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This paper is concerned with the dynamic stability and response of an inclined Euler–Bernoulli beam under a moving mass and a moving follower force. The extended Hamilton’s principle is used to derive the governing equation of motion and the boundary conditions for this general moving load/force problem. Considering a simply supported beam, one can solve the problem analytically by approximating the spatial part of the deflection with a Fourier sine series. Based on the formulation and method of solution, sample dynamic responses are determined for a beam that is inclined at 30${}^{\circ }$ with respect to the horizontal. It is shown that the dynamic response of the beam under a moving mass is rather different from an equivalent moving follower force. Also investigated herein are the dynamic stability of inclined beams under moving load/follower force which are described by four key variables, viz. the speed of the moving mass/follower force, concentrated mass to the beam distributed mass, vibration frequency and the magnitude of the moving mass/follower force. The critical axial load and the critical follower force are different when they are located at different positions in the beam; except for the special case when they are at the end of the beam.

The spatial-varying frequency of a vehicle-bridge interaction (VBI) system subjected to a moving mass is theoretically derived and numerically investigated through a three-dimensional VBI model, in which the effects of moving mass are introduced through the inertial force and centrifugal force in the equation of motion of the bridge. For a large vehicle-to-bridge mass ratio, it has been known that the frequency of a VBI system could change with respect to the location of a moving vehicle. As such, this study derives the analytical solution based on a moving mass-beam system to account for frequency variation and further builds the numerical model with detailed implementation for practical applications. The numerical results show the following findings: (1) The frequency of a VBI system is a function of velocity and location of a moving vehicle. (2) The reduction of spatial-varying frequency ratio for a particular mode decreases with respect to the mode of higher order. (3) The maximum reduction of spatial-varying frequency ratio of the first mode in a moving mass-beam system occurs in the location where the bridge has the maximum deflection as a result of local mode excitation. (4) For the VBI system with high suspension stiffness and large vehicle-to-bridge mass ratio, the absolute variation of spatial-varying frequency ratio of the first mode can be up to 30–40%.

As the frequency variation in a moving mass-plate system is seldom discussed in the previous work, this study first formulates the semi-analytical solutions of a system consisting of a moving mass on a square plate. Based on the moving mass-plate system, several finite element models with designed plate-type bridges are established to further investigate the effect of moving mass on a plate, the influence of bridge local modes, and the influence of moving loads on a bridge. The numerical results have shown that the frequency variation of a plate-type bridge can reach 30% for the first three modes. The unique phenomenon of wave propagation is exhibited in the acceleration responses of the plate-type bridge, while it is uncommon for a beam-type bridge. For design purpose, the following are remarked: (1) The use of Rayleigh damping on the stiffened bridge can effectively suppress the influence of local modes on the bridge deck. (2) The high stiffness in the moving load can be served as the damper to reduce the dynamic responses of a vehicle. (3) In the design stage of a plate-type bridge subjected to moving loads, the moving force is the most conservative one among the three moving loads.

This article extends a procedure that has been used to discretize the static physical system, following the assumption that a continuous flexible beam with torsional moments can be replaced by a system of rigid bars and joints, which resist relative rotation and twist of the attached bars. This model can fairly demonstrate the effect of bending, as well as warping action, caused by external transverse loading, torsional moments of the beam and moving mass. The object of this article is to present and formulate a new simple, practical and inexpensive approximate technique for determining the response of beams with different boundary conditions, carrying general transverse loading, moving mass and twisting moments. To verify the results, other solutions are obtained through the exact structural analysis and comparing of the results reveals very good agreement between both methods. However, this algorithm is shown to be much more efficient, computationally and the formulation can easily be adopted into two-dimensional structural networks and three-dimensional bodies.