Improved state augmentation method for buffeting analysis of structures subjected to non-stationary wind

https://doi.org/10.1016/j.probengmech.2022.103309Get rights and content

Abstract

Extreme winds such as hurricanes and thunderstorms often present non-stationary characteristics, having time-varying mean wind speeds and non-stationary wind fluctuations. When concerning the wind-induced vibrations under non-stationary wind, the excitation will be a non-stationary process, and the wind-structure coupled system can be represented by a linear time-varying (LTV) system. The aim of this study is to present a state augmentation method to investigate the non-stationary buffeting of a model bridge tower subjected to non-stationary wind with consideration of the aeroelastic damping. Based on the theory of stochastic differential equations and Itô’s lemma, the statistical moments of the non-stationary buffeting response are derived through solving a first-order ordinary differential equation system. The proposed method is validated by comparisons with the Monte Carlo method and the pseudo excitation method. The result shows that the state augmentation method has higher accuracy and efficiency than the well-accepted time–frequency techniques.

Introduction

In contrast with the stationary synoptic wind, extreme wind events such as hurricanes and thunderstorms always exhibit considerable non-stationary and non-gaussian characteristics [1], [2], [3], having time-varying mean wind speeds and non-stationary and non-gaussian wind fluctuating components. The rapid changes in the kinematics and dynamics of these flow fields can potentially amplify aerodynamic loads on structures and result in higher non-stationary buffeting responses [4]. Moreover, when considering aeroelastic effects, the aerodynamic damping and stiffness will be time-dependent due to the time-varying mean wind speed, and the wind-structure coupled system can be thus represented as a linear time-varying (LTV) system [5]. Brusco and Solari [6] investigated the aeroelastic effects induced by thunderstorms on the dynamic response of slender structures. These facts lead to difficulties in the calculation of the structural response by using the conventional buffeting analysis method [7], [8], which was originally developed for synoptic wind and stationary responses [9]. In view of these non-stationary wind effects, many attempts have been made to develop random vibration theory for non-stationary buffeting, which can be broadly classified into three categories.

The first category of methods is the Monte Carlo method (MCM). It uses multiple deterministic time–history samples generated, for instance, using the evolutionary power spectral density (EPSD) [10], [11] in order to statistically characterize the system response. Huang et al. [12] used this method to evaluate the along-wind responses of tall buildings induced by thunderstorm downbursts and investigated the transient wind load effects on various responses including the extreme value and peak factor. To obtain reliable statistical characteristics of the responses, sufficiently numerous samples should be employed, which results in computationally intensive simulations.

The second category of methods is the generalized frequency-domain method, also named as generalized wind loading chain proposed by Kareem et al. [9], which extends the well-established Davenport’s chain [7], [8] from a stationary buffeting analysis in frequency domain to a non-stationary format in time–frequency domain. Based on the Priestley’s Evolutionary Spectra theory [10], [11], the time–frequency response spectrum can be evaluated through a sequential product of the time–frequency wind spectrum, the instantaneous aerodynamic admittance function, and the structural transfer function. Chen [13] developed a frequency domain framework for calculating along-wind buffeting responses of tall buildings induced by transient non-stationary wind and studied the effects of wind speed profile, time-varying mean wind speed, and spatial correlations of wind turbulence on structural responses. This framework was later generalized [14] to predict multimodal coupled buffeting responses of long-span bridges under nonstationary wind with consideration of self-excited forces modeled as a frequency-dependent linear time-varying (LTV) system. Note that, to deal with the LTV system [15], [16], the generalized frequency-domain method assumes the system as time-invariant within a moving short window, so that the problem reduces to finding the time-independent frequency response function and stationary response of an equivalent linear time-invariant (LTI) system within each window. This is only applicable to the system with slowly time-varying characteristics.

The third category of methods for investigating the non-stationary buffeting in wind-structure coupled systems is the pseudo excitation method (PEM) developed by Lin et al. [17]. It can deal with the random vibration problem with non-stationary excitations and time-dependent system properties simultaneously. With this method, the nonstationary characteristics of the response can be conveniently obtained, including the EPSD and standard deviation. For instance, Hu et al. [5], [18] considered the self-excited forces characterized by flutter derivatives and computed the non-stationary buffeting response of long-span bridges subjected to typhoon-induced wind loads by means of the PEM. Note that these flutter derivatives are functions of time due to the effects of the time-varying mean wind speeds on the reduced frequencies. He et al. [19] considered the excitations due to stationary track irregularities and non-stationary wind loads simultaneously and calculated the non-stationary responses of the high-speed train-bridge coupled system based on the PEM. In essence, the PEM is a time–frequency technique: the pseudo excitations are expanded in frequency domain, and the responses are integrated in time domain. According to [20], the computation effort of the PEM is mainly taken in calculating the pseudo response. The Precise Integration Method [21] can be combined with this process to improve the efficiency when computing the corresponding time–history responses to the pseudo excitations, yet this is more applicable to the LTI system [22], and does not avoid the operations in the frequency content.

