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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.

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... For non-stationary excitations, the gradient forms a time-dependent curve, which is a set but not a subspace. See Der Kiureghian (2000) and Broccardo and Der Kiureghian (2018) for more geometrical insights. ...

This paper introduces the evolutionary tail-equivalent linearization method (ETELM) for stochastic dynamic analysis of nonlinear structures subjected to evolutionary input. A sequence of evolutionary tail-equivalent linear systems (ETELS) are defined by linearizing the response function in the space of discretized excitation for a selected threshold and at a sequence of time points. The ETELSs are used to compute the response statistics of interest, including the first-passage probability, in first-order approximation. Numerical examples demonstrate the accuracy of the proposed method.

... By virtue of this equivalence, TELS approximates the exceedance probability of the nonlinear system to the first order. TELM was later extended to processes discretized in the frequency-domain, [17], and further generalized to a broad range of basis functions [18][19][20]. In [21] TELM was applied to spatially correlated excitations for the analysis of multi-support structures, whereas in [22] TELM is extended with the Secant Hyperplane Method, leading to the so-called the Tail Probability Equivalent Method. ...

This paper estimates extreme midship wave bending moment of a ship in a specific sea state using First Order Reliability Method (FORM) and the Tail Equivalent Linearization Method (TELM). FORM and TELM are applied to the random vibration analysis of the nonlinear bending moment and results are compared and verified against experimental data obtained from model tests as well as numerical simulations. A case study of an LNG tanker is presented, demonstrating that the proposed methodology yields reasonable results and that the TELM approach provides a fast and computationally efficient way of analyzing nonlinear vertical bending moment of ships in stationary sea states.

... For non-stationary excitations, the gradient forms a time-dependent curve, which is a set but not a subspace. See Der Kiureghian (2000) and Broccardo and Der Kiureghian (2018) for more geometrical insights. The response of the ETELS is ...

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.

... in which s(t) and u are column vectors of deterministic basis functions (eg, Dirac Delta functions, linear filters, 14 and sinc function 40,41 in the time domain or sine and cosine in the frequency domain 18,27 ) and independent standard normal random variables, respectively. Particularly, in time-domain representation, u may represent the intensity of random pulses at the discretized time points while s(t) describes the linear filter(s) through which the pulses pass. ...

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.

The identification of patterns and underlying characteristics of natural or engineering time-varying phenomena poses a challenging task, especially in the scope of simulation models and accompanying stochastic models. Because of their complex nature, time-varying processes such as wind speed, seismic ground motion, or vibrations of machinery in the presence of degradation oftentimes lack a closed-form description of their underlying Evolutionary Power Spectral Density (EPSD) function. To overcome this issue, a wide range of measurements exist for these types of processes. This opens up the path to a data-driven stochastic representation of EPSD functions. Rather than solely relying on time-frequency transform methods like the familiar short-time Fourier transform or wavelet transform for EPSD estimation, a probabilistic representation of the EPSD can provide valuable insights into the epistemic uncertainty associated with these processes. To address this problem, the evolutionary EPSD function is relaxed based on multiple similar data to account for these uncertainties and to provide a realistic representation of the time data in the time-frequency domain. This results is the so-called Relaxed Evolutionary Power Spectral Density (REPSD) function, which serves as a modular probabilistic representation of the time-frequency content of stochastic signals. For this purpose, truncated normal distributions and kernel den-1 REVISED Manuscript (Clean version in word or latex Format) sity estimates are used to determine a probability density function for each time-frequency component. The REPSD function enables the sampling of individual EPSD functions, facilitating their direct application to the simulation model through stochastic simulation techniques like Monte Carlo simulation or other advanced methods. Even though the accuracy is highly dependant on the data available and the time-frequency transformation method used, the REPSD representation offers a stochastic representation of characteristics used to describe stochastic signals and can reduce epistemic uncertainty during the modelling of such time-varying processes. The method is illustrated by numerical examples involving the analysis of dynamic behaviour under random loads. The results show that the method can be successfully employed to account for uncertainties in the estimation of the EPSD function and represent the accuracy of the time-frequency transformation used.

