The effective period and damping of a class of hysteretic structures

This paper presents the results of a numerical investigation in which the maximum response of six hysteretic systems is calculated for an ensemble of twelve earthquakes. Inelastic response spectra are constructed for a range of response ductility. An effective linear period and damping are calculated for each system and ductility by determining those parameters which minimize an RMS response spectrum error. Conclusions are presented concerning the effects of deterioration, stiffness degradation, cracking and ductility on the effective linear system parameters.     

A three-stage identification approach for hysteretic systems

This paper deals with the identification of the parameters of a smoothed hysteretic model which was proposed by Bouc and Wen with emphasis on restoring force hysteresis. The problem of estimating the parameters of this system on the basis of input-output data, possibly noise corrupted, is considered. Through the application of various simulated time histories from the hysteretic model, a three-stage systematic method of system identification was proposed. Four different methods of identification are arranged and conducted in this three-stage system identification. The first stage, a sequential regressional analysis is used to identify the equivalent linear system from which elastic or inelastic response can be identified. The identified parameters can be used in the stage when the system is in elastic response. In the second stage, both time domain least-squares method and Gauss-Newton method are applied. The convergence of the Gauss-Newton method can be guaranteed if the identified results from least-squares method are adopted as the initial values for Gauss-Newton method. In the third stage, the extended Kalman filtering technique is needed to identify the noise-corrupt data. Application of this algorithm to a SDOF non-deteriorating system is verified.     

Lead-rubber hysteretic bearings suitable for protecting structures during earthquakes

Lead-rubber hysteretic bearings provide in a single unit the combined features of vertical load support, horizontal flexibility and energy absorbing capacity required for the base isolation of structures from earthquake attack. The lead-rubber hysteretic bearing is a laminated elastomeric bearing of the type used in bridge structures, with a lead plug down its centre. Since the invention of the lead-rubber bearing, a total of eleven bearings up to a diameter of 650mm, with lead plugs ranging from 50 to 170mm in diameter, have been tested under various conditions, including vertical loads to 3·15 MN, strokes to ρ 110 mm, rates from 1 mm/h to 100 mm/s, and temperatures of ?35°C to + 45°C. In all of these tests, the lead-rubber bearings behaved satisfactorily and the hysteresis loops could be described reasonably well by assuming that the lead behaved as an elastic-plastic solid with a yield stress in shear of 10·55 MPa. The bearings showed little rate dependence at ～100 mm/s, though at creep rates of ～1 mm/h the force due to the lead dropped to 30 per cent of that at typical earthquake frequencies. The effect of many small displacements has been tested with 11000 cycles at ±3 mm. A total of 92 lead-rubber bearings have been used in New Zealand to base isolate one building and three bridges. They have yet to be used overseas. This paper describes the tests on the lead-rubber bearings, the results and a design procedure for selecting the size of the lead plug.     

Resonance capacity method for earthquake response analysis of hysteretic structures

The Resonance Capacity Method is proposed for the earthquake response analysis of hysteretic structures. Resonance Capacity is a physical quantity of structures which is related to the hysteretic energy absorbed by structures in one cycle and is equated to the acceleration, velocity and displacement amplitudes α₀, d₀ and d₀ of earthquake ground motions at resonance.¹ According to the idealized trapezoidal approximation of earthquake ground motions in the logarithmic period–velocity plane as proposed by Veletsos and Newmark,⁸ the Resonance Capacity property applies in each period range, short, medium and long, where α₀, v₀ and d₀ respectively are approximately constant. In the medium range of periods, the energy dissipated in hysteretic loops and the deformation amplitudes of a single-degree system with elasto–plastic force–deformation relationships are calculated for the case of El Centro 1940, 18 May earthquake, by this Resonance Capacity Method. The result is compared with results from conventional numerical response analyses obtained by Berg and Thomaides,¹⁴ Kato and Akiyama¹² and Veletsos and Newmark,⁸ and the general agreement is seen to be good. Therefore, it may be possible to apply this Resonance Capacity Method over the entire range of periods. By means of this method the earthquake response analysis of hysteretic systems can be performed easily, and the hysteretic energy and fatigue characteristics of structures may be taken into account directly, up to the point of fracture.     

