Research on stiffness and consistent mass matrices for the tapered three-dimensional beam using a force-based method

For variable-section high-speed railway bridges, such as continuous girders and rigid-frame bridges, tapered 3D beam elements are frequently utilized to model bridge components. However, current high-speed railway design software and vehicle-bridge coupling analysis software tend to use prismatic beams instead, which leads to insufficient accuracy. In this paper, the cross-sectional stiffness matrix and flexibility matrix without rigid body displacement are established utilizing a generalized coordinate system with cantilever beam constraint and force interpolation function based on the equilibrium relation. Based on the force-based finite element method in combination with the virtual work principle, the shape function matrices for the cases with and without considering the shear effect are deduced, respectively. Then, the tapered 3D beam element consistent mass matrix is derived. To verify the accuracy of the proposed force-based finite element method, the derived matrix is degenerated into a prismatic beam to obtain the prismatic 3D beam element consistent mass matrix. Furthermore, compared with the commercial software Midas, the maximum natural frequency error for a linearly varying simply supported beam is less than 0.16 %. Both the theoretical degenerate solution and the numerical verification case prove that the stiffness matrix and consistent mass matrix of the tapered 3D beam element derived in this paper are highly accurate. Moreover, the force-based derivation method is proven to be reliable for deriving the tapered 3D beam element dynamic property matrix.