Statistical Equivalent Linearization Research Articles

Influence of the excitation frequency content on the efficiency of a tuned liquid column damper

The current study addresses the influence of the excitation frequency content on the efficiency of a tuned liquid column damper (TLCD) to control the displacement response in structures subjected to seismic excitations. A time-domain stochastic stationary analysis is performed, and an equivalent statistical linearization for the TLCD is obtained. The results show that the TLCD is more efficient in structures subjected to broadband processes. In narrowband processes, the maximum performance of the TLCD occurs when the linear equivalent period of the system TLCD-structure tunes the TLCD period, and the frequency of the structure matches the predominant frequency of the input motion. Results derived from this analysis are validated through two time-history seismic analyses.

Effect of earthquake statistically correlated vertical component on inelastic demand to regular reinforced-concrete frames

Post-earthquake survey and structural analysis prove that the vertical seismic action can damage buildings and bridges, even those with regular configuration. Moreover, there is evidence that one principal axis of ground motion may be not vertical. The inelastic demand to regular building structures is assessed here considering both uncorrelated and cross-correlated random seismic components, concurrent to deterministic weight. The method is conventional statistical equivalent linearization. Planar reinforced-concrete frames are analyzed parametrically varying the number of stories, the design ductility class, or the foundation ground type, as well as variance and cross-covariance of the vertical seismic component. Following a lumped plasticity approach, the bending moment-rotation relationship is a Bouc-Wen equation extended for asymmetry and interaction with axial force. Based on mean values, variances, and percentiles of response, the effect of uncorrelated vertical excitation is appreciable only on stress resultants of little practical consequence. Much more important is the effect of cross-correlated vertical excitation. As extreme results, the interstory drift demand increases up to four times. The rotation ductility demand to columns, less than 2 under uncorrelated excitations, rises to 5, meaning very likely yielding of columns designed as strong. The probability of axial force in the columns being tensile grows from negligible value to 8%. The beams may yield also around mid-span. The frame plastic mechanism worsens towards soft story, the peak deformation ductility being demanded more than the cyclic ductility. Not only variance of the vertical ground motion, but also cross-covariance with the horizontal one should be considered in the near field.

Analysis of a tuned liquid column damper in non-linear structures subjected to seismic excitations

Abstract The behavior of the tuned liquid column damper (TLCD) is analyzed in the control of non-linear structures subjected to random seismic excitations. The structure is modeled as a system of one degree of freedom with incursion in the non-linear range. The Bouc-Wen hysteretic model is used to model the non-linear behavior of the structure. A stationary stochastic analysis is performed in the domain of the frequency. An equivalent statistical linearization was used for the analysis of the main system and the TLCD. The TLCD parameters considered for the optimization process were the frequency and the head loss coefficient. Two target functions were considered, (i) reduction of the main displacement of the system, (ii) reduction of the hysteretic energy. Two random processes were considered as seismic excitation, first a broad bandwidth process and secondly a narrow bandwidth process. The results show that for a broad bandwidth process, the TLCD tends to tune with the linear equivalent frequency of the system in the case without TLCD, while for the narrow bandwidth process, it tunes (TLCD) with the dominant frequency of the input. It is seen that the TLCD becomes detuned with regard to the frequency of the structure as the structure becomes more non-linear. It is also seen that the optimal tuning ratio of the TLCD is unsensitive to the mass ratio of the device and the main damping ratio of the system. It is also concluded that in case of flexible structures, the optimal head loss coefficient tends to be lower and increases with regard to its length ratio. It is seen that the effectiveness of the TLCD is greater for higher mass ratios of the device. In addition, it is found that the optimal TLCD becomes less effective as the structure enters the non-linear range, showing lower efficiency than what is seen in the literature for optimal TLCDs in linear structures.