Based on the theory of Itô’s stochastic differential equation, Grigoriu [23], [24] proposed the state augmentation method to calculate the stochastic response of linear systems subjected to stationary non-Gaussian excitations expressed in the form of polynomials of Gaussian processes with time-dependent coefficients. With this method, the statistical moments of any order of the transient response can be directly obtained through solving a system of algebraic equations, i.e., moments equation, with high efficiency. This method [25] was then applied to predict the random vibration response of a flexible plate subjected to stationary Gaussian turbulence in which the wind pressure was assumed to be proportional to the square of the wind speed, so that the excitations were stationary non-Gaussian processes. Due to the orthogonality of structural modes, the coupling effect between each mode was not considered in this study, and the plate displacement was restricted to its first structural mode for clarity. Recently, Cui et al. [26] extended this method to study the multimodal buffeting response of long-span bridges subjected to stationary non-Gaussian turbulence, considering the coupling effect of aerodynamic damping between vertical and torsional motions. Although the state augmentation method has been applied in several wind engineering problems, such as quantification of wind loading uncertainties [27], [28] and non-Gaussian turbulence, these studies are primarily based on the stationary case with time-invariant system characteristics. It has not been reported for non-stationary case with time-dependent characteristics of the system, for which the methodology may change substantially, e.g., the statistical moment equations change from algebraic equations in the stationary case to differential equations in the non-stationary case.

The aim of this paper is to extend the state augmentation method to investigate the non-stationary along-wind buffeting of a bridge tower subjected to non-stationary wind loads with consideration of the time-dependent aeroelastic damping. First, the non-stationary wind speed is characterized as a time-varying mean and uniformly modulated wind fluctuations [1]. To formulate the non-stationary buffeting forces, the strip and quasi-steady theories are invoked [6], [29]. The aeroelastic term is included by considering the relative velocity as the combination of the structural velocity and the total wind speed. Due to the time-dependent characteristics of aerodynamic damping, the dynamic equation of the along-wind buffeting can be represented as a linear time-varying (LTV) system. By using the Ornstein–Uhlenbeck process to approximate wind fluctuations, the augmented states of the system and the excitation are written as an Itô-type stochastic differential equation. Based on Itô’s lemma, the moments equations of the non-stationary response are derived as a system of first-order ordinary differential equations. The proposed state augmentation method is first validated by comparing the time-varying standard deviation of the responses with those obtained from the Monte Carlo simulations and the pseudo excitation method. Then, the assets of the proposed method as well as the characteristics of non-stationary buffeting responses are discussed by comparing with conventional buffeting analyses. At last, the effect of the aerodynamic damping term on the non-stationary buffeting vibrations is investigated.

Section snippets

Non-stationary wind model

Analogously to the stationary wind model, which is comprised of the mean and fluctuating components, the velocity of a non-stationary extreme wind Ut can be usually characterized as the summation of a deterministic time-varying mean U¯t and a zero-mean non-stationary fluctuating wind component ut Ut=U¯t+ut

Note that the variation of U¯t is much slower than that of ut. Therefore, when subjected to a time-varying mean wind, the structure will present a static response so that the quasi-static

Augmented states of the system and the excitation

According to [23], [24], the stationary Gaussian process u0.6hst can be approximated by an Ornstein–Uhlenbeck (OU) process, i.e., Ztu0.6hst, which satisfies the stochastic differential equation dZt=αZtdt+σ2αdWtin which 1/α is the time relaxing coefficient, σ is the standard deviation of Zt, and Wt is a standard Wiener process satisfying EdWtdWt=dt where E indicates the expectation operator. The parameters α and σ can be found through fitting the auto-correlation function RZZτ=σ2eα|τ| or the

Validation

To illustrate the reliability and computation efficiency of the state augmentation method (SAM), the proposed method is first applied to calculate the buffeting response of a bridge tower subject to a non-stationary wind consisting of a time-varying mean and a stationary wind fluctuation. Notice that non-stationary fluctuations will be considered in Section 5. The obtained results are validated by comparing them with those obtained using the Monte Carlo method (MCM) and pseudo excitation method

Non-stationary wind field

In this section, the proposed state augmentation method is employed to investigate the along-wind buffeting responses of the bridge tower subjected to a more complex non-stationary wind, that consists of a time-varying mean and a uniformly modulated non-stationary fluctuation. This non-stationary wind speed model is obtained from a wind record measured in a mountainous area in China, of which the time history is shown in Fig. 6, with a sampling frequency of 10 Hz.

The time-varying mean wind

Conclusion

This study has investigated the non-stationary buffeting of a bridge tower subjected to non-stationary wind loads. The strip and quasi-steady assumptions are adopted to formulate the buffeting forces and taking the motion-induced force into account. Due to the time-varying mean wind speed, the dynamic equation of the along-wind buffeting presents time-dependent characteristics, and thus the responses are non-stationary even if the wind fluctuating components are stationary. Based on the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The study was supported by the National Natural Science Foundation of China (grant No. 51978527 and 52008314) and the China Scholarship Council (No. 202106260170).

References (37)

  • GrigoriuM. et al.

    A method for analysis of linear dynamic systems driven by stationary non-gaussian noise with applications to turbulence-induced random vibration

    Appl. Math. Model.

    (2014)
  • CuiW. et al.

    Non-gaussian turbulence induced buffeting responses of long-span bridges based on state augmentation method

    Eng. Struct.

    (2022)
  • CaracogliaL.

    A stochastic model for examining along-wind loading uncertainty and intervention costs due to wind-induced damage on tall buildings

    Eng. Struct.

    (2014)
  • SuY. et al.

    Derivation of time-varying mean for non-stationary downburst winds

    J. Wind Eng. Ind. Aerodyn.

    (2015)
  • HuangG. et al.

    Spectrum models for nonstationary extreme winds

    J. Struct. Eng.

    (2015)
  • CuiW. et al.

    Non-gaussian turbulence induced buffeting responses of long-span bridges

    J. Bridge Eng.

    (2021)
  • A. Kareem, The changing dynamics of aerodynamics: New frontiers, in: Proc. 7th Asia-Pacific Conf. on Wind Engineering...
  • DavenportA.G.

    Gust loading factors

    J. Struct. Div.

    (1967)
  • Cited by (0)

    View full text