Aeroelastic Analysis of Bridges under Multicorrelated Winds: Integrated State-space Approach

In this paper, an integrated state-space model of a system with a vector-valued white noise input is presented to describe the dynamic response of bridges under the action of multicorrelated winds. Such a unified model has not been developed before due to a number of innate modeling difficulties. The integrated state-space model is realized based on the state-space models of multicorrelated wind fluctuations, unsteady buffeting and self-excited aerodynamic forces, and the bridge dynamics. Both the equations of motion at the full order in the physical coordinates and at the reduced-order in the generalized modal coordinates are presented. This state-space model allows direct evaluation of the covariance matrix of the response using the Lyapunov equation, which presents higher computational efficiency than the conventional spectral analysis approach. This state-space model also adds time domain simulation of multicorrelated wind fluctuations, the associated unsteady frequency dependent aerodynamic forces, and the attendant motions of the structure. The structural and aerodynamic coupling effects among structural modes can be easily included in the analysis. The model also facilitates consideration of various nonlinearities of both structural and aerodynamic origins in the response analysis. An application of this approach to a long-span cable-stayed bridge illustrates the effectiveness of this scheme for a linear problem. An extension of the proposed analysis framework to include structural and aerodynamic nonlinearities is immediate once the nonlinear structural and aerodynamic characteristics of the bridge are established. INTRODUCTION Aerodynamic forces on bridges have conventionally been modeled as the sum of motion-induced self-excited and windinduced buffeting force components. These are in general functions of the geometric configurations of bridge sections, the incoming wind fluctuations, and the reduced frequency. In the wind velocity range of interest in structural design, the flow around bluff bridge sections is quite unsteady and not amenable to quasi-steady analysis techniques, which are only valid at very high wind velocities. The frequency dependent characteristics of aerodynamic forces are generally described in terms of experimentally quantified flutter derivatives for the self-excited forces and in terms of admittance and spanwise coherence functions for the buffeting forces. Incorporating these unsteady aerodynamic characteristics is essential for an accurate evaluation of the forces and the attendant response of the structures. These characteristics can be easily incorporated in the frequency domain analysis framework with and without the consideration of intermode coupling effects (Davenport 1962; Scanlan 1978a,b; Jain et al. 1996; Katsuchi et al. 1999; Chen et al. 2000a). To account for the structural and aerodynamic nonlinearities in the analysis, the equations of motion must be cast in the time domain and solved using a time domain scheme. Most previous time domain studies concerning bridge buffeting response have used quasi-steady assumption when modeling the aerodynamic forces. These assumptions manifest themselves by neglecting the frequency dependent flutter derivatives, admittance functions, and effects of spanwise correlation. This inconsistency with the frequency domain approach has been addressed by Chen et al. (2000b), in which the frequency dependent unsteady aerodynamic forces are accurately modeled in the time domain analysis framework. This time domain Postdoct. Res. Assoc., Dept. of Civ. Engrg. and Geological Sci., Univ. of Notre Dame, Notre Dame, IN 46556. Robert M. Moran Prof. and Chair, Dept. of Civ. Engrg. and Geological Sci., Univ. of Notre Dame, Notre Dame, IN 46556. Note. Associate Editor: George Deodatis. Discussion open until April 1, 2002. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on June 6, 2000; revised April 24, 2001. This paper is part of the Journal of Engineering Mechanics, Vol. 127, No. 11, November, 2001. qASCE, ISSN 07339399/01/0011-1124–1134/$8.00 1 $.50 per page. Paper No. 22359. 24 / JOURNAL OF ENGINEERING MECHANICS / NOVEMBER 2001 approach is regarded as a rigorous representation of the frequency domain analysis of linear problems as long as the aerodynamic forces are represented by rational function approximations (RFAs) exactly or with an acceptable level of error. Recent developments presented in this paper may be viewed as an extension of this time domain approach by utilizing a state-space modeling technique that is rooted in linear system theory. Much research has been performed in the area of linear state-space modeling of unsteady self-excited aerodynamic forces in the aeronautical field by using RFA technique (e.g., Roger 1977; Karpel 1982). Among these schemes, Roger’s RFA is the most widely utilized because of the accuracy, simplicity, and robustness of the method, although different forms of the approximation, such as the minimum state (MS) method, are available with a focus on reducing the dimensions of the augmented aerodynamic states (Karpel 1982). The application of RFAs to bluff body bridge aerodynamics can be found in the representation of the self-excited forces (Scanlan et al. 1974; Lin and Yang 1983; Xie et al. 1985; Bucher and Lin 1988; Matsumoto et al. 1994; Wilde et al. 1996; Boonyapinyo et al. 1999; Chen et al. 2000a,b). The modeling of frequency dependent buffeting forces is given in Matsumoto and Chen (1996), Matsumoto et al. (1996), and Chen et al. (2000b). The modeling of multicorrelated wind fluctuations in a state-space framework has also been addressed in Goßmann and Walter (1982), Suhardjo et al. (1992), Matsumoto et al. (1996), and Kareem (1997). This is based on factorization of the cross-power spectral density (XPSD) matrix of the wind fluctuations. The spectral matrix is first expressed in terms of RFAs and is then decomposed into a transfer function, which is then utilized to obtain the state-space matrices based on the realization of the transfer function. The modeling of a matrix transfer function and subsequent operations are nontrivial for a