In this study, a method for predicting the extreme value distribution of the Vertical Bending Moment (VBM) in a flexible ship under a given short-term sea state is presented. The First Order Reliability Method (FORM) is introduced to evaluate the Probability of Exceedances (PoEs) of extreme VBM levels. The Karhunen-Loeve (KL) representation of stochastic ocean wave is adopted in lieu of the normal wave representation using the trigonometric components, by introducing the Prolate Spheroidal Wave Functions (PSWFs) to formulate the wave elevations. By this means, reduction of the number of stochastic variables to reproduce ocean wave is expected, which in turn the number of computations required during FORM based prediction phases is significantly reduced. In this study, the Reduced Order Model (ROM), which was developed in our previous studies, is used to yield the time-domain VBMs along with the hydroelastic (whipping) component in a ship. Two different short-term sea states, moderate and severe ones, are assumed. The FORM based predictions using PSWF for normal wave-induced VBM are then validated by comparing with those using the normal trigonometric wave representation and Monte Carlo Simulations (MCSs). Through a series of numerical demonstrations, the computational efficiency of the FORM based prediction using PSWF is presented. Then, the validation is extended to the severe sea state where the whipping vibration contributes to the extreme VBM level to a large degree, and finally the conclusions are given.

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.

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.

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.

In the analysis of structural reliability, often a sequence of design points associated with a set of thresholds are sought in order to determine the tail distribution of a response quantity. In this paper, after a brief review of methods for determining the design point, an inverse reliability method named the λ-method is introduced for efficiently determining the sequence of design points. The λ-method uses Broyden's "good" method to solve a set of nonlinear simultaneous equations to find the design points for the values of an implicitly defined threshold that is associated with parameter λ. In a special parameter setting, the λ parameter equals the reliability index, thus allowing convenient implementation of the method. Three numerical examples illustrate the accuracy and efficiency of the proposed method.

A versatile, nonstationary stochastic ground-motion model accounting for the time variation of both intensity and frequency content typical of real earthquake ground motions is formulated and validated. An extension of the Thomson's spectrum estimation method is used to adaptively estimate the evolutionary power spectral density (PSD) function of the target ground acceleration record. The parameters of this continuous-time, analytical, stochastic earthquake model are determined by least-square fitting the analytical evolutionary PSD function of the model to the target evolutionary PSD function estimated. As application examples, the proposed model is applied to two actual earthquake records. In each case, model validation is obtained by comparing the second-order statistics of several traditional ground-motion parameters and the probabilistic linear-elastic response spectra simulated using the earthquake model with their deterministic counterparts characterizing the target record.

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 dynamic analysis of a deepwater floating production systems has many complexities, such as the dynamic
coupling between the vessel and the riser, the coupling between the first-order and second-order wave forces,
several sources of nonlinearities. These complexities can be captured by fully coupled time domain analyses.
Moreover, the sea state is random; hence the need of stochastic dynamic analysis. In this paper the evaluation of the non-Gaussian distributions of the responses of the systems is developed through the well-known First-Order Reliability Method (FORM) and the recently proposed Secant Hyperplane Method (SHM). They give rise to two stochastic Equivalent Linear Systems (ELS) allowing to determine any quantity of engineering interest: the TailEquivalent Linear System (TELS) based on FORM and the Tail Probability Equivalent Linear System (TPELS)
based on SHM. The TELS is the Equivalent Linear System (ELS) having the same design point of the original
nonlinear system. The Tail Probability Equivalent Linear System (TPELS) is the ELS where the difference in
terms of tail probability between the TPELS and the original system is minimized. A simplified 2-degrees-of freedom model is used to demonstrate how these methods can be effective for stochastic dynamic analysis of a marine riser.

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.

A parameterized stochastic model of near-fault ground motion in two orthogonal horizontal directions is developed. The major characteristics of recorded near-fault ground motions are represented. These include near-fault effects of directivity and fling step; temporal and spectral non-stationarity; intensity, duration, and frequency content characteristics; directionality of components; and the natural variability of ground motions. Not all near-fault ground motions contain a forward directivity pulse, even when the conditions for such a pulse are favorable. The proposed model accounts for both pulse-like and non-pulse-like cases. The model is fitted to recorded near-fault ground motions by matching important characteristics, thus generating an ‘observed’ set of model parameters for different earthquake source and site characteristics. A method to generate and post-process synthetic motions for specified model parameters is also presented. Synthetic ground motion time series are generated using fitted parameter values. They are compared with corresponding recorded motions to validate the proposed model and simulation procedure. The use of synthetic motions in addition to or in place of recorded motions is desirable in performance-based earthquake engineering applications, particularly when recorded motions are scarce or when they are unavailable for a specified design scenario. Copyright

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.

