100 M. Sheibani et al. Fig. 13.3 Surface fitted based on the andv values Table 13.1 Polynomial coefficients a b c d e f 0.02853 0.001208 0.008903 1.919e 5 1.202e 4 1.294e 3 lumped masses assigned to each node, the estimated matrix b mwill be diagonal and can be constructed using the following equation for each traffic situation. b m ;v Dh. ;v/ mL I (13.21) inwhich I is identity matrix and Lis the length of the entire beam. 13.3 Numerical Case Study A simply supported Euler-Bernoulli beam has been considered in order to simulate the bridge structure. The beam is consisted of 8 identical elements and each node has 2 degrees of freedom; transverse displacement and in-plane rotation. The mass matrix is assembled using consistent-mass matrices for each element, the stiffness matrix is assembled using cubic shape functions, and damping is neglected. The beam has one traffic lane and the vehicle velocities are the same during each simulation. Furthermore, each vehicle is assumed to impose a concentrated load on the beam and vehicle masses are the same and are assumed to be equal to 2% of the bridge mass. Natural frequencies of the FEM model are presented in Table 13.2. Two different conditions of the traffic has been considered. First, a free flow of the traffic is considered which has minor effects on the mass of the structure and can be assumed as the unmodified structure in the classical methods, and second, congested traffic situation is used which has significant effects on the mass properties and can be assumed as the modified structure [19]. Time period is 300 s for each simulation in which the traffic stream can be assumed stationary [10] and 0.001 s is taken as time step. Vehicle loads are then shared between successive nodes using Eqs. (13.11, 13.12) for each time step. A sample time history of excitation is depicted in Fig. 13.4 for one node of the structure The mass loading of the vehicles are applied to the global mass matrix of the finite element model in each time step. Since these quantities are shared between nodes and can be assumed to be lumped masses, they can be added to diagonal elements of the mass matrix that correspond with transvers degrees of freedom, i.e. the odd columns and rows of the mass matrix. The time-variant mass matrix will then be in the following form
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