An Algebraic Elimination by Substitution Algorithm for Vehicle–Bridge Interaction Problems

Abstract

This paper proposes an algorithm to solve vehicle–bridge interaction (VBI) problems. It describes the VBI system in a linear second-order differential-algebraic equation (DAE) manner, incorporating the motions of both the vehicle and the bridge subsystems, together with the contact forces. On account of the conjugation between the VBI system’s motions and contact forces, the key feature of the proposed algorithm is to solve the DAE with the the algebraic method of elimination by substitution, avoiding the singularity of the differential operator. It enables a versatile and consensus solution scheme for different types of contact mechanisms, including the rigid contact model with constraints, the elastic contact model with the compliance method, and the creep tangential contact model, as well as different wheel–rail profiles. Combined with an appropriate numerical integration scheme, it allows an explicit expression of the contact forces, subsequently leading to the partitioned non-iterative analysis of the motions of both the vehicle and the bridge subsystems. In addition, by incorporating the dynamic condensation of wheels’ degrees of freedom into those of upper components in a matrix form, i.e. a reduced system approach, the proposed algorithm can avoid the numerical drift associated with the kinematic constraints on the acceleration level. To demonstrate the accuracy of the proposed approach, this study first examines a benchmark example of a single degree of freedom sprung mass model traversing a simply supported bridge. Further, it considers a more realistic problem of one and eight train vehicles crossing three three-span continuous bridges. The comparisons and the agreements with existing algorithms prove the accuracy and computational efficiency of the proposed analysis scheme.