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.

Three different algorithms are presented for simulating a time series which is compatible with a given power spectrum of ocean waves. The first algorithm generates the current value of the time series as the sum of a linear combination of its past values and a white noise deviation. The second algorithm produces the values of the time series as a linear combination of white noise deviation. The third algorithm is a combination of the first and second algorithms. These algorithms are applied to the Pierson-Moskowitz (P-M) spectrum, exclusively. The third algorithm is associated with simple analogue filter approximations of the P-M spectrum. The advantages and disadvantages of each of the three algorithms are discussed in context with their applicability to offshore engineering problems. (A)

A new approach for first-order reliability analysis of structures with material parameters modeled as random fields is presented. The random field is represented by a series of orthogonal functions, and is incorporated directly in the finite-element formulation and first-order reliability analysis. This method avoids the difficulty of selecting a suitable mesh for discretizing the random field. A general continuous orthogonal series expansion of the random field is derived, and its relationship with the Karhunen-Loeve expansion used in recent stochastic finite- element studies is examined. The method is illustrated for a fixed-end beam with bending rigidity modeled as a random field. A set of Legendre polynomials is used as the orthogonal base to represent the random field. Two types of correlation models are considered. The Karhunen-Loeve expansion leads to a lower truncation error than does the Legendre expansion for a given number of terms, but one or two additional terms in the Legendre expansion yields almost the same results and avoids some of the computational difficulties associated with the use of the Karhunen-Loeve expansion.

A new method for efficient discretization of random fields (i.e., their representation in terms of random variables) is introduced. The efficiency of the discretization is measured by the number of random variables required to represent the field with a specified level of accuracy. The method is based on principles of optimal linear estimation theory. It represents the field as a linear function of nodal random variables and a set of shape functions, which are determined by minimizing an error variance. Further efficiency is achieved by spectral decomposition of the nodal covariance matrix. The new method is found to be more efficient than other existing discretization methods, and more practical than a series expansion method employing the Karhunen-Loeve theorem. The method is particularly useful for stochastic finite element studies involving random media, where there is a need to reduce the number of random variables so that the amount of required computations can be reduced.

Most of currently used algorithms for numerical generation of sea wave records which are compatible with a specified power spectrum are based on the superposition of several harmonic waves. An alternative method of simulation is presented. The basis of the method is the Linear Prediction Theory (LPT) which has been extensively used in processing digital data in other technical fields. Specifically, records of sea waves which are compatible with the target spectrum are obtained as the output of a recursive digital filter to a white noise input. A procedure for determining the filter parameters is discussed. Several numerical examples are presented.

