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This paper extends the Tail-Equivalent Linearization Method (TELM) to the case of a nonlinear mechanical system subjected to multiple stochastic excitations. Following the original formulation, the method employs a discrete representation of the stochastic inputs and the first-order reliability method (FORM). Each component of the Gaussian excitation is expressed as a linear function of standard normal random variables. For a specified response threshold of the nonlinear system, the Tail-Equivalent Linear System (TELS) is defined in the standard normal space by matching the design points of the equivalent linear and nonlinear systems. This leads to the identification of the TELS in terms of a frequency-response function or, equivalently, an impulse-response function relative to each component of the input excitation. The method is demonstrated through its application to an asymmetric, one-story building with hysteretic behavior and subjected to bi-component ground motion. The degree of asymmetry is controlled by the eccentricity of the center of stiffness with respect to the center of mass. The correlation between the probability of failure and the degree of asymmetry is studied in detail. The statistics of the response for stationary excitation obtained by TELM are in close agreement with Monte Carlo simulation results.

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... For other nonlinear models, the statistical linearization method is widely used [21][22][23][24]. Several new linearization methods were also proposed in recent years [25][26][27][28]. The linearization method for the CID based on the equivalent period and energy was discussed by Wang et al. [16]. ...

... For other nonlinear models, the statistical linearization method is widely used [21][22][23][24]. Several new linearization methods were also proposed in recent years [25][26][27][28]. The linearization method for the CID based on the equivalent period and energy was discussed by Wang et al [16]. ...

... in which t 1 and t 2 are obtained in Equation (26) and Equation (27), respectively. Substituting Equation (54) and Equation (55) into Equation (51), a more accurate solution of b eq is obtained: ...

Inerter-based dampers have gained great popularity in structural vibration control. In this paper, equivalent linearization methods (ELMs) for a single-degree-of-freedom (SDOF) system with a clutching inerter damper (CID) are studied. The comparison of a SDOF system with a CID and an inertial mass damper (IMD) shows the advantage of the CID. Considering that the system with the CID is nonlinear, which is problematic for its performance evaluation and the integrated design of the structure and control system, three equivalent linearization methods based on different principles are proposed and discussed in this paper. The CID is considered to be equal to a combination of an IMD and a viscous damper. The equivalent inertance and damping can be calculated using the obtained formulas for all methods. In addition, all methods are compared in a numerical study. Results show that the ELM based on period and energy is recommended for small inertance-mass ratios.

... The first one is on the sample level, namely Monte-Carlo simulation (MCS) and some improvement versions, including the important sampling [6], the subset simulation [7], [8], and the line sampling [9], [10], etc. The second one is on the level of moment, including the statistical linearizationbased technique [11] and moments approaches, such as higher-order moments [12], fractional moments [13], and linear moments [14], etc. ...

... Introducing one component of , i.e., , as an auxiliary process, for and , then the ABPs corresponding to and under can be constructed as (10) respectively, where is the first-passage time of [39], i.e., (11) in which denotes the infimum of the bracketed variable. In Fig. 1 shown is a schematic illustration of the sample paths of ABPs and their underlying processes. ...

Reliability analysis for engineering structures subjected to disastrous stochastic dynamical actions is of paramount importance for the performance-based decision-making of design, and has long been one of the major challenges in civil and various engineering fields. In the present paper, a novel method based on the globally-evolving-based generalized density evolution equation (GE-GDEE) is proposed to capture the time-variant first-passage reliability of high-dimensional nonlinear systems under non-white-noise excitations. The GE-GDEE has been established for the transient probability density function (PDF) of an arbitrary path-continuous process, e.g., one response of interest for a high-dimensional system, as a one-or two-dimensional partial differential equation. From this perspective, an absorbing-boundary process corresponding to the response of interest under a given safe domain can be constructed, and then its GE-GDEE can be developed. The equivalent drift coefficient in the GE-GDEE is a conditional expectation of the original drift function under the non-failure condition for the response of interest. It can be estimated by some feasible numerical approaches based on data from hundreds of representative deterministic dynamical analyses of the underlying high-dimensional system. Then, the GE-GDEE can be solved numerically to obtain the transient PDF of the absorbing boundary process and time-variant first-passage reliability further. A numerical example is illustrated to verify the efficiency and accuracy of the proposed method. It demonstrates remarkably the high accuracy of the failure probability even for rare events, which are achieved with only relatively small number of deterministic dynamic analyses for general high-dimensional nonlinear systems. Problems to be further studied are finally discussed.

... Recently, Der Kiureghian et al. (Fujimura and Der Kiureghian 2007;Der Kiureghian and Fujimura 2009;Garrè and Der Kiureghian 2010;Broccardo and Der Kiureghian 2015;Broccardo et al. 2016) have developed the tail equivalent linearization method (TELM), which is based on the first-order reliability method (FORM) ( Ditlevsen and Madsen 1996;Melchers 1999). The tail equivalent linear system (TELS) is defined as the ELS having the same design point as the original nonlinear system. ...

... Its extension to the GELS and TPELS will be the focus of future publications; it is here stressed that in any case GELS and TPELS do not require an increased number of dynamic responses for nonstationary analysis. Moreover, promising results have been obtained when the systems are subjected to multiple stochastic excitations ( Broccardo and Der Kiureghian 2015;Jensen et al. 2011). Therefore these methods are likely to represent a powerful tool for performance-based engineering methods of real-world engineering structures under multihazard analysis. ...

Three methods of stochastic equivalent linearizations defined in the broad framework of structural reliability analysis are presented. These methods are (1) the Gaussian equivalent linearization method (GELM), here defined for the first time as a linear response surface in terms of normal standard random variables; (2) the tail equivalent linearization method (TELM), here reinterpreted as a stochastic critical excitation method; and (3) a novel equivalent linearization called the tail probability equivalent linearization method (TPELM). The
Gaussian equivalent linear system (GELS) is the equivalent linear system (ELS) obtained by minimizing the difference between the variance of the GELS and the original nonlinear system. The tail equivalent linear system (TELS) is the ELS having the same critical excitation as the original system. The tail probability equivalent linear system (TPELS) is the ELS obtained by minimizing the difference between the tail probability of the equivalent system and the original nonlinear system. The knowledge of the ELS allows the evaluation of engineering
quantities of interest—e.g., first-passage probabilities—through the application of the random vibration analysis to these systems. Shortcomings and advantages of the three methods are presented and illustrated through applications to selected representative nonlinear oscillators. Finally, the methods are applied to an inelastic multi-degree-of-freedom (MDOF) system, showing their scalability to systems of higher complexity.

... The method is illustrated by investigating the response of an inelastic eccentric structure subjected to bi-directional seismic input. Part of this work has been published in [18]. Chapter 6 is dedicated to non-stationary stochastic processes, in particular to the broad family of modulated stochastic processes. ...

... The statistics of the response for stationary excitation obtained by TELM are in close agreement with Monte Carlo simulation results. Part of the material covered in this chapter has been published in [18]. ...

