A conforming unified finite element formulation for the vibration of thick beams and frames

Beams and frames are common features in many engineering structures and in this paper an approach is given to model their dynamic behaviour adequately. Whilst the eigen-frequencies of continuous systems comprising of slender beams can be identified, in most cases of practical interest, by means of Euler or Timoshenko beam theory, for structures comprising of thick beam models this is not necessarily true since such idealizations constrain the cross-sections to remain planar. This paper suggests an alternative approach by means of a unified fully conforming plane stress rectangular finite element which is believed to allow for more realistic representation of the shear effects and hence the strain field around the joints of such structures. The usefulness and functionality of this improved numerical approach is explored via comparison against a non-conforming two-dimensional plate as well as one-dimensional Euler–Bernoulli and Timoshenko finite element formulations corresponding to a variety of beam aspect ratios representing the structures of a rotor and a portal frame. The idealization is shown to be particularly advantageous for simulating the effects of shear distortion where beams join at right angles and the transverse forces in one member interact with the extensional forces of the adjoining structure