13 Mass Scaling of Mode Shapes Based on the Effect of Traffic on Bridges: A Numerical Study 99 Fig. 13.2 FEM model of the bridge [18] 13.2.4 Estimation of the Mass loading of Traffic In this article, the mass loading of traffic on the bridge is considered as the perturbation to the mass matrix of the structure, instead of additional lumped masses. The OMA techniques assume that the dynamic response of the structure is stationary in the testing period. Similarly, the effect of mass loading of traffic stream on the bridge in case of stationary traffic and sufficiently long observation periods, can also be assumed stationary. Accordingly, one can assume the average mass loading of the traffic to affect the structure with a uniform distribution during the test period. For that reason, expected value of the induced mass loading of traffic has been calculated for each node during the period of each test, and is used as the mass change needed in the Eqs. (13.5, 13.6). The expected value of the mass loading of traffic is related to several parameters such as traffic headway distribution, vehicle masses and velocities, driving behavior, etc. Nevertheless, different headway distributions and vehicle velocities have been studied in this article and other factors are neglected for simplicity of the method. In order to apply the mass changes due to traffic stream, shear values which are calculated for vehicle loads from Eq. (13.13) are transformed to mass quantities using Eq. (13.15) and moment values are assumed to have insignificant effect on the mass of the structure. Mi.t/ DF Qi .t/=g (13.15) where Mi (t) is the nodal mass of traffic in time t and g is the acceleration of gravity. The aim of this study is to propose a method to produce the mmatrix used in scaling factor Eqs. (13.5, 13.6) based on traffic characteristics. Accordingly, a reasonable range of traffic conditions which is virtually the case of congested traffic situations reported in [10] has been considered and artificial traffic is generated using Eq. (13.10). The following conditions has been considered p D1:35C p 20 p D1;2;:::;20 (13.16) vq D12C q 10 .km=h/ q D1;2;:::;100 (13.17) Shift value ˇ is considered to be 0.12 s and is assumed to be 0.9. Expected value of the time history of nodal masses which were mentioned earlier are then calculated for each traffic condition using the equation mi; ;v DE Mi; ;v.t/ (13.18) However, since the velocity of the vehicles are constant during each simulation and the testing period has been considered long enough, we can assume that mi; ;v ' mj; ;v ' m ;v (13.19) These values are normalized with respect to the total mass of the structure and a quadratic surface is fitted through them, as can be seen in Fig. 13.3. The governing polynomial is determined by Eq. (13.20) and the corresponding coefficients are presented in Table 13.1. m ;v Dh. ;v/ DaCb v Cc Cd v 2 Ce v Cf 2 (13.20) In order to use values generated from Eq. (13.20) in the scaling factor formulas, the mmatrix has to be constructed. However, due to the assumption that the average traffic mass loading is applied uniformly to the structure and it is converted to
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