This dissertation provides the foundation for an in-depth understanding and significant development of the tail-equivalent linearization method (TELM) to solve different classes of nonlinear random vibration problems. The TELM is a linearization method that uses the first-order reliability method (FORM) to define a tail-equivalent linear system (TELS) and to estimate the tail of the response distribution for nonlinear systems under stochastic inputs. The method was originally developed in the time domain for inelastic systems. It was later extended in the frequency domain for a specific class of nonlinear excitations, while the frequency domain version for inelastic systems is covered in the present work.
This dissertation mathematically formalizes and extends TELM analysis with different types of discretization of the input process. A general formulation for discrete representation of a Gaussian band-limited, white-noise process is introduced, which employs the sum of deterministic and orthogonal basis functions weighted by random coefficients. The selection of the basis functions completely defines the two types of discretizations used in the earlier works. Specifically, a train of equally spaced time delta-Dirac functions leads to the current time-domain discretization, while harmonic functions with equally spaced frequencies lead to the current frequency-domain discretization. We show that other types of orthogonal basis functions can be used with advantage to represent a Gaussian band-limited white noise and in particular we employ sinc basis functions, which are at the base of the Whittaker-Shannon interpolation formula. We demonstrate that this representation is suitable for reducing the total number of random variables that are necessary to describe the process, since it decouples the computational-time discretization from the band-limit of the process.
Next, the dissertation tackles the problem of a nonlinear system subjected to multi- component excitations by defining an augmented standard normal space composed of all the random variables that define the multiple components of the excitation. The tail-equivalent linearization and definition of the TELS is taken in this new space. Once the augmented TELS is defined, response statistics of interest are determined by linear random vibration analysis by superposition of responses due to each component of the excitation. The method is numerically examined for an asymmetric structure with varying eccentricity and subjected to two statistically independent components of excitation.
Several practical problems require analysis for non-stationary excitations. For this important class of problems the original TELM requires linearization for a series of points in time to study the evolution of response statistics. This procedure turns out to be computationally onerous. As an approximate alternative, we propose the evolutionary TELM, ETELM. In particular, we adopt the concepts of the evolutionary process theory, to de- fine an evolutionary TELS, ETELS. The ETELS approximately estimates the continuous time evolution of the design point by only one TELM analysis. This is the essence of its efficiency compared to the standard TELM analysis. Among response statistics of interest, the first-passage probability represents the most important one for this class of problems. This statistic is efficiently computed by using the Au-Beck important sampling algorithm, which requires knowledge of the evolving design points, in conjunction with the ETELS. The method is successfully tested for five types of excitation: (I) uniformly modulated white noise, (II) uniformly modulated broad-band excitation, (III) uniformly modulated narrow- band excitation, (IV) time- and frequency-modulated broad-band excitation, and (V) time- and frequency-modulated narrow-band excitation.

... Compared to traditional linearization strategies, such an approach proved to be more accurate (Kiureghian and Fujimura 2009) and is capable to address multicomponent (Broccardo and Der Kiureghian 2016) and multisupport (Wang and Der Kiureghian 2016) seismic actions as well as multi-objective responses . More in general, a representation of structural responses in frequency and time domain is a very powerful technique which can be also used to characterize even more complex problems . ...

This book presents a range of research projects focusing on innovative numerical and modeling strategies for the nonlinear analysis of structures and metamaterials. The topics covered concern various analysis approaches based on classical finite element solutions, structural optimization, and analytical solutions in order to present a comprehensive overview of the latest scientific advances. Although based on pioneering research, the contributions are focused on immediate and direct application in practice, providing valuable tools for researchers and practicing professionals alike.

... TELM was originally developed by Fujimura and Der Kiureghian (2007) in the time domain and later extended to the frequency domain by Garrè and Der Kiureghian (2009). Further developments and applications include a gradient-free formulation by Alibrandi and Der Kiureghian (2011), multi-component excitation analysis by Broccardo and Der Kiureghian (2016), and multi-support structures subjected to spatially varying ground motions by Wang and Der Kiureghian (2016). The first attempt to extend TELM to nonstationary problems is by Broccardo and Der Kiureghian (2013). ...

This study introduces the evolutionary tail-equivalent linearization method (ETELM) for nonlinear stochastic dynamic analysis. The method builds on the recently developed tail-equivalent linearization method (TELM) and it is designed for the class of evolutionary processes. The original TELM employs a tail-equivalent linear system (TELS) by equating the tail probability of a linear system response for a specified threshold to the first-order approximation of the tail probability of the nonlinear system response. For stationary problems, the TELS is time-independent and only one linear system needs to be defined for the specified threshold. However, for a transient input, the TELS is time dependent and an evolutionary tail-equivalent linear system (ETELS) must be defined to study the entire transient response. Algorithms are developed to determine a discrete-time ETELS based on a sequence of linearization points, and a continuous-time approximation based on Priestley’s evolutionary theory. The linearized evolutionary system is used to compute the response statistics of interest, including the first-passage probability, in first-order approximation. Numerical examples demonstrate the accuracy and limitations of the proposed method.

... This approach, which defines a linearized system as a set of impulse response functions relevant to fixed thresholds of the seismic response, has been successfully used in first-excursion analyses (Der Kiureghian and Fujimura, 2009). Moreover, several advancements, such as analyses of non-symmetrical (Sessa and Der Kiureghian, 2009) and multi-objective responses (Sessa, 2010) as well as extensions to multicomponent (Broccardo and Kiureghian, 2016), multisupport (Wang and Der Kiureghian, 2016) and marine excitations Garré and Der Kiureghian (2010), proved TELM to be very versatile in addressing a large variety of structural problems. ...

These are the proceedings of the nineteenth working conference of the International Federation of Information Processing (IFIP) Working Group 7.5 on Reliability and Optimization of Structural Systems, which took place in the historical main building of ETH Zurich, Switzerland, on June 26–29, 2018. This volume contains 21 papers presented at the conference.
The purpose of the WG7.5 Working Group is to promote modern theories and methods
of structural and system reliability and optimization, to stimulate research, development and applications of structural and system reliability and optimization, to foster the dissemination and exchange of information and to encourage education on those subjects.
The main themes of the conference were structural reliability methods, engineering risk
and resilience analysis, Bayesian methods, reliability-based design optimization and sensitivity analysis, as well as their applications in civil engineering, dynamics and natural hazards.

... Such includes the equivalence in terms of the moments of the probability density function (PDF), 16 mean up-crossing rates, 17 and the instantaneous failure probability for a given threshold. 14,18,19 There are also other branches such as quadratization 20 and wavelet-based linearization. 21,22 Among the various approaches, tail-equivalent linearization method (TELM) has been widely recognized. ...

Gaussian mixture–based equivalent linearization method (GM‐ELM) is a recently developed stochastic dynamic analysis approach which approximates the random response of a nonlinear structure by collective responses of equivalent linear oscillators. The Gaussian mixture model is employed to achieve an equivalence in terms of the probability density function (PDF) through the superposition of the response PDFs of the equivalent linear system. This new concept of linearization helps achieve a high level of estimation accuracy for nonlinear responses, but has revealed some limitations: (1) dependency of the equivalent linear systems on ground motion intensity and (2) requirements for stationary condition. To overcome these technical challenges and promote applications of GM‐ELM to earthquake engineering practice, an efficient GM‐ELM‐based fragility analysis method is proposed for nonstationary excitations. To this end, this paper develops the concept of universal equivalent linear system that can estimate the stochastic responses for a range of seismic intensities through an intensity‐augmented version of GM‐ELM. Moreover, the GM‐ELM framework is extended to identify equivalent linear oscillators that could capture the temporal average behavior of nonstationary responses. The proposed extensions generalize expressions and philosophies of the existing response combination formulations of GM‐ELM to facilitate efficient fragility analysis for nonstationary excitations. The proposed methods are demonstrated by numerical examples using realistic ground motions, including design code–conforming nonstationary ground motions.

