Abstract:
Conventional beam elements based on the mid-plane formulation are often used to predict vibration frequencies
of beams. Recently published work on the vibration of functionally graded materials (FGM) beams shows that if
the positions of immovable end supports are at a different plane to the middle-plane, beam elements based on
mid-plane formulation will yield inaccurate results. This paper extends this work further by critically examining
the effect of end support positions along the height of the beam in the context of vibration analysis. A formulation
for vibration analysis of beams based on reference-plane formulation is proposed. In order to verify the
proposed formulation, we considered FE modelling of beams based on generalized beam elements, composite
shell elements and 3D solid elements that are able to accurately capture the effect of end support height position
on vibration frequencies. Two vibrating beam examples are considered. The first problem deals with a single
layer beams and the second problem deals with a double layer beam. Both problems are analyzed using the
proposed method and FE models based on generalized beam elements, composite shell elements and 3D solid
elements for different end support conditions and heights. Single layer beam was also modelled in FE using
conventional beam elements. The results clearly showed that if one uses conventional beam elements for vibration
analysis of beams, the results will not be accurate if immovable end supports are applied at an eccentricity
to the mid-plane. Vibration frequencies predicted from the proposed reference-plane formulation were
shown to agree well with the results from FE models based on generalized beam elements, composite shell
elements, and 3D solid elements.