抗震结构的滞回耗能谱

本文在分析不同类型地面运动引起抗震结构不同类型破坏的基础上，指出结构滞回耗能总量是累积破坏的重要参数，以1735条实际地震记录作为输入计算了不同动力参数单自由度体系弹性及弹塑性总输入能量及滞回耗能量，得出计算弹塑性体系滞回耗能谱的简化公式。     

Response uncertainty and time-variant reliability analysis for hysteretic MDF structures

Response uncertainty evaluation and dynamic reliability analysis corresponding to classical stochastic dynamic analysis are usually restricted to the uncertainties of the excitation. The inclusion of the parameter uncertainties contained in structural properties and excitation characteristics has become an increasingly important problem in many areas of dynamics. In the present paper, a point estimate procedure is proposed for the evaluation of stochastic response uncertainty, and a response surface approach procedure in standard normal space is proposed for analysis of time-variant reliability analysis for hysteretic MDF structures having parameter uncertainties. Using the proposed procedures, the response uncertainties and time-variant reliability can be easily obtained by several repetitions of stochastic response analysis under given parameters without conducting sensitivity analysis, which is considered to be one of the primary difficulties associated with conventional methods. In the time-variant reliability analysis, the failure probability can be readily obtained by improving the accuracy of the first-order reliability method using the empirical second-order reliability index. The random variables are divided into two groups, those with CDF and those without CDF. The latter are included via the high-order moment standardization technique. A numerical example of a 15F hysteretic MDF structure that takes into account uncertainties of four structural parameters and three excitation characteristics is performed, based on which the proposed procedures are investigated and the effects of parameter uncertainties are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Predictive optimal control for seismic analysis of non-linear and hysteretic structures

In this paper, an effective active predictive control algorithm is developed for the vibration control of non-linear hysteretic structural systems subjected to earthquake excitation. The non-linear characteristics of the structural behaviour and the effects of time delay in both the measurements and control action are included throughout the entire analysis (design and validation). This is very important since, in current design practice, structures are assumed to behave non-linearly, and time delays induced by sensors and actuator devices are not avoidable. The proposed algorithm focuses on the instantaneous optimal control approach for the development of a control methodology where the non-linearities are brought into the analysis through a non-linear state vector and a non-linear open-loop term. An autoregressive (AR) model is used to predict the earthquake excitation to be considered in the prediction of the structural response. A performance index that is quadratic in the control force and in the predicted non-linear states, with two additional energy related terms, and that is subjected to a non-linear constraint equation, is minimized at every time step. The effectiveness of the proposed closed-open loop non-linear instantaneous optimal prediction control (CONIOPC) strategy is presented by the results of numerical simulations. Since non-linearity and time-delay effects are incorporated in the mathematical model throughout the derivation of the control methodology, good performance and stability of the controlled structural system are guaranteed. Copyright © 1999 John Wiley & Sons, Ltd.     

Stochastic seismic response analysis of hysteretic multi-degree-of-freedom structures

Analytical study of random vibration of non-linear mutli-degree-of-freedom (MDF) systems is generally difficult. This is particularly true for MDF inelastic systems due to the highly non-linear and hereditary behaviour of the restoring force. On the other hand, to obtain the response statistics using a step-by-step Monte Carlo simulation requires a large sample, and it could be very costly. The purpose of this paper is to present a practical analytical-empirical method for an MDF yielding system. The method is based on a substitute structure (SS) concept in which the SS parameters are determined from empirical results of single-degree-of-freedom systems, i.e. each element in the system is replaced by a linear counterpart with ductility-dependent stiffness and damping. Based on a linear random vibration response analysis, the statistics of the maximum response (ductility) of each element are obtained by iteration. Numerical examples are given for multi-storey buildings with deteriorating (reinforced concrete frame) or non-deteriorating (steel frame) restoring forces. Comparisons with empirical results are satisfactory qualitatively. The main advantage of this method is that it requires relatively insignificant computation time, e.g. 1 s of execution time on the IBM 360–75 system for an eight-storey steel frame.     

安装粘弹性阻尼器的滞变结构抗震可靠度分析

以Boue-Wen滞变恢复力模型模拟剪切型滞变结构的非线性特性，采用等价线性化建立了安装粘弹性阻尼器的滞变结构体系在地震作用下的随机响应计算方法，首当其冲在此基础上分别以结构的层间最大变形和阻尼器阻尼材料的最大剪切变形为结构和阻尼器打有限状态随机变量，探讨了剪切型滞变结构在安装粘弹性阻尼器后的抗震可靠度和阻尼器可靠度的计算方法。     

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.     

An alternative methodology for designing tuned mass dampers to reduce seismic vibrations in building structures

It is well established that small tuned mass dampers (TMDs) attached to structures are very effective in reducing excessive harmonic vibrations induced by external loads but are not as interesting within the context of earthquake engineering problems. For this reason, large mass ratio TMDs have been proposed with the objective of adding a significant amount of damping to structures, thus constituting a good means of reducing structural response in these cases. This solution has other important and attractive dynamic features such as robustness to system uncertainties and reduction of the motion of the inertial mass. In this context, this paper aims to describe an alternative methodology to existing procedures used to tune these devices to earthquake loads and to present some additional considerations regarding its performance in controlling seismic vibrations. The main feature of the proposed method consists of establishing a direct proportion between the damping ratios of the structure's first two vibration modes and the adopted mass ratio. By equalizing the damping ratios of the system's main vibration modes, this proposal also facilitates the use of simplified methods, such as modal analysis based on response spectra. To demonstrate the usefulness of this alternative methodology, an application example is presented, which was also used to perform a parametric study involving other tuning methods and to estimate mass ratio values from which there is no significant advantage in increasing the TMD mass. Copyright © 2012 John Wiley & Sons, Ltd.     