... An alternative philosophy for describing seismic excitations is to use simulated ground motions (Jalayer and Beck, 2008;Galasso et al., 2013). A specific modeling approach for the latter which has been steadily gaining increasing attention by the structural engineering community (Vetter and Taflanidis, 2014;Broccardo and Der Kiureghian, 2015) is the use of stochastic ground motion models (Rezaeian and Der Kiureghian, 2010; Gavin and Dickinson, 2010;Yamamoto and Baker, 2013;Vlachos et al., 2016;Boore, 2003;Atkinson and Silva, 2000). These models are based on modulation of a stochastic sequence, through functions (filters) that address spectral and temporal characteristics of the excitation. ...

The recent advances in computational efficiency and the scarcity/absence of recorded ground motions for specific seismicity scenarios have led to an increasing interest in the use of ground motion simulations for seismic hazard analysis, structural demand assessment through response-history analysis, and ultimately seismic risk assessment. Two categories of ground motion simulations, physics-based and stochastic site-based are considered in this study. Physics-based ground motion simulations are generated using algorithms that solve the fault rupture and wave propagation problems and can be used for simulating past and future scenarios. Before being used with confidence, they need to be validated against records from past earthquakes. The first part of the study focuses on the development of rating/testing methodologies based on statistical and information theory measures for the validation of ground motion simulations obtained through an online platform for past earthquake events. The testing methodology is applied in a case-study utilising spectral-shape and duration-related intensity measures (IMs) as proxies for the nonlinear peak and cyclic structural response. Stochastic site-based ground motion simulations model the time-history at a site by fitting a statistical process to ground motion records with known earthquake and site characteristics. To be used in practice, it is important that the output IMs from the developed time-histories are consistent with these prescribed at the site of interest, something that is not necessarily guaranteed by the current models. The second part of the study presents a computationally efficient framework that addresses the modification of stochastic ground motion models for given seismicity scenarios with a dual goal of matching target IMs for specific structures, while preserving desired trends in the physical characteristics of the resultant time-histories. The modification framework is extended to achieve a match to the full probability model of the target IMs. Finally, the proposed modification is validated by comparison to seismic demand of hazard-compatible recorded ground motions. This study shows that ground motion simulation is a promising tool that can be used for many engineering applications.

... The presented procedures based on random vibration theory can be applied to fragility analysis when the ground motion is specified as a stochastic process. More recently, time-and frequency-domain TELMs were developed for analysis of stochastic dynamical systems (Alibrandi and Der Kiureghian 2012, Broccardo and Der Kiureghian 2013, 2015, Raoufi and Ghafory-Ashtiany 2016. During the last decade, seismic fragility analyses have been investigated by numerous researchers and have been developed for a large number of structural and nonstructural systems (Schotanus et al. 2004, Sung Kwon and Elnashai 2006, Ellingwood et al. 2007, Mitropoulou and Papadrakakis 2011, liu et al. 2010, Ju et al. 2013, Mehani et al. 2013, Lallemant et al. 2015, Mandal et al. 2016, Khorami et al. 2017. ...

This study presents the reliability-based analysis of nonlinear structures using the analytical fragility curves excited by random earthquake loads. The stochastic method of ground motion simulation is combined with the random vibration theory to compute structural failure probability. The formulation of structural failure probability using random vibration theory, based on only the frequency information of the excitation, provides an important basis for structural analysis in places where there is a lack of sufficient recorded ground motions. The importance of frequency content of ground motions on probability of structural failure is studied for different levels of the nonlinear behavior of structures. The set of simulated ground motion for this study is based on the results of probabilistic seismic hazard analysis. It is demonstrated that the scenario events identified by the seismic risk differ from those obtained by the disaggregation of seismic hazard. The validity of the presented procedure is evaluated by Monte-Carlo simulation.

... [4][5][6] Though the most popular methodology for performing this task is the selection of real (ie, recorded from past events) ground motions, 7-10 potentially scaled based on a target intensity measure (IM), an alternative philosophy is the use of simulated ground motions. 11,12 A specific modeling approach for the latter which has been steadily gaining increasing attention by the structural engineering community, [13][14][15] is the use of stochastic ground motion models. [16][17][18][19][20][21][22] These models are based on a parametric description of the spectral and temporal characteristics of the excitation, with synthetic time-histories obtained by filtering a stochastic sequence through the resultant frequency and time domain modulating functions. ...

A computationally efficient framework is presented for modification of stochastic ground motion models to establish compatibility with the seismic hazard for specific seismicity scenarios and a given structure/site. The modification pertains to the probabilistic predictive models that relate the parameters of the ground motion model to seismicity/site characteristics. These predictive models are defined through a mean prediction and an associated variance, and both these properties are modified in the proposed framework. For a given seismicity scenario, defined for example by the moment magnitude and source‐to‐site distance, the conditional hazard is described through the mean and the dispersion of some structure‐specific intensity measure(s). Therefore, for both the predictive models and the seismic hazard, a probabilistic description is considered, extending previous work of the authors that had examined description only through mean value characteristics. The proposed modification is defined as a bi‐objective optimization. The first objective corresponds to comparison for a chosen seismicity scenario between the target hazard and the predictions established through the stochastic ground motion model. The second objective corresponds to comparison of the modified predictive relationships to the pre‐existing ones that were developed considering regional data, and guarantees that the resultant ground motions will have features compatible with observed trends. The relative entropy is adopted to quantify both objectives, and a computational framework relying on kriging surrogate modeling is established for an efficient optimization. Computational discussions focus on the estimation of the various statistics of the stochastic ground motion model output needed for the entropy calculation.

... The growing popularity the past decade in simulation-based probabilistic seismic risk assessment [9,10] has increased the importance of ground motion modeling. Though undoubtedly scaling of ground motions [12] is the most popular methodology to do so, an approach that has been gaining increasing interest within the structural engineering community [5,6] is the use of stochastic ground motion models [2,3,8,13,18]. These models are based on modulation of a stochastic sequence, through functions (filters) that address spectral and temporal characteristics of the excitation. ...

This paper discusses a computationally efficient framework for the hazard-compatible tuning of existing stochastic ground motion models. The tuning pertains to the modification of the probabilistic predictive relationships that relate the ground motion model parameters to seismicity characteristics, whereas the seismic hazard is quantified through ground motion prediction equations (GMPEs), which for a specified earthquake scenario and period range provide information for both the conditional mean and the dispersion (variability) of the resultant spectral accelerations. The proposed modification is defined as an optimization problem with a dual objective. The first objective corresponds to comparison for a chosen earthquake scenario between the regional conditional hazard and the predictions established through the stochastic ground motion model. The second objective corresponds to comparison of the new predictive relationships with the pre-existing predictive relationships, developed considering regional data. This second objective guarantees that the resultant ground motions not only match the regional hazard (objective one) but are also compatible with observed trends. The relative entropy is adopted to quantify both objectives since they are both related to comparison between probability distributions, and a computational framework relying on Kriging surrogate modeling is established for an efficient optimization.

... 8,9 An alternative philosophy for describing seismic excitations is to use simulated ground motions. 10,11 A specific modeling approach for the latter, which has been steadily gaining increasing attention by the structural engineering community, [12][13][14] is the use of stochastic ground motion models. [15][16][17][18][19][20] These models are based on modulation of a stochastic sequence, through functions (filters) that address spectral and temporal characteristics of the excitation. ...

