98 M. Sheibani et al. Fig. 13.1 Model of moving load on the bridge [18] Fx.t/ D N.t/ XiD1 Piı Œx v.t i/ (13.9) where Fx(t) is the time history of traffic excitation at location x on the bridge, N(t) is the number of vehicles on the bridge at the time t, and 1 , 2 , . . . i , . . . N are the sequence of headway times. The shifted lognormal distribution is used to model headway times in various studies and in this study is used to generate headway times of the artificial traffic [10]. The probability distribution equation is f .tjˇ; ; / D 1 .t ˇ/ p2 exp .ln.t ˇ/ /2 2 2 !I t > (13.10) inwhicht is the time headway, ˇis the shift value in seconds, and and are parameters of lognormal distribution; location and scale parameter, respectively. Dynamic nodal loading (DNL) is used to convert time-varying moving forces into load histories at each node of the Finite Element Method (FEM) model based on the equivalent nodal forces (ENFs) method [18]. The bridge is modeled by beam elements and each node of the beam has two degrees of freedom; yi, the transvers displacement and i, the in-plane rotation. The vertical force Pis then applied to nodes of the structure as nodal shear WQi and nodal moment WMi using the following equations WQi .t/ D 8ˆˆˆ< ˆˆ : 0 .li 1 xiCvt/ 2 l2 i 1 h 1C2.xi vt/ li 1 i .liCxi vt/ 2 l2 i h 1C2.vt xi/ li i 0 t xi li 1 v xi li 1 v <t xi v xi v <t xiCli v xiCli v <t (13.11) WMi .t/ D 8ˆˆˆ< ˆˆ : 0 .li 1 xiCvt/ 2 l2 i 1 .xi vt/ .liCxi vt/ 2 l2 i .vt xi/ 0 t xi li 1 v xi li 1 v <t xi v xi v <t xiCli v xiCli v <t (13.12) inwhich v, li and li-1 are illustrated in Fig. 13.2, and xi is the coordinate of node i. Therefore, the equivalent forces at node i are presented by FQi .t/ D N.t/ XjD1 PjWQi t j (13.13) FMi .t/ D N.t/ XjD1 PjWMi t j (13.14) These forces are then used to simulate the traffic excitation for each node.
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