Perturbation methods in evolutionary spectral analysis for linear dynamics and equivalent statistical linearization

The analysis of large-scale structures subject to transient random loads, coherent in space and time, is a classic problem encountered in earthquake and wind engineering. The simulation-based framework is usually seen as the most convenient approach for both linear and nonlinear dynamics. However, the generation of statistically consistent samples of an excitation field remains a heavy computational task. In light of this, perturbation techniques are applied to develop and improve evolutionary spectral analysis.Advantageously performed in a standard modal basis, this evolutionary spectral analysis for linear structures requires the computation of the modal impulse response matrix. However, this matrix has no general closed-form expression in the presence of modal coupling. We propose therefore to model it by an asymptotic approximation, obtained by the inverse Fourier transform of an asymptotic expansion of the modal transfer matrix of the structure. This latter expansion considers the modal coupling as a perturbation of a main decoupled system. This strategy leads to an expansion known in a closed-form. Finally, the semi-group property allows the use of an efficient recurrence relation to approximate the modal evolutionary transfer matrix, i.e. the evolutionary extension of the transfer matrix.The asymptotic expansion-based method and the recurrence relation are then applied to nonlinear transient dynamics by using Gaussian equivalent linearization. This extension is formalized by a multiple timescales approach, allowing to consider a linearized structure, namely a time variant system, as piecewise linear time invariant depending on a statistical timescale. The proposed developments are finally illustrated on realistic civil engineering applications.

An asymptotic expansion-based method for a spectral approach in equivalent statistical linearization

Equivalent linearization consists in replacing a nonlinear system with an equivalent linear one whose parameters are tuned with regard to the minimization of a suitable function. In particular, the Gaussian equivalent linearization expresses the properties of an equivalent linear system in terms of the mean vector and the covariance matrix of the responses, which are the unknowns of the optimization problem in a spectral approach. Even though the system has been linearized, the resulting set of equations is nonlinear. The computational effort in this method pertains to the solution of a possibly large set of nonlinear algebraic equations involving integrals and inversions of full matrices. This work proposes to develop and apply an asymptotic expansion-based method to facilitate and to improve the statistical linearization for large nonlinear structures. The proposed developments demonstrate that for slightly to moderately coupled nonlinear systems, the equivalent linearization can be applied with an appropriate modal approach and eventually seen as a convergent series initiated with the stochastic response of a main decoupled linear system. With this method, the computational effort is attractively reduced, the conditioning of the set of nonlinear algebraic equations is improved and inversion of full transfer matrices and repeated integrations are avoided. The paper gives a formal description of the method and illustrates its implementation and performances with the computation of stationary responses of nonlinear structures subject to coherent random excitation fields.

Further Investigation of a Gaussian Sum Filter Based on Stochastic Equivalent Linearization

We proposed a stochastic (or statistical) Equivalent linearization - Gaussian Sum Filter(: EqGS Filter) for discrete time nonlinear Systems. Subsequently, in this paper, we investigate and show the further results related to the EqGS Filter. Especially we discuss a method to apply GaussHermite quadrature rules for evaluation of the conditional expected values of the quantities required to design the EqGS filter. Finally, we show the estimation results of AR modeling comparison with the extended Kalman and the equivalent linearization filters.

Further Investigation of a Gaussian Sum Filter Based on Stochastic Equivalent Linearization

We proposed a stochastic (or statistical) Equivalent linearization - Gaussian Sum Filter(: EqGSFilter) for discrete time nonlinear Systems. Subsequently, in this paper, we investigate and showthe further results related to the EqGS Filter. Especially we discuss a method to apply Gauss-Hermite quadrature rules for evaluation of the conditional expected values of the quantities requiredto design the EqGS filter. Finally, we show the estimation results of AR modeling comparison withthe extended Kalman and the equivalent linearization filters.

Structural optimization of non-linear systems under stochastic excitation

This work concentrates on the structural optimization of a class of non-linear systems with deterministic structural parameters subject to stochastic excitation. The optimization problem is formulated as the minimization of an objective function subject to constraints on the response level. The stochastic response is characterized by its first two statistical moments, which are computed by a statistical equivalent linearization technique. The implicit structural optimization problem is replaced by a sequence of explicit sub-optimization problems. The sub-problems are constructed by using a conservative first-order approximation of the objective and constraint functions. The applicability of the proposed design process is demonstrated in three numerical examples where the methodology is applied to systems with nonlinearity of hardening and hysteretic type. The effects of the nonlinearity on the general performance of the final designs are discussed. At the same time, some engineering implications of the results obtained in this work are addressed.