Stochastic ground motion models produce synthetic time-histories by modulating a white noise sequence through functions that address spectral and temporal properties of the excitation. The resultant ground motions can be then used in simulation-based seismic risk assessment applications. This is established by relating the parameters of the aforementioned functions to earthquake and site characteristics through predictive relationships. An important concern related to the use of these models is the fact that through current approaches in selecting these predictive relationships, compatibility to the seismic hazard is not guaranteed. This work offers a computationally efficient framework for the modification of stochastic ground motion models to match target intensity measures (IMs) for a specific site and structure of interest. This is set as an optimization problem with a dual objective. The first objective minimizes the discrepancy between the target IMs and the predictions established through the stochastic ground motion model for a chosen earthquake scenario. The second objective constraints the deviation from the model characteristics suggested by existing predictive relationships, guaranteeing that the resultant ground motions not only match the target IMs but are also compatible with regional trends. A framework leveraging kriging surrogate modeling is formulated for performing the resultant multi-objective optimization, and different computational aspects related to this optimization are discussed in detail. The illustrative implementation shows that the proposed framework can provide ground motions with high compatibility to target IMs with small only deviation from existing predictive relationships and discusses approaches for selecting a final compromise between these two competing objectives.

... The original method was developed in the time domain for inelastic systems and, in 2008, was extended in frequency domain in the context of marine structures for a nonlinear type of loading (Garrè and Der Kiureghian 2010). Further developments includes: extension of the frequency domain for inelastic structures (Broccardo 2014), multicomponent excitations (Broccardo and Der Kiureghian 2015), nonstationary excitation (Broccardo and Der Kiureghian 2013), sinc expansion (Broccardo and Der Kiureghian 2012), multi-supported structures (Wang and Der Kiureghian 2016), the secant hyperplane method , and the tail probability equivalent linearization method. Among these, this chapter presents a review of the TELM with sinc expansion, the TELM for multi-supported structures, and the secant hyperplane method. ...

This chapter aims to provide a general prospective of the Tail Equivalent Linearization Method, TELM, by offering a review that starts with the original idea and covers a broad array of developments, including a selection of the most recent developments. The TELM is a linearization method that uses the first-order reliability method (FORM) to define a tail-equivalent linear system (TELS) and estimate the tail of the response distribution for nonlinear systems under stochastic inputs. In comparison with conventional linearization methods, TELM has a superior accuracy in estimating the response distribution in the tail regions; therefore, it is suitable for high reliability problems. Moreover, TELM is a non-parametric method and it does not require the Gaussian assumption of the response. The TELS is numerically defined by a discretized impulse-response function (IRF) or frequency-response function (FRF), thus allowing higher flexibility in linearizing nonlinear structural systems. The first part of the chapter focuses on the original idea inspiring TELM. The second part offers fourth developments of the method, which were studied by the authors of this chapter. These developments include: TELM in frequency domain, TELM with sinc expansion formula, TELM for multi-supported structures, and the secant hyperplane method giving rise to an improved TELM.

... The tail-equivalent linear system (TELS) is numerically obtained in terms of a discretized impulse-response function (IRF) or frequency-response function (FRF), allowing more flexibility in linearizing the nonlinear system. More recently, time-and frequency-domain TELMs have been applied to problems with statistically independent, multicomponent seismic excitations (Broccardo and Der Kiureghian 2015). Furthermore, the frequency-domain TELM has been applied to marine structures with multiple supports and subjected unidirectional wave excitation (Garrè and Der Kiureghian 2010). ...

After a brief review of time- and frequency-domain tail-equivalent linearization methods (TELM) for uniform excitation problems, this paper extends TELM for application to nonlinear systems subjected to multisupport seismic excitations. The spatial variability of the ground motion is represented by a coherency function that characterizes the incoherence, wave-passage, and site-response effects. It is found that for multisupport excitation problems, it is most convenient to formulate TELM by using the ground displacement as input. The resulting tail-equivalent linear system (TELS) is defined by frequency-response functions relating the response quantity of interest to each support displacement. A method to reduce the number of random variables in the TELM analysis is introduced. The proposed method is demonstrated through numerical examples with varying structural properties and ground motion coherency in order to investigate various aspects of TELM and the major influences of differential support motions on a nonlinear system.

高维非线性随机动力系统的响应和可靠度分析长期以来是科学和工程领域的重要挑战性难题。近年来国内外研究者们发展了许多针对实际复杂结构系统的可靠度评估方法，然而大多方法对于罕遇灾害性动力作用下结构的小失效概率仍难以精确估计。随着向第三代结构设计理论的迈进，这一问题愈显重要而迫切。
概率密度演化理论及其群演化路径的提出为这一问题的精细化高效实现提供了新的视角。遵循这一路径，本文建立并发展了广义概率密度全局演化方程，为一般连续随机过程的概率密度演化分析建立了统一的理论框架。在此框架下，各类经典的概率密度演化方程（如Liouville方程、Fokker-Planck-Kolmogorov（FPK）方程等）均可以视为广义密度全局演化方程针对某些具体物理系统的特殊形式。值得指出的是，这一方程不需要随机过程的Markov性。
在这一理论框架下，本文首先将广义密度全局演化方程应用于参数和激励均具有随机性的高维非线性系统的随机响应分析，其中系统参数可以考虑为若干独立或相关的随机变量，而系统遭受的激励可以视为Gauss白噪声或非平稳非白噪声过程。若仅关心系统中的某一感兴趣响应量，则可以建立该响应瞬时概率密度所满足的广义密度全局演化方程。它是一个一维或二维偏微分方程，可以方便地实现数值求解。方程中的本征漂移系数是响应概率密度演化的物理机制驱动综合效应，它是原高维系统运动方程中漂移力的条件期望函数，由系统物理（或动力学）机制决定。对于一般高维非线性系统，本文发展了根据系统有限次代表性确定性分析（即物理方程求解）的集合数据，构造本征漂移系数数值表达的方法，包括局部加权回归和基于藤式连接函数的参数化方法。将获得的本征漂移系数数值解代入全局演化方程，即可数值求解以获得感兴趣响应量的瞬时概率密度。
对于随机系统可靠度分析问题，本文从极值分布与吸收边界两方面进行了新的探索。为此，本文首先建立了一维连续Markov过程时变极值分布的概率演化积分方程，进而提出了Gauss或Poisson白噪声等各类随机激励下低维Markov系统极值分布数值求解的增广Markov向量方法。结合增广Markov向量方法和概率密度全局演化降维，可以给出高维Markov系统时变极值过程的广义密度全局演化方程，从而实现高维系统极值分布、以及不同阈值下时变可靠度的数值求解。数值算例表明，本文提出的几类方法对低维和高维Markov系统的极值分布求解，均具有理想的数值精度和有效性。
另一方面，若进一步构造响应量在给定安全域下的吸收边界过程，并建立吸收边界过程的广义密度全局演化方程，则可以实现高维随机动力系统的首次超越可靠度分析。文中严格证明了施加吸收边界以求解首次超越问题的正确性。论证了降维过程与吸收边界施加操作的不可交换性，奠定了降维系统求解系统可靠性的理论基础。通过数值算例与蒙特卡罗（Monte Carlo）模拟结果的对比，验证了概率密度全局演化方法对于高维随机动力系统响应和可靠度分析具有很高的数值精度和计算效率。特别地，该方法对于概率密度尾部精度、以及罕遇事件下小失效概率的计算具有相当的优势。本文还进一步将概率密度全局演化方程应用于工程实践，分析了具有参数随机性的实际高层钢筋混凝土结构在随机地震动作用下的抗震可靠性，实现了这一具有近280000个自由度的复杂非线性系统同时具有参数与激励随机性情况下的随机响应与动力可靠性精细化分析。这对于进一步指导工程防灾和基于可靠度的工程设计具有重要意义。
文中最后对需要进一步开展的研究工作进行了讨论。

Influence of non-structural elements on the seismic response of nonlinear systems is hereby investigated. In particular, tail-equivalent linearization has been adopted because of its capability of determining the statistics of stochastic response processes in order to characterize secondary excitation. A random vibration analysis determined a parameter range, characterizing the dynamic properties of non-structural components, for which the linearized system is not influenced by the presence of such devices. Numerical results show that tail equivalent linearization is an appealing strategy for the characterization of secondary seismic excitations.

