A Dynamic Stiffness Method for Determining Natural Frequencies of Beams With Allowance for Dead Load

The initial stresses due to dead loads have an influence on the natural frequencies of bridges. In this paper, a dynamic stiffness-based method is proposed for determining the natural frequencies of uniform elastic beams with allowance for the dead load effect. Firstly, the governing differential equation including the effect of dead loads is derived. Next, the analytical dynamic stiffness matrix is obtained by applying the displacements and forces boundary conditions at the ends of the beam. In order to solve analytically the governing differential equation, the modified dynamic stiffness matrix is defined by converting the governing quasi-static boundary value problem into an equivalent set of initial value problems. Finally, the Wittrick–Williams algorithm is implemented to extract the natural frequencies from the modified dynamic stiffness matrix. Numerical examples are presented and corresponding parameter studies have been performed to illustrate the applicability and reliability of the proposed method. It is demonstrated that the proposed dynamic stiffness matrix-based method is effective even though the beam is considered as a single element without adding additional nodes.