This is a book for engineers about an approximation method for the design of structures under random actions such as wind, waves, earthquakes. This method, statistical linearization, consists in changing the mechanical equation of motion of the structure, which is usually nonlinear, into a linear one which is chosen in such a way that it minimizes the difference to the nonlinear term in the sense of quadratic mean. This method is not rigorous by the fact that this quadratic mean is taken for the probability which governs the linearized response, the only one which is computable, instead of the true response of the structure. It would be a mistake to believe that this point disqualifies the method, on the contrary it is a very convenient and useful tool during the first step of the design process (predimensioning) provided that the user be well warned about its field of application. It is precisely what the authors are doing quite clearly in the first chapter discussing the place of the method among the available procedures for dealing with nonlinear responses of structures to stochastic inputs. The second chapter is a presentation of the mechanical arguments yielding the equations of the motion of structures. An interesting and well organized survey of the different types of nonlinearities is proposed with a discussion of the most convenient mathematical representations including a detailed treatment of hysteresis. The two following chapters give an elementary account of probability theory and second order stationary processes. The explanation of the statistical linearization method begins in chapter 5 by the case of systems of single degree of freedom. Numerous examples are given corresponding to the mechanical classification of nonlinearities precedingly stated. The multivariate case, the most usual for engineers, is then detailed with an emphasis on effective procedures for getting numerically the linearized equations. The case where the coefficients of the linearized equation depend on time allows to reach nonstationary systems. A whole chapter is devoted to the case of systems with hysteretic nonlinearity. The book ends with an analysis of the accuracy of the method obtained by comparison with exact analytical solutions of the nonlinear equations when it is possible or with results of Monte Carlo simulations. This is quite enlightening on the size of the errors which remain small, generally. The point of relaxing the Gaussian hypotheses and extending the method by other closure techniques or by using explicitely solvable nonlinear equations, or by direct optimization on paths of the response, is the subject of chapter 9. It were be worth, when discussing the closure methods by expansion on Hermite polynomials or by truncating the sequence of cumulants to quote that the positivity of the density is not preserved in general by these procedures yielding sometimes nonpositive expectation of positive quantities. This book, which is easy to read and well written, is not only a reference book for engineers but a quite motivating reading for mathematicians interested by improvement of the correctness and research of effective bounds in the asymptotic expansions related to this method or its extensions.

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).

The aim of the paper is to advocate effective stochastic procedures, based on the First Order Reliability Method (FORM) and Monte Carlo simulations (MCS), for extreme value predictions related to wave and wind-induced loads.Due to the efficient optimization procedures implemented in standard FORM codes and the short duration of the time domain simulations needed (typically 60–300s to cover the hydro- and aerodynamic memory effects in the response) the calculation of the mean out-crossing rates of a given response is fast. Thus non-linear effects can be included. Furthermore, the FORM analysis also identifies the most probable wave episodes leading to given responses.Because of the linearization of the failure surface in the FORM procedure the results are only asymptotically exact and thus MCS often also needs to be performed. In the present paper a scaling property inherent in the FORM procedure is investigated for use in MCS in order to reduce the necessary simulation time. Thereby uniform accuracy for all exceedance levels can be achieved by a modest computational effort, even for complex non-linear models. As an example extreme responses of a floating offshore wind turbine are analyzed taking into consideration both stochastic wave and wind-induced loads.

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.

The subject of this paper is the simulation of multi-dimensional, homogeneous, Gaussian stochastic fields using the spectral representation method. Following this methodology, sample functions of the stochastic field can be generated using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic field when the number of 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 spatially-averaged mean value, autocorrelation function and power spectral density function are identical with the corresponding targets, when the averaging takes place over the multi-dimensional domain associated with the fundamental period of the cosine series. Another property of the simulated stochastic field is that it is asymptotically Gaussian as the number of terms in the cosine series approaches infinity. The most important feature of the method is that the cosine series formula can be numerically computed very efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in structural engineering, engineering mechanics and physics. 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).

Two models and are considered for generating samples of stationary band-limited Gaussian processes. The models are based on the spectral representation method and consist of a superposition of n harmonics. The harmonics of have random phase and amplitude while the harmonics of only have random phase. It is shown that the two models are equal in the second-moment sense. However, has stronger ergodic properties than . On the other hand, is a Gaussian process for any value of n while is asymptotically Gaussian as n approaches infinity. It is demonstrated that the rejection of , because of its weak ergodic property, or of model , because of its nonGaussian distribution, it is not generally justified. One special case in which should not be used is that of Gaussian processes with power concentrated at a few discrete frequencies.

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.

S ummary
We develop an approach to the spectral analysis of non‐stationary processes which is based on the concept of “evolutionary spectra”; that is, spectral functions which are time dependent, and have a physical interpretation as local energy distributions over frequency. It is shown that the notion of evolutionary spectra generalizes the usual definition of spectra for stationary processes, and that, under certain conditions, the evolutionary spectrum at each instant of time may be estimated from a single realization of a process. By such means it is possible to study processes with continuously changing “spectral patterns”.