In this research the tail equivalent linearization method (TELM) has been extended to study structures with degrading materials. The responses of such structures to excitations are non-stationary, even if the excitations are stationary. Non-stationary behavior of the system cannot be considered by conventional TELM. Applying the conventional TELM, the only distinction in the design point excitation for two stationary excitations with different durations is in the addition of a zero value part at the beginning of the design point of the longer excitation. This means that the failure probability is the same for the non-stationary systems under excitations with different durations. Therefore, this solution cannot be correct. In this study, in using TELM for systems with degrading materials, hysteretic energy is replaced by average hysteretic energy, calculated by averaging the obtained hysteretic energy of the structure subjected to a few random sample load realization. In this way, the degradation parameters under design point coincide with those under sample load realizations. Since the average of the hysteretic energy is converges very fast, the modified TELM only requires about tens to hundreds solutions of the response in addition to the ordinary calculations of conventional TELM.

A conventional approach for nonlinear random vibration analysis is using equivalent linearization method. Tail-Equivalent Linearization Method (TELM) is one the best proposed methods in the recent decade for determination of equivalent linear model. In TELM, the design point is obtained using first-order reliability method. In the current research, a non-gradient-based method is applied for determination of the design point. One of the main advantages of this method is non-application of limit-state function gradient for calculation of the design point. In the implemented method, n arbitrary points in n-dimension standard normal space are selected and limit-state function in these points is estimated. Then, these points converge to the design point using a convergent algorithm. Since many random variables are produced in TELM for discretization of seismic excitation, iterative algorithms for determination of design point numerical instability would be encountered. By modification of step length for each iteration and application of a magnification coefficient for each step, an appropriate non-gradient method is proposed for analysis of the problems with many random variables. The efficiency of this method was investigated by solving numerical examples. Moreover, the convergence of this method for finding the design point was presented. It was also indicated that results obtained using this method are in good agreement with results obtained by gradient methods.

Inelastic seismic analysis of buildings should consider the interaction of resisting axial force and bending moment in the columns, second order effects apart. Many fiber-based models are available but unsuitable for stochastic analysis, except Monte Carlo simulation. In contrast, a lumped plasticity frame model based on Bouc-Wen hysteresis, as recently extended to introduce interaction in a simple fashion, is straightforward to implement within stochastic equivalent linearization. Herein the interaction effect on the nonzero mean response is discussed. The model parameters are tuned for engineering structural analysis. Application is to eight reinforced concrete frames under gravity load and horizontal seismic excitation. The interstory drift appears to be almost insensitive to interaction. The rotation and energy ductility demands may change significantly, not only in the columns but also in the beams. All in all, increase prevails on decrease; the interaction effect is negative. For instance, decrease in the 95th percentile of rotation ductility demand is at most 10% or, in very few cases, 30%, whereas increase is up to 60% in most frames, and even two and three times in two frames. The bending moment statistics may change as well. Mean value is affected more than variance. Results are consistent with established outcome on nonzero mean random vibration. The interaction effect is marked on the outer members at bottom stories, tall frames, and soft ground. The lumped plasticity model seems to be suitable for practical structural analysis.

This study serves three purposes: (i) to review a synthesis formula for simulation of band-limited stochastic processes based on the sinc expansion; (ii) to implement this synthesis formula in the tail-equivalent linearization method (TELM); and (iii) to demonstrate increased computational efficiency when the sinc expansion is implemented in this context. The proposed representation enables the reduction and control of the number of random variables used in the simulation of band-limited stochastic processes. This is of great importance for gradient-based reliability methods, including TELM, for which the computational cost is proportional to the total number of random variables. A direct application of the representation is used in TELM analysis. Examples of single-and multiple-degrees-of-freedom nonlinear systems subjected to Gaussian band-limited white noise simulated by use of sinc expansion are presented. The accuracy and efficiency of the representation are compared with those of the current time-domain discretization method. The analysis concludes by shedding light on the specific cases for which the introduced reduction technique is beneficial.

In this research the tail-equivalent linearization method (TELM) has been applied to a structure with biaxial behavior of materials, using the biaxial Bouc-Wen material model. The modeled structure has been subjected to independent bidirectional excitation with the incident angle θ with major axes of structure. The direct differentiation method (DDM) has been developed for calculating the response and its derivatives for the first time for the biaxial Bouc-Wen material model, where the application of DDM is more difficult compared with uniaxial Bouc-Wen models due to its coupled constitutive law of material. The method is applied to a structure with a rigid diaphragm, supported by four different columns. The structure is subjected to bidirectional and modulated filtered white noise excitations. The cumulative probability distribution function (CDF), probability density function (PDF), average rate of crossing, and first passage probability of displacement response are calculated for a column in the roof level of the structure. The results have been compared with those of Monte Carlo simulation presenting good agreement. The effects of nonlinearity degree and the levels of threshold have been investigated on the tail-equivalent linear system (TELS). The importance and effects of considering biaxial nonlinear behavior have been assessed by changing its relevant parameter in the Bouc-Wen model to obtain and compare different TELSs. The effects of incident angle have been investigated for independent components of excitation to find the most critical angle related to the minimum reliability index in TELM. Furthermore, system eccentricity, most critical incident angle for different responses, nonlinearity, and spectral intensity ratio of bidirectional excitation are considered.

