. The pushover curves of sample buildings were obtained via pushover analyses. Using 60 ground motion records, nonlinear time-history analyses of equivalent single degree of freedom systems were performed using bilinear hysteretic model and peak-oriented hysteretic model with stiffness - strength deterioration for each scaled elastic spectral displacement. The damage measure is maximum inter-story drift ratio and each performance level considered in this study has an assumed limit value of damage measure. Discrete damage probabilities were calculated using statistical methods for each considered performance level and elastic spectral displacement. Consequently, continuous fragility curves have been constructed based on the lognormal distribution assumption. Furthermore, the effect of hysteresis model parameters on the damage probability is investigated.

Simplified multi‐degree‐of‐freedom model for estimation of seismic response of regular wall‐frame structures

AbstractA simplified multi‐degree‐of‐freedom (MDOF) model is developed for estimation of seismic response of tall wall‐frame structures. By using the continuum technique for the structure and adopting the bilinear hysteretic model for material properties, procedure for the development of the simplified MDOF model is derived. The numerical study for a 20‐storey reinforced concrete (RC) wall‐frame structure is conducted to investigate the accuracy of seismic response predicted by the proposed model. Results from the nonlinear response history analyses based on the proposed MDOF model and the detailed structural model with member‐by‐member representation are compared and show very good agreement. The proposed simplified MDOF model is shown to provide a simple, efficient and accurate method for estimation of seismic performance of tall wall‐frame structures. Copyright © 2009 John Wiley & Sons, Ltd.

Tendons Arrangement Effects on Reinforced Concrete Frames under Blast Loading

The tendons arrangement effects on the response of reinforced concrete frame experiencing blast loads is investigated in this paper. The structure is modeled using nonlinear finite elements employing a bilinear hysteretic model. So that elements are used so that yielding of the structures could be accurately modeled and captured. The frame is analyzed using a non-linear, elastic-plastic finite element program written in code MATLAB. Six tendon architectures were investigated. A single tendon was placed between different floors and its effects investigated. From the obtained results, the ideal case which causes a reduction in the maximum displacement and the amount of permanent deflection without increase in the maximum structural shear forces greatly is obtained.

Physical-Parameter Identification of Base-Isolated Buildings Using Backbone Curves

In this paper, a simplified system identification procedure is developed to investigate the dynamic characteristics of base-isolated buildings equipped with lead-rubber bearings (LRBs). The multistory superstructure is assumed to be linear on the account of substantial reduction in seismic forces due to the installation of LRBs for which a bilinear hysteretic model is considered. The hysteretic model is in turn characterized by a backbone curve by which the multivalued restoring force is transformed into a single-valued function. With the introduction of backbone curves, the system identification analysis of inelastic structures is simplified to a large extent. The proposed algorithm extracts individually the physical parameters of each floor and the bearing system that are considered useful information in the structural health monitoring. A numerical example is conducted to demonstrate the feasibility of using the proposed technique for physical-parameter identification of partially inelastic based-isola...

Physical system identification of an isolated bridge using seismic response data

This paper develops a system identification process based on seismic response data to investigate the physical parameters of an isolated highway bridge equipped with the lead–rubber bearings (LRBs). The effects of soil–structure interaction (SSI) of the highway bridge built on pile foundations are taken into account in the analysis. Unlike many other identification methods that generally seek for the equivalent system parameters or modal parameters of the target structure, the proposed method aims at identifying physical parameters such as the pier stiffness or hysteresis of the LRB that are considered more critical and useful for health monitoring of bridges. In this study, a linear elastic model is used for the pier while a bilinear model is considered to characterize the hysteretic behavior of the LRB. The bilinear hysteretic model is in turn reduced to a backbone curve based on the Masing criterion by which the multi-valued restoring force is transformed into a single-valued function so that the computational effort of the identification process can be minimized. Numerical examples have been conducted, including the identification of Bai-Ho Bridge, which is the first seismic isolation bridge in Taiwan, to demonstrate the feasibility of the proposed technique for dynamic systems with inelastic components and SSI. Copyright © 2008 John Wiley & Sons, Ltd.