large size wind field simulation. In some cases, the mathematical difficulty and numerical error introduced by the calculation procedure precluded the use of this technique to realistic problems (Matsumoto et al. 1996). Kareem (1997) has suggested some simplifications, but the approach remains tedious as attested by a lack of realistic feed-forward modeling of wind in the literature. In Kareem and Mei (1999) and Benfratello and Muscolino (1999), the stochastic decomposition technique is utilized to decompose a multicorrelated random process into a set of independent random subprocesses. For each subprocess the state-space model is derived, and then through a transformation the state-space model of the original multicorrelated process is composed. For simplification of the state-space modeling in the original coordinate space, an eigenvector matrix value at a fixed frequency is chosen based on the observation that the eigenvectors of the XPSD matrix change very slowly with respect to the frequency. This technique requires the eigenvalue analysis of the XPSD matrix at each discrete frequency, which may result in large computational efforts for a large size wind-field simulation. In addition, the assumption of constant eigenvectors may introduce errors in some cases depending on the spectral matrix of wind fluctuations. Chen and Kareem (2000) pointed out that, with the truncation of higher modes of wind fluctuations, this stochastic decomposition technique provides an efficient tool for statespace modeling of well-correlated random processes. Its effectiveness in modeling poorly correlated random processes is rather limited, particularly, for representing high-frequency wind fluctuations. In this paper, an integrated state-space model of a multiinput and multioutput system with a vector-valued white noise input is presented to model the dynamic response of bridges under the multicorrelated winds. Such a unified model has not been developed before due to a number of innate modeling difficulties. This integrated state-space model is realized based on the state-space models of the multicorrelated wind fluctuations, the unsteady aerodynamic forces, and the structure. Both the equations of motion at the full order in the physical coordinates and at reduced order in the generalized modal coordinates are presented. The full-order form is more appropriate for nonlinear problems by using time-variant system models, whereas the reduced form is computationally more efficient for the linear problems. An application of this approach to a long span cable-stayed bridge demonstrates its effectiveness. STATE-SPACE REPRESENTATION OF RESPONSE UNDER WINDS The mathematical model for describing the response of wind-excited structure based on linear system theory is schematically shown in Fig. 1. The wind-induced motions of the structure can be represented as the outputs of an integrated multiinput and multioutput system excited by a vector-valued white noise process. The multicorrelated wind fluctuations are considered as the output of a system with a vector-valued white noise excitation, whose transfer functions can be derived by factorizing their power spectral density matrix. Similarly, the buffeting forces are derived as the output of a system with wind fluctuations as input. Their transfer functions are described in terms of the admittance function and spanwise coherence of unsteady buffeting forces. Similarly, the self-excited forces are modeled as the output of a system with the structural response as input. Their transfer functions are defined in terms of the flutter derivatives. By augmenting the state-space equations of structural motion with the corresponding state-space representation of the loading components and wind fluctuations, as stated above, an integrated state-space model is established that synthesizes the unsteady characteristics of multicorrelated wind field, frequency dependent unsteady aerodynamic forces, and the dynamics of the bridge. The integrated state-space model for describing the response of a structure under winds has several significant mathematical advantages. The recasting of the overall system equations in the state-space format allows the use of tools based on linear system theory for response analysis, optimization, and design of active control devices to suppress flutter and buffeting. By using this model, the wind load information can be incorporated in a structural control design as a feed-forward link with the potential to enhance the control effectiveness (Suhardjo et al. 1992). In addition, the structural and aerodynamic coupling effects can be automatically included in the computation. For linear problems, conventional spectral analysis approach requires intensive computational efforts in the estimation of the transfer function and response spectral density matrix at each discrete frequency with an interval that must be very small for bridges with closely spaced natural frequencies. Subsequent integration of the spectral matrix needed to determine the response covariance requires additional computational effort. An integrated linear time-invariant state-space model of the response with a vector-valued white noise input facilitates direct estimation of the covariance matrix of the response through the Lyapunov equation and allows higher computational efficiency. MODELING OF MULTICORRELATED WINDS Consider a structure represented by a finite-element discretization. The longitudinal and vertical components of wind fluctuations at the centers of elements, W(t), are represented by a multicorrelated random process. These can be represented as the output of a linear system with input of a vector-valued Gaussian white noise process N(t) with a zero mean and identity covariance matrix. In this study, an autoregressive (AR) model is used to describe this linear system for accurate modeling and simplicity. The AR model is considered as a special FIG. 1. Integrated Modeling of Dynamic Response of Wind-Excited Structure JOURNAL OF ENGINEERING MECHANICS / NOVEMBER 2001 / 1125 case of a general autoregressive moving-average (ARMA) model (e.g., Samaras et al. 1985; Mignolet and Spanos 1987; Li and Kareem 1990a,b). The AR model is expressed as