This dissertation provides the foundation for an in-depth understanding and significant development of the tail-equivalent linearization method (TELM) to solve different classes of nonlinear random vibration problems. The TELM is a linearization method that uses the first-order reliability method (FORM) to define a tail-equivalent linear system (TELS) and to estimate the tail of the response distribution for nonlinear systems under stochastic inputs. The method was originally developed in the time domain for inelastic systems. It was later extended in the frequency domain for a specific class of nonlinear excitations, while the frequency domain version for inelastic systems is covered in the present work.
This dissertation mathematically formalizes and extends TELM analysis with different types of discretization of the input process. A general formulation for discrete representation of a Gaussian band-limited, white-noise process is introduced, which employs the sum of deterministic and orthogonal basis functions weighted by random coefficients. The selection of the basis functions completely defines the two types of discretizations used in the earlier works. Specifically, a train of equally spaced time delta-Dirac functions leads to the current time-domain discretization, while harmonic functions with equally spaced frequencies lead to the current frequency-domain discretization. We show that other types of orthogonal basis functions can be used with advantage to represent a Gaussian band-limited white noise and in particular we employ sinc basis functions, which are at the base of the Whittaker-Shannon interpolation formula. We demonstrate that this representation is suitable for reducing the total number of random variables that are necessary to describe the process, since it decouples the computational-time discretization from the band-limit of the process.
Next, the dissertation tackles the problem of a nonlinear system subjected to multi- component excitations by defining an augmented standard normal space composed of all the random variables that define the multiple components of the excitation. The tail-equivalent linearization and definition of the TELS is taken in this new space. Once the augmented TELS is defined, response statistics of interest are determined by linear random vibration analysis by superposition of responses due to each component of the excitation. The method is numerically examined for an asymmetric structure with varying eccentricity and subjected to two statistically independent components of excitation.
Several practical problems require analysis for non-stationary excitations. For this important class of problems the original TELM requires linearization for a series of points in time to study the evolution of response statistics. This procedure turns out to be computationally onerous. As an approximate alternative, we propose the evolutionary TELM, ETELM. In particular, we adopt the concepts of the evolutionary process theory, to de- fine an evolutionary TELS, ETELS. The ETELS approximately estimates the continuous time evolution of the design point by only one TELM analysis. This is the essence of its efficiency compared to the standard TELM analysis. Among response statistics of interest, the first-passage probability represents the most important one for this class of problems. This statistic is efficiently computed by using the Au-Beck important sampling algorithm, which requires knowledge of the evolving design points, in conjunction with the ETELS. The method is successfully tested for five types of excitation: (I) uniformly modulated white noise, (II) uniformly modulated broad-band excitation, (III) uniformly modulated narrow- band excitation, (IV) time- and frequency-modulated broad-band excitation, and (V) time- and frequency-modulated narrow-band excitation.

The tail-equivalent linearization method (TELM) is a recently developed computational method to solve nonlinear stochastic dynamic problems by the first-order reliability method (FORM). TELM employs a tail-equivalent linear system (TELS) by equating the tail probability of a linear system to the first-order approximation of the tail probability of the nonlinear system. For stationary problems, the TELS is time- independent and only one linear system needs to be defined to study the statistics of the response. However, for a transient input, the TELS is time-dependent. Thus, TELSs for different time points must be defined to study the non-stationary response. Since each TELS is obtained from the solution of an optimization problem, the computational cost required to solve the non-stationary problem can be prohibitive. This paper tackles the class of non-stationary problems described via evolutionary power spectral density by defining an evolutionary TELS (ETELS) in place of a series of point-in-time TELSs. An example shows the accuracy and effectiveness of the method.

Incremental dynamic analysis has recently emerged to offer comprehensive evaluation of the seismic performance of structures using multiple nonlinear dynamic analyses under scaled ground-motion records. Being computer-intensive, it can benefit from parallel processing to accelerate its application on realistic structural models. While the task-farming master–slave paradigm seems ideal, severe load imbalances arise due to analysis non-convergence at structural instability, prompting the examination of task partitioning at the level of single records or single dynamic runs. Combined with a multi-tier master–slave processor hierarchy employing dynamic task generation and self-scheduling we achieve a flexible and efficient parallel algorithm with excellent scalability.

Incremental dynamic analysis (IDA) is a parametric analysis method that has recently emerged in several different forms to estimate more thoroughly structural performance under seismic loads. It involves subjecting a structural model to one (or more) ground motion record(s), each scaled to multiple levels of intensity, thus producing one (or more) curve(s) of response parameterized versus intensity level. To establish a common frame of reference, the fundamental concepts are analysed, a unified terminology is proposed, suitable algorithms are presented, and properties of the IDA curve are looked into for both single-degree-of-freedom and multi-degree-of-freedom structures. In addition, summarization techniques for multi-record IDA studies and the association of the IDA study with the conventional static pushover analysis and the yield reduction R-factor are discussed. Finally, in the framework of performance-based earthquake engineering, the assessment of demand and capacity is viewed through the lens of an IDA study. Copyright © 2001 John Wiley & Sons, Ltd.

Thesis (Ph. D. in Engineering-Civil and Environmental Engineering)--University of California, Berkeley, Fall 2003. Includes bibliographical references (leaves 352-363).

This paper presents a comprehensive representation of band-limited stochastic processes. The approach employs a unified framework for such representations by use of con- cepts of Hilbert space and general decomposition of a signal into basis functions. The proposed representation, in general, allows reducing the number of random variables used in the simula- tion of stochastic processes. Direct application of the representation is employed to the Tail- Equivalent Linearization Method (TELM) analysis. An example of a single degree of freedom hysteretic oscillator subjected to a Gaussian band-limited white noise simulated by use of sinc basis functions is presented. The accuracy and the efficiency of the representation are compared with the current discretizations.

The method of equivalent linearization of Kryloff and Bogliubov is generalized to the case of nonlinear dynamicsystems with random excitation. The method is applied to a variety of problems, and the results compared with exact solutions of the Fokker?Planck equation for those cases where the Fokker?Planck technique may be applied. Alternate approaches to the problem are discussed, including the characteristic function method of Rice.

In this study a generalization of the stochastic linearization method is proposed; namely the nonlinear system is suggested to be replaced by a linear system equivalent to the original one in the following sense: The two systems should share common mean-square values of potential energies, as well as have coincident mean square values of energy dissipation function. An example of a system with nonlinear damping and nonlinear stiffness is numerically evaluated, to elucidate the proposed method.

Improved versions of two optimization algorithms commmonly used in first-order reliability analysis are developed. One is for determining the first-order reliability index β. The other is for inverse reliability analysis, i.e., determining the value of a deterministic parameter such that the reliability index equals a target value β
t
. Besides being mathematically more rigorous, these new versions are much simpler than earlier versions of these algorithms.

This self-contained volume explains the general method of statistical, or equivalent, linearization and its use in solving random vibration problems. Subjects include general equations of motion and representation of non-linearities, probability theory and stochastic processes, elements of linear random vibration theory, statistical linearization for simple systems with stationary response, more. 1990 edition.

In this paper we discuss certain questions related to the method of statistical linearization as an approximation to the response of non-linear systems. We discuss its second moment properties, its relationship to parameter estimation for linear systems as well as possible connections with recursive estimators. Finally, we show by example that for systems with randomly varying coefficients statistical linearization will yield sample properties that differ significantly from the sample solution properties of the original non-linear system.

The subject of this paper is the simulation of one-dimensional, uni-variate, stationary, Gaussian stochastic processes using the spectral representation method. Following this methodology, sample functions of the stochastic process can be generated with great computational efficiency using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number N of the terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the temporally-averaged mean value and the autocorrelation function are identical with the corresponding targets, when the averaging takes place over the fundamental period of the cosine series. The most important property of the simulated stochastic process is that it is asymptotically Gaussian as N -> oo. Another attractive feature of the method is that the cosine series formula can be numerically computed efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in engineering mechanics and structural engineering. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response, variability due to system stochasticity).

