Equivalent Static Wind Loads for Buffeting Response

In current design practice, the dynamic wind loads are described in terms of the equivalent static wind loads based on the gust response factor. This approach results in a distribution of the equivalent static loading similar to the mean static wind load distribution, which may not always be a physically meaningful and realistic load description. In this paper, the equivalent static load representation for multimode buffeting response of bridges is formulated in terms of either a weighted combination of modal inertial load components, or the background and resonant load components. The focus of the present study is on the determination of weighting factors of equivalent static load components in which the correlation among modal response components due to structural and aerodynamic coupling effects is taken into consideration. It is noteworthy that the equivalent static load distributions vary for each response component. The proposed approach particularly helps in extracting design loads from full aeroelastic model test results by expressing the dynamic loads in terms of the equivalent static loads. This facilitates in drawing useful design input from full aeroelastic tests, which have been employed mostly for monitoring the response of bridge models at selected locations. A simplified formulation is also presented in a closed form when wind loading information is available and coupling in modal response components is negligible, which can be very attractive for the preliminary design application. Examples are presented to illustrate modeling of the equivalent static loading and to demonstrate its effectiveness in bridge design. INTRODUCTION In current wind resistant design practice, the dynamic wind loads are generally represented in terms of equivalent static wind loads expressed as the mean static wind loads multiplied by the gust response factor (GRF). The gust response factor, or gust loading factor, was originally introduced by Davenport (1967) and is defined as the ratio of the maximum expected wind load or response to its corresponding mean value. The GRFs are generally different for different response components and may vary in a wide range, depending on the structure system, the wind load characteristics, and the influence functions related to the response components. This GRF approach does not provide useful information in cases with zero mean load or response. In contrast to the GRF approach, an equivalent static load representation in terms of background and resonant load distributions leads to a physically meaningful and realistic load description (Davenport 1985; Holmes 1992; Kasperski 1992; Holmes and Kasperski 1996; Irwin 1998; Zhou et al. 2000). The background component of the wind load can be treated as a quasi-static load, and its static load distribution for a specific dynamic response depends on the influence function and the distribution of the external wind load. It can be determined based on the load-response-correlation (LRC) approach (Kasperski 1992; Kasperski and Niemann 1992). The resonant load component follows the distribution of the inertial load and can be expressed in terms of modal inertial loads (Davenport 1985; Irwin 1998; Holmes 1999; King 1999). For design use, the equivalent static load can be expressed in a separated form in terms of the background component and the resonant components of structural modes. The total response is then calculated by combining the background and resonant responses utilizing the square root of the sum of 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: Bogusz Bienkiewicz. Discussion open until May 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 August 24, 2000; revised February 21, 2001. This paper is part of the Journal of Structural Engineering, Vol. 127, No. 12, December, 2001. qASCE, ISSN 0733-9445/01/0012-1467–1475/$8.00 1 $.50 per page. Paper No. 22534. squares (SRSS) combination approach or the complete quadratic combination (CQC) approach. The application of this approach in combining the section model tests for the equivalent wind loads on bridges has been presented by Davenport and King (1984). Alternatively, the equivalent static load can be provided as a linear combination of its background and resonant components using a set of load weighting factors (Irwin 1998; Holmes 1999; King 1999). The resulting structural response can be estimated by means of a static analysis. Such a format facilitates the combination of wind load with other loads and is more appropriate for current design procedures. Instead of using arbitrarily selected load weighting factors (Irwin 1998), a numerical iteration scheme for calculating these weighting factors has been used in King (1999). A methodology based on the LRC approach has been given by Holmes (1999) for the resonant equivalent static load associated with multimode response of bridges without modal response correlations. Representing dynamic load in terms of equivalent static load is particularly suitable for providing the design loads based on wind tunnel tests. Most wind loading information has been derived from section model tests instead of using full bridge aeroelastic model tests, which are traditionally used as a final confirmation of the performance of important bridges. Using the equivalent static load approach, full bridge aeroelastic model tests can be used to gain useful insight into the description of the wind loads (King 1999). This equivalent static load approach can also be used to aid the wind tunnel tests in predicting the response components not directly measured during the test. Modal response coupling due to closely spaced frequencies and coupled self-excited wind loads may result in significant modal response correlation. For long span suspension bridges, significant aerodynamic coupling between vertical bending and torsional modes exists at higher wind velocities. Neglecting the contribution of these correlations will result in predictions that underestimate the responses (Chen et al. 2000a). In this paper, the equivalent static load distribution for the multimode buffeting response of bridges is formulated in terms of either a weighted combination of modal inertial load components or the background and resonant load components. The focus of the present study is on the determination of load weighting factors of equivalent static load components, in which the correlation among modal response components due to structural and aerodynamic coupling effects is taken into consideration. The background load component has been exJOURNAL OF STRUCTURAL ENGINEERING / DECEMBER 2001 / 1467 pressed based on inertial loads and based on external wind load distribution. A simplified formulation is presented in a closed form when wind loading information is available and coupling in modal response components is negligible. Examples are presented to illustrate the modeling of the equivalent static loading and to demonstrate its effectiveness in bridge design. METHODOLOGY The dynamic response of a bridge to turbulent wind excitation can be expressed in terms of the matrix equations below: ̈ ̇ MY 1 CY 1 KY = F (1) where M, C, and K = mass, damping, and stiffness matrices, respectively; Y = dynamic displacement vector; and F = external stochastic wind load vector, including turbulence induced buffeting and motion induced self-excited components. Using the modal coordinates, the dynamic displacement and elastic force vectors can be represented as