Non-linear seismic behavior of structures with limited hysteretic energy dissipation capacity

This paper investigates the non-linear seismic behavior of structures such as slender unreinforced masonry shear walls or precast post-tensioned reinforced concrete elements, which have little hysteretic energy dissipation capacity. Even if this type of seismic response may be associated with significant deformation capacity, it is usually not considered as an efficient mechanism to withstand strong earthquakes. The objective of the investigations is to propose values of strength reduction factors for seismic analysis of such structures. The first part of the study is focused on non-linear single-degree-of-freedom (SDOF) systems. A parametric study is performed by computing the displacement ductility demand of non-linear SDOF systems for a set of 164 recorded ground motions selected from the European Strong Motion Database. The parameters investigated are the natural frequency, the strength reduction factor, the post-yield stiffness ratio, the hysteretic energy dissipation capacity and the hysteretic behavior model (four different hysteretic models: bilinear self-centring, with limited or without energy dissipation capacity, modified Takeda and Elastoplastic). Results confirm that the natural frequency has little influence on the displacement ductility demand if it is below a frequency limit and vice versa. The frequency limit is found to be around 2 Hz for all hysteretic models. Moreover, they show that the other parameters, especially the hysteretic behavior model, have little influence on the displacement ductility demand. New relationships between the displacement ductility demand and the strength reduction factor for structures having little hysteretic energy dissipation capacity are proposed. These relationships are an improvement of the equal displacement rule for the considered hysteretic models. In the second part of the investigation, the parametric study is extended to multi-degree-of-freedom (MDOF) systems. The investigation shows that the results obtained for SDOF systems are also valid for MDOF systems. However, the SDOF system overestimates the displacement ductility demand in comparison to the corresponding MDOF system by approximately 15%.     

Optimum hysteretic damper design for multi-story timber structures represented by an improved pinching model

Timber structures are characterized by a pinching phenomenon that leads to reduced dissipative capability. A few hysteretic models have been proposed to simulate the mechanical behavior of timber structures, among which the one composed of a bilinear element and a slip element in parallel has been popular in practice. Based on this model, this paper expands on the existing seismic control design methodology to determine the capacity of hysteretic dampers for multi-story timber structures. The equivalent linearization method for a single-degree-of-freedom timber structure with added hysteretic damper is established and is verified through nonlinear timber history analysis over a wide range of structural parameters. The design formulas for determining the damper capacity for a multi-degree-of-freedom system are derived, based on the concept of adjusting the distribution of equivalent stiffness of structure. The seismic control design is applied to many buildings with randomly generated parameters and the effectiveness is confirmed through a nonlinear time history analysis with four sets of seismic excitations. An extended study has shown that the shear force pattern plays an important role in the seismic control design results and thus the performance of structures. The effectiveness of the control of residual deformations by adding dampers is also studied.     

Nonlinear seismic analysis of unsymmetric-plan structures retrofitted by hysteretic damped braces

A displacement-based design procedure using hysteretic damped braces (HYDBs) is proposed for the seismic retrofitting of unsymmetric-plan structures. An expression of the viscous damping equivalent to the hysteretic energy dissipated by the damped braced frame is proposed under bidirectional seismic loads, where corrective factors are assumed as a function of design parameters of the HYDBs. To this end, the nonlinear dynamic analysis of an equivalent two degree of freedom system is firstly carried out on seven pairs of real ground motions whose displacement response spectra match, on average, the design spectrum proposed by the Italian seismic code for a high-risk seismic zone and a medium subsoil class. Then, the extended N2 method considered by the European seismic code, which combines the nonlinear static analysis along the in-plan principal directions of the structure with elastic modal analysis, is adopted to evaluate the higher mode torsional effects. The town hall of Spilinga (Italy), a reinforced concrete (r.c.) framed building with an L-shaped plan, is supposed to be retrofitted with HYDBs. Six structural solutions are compared considering two alternative in-plan distributions of the HYDBs, to eliminate (elastic) torsional effects, and three design values of the frame ductility combined with a constant design value of the damper ductility. To check the effectiveness and reliability of the DBD procedure, the nonlinear static analysis of the test structures is carried out, by evaluating the vulnerability index of r.c. frame members and the ductility demand of HYDBs for different in-plan directions of the seismic loads.     

