Hysteretic Structural Systems Research Articles

An approximate stochastic dynamics approach for design spectrum based response analysis of nonlinear structural systems with fractional derivative elements

A novel approximate approach is developed for determining, in a computationally efficient manner, the peak response of nonlinear structural systems with fractional derivative elements subject to a given seismic design spectrum. Specifically, first, an excitation evolutionary power spectrum is derived that is compatible with the design spectrum in a stochastic sense. Next, relying on a combination of statistical linearization and stochastic averaging yields an equivalent linear system (ELS) with time-variant stiffness and damping elements. Further, the values of the ELS elements at the most critical time instant, i.e., the time instant associated with the highest degree of nonlinear/inelastic response behavior exhibited by the structural system, are used in conjunction with the design spectrum for determining approximately the nonlinear system peak response displacement. The herein developed approach can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the governing equations of motion. Furthermore, the approach exhibits the significant novelty of exploiting the localized time-dependent information provided by the derived time-variant ELS elements. Indeed, the values of the ELS stiffness and damping elements at the most critical time instant capture the system dynamics better than an alternative standard time-invariant statistical linearization treatment. This leads to enhanced accuracy when determining nonlinear system peak response estimates. An illustrative numerical example is considered for assessing the performance of the approximate approach. This pertains to a bilinear hysteretic structural system with fractional derivative elements subject to a Eurocode 8 elastic design spectrum. Comparisons with pertinent Monte Carlo simulation data are included as well, demonstrating a high degree of accuracy.

Rocking incremental dynamic analysis

AbstractThe seismic response assessment of rocking systems via Incremental Dynamic Analysis (IDA) is investigated, focusing on the issues that arise in the analysis and postprocessing stages. Rocking IDA curves generally differ from those of hysteretic structural systems due to (i) the frequent appearance of resurrections; (ii) their highly weaving non‐monotonic behavior; and (iii) their overall high variability. Hence, including or ignoring analysis results above the first resurrection level, deriving statistics given a response level versus an intensity measure level, as well as selecting an adequate number of ground motion records and runs per record, become challenging issues with non‐trivial impact on the probabilistic characterization of rocking response. This necessitates a fresh view on analysis choices and post‐processing techniques, aiming to assure the accuracy and fidelity of rocking IDA results. As an example, the effect of different choices and techniques are showcased on two‐dimensional rigid blocks that are assumed to represent simplified models of monolithic ancient columns of different slenderness.

Experimental verification of constrained minimum variance unbiased estimator for simultaneous input and state estimation of Bounded Input and Bounded Output (BIBO) type Bouc–Wen hysteretic structural system

Constraint minimum variance unbiased estimator is improvised for simultaneous input and parameter estimation of the Bouc–Wen hysteretic system of a full-scale bridge pier. The essential structural properties in this case (i.e., bounded input and bounded output) are incorporated in the form of constraint conditions along with degradation and pinching effects. To demonstrate the efficiency of this algorithm, two different types of sigma points generation schemes (viz., unscented transformed points and cubature quadrature points) are presented here. First, the algorithm is validated using simulated response obtained from a single-bay single-storey reinforced concrete frame, whose hysteretic parameters are obtained by minimizing the error between the model and experimentally observed load-deformation characteristics. The rationale behind the use of experimentally observed hysteresis in simulation is to replicate the actual structure so that the realistic material behaviour is maintained. Once the algorithm is validated, its performance is further demonstrated using experimental results of a full-scale bridge pier, which is the main objective of this study. The numerical results presented in this paper clearly show the robustness of the proposed algorithm using unscented sigma points, which can identify both hysteretic properties and input excitation with sufficient level of accuracy.

First-excursion stochastic incremental dynamics methodology for hysteretic structural systems subject to seismic excitation

A novel efficient stochastic incremental dynamics methodology considering first-excursion probability for nonlinear structural systems subject to stochastic seismic excitations in alignment with contemporary aseismic codes provisions is developed. To this aim, an approximate nonlinear stochastic dynamics technique for conducting first-passage probability density function (PDF) based stochastic incremental dynamic analysis is developed. Firstly, an efficient stochastic iterative linearization methodology is devised achieving convergence of the equivalent system damping ratios with the damping premises of the excitation response spectrum leading to a coherent determination of a robust scalable intensity measure (IM) which bears direct relation to its damaging potential. Subsequently, utilizing the stochastically derived time-varying forced vibrational system properties in conjunction with a combination of deterministic and stochastic averaging treatment the first-excursion PDF is efficiently determined for each and every of the considered limit-state rules (LSs). Lastly, an incremental mechanization analogous to the one used in normal incremental dynamic analysis (IDA) is proposed to ensure the necessary compatibility for applications in the fields of structural and earthquake engineering. The back-and-forth twisting pattern of IDA curves which is related with the multiple points satisfaction of the very same limit-state rule encourages the study of the problem from a first-passage perspective considering timing in addition to the intensity of the excitation variable. The selected engineering demand parameter (EDP) of the first-excursion time constitutes an excellent response related variable with twofold meaning; it performs structural behavior monitoring considering intensity and timing information whereas it is inherently coupled with limit-state requirements. The developed stochastic dynamics technique provides with reliable higher order statistics (i.e., PDF) of the chosen EDP. A structural system comprising the bilinear hysteretic model serves as a numerical example for demonstrating the reliability of the proposed first-excursion PDF-based stochastic incremental dynamics methodology. Nonlinear response time-history analysis involving a large ensemble of Eurocode 8 spectrum compatible accelerograms is conducted to assess the accuracy of the proposed methodology in a Monte Carlo-based context.

