Vibration and stability of an Euler–Bernoulli beam with up to three-step changes in cross-section and in axial force

The paper treats the vibration of beams with up to three-step changes in cross-section and in which the axial force in each portion is constant but different. The system parameters are the step positions, the flexural rigidity, the mass per unit length and the axial force in the beam portions—all of which were normalized. The frequency equation for 16 combinations of classical boundary conditions are expressed as fourth-order determinants equated to zero. The first three frequency parameters are tabulated for sets of system parameters (arbitrarily chosen and which includes a stepped beams under tensile or compressive axial end force). Critical compressive end force which causes a stepped beam to buckle are tabulated. Buckling under a system of axial forces, one of which is critical is discussed and several critical combinations of the system parameters are tabulated. Beams of constant depth and step change in breadth, of constant breadth and step change in depth and shafts with step change in diameter are considered. It is shown that stepped shafts are inferior machine elements if dynamic properties are the prime consideration.