A Highly Accurate and Efficient Analytical Approach to Bridge Deck Free Vibration Analysis

Shock and Vibration 1070-9622 A highly accurate and efficient analytical approach to bridge deck free vibration analysis D.J. Gorman 1 L. Garibaldi 0 0 Dipartimento di Meccanico, Politecnico di Torin , Torino , Italy 1 Department of Mechanical Engineering, University of Ottawa , 770 King Edward Avenue, Ottawa , Canada - The superposition method is employed to obtain an accurate analytical type solution for the free vibration frequencies and mode shapes of multi-span bridge decks. Free edge conditions are imposed on the long edges running in the direction of the deck. Inter-span support is of the simple (knife-edge) type. The analysis is valid regardless of the number of spans or their individual lengths. Exact agreement is found when computed results are compared with known eigenvalues for bridge decks with all spans of equal length. Mode shapes and eigenvalues are presented for typical bridge decks of three and four span lengths. In each case torsional and non-torsional modes are studied. 1. Introduction At the Dipartimento di Meccanica of the Polytecnico di Torino, of Turin Italy, an extended study of the dynamic response of bridge decks to vehicular excitation is in progress. The study involves the taking of vibration measurements on actual bridges and development of theoretical models. The long term goal is to develop accurate modeling of bridge dynamic behavior in response to the above excitation. Ultimately, knowledge gained thereby will be utilized to achieve optimization in bridge design. As a first step it was desired to develop analytical means for accurately predicting free vibration frequencies and mode shapes of the bridge decks. The bridge response to forced excitation would then be represented in terms of these mode shapes. It was recognized that in a multi-span deck many closely spaced frequencies (eigenvalues) must be anticipated and a careful delineation must be made between adjacent eigenvalues and their associated mode shapes. For these reasons it was concluded that a highly accurate analytical type analysis based on continuum mechanics was essential. Furthermore, results obtained in this manner could augment, and help verify, results of numerical studies carried out by a finite element method. Such an analysis, based on the superposition method, is described in this paper for the first time. It will be seen that the governing differential equation is satisfied exactly throughout the structure. Boundary conditions are satisfied to any desired degree of accuracy. Convergence is rapid and the method is found to be highly efficient from a computational point of view. While there is a massive literature available related to single span plate vibration (the classical rectangular plate) and a limited literature related to double span plates there does not appear to be publications of the type discussed here dealing with vibration of plates of any arbitrary number of spans. 2. Mathematical procedure 2.1. Development of building block solutions For illustrative purposes we examine the behavior of a three-span deck. It will be obvious to the reader that the same analytical technique is easily extended to handle decks of any number of spans. The three-span deck utilized for illustrative purposes is represented schematically in Fig. 1. Each span has a common width ‘b’. The first and second spans have lengths a1, and a2, respectively. Edges along the deck entrance and exit are given simple support. Simple line support is provided along the interface of the spans. We may think of this latter support as being of the knifeedge type. All edges running in the direction of the long axis of the deck are considered to be free. A solution for the free vibration frequencies and mode shapes of the deck as described above is obtained by superimposing a set of rectangular plate forced vibration problems, called building blocks, and constraining constants appearing in these solutions so that prescribed boundary conditions are satisfied by the superimposed set. The building blocks to be superimposed here are shown schematically in Fig. 2. Each building block has a rectangular area coinciding with the entire area of the bridge deck. Overall length of the deck is denoted by the symbol ‘a’. The first building block has simple support conditions imposed along the edges ξ = 0, and ξ = 1. Slip-shear conditions are imposed along the edge, η = 0. This condition is denoted by two small circles adjacent to the edge. It implies that vertical edge reaction and slope taken normal to the edge are everywhere zero. The edge, η = 1, is also free of vertical edge reaction but it is driven by a distributed harmonic edge rotation of circular frequency ω.Our first task is to determine the response of this building block to the harmonic excitation. We represent the distribution of amplitude of the imposed rotation as, ∂W (ξ, η) ∂η ∞ m=1,2 η=1 = Em sin mπξ (1) It will be obvious that the amplitude of the response of (...truncated)