Vibration and Stability of Uniform Euler–Bernoulli Beams with Step Change in Axial Force

Abstract

Beams which carry a constant axial force in one portion of the beam and a different axial force in the rest of the beam occur in engineering applications. The system parameters are the position where the axial force change occurs and the two axial force parameters. Transverse vibration of such systems (based on the Euler—Bernoulli theory of bending) is considered in this paper. Tables of the first three frequency parameters are given for various combinations of classical boundary conditions and system parameters. High-precision computing is necessary to handle large axial forces. It is possible for a natural frequency to be zero for certain combinations of the system parameters. This is Euler buckling and a table of combinations of critical system parameters is given. Transverse vibration of tie-bars (a special case of the title problem) is covered in textbooks but benchmark frequencies are not documented. In the paper, a table of frequency parameters of beams carrying tensile or compressive axial forces is given for all the combinations of classical boundary conditions.