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.
- Beam theory
- Laminated beams
- Neutral plane
- Support positions along beam height