An extended model-based observer for state estimation in nonlinear hysteretic structural systems

This paper proposes a model-based observer for state estimation in nonlinear hysteretic structural systems. The observer combines a nonlinear model and response measurements to estimate the complete dynamic response of the structure of interest. The feedback gain of the observer is computed by an iterative algorithm that accounts for the nonlinearity of the system and minimizes the trace of the state error covariance matrix. The main feature of the proposed observer (and its main advantage with respect to existing nonlinear state estimators) is that it is designed to be physically realizable as a nonlinear structural model, which allows the user to benefit from modeling capabilities available in finite element solvers. The performance of the proposed observer is studied in the context of a numerical example consisting of a nonlinear chain model with a Bouc-Wen model of hysteresis subjected to a base motion; the estimation results are compared to those obtained using an unscented Kalman filter. The observer is also validated experimentally using data from a six-story full-scale benchmark wood frame building tested by the NEESWood project at the E-Defense facility in Miki, Japan. The results show the effectiveness of the proposed approach to estimate the dynamic response of large-scale structural systems exhibiting a strong nonlinear behavior.

Bivariate Gaussian mixture–based equivalent linearization method for stochastic seismic analysis of nonlinear structures

SummaryTo address challenges in stochastic seismic analysis of nonlinear structures, this paper further develops a recently proposed Gaussian mixture–based equivalent linearization method (GM‐ELM). The GM‐ELM uses a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. Using properties of the Gaussian mixture model, GM‐ELM can decompose the non‐Gaussian response of a nonlinear system into multiple Gaussian responses of linear single–degree of freedom oscillators. With the set of the equivalent linear systems identified by GM‐ELM, response statistics as crossing rate and first‐passage probability can be computed conveniently using theories of linear random vibration analysis. However, the original version of GM‐ELM may lead to an inaccurate estimate because of the heuristic parameters of the linear system introduced to supplement insufficient information. To overcome this limitation and define unique equivalent linear systems, this paper proposes a further developed version of GM‐ELM, which uses a mixture of bivariate Gaussian densities instead of univariate models. Moreover, to facilitate the use of elastic response spectra for estimating the mean peak responses of a nonlinear structure, a new response spectrum combination rule is proposed for GM‐ELM. Two numerical examples of hysteretic structural systems are presented in this paper to illustrate the application of the bivariate GM‐ELM to nonlinear stochastic seismic analysis. The analysis results obtained by the bivariate GM‐ELM are compared with those obtained by the univariate GM‐ELM, the conventional equivalent linearization method, the tail equivalent linearization method, and Monte Carlo simulation. The supporting source code and data are available for download at https://github.com/yisangri/GitHub‐bGM‐ELM‐code.git

Adaptive Dynamic Surface Control of Bouc–Wen Hysteretic Systems

This paper develops an adaptive dynamic surface algorithm for designing the control law for uncertain hysteretic structural systems with seismic disturbances that can be converted to a semi strict feedback form. Hysteretic behavior is usually described by Bouc–Wen model for hysteretic structural systems like base isolation systems. Adaptive sliding mode and adaptive backstepping algorithms are also studied and simulated for comparison purposes. The presented simulation results indicate the effectiveness of the proposed control law in reducing displacement, velocity and acceleration responses of the structural system with acceptable control force. Moreover, using dynamic surface control (DSC), the study analyzes the stability of the controlled system based on the Lyapunov theory.

Comparison of restoring force models for the identification of structures with hysteresis and degradation

Abstract When subjected to events such as earthquakes, engineering structures typically exhibit a nonlinear and hysteretic behaviour with stiffness and strength degradations. Though a reliable evaluation of safety conditions should take into account the nonlinear dynamic and evolutionary nature of the structural response, the experimental identification of a nonlinear behaviour under dynamic and seismic loading is, to date, an open problem. The present research aims at evaluating the potential of different restoring force models for simulating the seismic response of hysteretic structural systems, with special emphasis on the two main problems encountered when applying this approach to full-scale structures under intense excitation: (a) a markedly time-dependent behaviour; (b) need to compare among different restoring force models, either expressed in a parametric or polynomial form. In particular, polynomial models will be formulated both in terms of restoring force and its derivative, in order to present a comprehensive discussion of different strategies. The nonlinear identification technique employed in this paper is required to account for a time-dependent behaviour. In fact, in presence of degradation or any other time-varying characteristics, instantaneous identification certainly constitutes an enhancement of the classical restoring force based approach, and may as well provide checks on the consistency of the assumed models.

