On Multiple Support Excitation Analysis of Bridges
Multiple support excitations of elastic multi-span beams are studied. Based on the common set of equations of motion an efficient formulation is developed in order to reduce the degrees of freedom. The resulting equations are formally identical to those t
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Abstract Multiple support excitations of elastic multi-span beams are studied. Based on the common set of equations of motion an efficient formulation is developed in order to reduce the degrees of freedom. The resulting equations are formally identical to those that are valid for structures under uniform support excitations. Applying classical modal analysis results in a set of uncoupled differential equations with time-dependent participation factor. A numerical example is given for a twospan railway bridge.
1 Introduction Long extended structures such as bridges or structures supported on several foundations behave very complex when subjected to ground motions, e.g., earthquakes. Analysis of seismic response cannot be based on the single assumption that free-field ground motions are spatially uniform. Therefore common discretization procedures, originally derived for structures under uniform support excitations, must be extended accordingly resulting in a larger system of equations of motion, see, e.g., [1] and [2]. The structural response of bridges subjected to deterministic multiple support excitation has been investigated by various researchers [3–5]. The dynamic behavior of railway bridges under seismic excitation is studied in [6]. Random vibrations of bridges have been analyzed generally by spectral analysis approach in the last two decades. In [7] the response of continuous two- and three-span beams to varying ground motions is evaluated and the validity of the commonly used assumption of equal support motion is examined. An extensive comparison of random vibration methods for multiple support seismic excitation analysis of long-span bridges can be found in [8].
R. Heuer () TU Wien, Karlsplatz 13/E2063, A-1040 Vienna, Austria e-mail: [email protected] D. Watzl Zwinzstraße 7/379, A-1160 Vienna, Austria e-mail: [email protected] © Springer International Publishing Switzerland 2017 H. Irschik et al. (eds.), Dynamics and Control of Advanced Structures and Machines, DOI 10.1007/978-3-319-43080-5_15
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R. Heuer and D. Watzl
In this contribution a new formulation for linear elastic multi-span beams under multiple support excitation is proposed in order to reduce the degrees of freedom in a mechanically consistent manner. The resulting differential equations are formally identical to those of structures under uniform support excitations. To demonstrate the application of the introduced method, an example is given where the discretized system of a continuous two-span railway bridge is evaluated by means of modal analysis where it becomes necessary to introduce time-dependent participation factors.
2 Basic Equations of Motion The equation of motion of a discretized linear elastic beam subjected to uniform support excitation, wg1 .t/ D wg2 .t/ D : : : D wgM .t/ D wg .t/;
(1)
R g; muR C cuP C ku D mes w
(2)
reads, compare [1],
where m, c, and k stand for the mass, damping, and stiffness matrix, respectively (Fig. 1). u.t/ denotes the vector of the nodal transverse deflections wi .t/ ;
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