The Karhunen-Loeve, spectral, and sampling representations, referred to as the KL, SP, and SA representations, are defined and some features/limitations of KL-, SP, and SA-based approximations commonly used in applications are stated. Three test applications are used to evaluate these approximate representations. The test applications include (1) models for non-Gaussian processes; (2) Monte Carlo algorithms for generating samples of Gaussian and non-Gaussian processes; and (3) approximate solutions for random vibration problems with deterministic and uncertain system parameters. Conditions are established for the convergence of the solutions of some random vibration problems corresponding to KL, SP, and SA approximate representations of the input to these problems. It is also shown that the KL and SP representations coincide for weakly stationary processes.

In this paper, a simulation methodology is proposed to generate sample functions of a stationary, non-Gaussian stochastic process with prescribed spectral density function and prescribed marginal probability distribution. The proposed methodology is a modified version of the Yamazaki and Shinozuka iterative algorithm that has certain difficulties matching the prescribed marginal probability distribution. Although these difficulties are usually sufficiently small when simulating non-Gaussian stochastic processes with slightly skewed marginal probability distributions, they become more pronounced for highly skewed probability distributions (especially at the tails of such distributions). Two major modifications are introduced in the original Yamazaki and Shinozuka iterative algorithm to ensure a practically perfect match of the prescribed marginal probability distribution regardless of the skewness of the distribution considered. First, since the underlying "Gaussian" stochastic process from which the desired non-Gaussian process is obtained as a translation process becomes non-Gaussian after the first iteration, the empirical (non-Gaussian) marginal probability distribution of the underlying stochastic process is calculated at each iteration. This empirical non-Gaussian distribution is then instead of the Gaussian to perform the nonlinear mapping of the underlying stochastic process to the desired non-Gaussian process. This modification ensures that at the end of the iterative scheme every generated non-Gaussian sample function will have exact prescribed non-Gaussian marginal probability distribution. Second, before the start of the iteractive scheme, a procedure named "spectral preconditioning" is carried out to check the compatibility between the prescribed spectral density function and prescribed marginal probability distribution. If these two quantities are found to be incompatible, then the spectral density function can be slightly modified to make it compatible with the prescribed marginal probability distribution. Finally, numerical examples (including a stochastic process with a highly skewed marginal probability distribution) are provided to demonstrate the capabilities of the proposed algorithm.

Probability density functions, mean crossing rates, and other descriptors are developed for the quasi-static and dynamic responses of offshore platforms to wave forces. It is assumed that offshore platforms can be modeled by simple oscillators, the wave particle velocity is a stationary differentiable Gaussian process, Morison's equation can be applied, and wave forces are perfectly correlated spatially. Results show that both the quasi-static response and the dynamic response of offshore platforms to wave forces are generally underestimated if the drag force is linearized. Estimates are developed for probabilistic characterictics of these responses based on the crossing theory of random processes and time-discretization of the wave force process. Simulation studies indicate that these estimates are satisfactory

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 gradient-free method is developed for finding the design point in nonlinear stochastic dynamic analysis, where the input excitation is discretized into a large number of random variables. This point defines the realization of the excitation that is most likely to give rise to a specific response threshold at a given time. The design point is the essential information in the recently developed tail-equivalent linearization method. The proposed approach employs a variant of the model correction factor method developed by O. Ditlevsen, which is further improved by the use of a novel response surface technique. Example applications to single- and multi-degree-of-freedom hysteretic systems demonstrate the efficiency and accuracy of the method.

This book addresses random vibration of mechanical and structural
systems commonly encountered in aerospace, mechanical, and civil
engineering. Techniques are examined for determining probabilistic
characteristics of the response of dynamic systems subjected to random
loads or inputs and for calculating probabilities related to system
performance or reliability. Emphasis is given to applications.

SUMMARYA method is presented for simulating arrays of spatially varying ground motions, incorporating the effects of incoherence, wave passage, and differential site response. Non-stationarity is accounted for by considering the motions as consisting of stationary segments. Two approaches are developed. In the first, simulated motions are consistent with the power spectral densities of a segmented recorded motion and are characterized by uniform variability at all locations. Uniform variability in the array of ground motions is essential when synthetic motions are used for statistical analysis of the response of multiply-supported structures. In the second approach, simulated motions are conditioned on the segmented record itself and exhibit increasing variance with distance from the site of the observation. For both approaches, example simulated motions are presented for an existing bridge model employing two alternatives for modeling the local soil response: i) idealizing each soil-column as a single-degree-of-freedom oscillator, and ii) employing the theory of vertical wave propagation in a single soil layer over bedrock. The selection of parameters in the simulation procedure and their effects on the characteristics of the generated motions are discussed. The method is validated by comparing statistical characteristics of the synthetic motions with target theoretical models. Response spectra of the simulated motions at each support are also examined. Copyright © 2011 John Wiley & Sons, Ltd.