A method is presented whereby a non-linear second order dynamical system is replaced by a linear system in such a way that an average of the difference between the two systems is minimized. Provided the averaging operator possesses certain properties, it is shown that the replacement is unique and can be accomplished in a straightforward manner. The parameters of the replacement linear system are expressed in terms of averages of functions of the linearized solution.RésuméOn présente une méthode par laquelle on remplace un système dynamique non linéaire du deuxième ordre par un système linéaire de telle sorte que la moyenne de la différence entre les deux systèmes soit minimum. A condition que l'opérateur servant au calcul de la moyenne possède certaines propriétés, on montre que le remplacement est unique et peut être accompli de façon immédiate. Les paramètres du système linéaire de remplacement s'expriment en termes de moyennes de fonctions de la solution linéarisée.ZusammenfassungEs wird eine Methode dargestellt, mit deren Hilfe ein nichtlineares dynamisches System zweiter Ordnung so durch ein lineares System ersetzt wird, dass ein Mittelwert der Differenz zwischen beiden Systemen zum Minimum gemacht wird. Vorausgesetzt dass der mittelnde Operator bestimmte Eigenschaften hat, so kann gezeigt werden, dass der Ersatz eindeutig ist und in einfacher Weise durchgeführt werden kann. Die Parameter des linearen Ersatzsystems werden in Abhängigkeit von Funktionsmittelwerten der linearisierten Lösung ausgedrückt.РефератПpeдcтaвлeн мeтoд, пpи пoмoщи кoтopoгo нeлинeйнyю динaмичecкyю cиcтeмy втopoгo пopядкa мoжнo зaмeнить линeйнoй cиcтeмoй тaким oбpaзoм, чтo cpeдняя вeличинa oтклoнeния oбeич cиcтeм пpивeдeнa к минимyмy. Ecли пpeдпoлoжить, чтo ocpeдняющий oпepaтoь oблaдaeт нeкoтopыми дoпoлнитeльными cвoйcтвaми, мoжнo дoкaзaть, чтo дaннaя зaмeнa являeтcя eдинcтвeннoй и нeпocpeдcтвeннo выпoлнимoй. Пapaмeтpы зaмeщaющeй линeйнoй cиcтeмы выpaжeны чepeз cpeдниe знaчeния фyнкций, вчoдящич в линeapизoвaннoe peшeниe.

A statistical linearization approach is applied to problems of the stationary random response of nonlinear multidegree-of-freedom dynamical systems. The approach is formulated in such a way that the resulting linear system parameters have a simple physical interpretation and can often be determined analytically. The accuracy of the approach is investigated by means of examples.

This book is designed for use as a text for graduate courses in random vibrations or stochastic structural dynamics, such as might be offered in departments of civil engineering, mechanical engineering, aerospace engineering, ocean engineering, and applied mechanics. It is also intended for use as a reference for graduate students and practicing engineers with a similar level of preparation. The focus is on the determination of response levels for dynamical systems excited by forces that can be modeled as stochastic processes. The choice of prerequisites, as well as the demands of brevity, sometimes makes it necessary to omit mathematical proofs of results. We do always try to give mathematically rigorous definitions and results even when mathematical details are omitted. This approach is particularly important for the reader who wishes to pursue further study. An important part of the book is the inclusion of a number of worked examples that illustrate the modeling of physical problems as well as the proper application of theoretical solutions. Similar problems are also presented as exercises to be solved by the reader.

Based on a Markov-vector formulation and a Galerkin solution procedure, a new method of modeling and solution of a large class of hysteretic systems (softening or hardening, narrow or wide-band) under random excitation is proposed. The excitation is modeled as a filtered Gaussian shot noise allowing one to take the nonstationarity and spectral content of the excitation into consideration. The solutions include time histories of joint density, moments of all order, and threshold crossing rate; for the stationary case, autocorrelation, spectral density, and first passage time probability are also obtained. Comparison of results of numerical example with Monte-Carlo solutions indicates that the proposed method is a powerful and efficient tool.

A method of equivalent linearization for smooth hysteretic systems under random excitation is proposed. The hysteretic restoring force is modeled by a nonlinear differential equation and the equation of motion is linearized directly in closed form without recourse to Krylov-Bogoliubov technique. Compared with previously proposed similar methods, the formulation of the present method is versatile and considerably simpler. The accuracy of this method is verified against Monte-Carlo simulation for all response levels. It has a great potential in the analysis of multidegree-of-freedom and degrading systems.

Recent state-of-the-art reports emphasize the generality of stochastic equivalent linearization techniques in the nonlinear dynamic analysis of stochastically excited structures. When a three-dimensional frame is considered, it cannot be studied by decomposing the structure into several plane frames due to the impossibility of summing the effects that characterize nonlinear analyses. The equations of motion must be written for the whole structure. This is made possible by the knowledge of the constitutive law for the single story of the frame. Such a constitutive law can be identified from experimental data for regular buildings; otherwise, its dependence on the geometrical and mechanical properties of each structural element must be specified. This paper shows how the equations of motion can be written starting from the hysteretic constitutive law in the single potential plastic hinge in terms of the two bending moments and of the associate inelastic rotations. The generalization, including axial and shear forces and twisting moment in every beam section, is straightforward, but the dimension of the solving system for the equations quickly becomes impressive. Limitation to the bending moments provides the general features of the response and permits the analyzer to determine possible global torsional effects.

An extension of the Tail-Equivalent Linearization Method (TELM) to the frequency domain is presented. The extension defines the Tail-Equivalent Linear System in terms of its frequency-response function. This function is obtained by matching the design point of the nonlinear response with that of the linearized response, thus guaranteeing the equivalence of the tail probability of the latter and the first-order approximation of the tail probability of the nonlinear response. The proposed approach is particularly suitable when the input and response processes are stationary, as is usually the case in the analysis of marine structures. When linear waves are considered, the Tail-Equivalent Linear System possesses a number of important properties, such as the capability to account for multi-support excitations and invariance with respect to scaling of the excitation. The latter property significantly enhances the computational efficiency of TELM for analysis with variable sea states. Additionally, the frequency-response function of the Tail-Equivalent Linear System offers insights into the geometry of random vibrations discretized in the frequency domain and into the physical nature of the response process. The proposed approach is applied to the analysis of point-in-time and first-passage statistics of the random sway displacement of a simplified jack-up rig model.

The geometry of random vibration problems in the space of standard normal random variables obtained from discretization of the input process is investigated. For linear systems subjected to Gaussian excitation, the problems of interest are characterized by simple geometric forms, such as vectors, planes, half spaces, wedges and ellipsoids. For non-Gaussian responses, the problems of interest are generally characterized by non-linear geometric forms. Approximate solutions for such problems are obtained by use of the first- and second-order reliability methods (FORM and SORM). This article offers a new outlook to random vibration problems and an approximate method for their solution. Examples involving response to non-Gaussian excitation and out-crossing of a vector process from a non-linear domain are used to demonstrate the approach.

A new approach for computing seismic fragility curves for nonlinear structures for use in performance-based earthquake engineering analysis is proposed. The approach makes use of a recently developed method for nonlinear stochastic dynamic analysis by tail-equivalent linearization. The ground motion is modeled as a discretized stochastic process with a set of parameters that characterizes its evolutionary intensity and frequency content. For each selected response (seismic demand) threshold, a linear system is defined that has the same tail probability as the nonlinear response in first-order approximation. Simple linear random vibration analysis with the tail-equivalent linear system then yields the fragility curve. The approach has the advantage of avoiding the selection and scaling of recorded accelerograms and repeated time-history analyses, which is the current practice for developing fragility curves. Copyright © 2009 John Wiley & Sons, Ltd.

An equivalent linearization technique to obtain the response of non-linear multi-degree-of-freedom dynamic systems to stationary gaussian excitations is developed. The non-linearities are assumed to be single-valued functions of accelerations, velocities and displacements. Using a property of gaussian vector processes, the closed forms of the coefficients of the equivalent linear system are obtained by the direct application of partial differentiation and expectation operators to the non-linear terms. It is shown that when the non-linearities possess potentials, the linear system has symmetric coefficient matrices. A geometrical interpretation of the linear coefficients, in connection with the original non-linearities, is presented. The accuracy is investigated by means of examples.

