Vibration of Complex Euler–Bernoulli and Timoshenko–Ehrenfest Beams Through Affine GPSFs

Abstract

Vibrations of thin and thick beams containing internal complexities are analyzed through generalized bases made of global piecewise-smooth functions (GPSFs). Such functional bases allow us to globally analyze multiple domains as if these latter were only one, such that a unified formulation can be used for different mechanical systems. Such bases were initially introduced to model a specific part of stress and displacement components through the thickness of multi-layered plates; subsequent extensions were introduced in the literature to allow the modeling of thin-walled beams and plates. However, in these latter cases, certain analytical difficulties were experienced when inner boundary conditions needed to be englobed into the GPSFs; in this work, such mentioned difficulties are successfully overcome through certain affine transformations which allow the analyses of vibrating complex beam systems through a straightforward analytical procedure. The complex mechanical components under investigation are Euler and/or Timoshenko models containing inner complexities (stepped beams, concentrated mass or stiffness, internal constraints, etc.). The ability of the models herein analyzed is shown through the comparison of the resulting solutions to exact counterparts, if existing, or to finite elements solutions.