Finite element formulae for internal forces of Bernoulli–Euler beams under moving vehicles

This paper presents finite element formulae for calculating accurately the internal forces, namely bending moment and shear force, of Bernoulli–Euler beams under moving vehicles. The formulae for evaluating these internal forces are derived based on the dynamic equilibrium conditions and the solution procedure is also given. The correctness of the proposed formulae is verified by comparing with available closed-form solutions. The internal forces of simply supported and continuous beams subjected to moving vehicles are obtained by several methods. The numerical results show that the proposed formulae are efficient and accurate in predicting the internal forces.