강도한계 이선형 단자유도 시스템의 동적 불안정

강도한계 이선형 단자유도 시스템의 지진 하중 하에서의 동적 불안정에 대해 연구하였다. 강도한계 이선형 이력 모델은 철골 모멘트 골조의 이력거동을 가장 잘 모사한다. 단자유도 시스템의 동적 불안정을 판단하기 위해 붕괴 강도비를 사용하였고, 이것은 붕괴가 일어날 때의 항복강도 저감계수이다. 단단한 지반에서 측정된 240개의 지진을 이용하고 고유주기, 강성 경화 기울기, 음강성 기울기, 연성 그리고 <TEX>$2{\sim}20%$</TEX>의 감쇠비를 변수로 하여 강도한계 이선형 단자유도 시스템의 붕괴 강도비의 평균과 편차 값들을 구할 수 있도록 통계 분석을 하였다. 비선형 회귀분석을 통해 강도한계 이선형 단자유도 시스템의 붕괴 강도비의 평균과 표준편차를 계산할 수 있는 식을 구하였다. 제안한 식을 이용하여 붕괴 강도비의 확률적 분포를 구하였고, 실제 값과 비교하여 제안한 식의 정확성을 입증하였다. This study investigates the dynamic instability of strength-limited bilinear single degree of freedom (SDF) systems under seismic excitation. The strength-limited bilinear hysteretic model best replicates the hysteretic behavior of the steel moment resisting frames. To estimate the dynamic instability of SDF systems, the collapse strength ratio is used, which is the yield-strength reduction factor when collapse occurs. Statistical studies are carried out to estimate median collapse strength ratios and those dispersions of strength-limited bilinear SDF systems with given natural periods, hardening stiffness ratios, post-capping stiffness ratios, ductility and damping ratios ranging from 2 to 20% subjected to 240 earthquake ground motions recorded on stiff soil sites. Equations to calculate median and standard deviation of collapse strength ratios in strength-limited bilinear SDF systems are obtained through nonlinear regression analysis. By using the proposed equations, this study estimated the probabilistic distribution of collapse strength ratios, and compared this with the exact values from which the accuracy of the proposed equations was verified.

Effect of Soil–Structure Interaction on Seismic Isolated Bridges

The role of soil–structure interaction (SSI) on the response of seismically isolated bridges is studied. A generic bilinear hysteretic model is utilized to model the isolation system. The behavior of the pier is assumed to be linear and the foundation system is modeled with frequency-dependent springs and dashpots. Two bridge systems were considered, one representative of short stiff highway overpass systems and another representative of tall flexible multispan highway bridges. Nonlinear time history analyses were employed with two sets of seismic motions; one containing 20 far-field accelerograms and one with 20 near-fault accelerograms. The results from these comprehensive numerical analyses show that soil–structure interaction causes higher isolation system drifts as well as, in many cases, higher pier shears when compared to the fixed-pier bridges (no SSI).

Drift response of a bilinear hysteretic system to periodic excitation under sustained load effects

This paper presents an investigation of response characteristics for hysteretic systems idealized as a bilinear hysteretic model subjected to period excitations composed of a harmonic function and a sustained load. It is shown that the displacement solution can exhibit a drift sequence persistently repeated at a frequency identical to the excitation frequency in the case of zero post-yielding stiffness. The periodic-like drift sequence is further classified into three major types according to their different hysteretic looping behaviors. An approximate solution approach based on the method of weighted residuals is proposed to analyze the drift amplitude per response cycle. The assumed response shape is composed of two concatenated harmonic functions each with a frequency slightly detuned from the excitation frequency. The method is accompanied with a subsequent first-order analysis to obtain a closed-form approximation for the drift response. Good response predictions of the proposed solution method are demonstrated through both undamped and damped drift–frequency analyses.