Non-stationary response of large, non-linear finite element systems under stochastic loading

A very efficient and straightforward numerical procedure for the computation of statistical second moment characteristics of large, non-linear finite element systems under stochastic loading is presented. For the modeling of both the loading and the response of the system an orthogonal series expansion of the corresponding covariance matrix, the so-called Karhunen–Loève expansion is applied, allowing to incorporate potentially available statistical data of an excitation process directly into the analysis. The non-linear equation of motion is linearized by the method of equivalent statistical linearization. According to the present capabilities of this linearization technique, one-dimensional hysteretic elements are used for modeling the non-linear system behavior. The mode acceleration method is applied in order to reduce significantly the size of the system equation and thus increasing the computational efficiency of the proposed procedure. Contrary to methodologies based on state space formulations, this procedure relies on deterministic step by step integration, implying that the dimension of the system equation is the same as in a purely deterministic analysis.

A computational procedure to estimate the stochastic dynamic response of large non-linear FE-models

An algorithm for the computation of the stochastic non-stationary non-linear response of large FE-models is presented. In stochastic analysis, the response is described by the mean and covariance function and possibly by the probability distribution of the non-linear response. To estimate the covariance matrix, the method of equivalent statistical linearization is applied for linearizing all non-linear elements. The large fully populated and symmetric covariance matrix of dimension ⩾2 n is described by the so called Karhunen–Loéve expansion representation, which allows one to employ a feasible description. In this context, an efficient procedure is suggested to determine the minimal number of Karhunen–Loéve vectors necessary to assure a sufficiently accurate representation. This method in fact allows one to employ any available deterministic integration scheme to compute the Karhunen–Loéve vectors. To increase the computational efficiency, modal coordinates are used to represent the linear sub-system. Special attention is given to the effects of mode truncation which are generally not negligible for large FE-systems. Moreover, the suggested approach has no limitation w.r.t. the size of the model in terms of degrees of freedom (DOFs) or the number of non-linear elements. The feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25, 50 and 100 thousand DOFs containing hysteretic damping elements.

Statistical Seismic Response Analysis of Piping System with a Teflon Friction Support

A shaking table experiment for the piping system with a friction support has been conducted to investigate the reduction characteristics of vibration responses of the piping system due to frictional effect against earthquake type excitations. The piping system has been so designed that the contact point of the piping moves in-plane 2 dimensional motion on the teflon bearing plate supported by the shaking table. The experimental results were well explained by a proposed deterministic approximate modal analysis theory which were already reported in the separate paper. Here, we try to evaluate the seismic responses by means of the non-deterministic method, i. e. the statistical equivalent linearization method, for friction forces at the friction support of the piping. It has been found that the calculated responses by the linearization method show in good agreement with the experimental ones. The equivalent modal damping is also estimated in this process. From the above results, seismic performance of the frictional effect is discussed.

Critical excitation for elastic–plastic structures via statistical equivalent linearization

Since earthquake ground motions and their effects on structural responses are very uncertain even with the present knowledge, it is desirable to develop a robust structural design method taking into account these uncertainties. Critical excitation approaches are promising and a new random critical excitation method for single-degree-of-freedom (SDOF) elastic–plastic structures is proposed. The power (area of power spectral density (PSD) function) and the intensity (magnitude of PSD function) are fixed and the critical excitation is found under these restrictions. In contrast to linear elastic structures, transfer functions and related simple expressions for response evaluation cannot be defined in elastic–plastic structures and difficulties arise in describing the peak responses except elastic–plastic time-history response analysis. Statistical equivalent linearization is utilized to estimate the elastic–plastic stochastic peak responses approximately. The critical excitations are obtained for two examples and compared with the corresponding recorded earthquake ground motions.

Probabilistic critical excitation for MDOF elastic–plastic structures on compliant ground

AbstractEarthquake ground motions and their effects on structural responses are very uncertain even with the present knowledge. It is therefore desirable to develop a robust structural design method taking into account these uncertainties. Critical excitation approaches are promising and a new random critical excitation method is proposed for MDOF elastic–plastic shear‐building structures on compliant ground. The power (area of power spectral density (PSD) function) and the intensity (magnitude of PSD function) are fixed and the critical excitation is found under these restrictions. In contrast to linear elastic structures, transfer functions and simple expressions for response evaluation cannot be defined in elastic–plastic structures and difficulties arise in describing the peak responses except by laborious elastic–plastic time‐history response analysis. Statistical equivalent linearization is used to estimate the elastic–plastic stochastic peak responses approximately. The critical excitation responses are obtained for several examples and compared with those of the corresponding recorded earthquake ground motion. Copyright © 2001 John Wiley & Sons, Ltd.