Optimal sliding mode control of single degree-of-freedom hysteretic structural system

This paper addresses the controller design problem of a nonlinear single degree-of-freedom structural system excited by the earthquake. Bouc–Wen model, as an efficient hysteresis modeling method, is used to model the system nonlinearity. Sliding mode control (SMC), due to its robustness in dealing with uncertainty, is utilized as the main control strategy. An optimal sliding surface is presented which minimizes the displacement and control force in terms of a quadratic cost function. Two numerical examples are given to illustrate the effectiveness of the proposed strategy subject to three earthquakes of El-Centro, Rinaldi and Kobe. Simulation results show a significant and considerable reduction in structural response and indicate that the performance of suggested optimal SMC strategy is remarkable.

Order-independent optimal polynomial control of stochastic dynamical systems

As the traditional knowledge of optimal polynomial controls, the control effect of non-linear control strategy hinges on the order of the polynomial controllers either for linear systems or non-linear systems, In the present paper, a novel optimal polynomial control strategy against this knowledge, based on the framework of physical stochastic optimal control, for stochastic dynamical systems is proposed. The control criterion relies on the minimum of a performance function in an energy trade-off sense, as gauged by exceedance probability of system quantities of interest, involving an optimisation programme. For illustrative purposes, non-linear stochastic optimal controls of base-excited multi-degree-of-freedom linear and hysteretic structural systems are carried out, respectively. Numerical results reveal that the optimal polynomial control exhibits the order-independent behaviour that the linear control with the first-order controller suffices even for the hysteretic systems when the exceedance probability based control criterion is employed. This is practically meaningful since it bypasses the need to utilise non-linear controllers which might be associated with dynamical instabilities due to time delay and computational dynamics. It is noted, meanwhile, that the response performance of the controlled structural systems is improved significantly.

Nonlinear stochastic optimal control strategy of hysteretic structures

Referring to the formulation of physical stochastic optimal control of structures and the scheme of optimal polynomial control, a nonlinear stochastic optimal control strategy is developed for a class of structural systems with hysteretic behaviors in the present paper. This control strategy provides an amenable approach to the classical stochastic optimal control strategies, bypasses the dilemma involved in It<TEX>$\hat{o}$</TEX>-type stochastic differential equations and is applicable to the dynamical systems driven by practical non-stationary and non-white random excitations, such as earthquake ground motions, strong winds and sea waves. The newly developed generalized optimal control policy is integrated in the nonlinear stochastic optimal control scheme so as to logically distribute the controllers and design their parameters associated with control gains. For illustrative purposes, the stochastic optimal controls of two base-excited multi-degree-of-freedom structural systems with hysteretic behavior in Clough bilinear model and Bouc-Wen differential model, respectively, are investigated. Numerical results reveal that a linear control with the 1st-order controller suffices even for the hysteretic structural systems when a control criterion in exceedance probability performance function for designing the weighting matrices is employed. This is practically meaningful due to the nonlinear controllers which may be associated with dynamical instabilities being saved. It is also noted that using the generalized optimal control policy, the maximum control effectiveness with the few number of control devices can be achieved, allowing for a desirable structural performance. It is remarked, meanwhile, that the response process and energy-dissipation behavior of the hysteretic structures are controlled to a certain extent.

Evaluation of current criteria in predicting the separation necessary to prevent seismic pounding between nonlinear hysteretic structural systems

This study evaluates the accuracy of four different criteria to calculate the separation necessary to prevent seismic pounding between nonlinear hysteretic structural systems. All of the criteria considered in this paper make use of the same basic equation of the Double Difference Combination rule, but they adopt different procedures to estimate the correlation between displacement responses of nonlinear hysteretic systems. The seismic excitation was modeled as a nonstationary random process, and adjacent structures were modeled as 5%-damped bilinear hysteretic SDOF systems having a 5% post-yielding stiffness ratio. Results obtained through Monte Carlo simulations indicate that none of the criteria evaluated in this study is completely satisfactory in the sense that none of them can consistently provide “exact” or “somewhat conservative” results.

