Equivalent representations of beams with periodically variable cross-sections

This paper examines representations of a beam with periodically variable cross-sections. First, the rotational equivalence of a beam with two periodical steps to a uniform beam is considered when a constant bending moment is applied along the lengths of the two beams. The representation is then extended to a beam with several different steps and then to a beam with a variable cross-section. Variable bending moments are also taken into account in the equivalent representation. Equivalent representations based upon the same maximum deflection and fundamental natural frequency are derived. If a beam consists of several periodically variable units and the bending moment in each unit is considered to be constant, the various equivalent criteria can be unified. Detailed finite element analysis (FEA) shows that the plane-deformation assumptions no longer hold in the transition regions between adjacent steps. The transition area is defined using a straight line with an angle determined by curve-fitting and the results are provided in a tabular form considering all related parameters. A comparison of the maximum displacement and fundamental natural frequency shows good agreement for a number of beams with periodically variable cross-sections and their equivalent uniform beams. The study demonstrates that the equivalent representation of a beam with periodical variable cross-sections is accurate and simple to apply to engineering practice.