Seismic performance evaluation of steel arch bridges against major earthquakes. Part 2: simplified verification procedure

AbstractThe performance‐based philosophy has been accepted as a more reasonable design concept for engineering structures. For this purpose, capacity evaluation and demand prediction procedures for civil engineering structures under earthquake excitations are of great significance. This work presents a displacement‐based seismic performance verification procedure including capacity and seismic demand predictions for steel arch bridges and investigates its applicability. Pushover analyses is employed as a basis in this method to investigate the structure's behaviors. A failure criterion for steel members accounting for the effect of local buckling is involved and an equivalent single‐degree‐of‐freedom (ESDOF) system with a simplified bilinear hysteretic model formulated using pushover analyses results is introduced to estimate the displacement capacity and maximum demand of steel arch bridges under major earthquakes. To check the accuracy of the proposed method, seismic capacities and demands from multi‐degree‐of‐freedom (MDOF) time‐history analyses with Level‐II design earthquake record inputs modeling major earthquakes are used as benchmarks for comparison. By a case study, it is clarified that the proposed prediction procedure can give accurate estimations of displacement capacities and demands of the steel arch bridge in the transverse direction, while insufficient for the longitudinal direction, which confirms the conclusion drawn in other structure types about the applicability of pushover analyses. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Nonlinear Frequency Spectrum Analysis of Vibration System with ER Damper

The Electro-rheological (ER) fluid is a new kind of smart material which has been applied to various engineering problems. Using the viscous and bilinear hysteretic damping model, this paper discusses the nonlinear frequency spectrum by iterative perturbation method for nonlinear vibration system with ER damper. Iterative perturbation is based on the matrix perturbation. The initial frequencies are found by subspace iteration. The subsequent frequency spectrum of nonlinear states, which is based upon the tangent stiffness of the structure, is calculated by the iterative perturbation method. A numerical example is given to illustrate the method presented by this paper.

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...

A simple approach to the seismic design of friction damped braced medium-rise frames

This paper is concerned with the seismic design of steel frames using friction damped slotted bolted connections (SBCs) in the diagonal braces as a means for passive control. A dynamic model that uses bilinear hysteretic behavior for the damper describes the response of a single degree of freedom (SDOF) steel frame and constitutes the basis for the design. A design procedure that attains the stiffness of the individual braces and their elongation at the threshold of activation is then applied to 10-story and 3-story steel frames. This design procedure is a two-phase iterative scheme of the analysis/redesign type, the analysis of which is modal. Finally, a full nonlinear dynamic analysis that uses a generalized version for MDOF's of the bilinear hysteretic SDOF model assesses the design.

Equivalent Viscous Damping for a Bilinear Hysteretic Oscillator

A bilinear hysteretic model is commonly used to study elastoplastic structures. In this paper, a damped, bilinear hysteretic oscillator is studied under harmonic loading. We show the existence of an equivalent viscous damping for small values of a loading parameter such that the associated linear structure and the hysteretic structure have the same frequency response curves. We use the Kryloff-Bogoliuboff method of averaging to find the equivalent viscous damping as a function of the steady state amplitude. We present a model of a bilinear elastic oscillator which captures the steady-state dynamics of the hysteretic oscillator for low values of the loading parameter. We also study the nature of the dependence of the equivalent viscous damping on the kinematic hardening parameter.

Optimal Structural Control Using Neural Networks

An optimal control algorithm using neural networks is proposed. The controller neural network is trained by a training rule developed to minimize cost function. Both the linear structure and the nonlinear structure can be controlled by the proposed neurocontroller. A bilinear hysteretic model is used to simulate nonlinear structural behavior. Three main advantages of the neurocontroller can be summarized as follows. First, it can control a structure with unknown dynamics. Second, it can easily be applied to nonlinear structural control. Third, external disturbances can be considered in the optimal control. Examples show that structural vibration can be controlled successfully.