A fully nonstationary stochastic model for strong earthquake ground motion is developed. The model employs filtering of a discretized white-noise process. Nonstationarity is achieved by modulating the intensity and varying the filter properties in time. The formulation has the important advantage of separating the temporal and spectral nonstationary characteristics of the process, thereby allowing flexibility and ease in modeling and parameter estimation. The model is fitted to target ground motions by matching a set of statistical characteristics, including the mean-square intensity, the cumulative mean number of zero-level up-crossings and a measure of the bandwidth, all expressed as functions of time. Post-processing by a second filter assures zero residual velocity and displacement, and improves the match to response spectral ordinates for long periods. Copyright © 2008 John Wiley & Sons, Ltd.

A method for generating a suite of synthetic ground motion time-histories for specified earthquake and site characteristics defining a design scenario is presented. The method employs a parameterized stochastic model that is based on a modulated, filtered white-noise process. The model parameters characterize the evolving intensity, predominant frequency, and bandwidth of the acceleration time-history, and can be identified by matching the statistics of the model to the statistics of a target-recorded accelerogram. Sample ‘observations’ of the parameters are obtained by fitting the model to a subset of the NGA database for far-field strong ground motion records on firm ground. Using this sample, predictive equations are developed for the model parameters in terms of the faulting mechanism, earthquake magnitude, source-to-site distance, and the site shear-wave velocity. For any specified set of these earthquake and site characteristics, sets of the model parameters are generated, which are in turn used in the stochastic model to generate the ensemble of synthetic ground motions. The resulting synthetic acceleration as well as corresponding velocity and displacement time-histories capture the main features of real earthquake ground motions, including the intensity, duration, spectral content, and peak values. Furthermore, the statistics of their resulting elastic response spectra closely agree with both the median and the variability of response spectra of recorded ground motions, as reflected in the existing prediction equations based on the NGA database. The proposed method can be used in seismic design and analysis in conjunction with or instead of recorded ground motions. Copyright © 2010 John Wiley & Sons, Ltd.

A new, non-parametric linearization method for nonlinear random vibration analysis is developed. The method employs a discrete representation of the stochastic excitation and concepts from the first-order reliability method, FORM. For a specified response threshold of the nonlinear system, the equivalent linear system is defined by matching the “design points” of the linear and nonlinear responses in the space of the standard normal random variables obtained from the discretization of the excitation. Due to this definition, the tail probability of the linear system is equal to the first-order approximation of the tail probability of the nonlinear system, this property motivating the name Tail-Equivalent Linearization Method (TELM). It is shown that the equivalent linear system is uniquely determined in terms of its impulse response function in a non-parametric form from the knowledge of the design point. The paper examines the influences of various parameters on the tail-equivalent linear system, presents an algorithm for finding the needed sequence of design points, and describes methods for determining various statistics of the nonlinear response, such as the probability distribution, the mean level-crossing rate and the first-passage probability. Applications to single- and multi-degree-of-freedom, non-degrading hysteretic systems illustrate various features of the method, and comparisons with results obtained by Monte Carlo simulations and by the conventional equivalent linearization method (ELM) demonstrate the superior accuracy of TELM over ELM, particularly for high response thresholds.

It has been shown in recent years that certain non-linear random vibration problems can be solved by well established methods of time-invariant structural reliability, such as FORM and importance sampling. A key step in this approach is finding the design-point excitation, which is that realization of the input process that is most likely to give rise to the event of interest. It is shown in this paper that for a non-linear, elastic single-degree-of-freedom oscillator subjected to white noise, the design-point excitation is identical to the excitation that generates the mirror image of the free-vibration response when the oscillator is released from a target threshold. This allows determining the design-point excitation with a single non-linear dynamic analysis. With a slight modification, this idea is extended to non-white and non-stationary excitations and to hysteretic oscillators. In these cases, an approximate solution of the design-point excitation is obtained, which, if necessary, can be used as a ‘warm’ starting point to find the exact design point using an iterative optimization algorithm. The paper also offers a simple method for computing the mean out-crossing rate of a response process. Several examples are provided to demonstrate the application and accuracy of the proposed methods. The methods proposed in this paper enhance the feasibility of approximately solving non-linear random vibration problems by use of time-invariant structural reliability techniques.

It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative J2 flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a class of non-associative flow rules are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton's method, whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.

Several optimization algorithms are evaluated for application in structural reliability, where the minimum distance from the origin to the limit-state surface in the standard normal space is required. The objective is to determine the suitability of the algorithms for application to linear and nonlinear finite element reliability problems. After a brief review, five methods are compared through four numerical examples. Comparison criteria are the generality, robustness, efficiency, and capacity of each method.

Efficient methods are presented for digital simulation of a general homogeneous process (multidimensional or multivariate or multivariate-multidimensional) as a series of cosine functions with weighted amplitudes, almost evenly spaced frequencies, and random phase angles. The approach is also extended to the simulation of a general non-homogeneous oscillatory process characterized by an evolutionary power spectrum. Generalized forces involved in the modal analysis of linear or non-linear structures can be efficiently simulated as a multivariate process using the cross-spectral density matrix computed from the spectral density function of the multidimensional excitation process. Possible applications include simulation of (i) wind-induced ocean wave elevation, (ii) spatial random variation of material properties, (iii) the fluctuating part of atmospheric wind velocities and (iv) random surface roughness of highways and airport runways.

An extension, to the time-dependent situation, of what is known in static structural reliability as the ‘normal tail approximation’ is presented. This is pursued within the classical stochastic equivalent linearization scheme. Duffing and hysteretic oscillators are studied in detail.

A general finite element solution method for the dynamic response sensitivity of inelastic structures is developed. Employing a direct differentiation method, the gradient equation of motion is solved without iteration and by taking advantage of the available solution of the response. Special attention is given to sensitivities with respect to inelastic material parameters and detailed derivations are made for theJ2 plasticity model with a linear hardening rule. The method can be applied to any other inelastic material model that has an analytically defined yield function and flow rule. The formulation is easily incorporated in existing finite element codes. Numerical examples demonstrate the accuracy and efficiency of the method.

An approximate analytical technique is developed for obtaining the non-stationary response of a single-degree-of-freedom mechanical system with bilinear hysteresis subjected to amplitude modulated non-white random excitations. The hysteretic behavior is described by introducing an additional state variable and non-linear functions. The equivalent linear coefficients and also the moment equations of the equivalent linear systems are derived using a non-Gaussian probability density function which is composed of a truncated Gaussian probability density function and a couple of delta functions. The mean square responses and the mean dissipated hysteretic energy are calculated by solving the moment equations and are compared with the corresponding digital simulation results.

The method of equivalent linearization of Kryloff and Bogoliubov is generalized to the case of nonlinear dynamic systems with random excitation. The method is applied to a variety of problems, and the results are compared with exact solutions of the Fokker-Planck equation for those cases where the Fokker-Planck technique may be applied. Alternate approaches to the problem are discussed, including the characteristic function method of Rice.

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Structural analysis theory and applications

- F C Filippou

Filippou, F. C. (2009). " Structural analysis, theory and applications. " Univ.
of California, Berkeley, CA.

Stochastic simulation of near-fault ground motions for specified earthquake and site characteristics

- M Dabaghi
- S Rezaeian
- Der Kiureghian