Vibrations of bridges without conservation of mass

In structural mechanics there are various problems where material is carried by some mechanism. Frequently, it is impossible to model the motion of the material transported far away from the mechanism. It is then necessary to enclose the interesting portions of these material bodies by means of a non-material volume. This applies also if a singular surface is present in the volume under consideration. The total volume then must be subdivided into non- material control volumes where the singular surface is a part of the boundary. A flow of mass takes place through the control surface enclosing the non-material volume. Hence, the total mass included in the control volume is generally not conserved, but is varying in time. In this project special emphasis will be laid on deformable, vibrating bridge-type carrier structures, where the variable mass is the mass of trains moving at a high speed through the spatial control volume including the bridge. Another topic of interest will be the fast dynamic deployment of bridge-type structures from an undeformed configuration into a deformed position. The fundamental laws of balance of mass, of linear and angular momentum and of kinetic energy were extended to problems without conservation of mass in the last century. Since the vibrations and deformations of solid structures usually have to be tackled by numerical procedures, however, the fundamental equations should be replaced by equivalent variational statements or formulations stemming from variational considerations. A rather small progress has been achieved in this direction for problems without conservation of mass. That is why the available powerful commercial computer codes are not able to tackle vibrations of structures without conservation of mass. As a step towards more general formulations, we intend to work out a numerical strategy for describing the vibrations of bridge-type structures without conservation of the carried masses in a spatial control volume, or without conservation of the deploying bridge mass contained in a nonmaterial volume. In particular, we intend to extend and combine a formulation of the equations of Lagrange for non-material volumes, which has been recently developed by the group of the principal investigator, with the Ritz approximation method for these special problems. Alternatively, the Ritz-Galerkin numerical procedure will be extended to the vibrations of bridge-type structures without conservation of mass. In this first part of the project, the experience of the principal investigator and his group concerning non-linear vibrations of structures will be combined with the experience of the proposed project co-worker Mrs. Dr. Cojocaru in the analysis of vibrating railway bridges. In the applied part of the project, we plan to study the implications of our theoretical developments and numerical simulations on the practical design of bridges. During the last decades, high-speed trains have been developed and brought into the traffic, while the international construction codes are often guided by the ideal of slender, light- weighted bridges. Therefore, the influence of the moving mass of the high speed trains on the vibrations and the strength of railway bridges should be modelled carefully. Such studies would be enabled by the computational tools developed in the present project. The experience of Mrs. Dr. Cojocaru in design, inspection and maintainance of railway bridges in Romania and in France should ensure a rapid practical application of the theoretical results of the proposed project.