高层混合结构滞回耗能比的研究

基于结构层间弯曲屈服强度的概念,提出了高层混合结构滞回耗能比的简化计算公式,该公式综合体现了结构以及地震动特征参数对结构滞同耗能比的影响。研究表明,结构滞回耗能比随地震动的峰值速度与峰值加速度的比值的增大而增大;随着地震动幅值的增大,滞回耗能比也线性增加;对于短持时地震动,滞回耗能比与强震持时之间没有一定的规律性,但对于长持时地震动,结构滞回耗能比会随强震持时的增大而线性增加。随着结构弯曲屈服强度系数的增大,结构滞回耗能比呈凹函数下降,结构的自振周期越大,同一弯曲屈服强度系数对应的滞回耗能比越小;钢框架与混凝土剪力墙承载力比值的增加能够降低结构的滞回耗能比。     

A memory-matrix-based identification methodology for structural and mechanical systems

This paper uses an associative memory approach to identify the properties of structural and mechanical systems. The methodology differs from standard identification methods in that it uses a set of parameter vectors simultaneously to generate the estimated parameter vector. The method develops a technique for sequentially generating genetically engineered relevant parameter vectors whose use results in accurate identification, while still using small data sets. This makes the approach promising for on-line, rapid, identification of structures and their health monitoring. © 1998 John Wiley & Sons, Ltd.     

An energy-based method for seismic retrofit of existing frames using hysteretic dampers

This paper proposes a method for the seismic retrofitting of existing frames by adding hysteretic energy dissipating devices (EDDs). The procedure is based on the energy balance of the structure, and it is used to determine the lateral strength, the lateral stiffness and the energy dissipation capacity of the EDDs needed in each story to achieve prescribed target performance levels for a given earthquake hazard. The performance levels are governed by the maximum lateral displacement. The earthquake hazard is characterized in terms of input energy and several seismological parameters, and further takes into account the proximity of the earthquake to the source. The proposed method deals with the effect of the EDDs explicitly in terms of hysteretic energy, bypassing equivalent viscous damping approximations, and directly quantifies the cumulative damage induced in the EDDs. The validity of the method is assessed numerically through nonlinear dynamic response analyses with near-fault and far-field ground motions, as well as experimentally through dynamic shaking table tests.     

An equivalent SDOF system model for estimating the response of R/C building structures with proportional hysteretic dampers subjected to earthquake motions

For the purpose of estimating the earthquake response, particularly the story drift demand, of reinforced concrete (R/C) buildings with proportional hysteretic dampers, an equivalent single‐degree‐of‐freedom (SDOF) system model is proposed. Especially in the inelastic range, the hysteretic behavior of an R/C main frame strongly differs from that of hysteretic dampers due to strength and stiffness degradation in R/C members. Thus, the proposed model, unlike commonly used single‐spring SDOF system models, differentiates the restoring force characteristics of R/C main frame and hysteretic dampers to explicitly take into account the hysteretic behavior of dampers. To confirm the validity of the proposed model, earthquake responses of a series of frame models and their corresponding equivalent SDOF system models were compared. 5‐ and 10‐story frame models were studied as representative of low‐ and mid‐rise building structures, and different mechanical properties of dampers—yield strength and yield deformation—were included to observe their influence on the effectiveness of the proposed model. The results of the analyses demonstrated a good correspondence between estimated story drift demands using the proposed SDOF system model and those of frame models. Moreover, the proposed model: (i) led to better estimates than those given by a single‐spring SDOF system model, (ii) was capable of estimating the input energy demand and (iii) was capable of estimating the total hysteretic energy and the participation of dampers into the total hysteretic energy dissipation, in most cases. Results, therefore, suggest that the proposed model can be useful in structural design practice. Copyright © 2010 John Wiley & Sons, Ltd.     

A component approach for the hysteretic behaviour of connections in cross‐laminated wooden structures

In this paper, a numerical model to estimate the dissipative capacity and describe the cyclic response of cross‐laminated (X‐lam) timber buildings is presented. The connections between panels and to the foundation (metal hold‐downs and angle brackets, and screwed connectors) are modelled with nonlinear hysteretic multispring elements taking into account the strength interaction between different degrees of freedom according to a predefined domain. The timber components (solid X‐lam floors and wall panels) are modelled using elastic shell elements. By calibration on experimental cyclic tests carried out on each degree of freedom, important features of timber connection behaviour such as post‐peak strength, pinching and stiffness degradation can all be considered. In addition, the effect of friction at the interface between panels and with foundation can be taken into account. These springs have been implemented as external subroutines in a widespread software package such as Abaqus. By comparison with the experimental results of cyclic tests carried out on single X‐lam walls, coupled X‐lam walls and a single‐storey X‐lam building, the accuracy of the proposed model is demonstrated. Copyright © 2013 John Wiley & Sons, Ltd.