Numerical modeling of the dynamic response of a beam under moving load

Resumo

Bridges are fundamental elements for the infrastructure of any nation. In Brazil, the structural arrangement adopted in the design of these systems has been the use of stringers to support the deck and cross members to laterally lock the main beams. The analysis of the stringers is usually done using the Bernoulli-Euler beam theory, since they have large spans in relation to the height of the cross section. Therefore, bending predominates in the working condition. In determining the demands induced by vertical loading, the passage of a time-varying force, called a train-type, is the recommendation adopted. In addition to the static analysis, the dynamic response of the structure to this loading must also be verified, since it directly influences the comfort of the users. Due to the beam-column support conditions, provided by the use of Neoprene, the structure can be analyzed considering the longitudinal parts as isolated elements. This article analyzes the dynamic response of a typical bridge beam, considering traffic in one direction. The reference structure consists of a central slab, lateral beams and columns at the ends. The simulation follows the passage of half of the train-type along one of the beams, assuming that the vehicle travels through the center of the bridge, distributing the forces equally on the supports. To determine the dynamic response, the approximated elastic line equation is assumed as the vibration form. At the moments of interest, during the vehicle's transit, successive integrations of the bending moment and rotation angle equations are performed, leading to the beam's elastic, under a linear geometric hypothesis. With the vibration form defined, the Rayleigh method is applied to determine the beam's natural frequency. To this end, a computational routine in Python language is developed to define the beam's configuration under the effect of the load, at each increment of the vehicle's position. In this context, the bending moments and rotation angles are calculated, necessary to obtain the beam's deformation and its natural vibration frequency. Additionally, the shear force and bending moment diagrams for the vehicle's passage are presented. Finally, the temporal displacements generated by the passage of the vehicle are obtained considering a typical damping rate for these structures. The results are compared with normative recommendations and comfort standards.