Viscoelastic Dampers at Expansion Joints for Seismic Protection of Bridges

This paper presents results of a study on the use of viscoelastic dampers at expansion joints of highway bridges for preventing superstructure decks from falling off the seats and/or from colliding with each other in the event of a severe earthquake. The Kelvin and Maxwell models, consisting of an elastic spring and a linear viscous damper combined in parallel and in series, respectively, are considered for analysis. A 2-D finite-element analysis using bilinear hysteretic models for bridge substructure joints was performed on example bridges constructed with 1 or 2 expansion joints. It was demonstrated that the damper is effective in suppressing the relative displacements at the expansion joints without introducing a significant increase in ductility demands for the substructures. Results also showed that the spring component of the Kelvin and Maxwell models has little effect on the performance of the damper component. This study clearly indicated that the use of linear viscous dampers offers a practical solution to the seismic problem often arising in bridges with expansion joints.

Energy dissipating restrainers for highway bridges

Recent destructive earthquakes have demonstrated the vulnerability of highway bridges to collapse due to excessive movement beyond the available seat widths at expansion joints. This paper investigates the efficacy of using energy dissipating restrainers at expansion joints for preventing collapse of highway bridges in the event of a severe earthquake. The restrainer consists of a nonlinear viscous damper and an elastic spring connected in parallel or in series. Two-dimensional finite element analysis using bilinear hysteretic models for bridge substructure joints and nonlinear gap elements for expansion joints is performed on example bridges with one or two expansion joints. The analytical study demonstrates that the energy dissipating restrainers are effective in reducing the relative opening displacements and impact forces due to pounding at the expansion joints, without significantly increasing ductility demands in the bridge substructures.

System identification of a bridge with lead–rubber bearings

This study develops an identification algorithm to investigate the dynamic properties of a base-isolated highway bridge equipped with lead–rubber bearings. A linear model is used for the substructure while a bilinear hysteretic model is chosen for the bearing system. The nonlinear hysteresis in the latter is characterized through a backbone curve. Application of Masing criterion enables the multivalue restoring force to be transformed into a single-value function. Consequently, the identification scheme which is effective for a linear system can be applied. A numerical example illustrates the proposed identification procedure and its applicability.

Shaking table and pseudodynamic tests for the evaluation of the seismic performance of base-isolated structures

This paper describes a series of shaking table and pseudodynamic tests for the evaluation of seismic performance of base-isolated structures subjected to various seismic earthquake inputs. The main objectives of this study are: (1) evaluation of the effectiveness of base isolation systems for low-rise structures against severe seismic loads through shaking table tests; (2) verification of the substructuring pseudodynamic test method for base-isolated structures in comparison with the shaking table test results; and (3) development of an analytical method for predicting earthquake responses of base isolation systems. In the shaking table test, a quarter-scale three-storey structure base-isolated by laminated rubber bearings is tested. In the pseudodynamic test, only the laminated rubber bearings are tested using the substructuring technique, whereas the concurrent seismic responses of the superstructure are computed using on-line numerical integration. Comparison with the shaking table test results indicates that the substructuring pseudodynamic test method is very effective for determining the dynamic responses of the base-isolated structure. It has also been found that the analytical method with a bi-linear hysteretic model for the base isolator can reasonably predict the earthquake responses of base isolation systems.

A faster simulation method for the stochastic response of hysteretic structures subject to earthquakes

A semi-analytical forward-difference Monte Carlo simulation procedure is proposed for the determination of the lower order statistical moments and the joint probability density function of the stochastic response of hysteretic non-linear multi-degree-of-freedom structural systems subject to nonstationary gaussian white noise excitation, as an alternative to conventional direct simulation methods. The method generalizes the so-called Ermak-Allen algorithm developed for simulation applications in molecular dynamics to structural hysteretic systems. The proposed simulation procedure rely on an assumption of local gaussianity during each time step. This assumption is tantamount to various linearizations of the equations of motion. The procedure then applies an analytical convolution of the excitation process, hereby reducing the generation of stochastic processes and numerical integration to the generation of random vectors only. Such a treatment offers higher rates of convergence, faster speed and higher accuracy. The procedure has been compared to the direct Monte Carlo simulation procedure, which uses a fourth-order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process. The considered system was a multi-dimenensional hysteretic shear frame, where the constitutive equation of the hysteretic shear forces are described by a bilinear hysteretic model. The comparisons show that significant savings in computer time and accuracy can be achieved.