H∞ Filtering in Neural Network Training and Pruning with Application to System Identification

An efficient training and pruning methodology based on the H∞ filtering algorithm is proposed for artificial neural networks (ANNs). ANNs are first trained by the H∞ filtering algorithm and then some unimportant weights are removed based on the training. The results presented in the paper show that the proposed method provides better pruning results of the network without losing its generalization capacity. It also provides a robust training algorithm for given arbitrary network structures. The usefulness and effectiveness of the proposed methodology are demonstrated in developing an ANN model of a hysteretic structural system.

Adaptive Control of a Base Isolated System for Protection of Building Structures

In this paper, we present two adaptive backstepping control algorithms for a second-order uncertain hysteretic structural system found in base isolation scheme for seismic active protection of building structures. The hysteretic nonlinear behavior is described by a Bouc–Wen model. The structural parameters and isolation parameters are all uncertain parameters. In the first scheme, there is no apriori information required from these parameters and the residual effect of the hysteresis is treated as a bounded disturbance. An update law is used to estimate the bound involving this partial hysteresis effect and external disturbance. In the second scheme, we further take the structure of the Bouc–Wen model describing the hysteresis into account in the controller design, if apriori knowledge on some parameters of the model is available. It is shown that not only is global stability guaranteed by the proposed controller, but also both transient and asymptotic performances are quantified as explicit functions of the design parameters so that designers can tune the design parameters in an explicit way to obtain the required closed loop behavior.

Adaptive control of a hysteretic structural system

This paper addresses the problem of controlling a second-order uncertain structural hysteretic system. The hysteretic behavior is represented by a Bouc–Wen model whose parameters lie within intervals. It is shown that the nonlinear response of the model acts as a bounded disturbance. This fact is used to elaborate an adaptive controller, to prove the closed-loop stability and to obtain performance bounds.

Development of adaptive modeling techniques for non-linear hysteretic systems

Adaptive estimation procedures have gained significant attention by the research community to perform real-time identification of non-linear hysteretic structural systems under arbitrary dynamic excitations. Such techniques promise to provide real-time, robust tracking of system response as well as the ability to track time variation within the system being modeled. An overview of some of the authors’ previous work in this area is presented, along with a discussion of some of the emerging issues being tackled with regard to this class of problems. The trade-offs between parametric-based modeling and non-parametric modeling of non-linear hysteretic dynamic system behavior are discussed. Particular attention is given to (1) the effects of over- and under-parameterization on parameter convergence and system output tracking performance, (2) identifiability in multi-degree-of-freedom structural systems, (3) trade-offs in setting user-defined parameters for adaptive laws, and (4) the effects of noise on measurement integration. Both simulation and experimental results indicating the performance of the parametric and non-parametric methods are presented and their implications are discussed in the context of adaptive structures and structural health monitoring.

On‐line identification of non‐linear hysteretic structural systems using a variable trace approach

AbstractIn this paper, an adaptive on‐line parametric identification algorithm based on the variable trace approach is presented for the identification of non‐linear hysteretic structures. At each time step, this recursive least‐square‐based algorithm upgrades the diagonal elements of the adaptation gain matrix by comparing the values of estimated parameters between two consecutive time steps. Such an approach will enforce a smooth convergence of the parameter values, a fast tracking of the parameter changes and will remain adaptive as time progresses. The effectiveness and efficiency of the proposed algorithm is shown by considering the effects of excitation amplitude, of the measurement units, of larger sampling time interval and of measurement noise. The cases of exact‐, under‐, over‐parameterization of the structural model have been analysed. The proposed algorithm is also quite effective in identifying time‐varying structural parameters to simulate cumulative damage in structural systems. Copyright © 2001 John Wiley &amp; 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.

Nonlinear behaviour of a coupled shear wall and shear core system under random ground excitation

Nonlinear behaviour of a coupled shear wall and shear core system under random ground excitation is presented. Random vibrational theory is used for the response analysis of hysteretic structural systems under stochastic earthquake excitation. An equivalent linearization technique, based on Wen's model, is used to convert the nonlinear differential equations to the equivalent linear differential equations, under bi-axial hysteretic restoring forces. The damping is assumed to be linear and a Clough–Penzien double filter is used to calculate the input ground excitation forces on the structure. A comparison has been made between coupled shear wall and shear core systems, to get a best model for a 15 storey building. © 1997 John Wiley & Sons, Ltd.

Nonlinear behaviour of a coupled shear wall and shear core system under random ground excitation

Nonlinear behaviour of a coupled shear wall and shear core system under random ground excitation is presented. Random vibrational theory is used for the response analysis of hysteretic structural systems under stochastic earthquake excitation. An equivalent linearization technique, based on Wen's model, is used to convert the nonlinear differential equations to the equivalent linear differential equations, under bi-axial hysteretic restoring forces. The damping is assumed to be linear and a Clough–Penzien double filter is used to calculate the input ground excitation forces on the structure. A comparison has been made between coupled shear wall and shear core systems, to get a best model for a 15 storey building. © 1997 John Wiley & Sons, Ltd.