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.

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 random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen-decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non-stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second-order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen-solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non-stationary processes with little additional effort. Copyright © 2001 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.

In on-board decision support systems, efficient procedures are needed for real-time estimation of the maximum ship responses
to be expected within the next few hours, given online information on the sea state and user-defined ranges of possible headings
and speeds. For linear responses, standard frequency domain methods can be applied. For non-linear responses, as exhibited
by the roll motion, standard methods such as direct time domain simulations are not feasible due to the required computational
time. However, the statistical distribution of non-linear ship responses can be estimated very accurately using the first-order
reliability method (FORM), which is well known from structural reliability problems. To illustrate the proposed procedure,
the roll motion was modelled by a simplified non-linear procedure taking into account non-linear hydrodynamic damping, time-varying
restoring and wave excitation moments, and the heave acceleration. Resonance excitation, parametric roll, and forced roll
were all included in the model, albeit with some simplifications. The result is the mean out-crossing rate of the roll angle
together with the most probable wave scenarios (critical wave episodes), leading to user-specified specific maximum roll angles.
The procedure is computationally very effective and can thus be applied to real-time determination of ship-specific combinations
of heading and speed to be avoided in the actual sea state.

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.

An efficient procedure for the derivation of mean outcrossing rates for non-linear wave-induced responses in stationary sea states is presented and applied to an analysis of the horizontal deck sway of a jack-up unit. The procedure is based on the theory of random vibrations and uses the first order reliability method (FORM) to estimate the most probable set of wave components in the ocean wave system that will lead to exceedance of a specific response level together with the associated mean outcrossing rate. The procedure bears some resemblance to the Constrained NewWave methodology, but is conceptually simpler and makes efficient use of the optimisation procedures implemented in standard FORM software codes.Due to the fast calculation procedure the analysis can be carried out taking into account all relevant non-linear effects. Specifically, the present analysis accounts for second order stochastic waves, not previously included in the analysis of jack-up units in stochastic seaways.

This paper is concerned with the time domain simulation of the second order motions of a moored vessel when the random seastate is represented as a sum of harmonic components. It is known that in these circumstances successive runs of a simulation program produce different results for the statistical moments of the response. Here, the variation of the first four statistical moments of the response over an ensemble of program runs is investigated, leading to an assessment of the likely accuracy of these quantities as predicted by a limited number of program runs. Also, it is shown that an approximate simulation method which uses deterministic wave amplitudes and random phase angles does not correctly predict the fourth moment of the response.

An efficient algorithm to simulate turbulent, atmospheric or wind tunnel generated wind fields is devised. The method is based on a model of the spectral tensor for atmospheric surface-layer turbulence at high wind speeds and can simulate two- or three-dimensional fields of one, two or three components of the wind velocity fluctuations. The spectral tensor is compared with and adjusted to several spectral models commonly used in wind engineering. Compared to the Sandia method (see Veers, P. S., Three-dimensional wind simulation. Technical Report SAND88-0152, Sandia National Laboratories, 1988) the algorithm is more efficient, simpler to implement, and in some respects more physical. The simulation method is currently used for load calculations on wind turbines and bridges.

In this paper we present a review of stochastic process models proposed for the simulation of seismic ground motion. The models reviewed include those based on filtered white noise processes, filtered Poisson processes, spectral representation of stochastic processes, and finally those based on stochastic wave theory. Mathematical expressions are provided for all models along with comments on their usefulness, advantages and disadvantages.Together with the review of auto-regressive moving-average models by F. Kozin in this PEM review series on earthquake engineering (June issue), this paper represents an overview of stochastic models of earthquake ground motion, which is hopefully of some use to researchers as well as practitioners.

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.