Non-linear stochastic response distributions by local statistical linearization

A novel computational approach based on the statistical equivalent linearization and Gaussian superposition is introduced. The methodology allows to extend the concept of statistical equivalent linearization to proceed from estimates of the second moment properties of the stochastic response to estimates for the non-Gaussian probability distribution. This paper introduces the concept of local equivalent linearization designed to compute a close approximation of the non-stationary probability density function p( x ,t) of a non-linear system exited by random excitation. Non-Gaussian distributions are represented as a sum of normal distributions. The suggested approach employs the well-developed equivalent linearization procedure or Gaussian closure to compute the non-Gaussian distribution of the response of a non-linear system. Since equivalent linearization is applicable for higher dimensions and finite element models, the suggested approach lends itself to providing numerical solutions for higher-dimensional cases.

A computational procedure for the implementation of equivalent linearization in finite element analysis

This paper deals with the practical implementation of the statistical equivalent linearization method (EQL) in conjunction with general FE-analysis to evaluate non-linear structural response under random excitation. A computational procedure is presented which requires the non-linear part of the system to be subdivided into suitable sub-domains (elements). Each element is independently linearized using only a minimum number of co-ordinates. A local co-ordinate system is introduced using linear transformations of the global (master) degrees of freedom. Restoring forces and non-linear constitutive laws are defined by the local co-ordinates of each element. The linearization coefficients are further transformed back to establish the global linearized system. The procedure has, on one hand, the ability to use any desired linearization criterion and, on the other hand, it can be combined with highly developed procedures to determine the response of arbitrary large FE-models. To illustrate the applicability of the procedure, two different non-linear systems are analysed under bi-directional earthquake excitation. Copyright © 2000 John Wiley & Sons, Ltd.

Statistical Seismic Response Analysis of Piping System with Friction Support.

A large shaking table experimental study for a piping system with a friction support has been conducted for many years to investigate the reduction capability of responses of piping system due to frictional effect against earthquake type excitation. Here, we try to evaluate the vibration responses by means of a non-deterministic method, i.e. the statistical equivalent linearization method, for the piping system. The friction forces at a friction support of the piping for modal equations are calculated by means of the statistical linearization method. It has been found that the calculated responses by the linearization method show good agreement with the experimental ones. The equivalent modal damping were also estimated in this process. Seismic performance of the frictional effect is also discussed.

On the stochastic response of nonlinear FE models

This paper focuses on the stochastic dynamic response of structures modeled by finite elements with a relatively large number of degrees of freedom. FE models with nonlinearities and uncertain (stochastic) system properties are discussed. It is shown that component mode synthesis can be used most advantageously to solve the issue of computational efficiency and feasibility. The stochastic response due to stochastic loading of large FE models with nonlinear elements is determined by statistical equivalent linearization (EQL). The developed component-mode synthesis allows to determine the complex modal properties of arbitrary large linearized finite element models. Nonsymmetric structural matrices, as a result of the EQL, and filters for modeling of filtered white noise can be treated by the suggested approach. Since the efficiency of the procedure is nearly independent of the number of degrees-of-freedom (DOF) involved, statistical equivalent linearization becomes applicable for arbitrary detailed FE models. Furthermore, the dynamic response of FE models with uncertain or stochastic system properties is discussed. In this case, Monte Carlo simulation is suggested as the most appropriate approach for FE models. The paper focuses on the random eigenvalue problem for large FE systems as the computationally most demanding part of the dynamic analysis. Component-mode synthesis is used to provide in an efficient manner all the eigenvalue solutions of the FE system needed by the Monte Carlo simulation.

Stochastic linearization: the theory

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.

Stochastic linearization: the theory

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.

Error analysis of statistical linearization with Gaussian closure for large-degree-of-freedom systems

This paper contains an analysis of the error induced by applying the method of equivalent statistical linearization (ESL) to randomly excited multi-degree-of-freedom (m.d.f.) geometrically nonlinear shear-frame structures as the number of degrees of freedom increases. The quantity that is analyzed is the variance of the top-story displacement. The m.d.f. systems under consideration obtain their nonlinearity through cubic polynomial interstory restoring forces and the external excitation is modeled as the stationary output of a Kanai-Tajimi filter. Parameters of the filter and the m.d.f. structures, as well as the intensity of the gaussian white noise, are calibrated such that quantitative comparisons of the error between the exact solutions, estimated from Monte Carlo simulations, and the ESL solutions are possible among systems of different dimensions.