Critical examination of midplane and neutral plane formulations for vibration analysis of FGM beams

- Vibration of FGM beam is investigated using Euler Bernoulli beam theory.
- The Mid-plane (MP) and neutral plane (NP) formulations are used.
- The MP formulation is perfectly fine for the analysis of FGM beams.
- The NP formulation is useful when the supporting lines are at the NP.
- For immovable simple support (SS) the MP and NP formulations give different results.
- The positioning of the immovable SS leads to the difference in results.
- For the clamped boundary conditions, MP and NP formulations give identical results.

There has been a controversial claim that the beam model formulation for functionally graded materials (FGM) beams must be based on the neutral plane for correct solutions. This claim cuts across well accepted mid-plane formulation for FGM beams. Presented herein is a critical examination of the mid-plane and neutral plane formulations for the vibration analysis of FGM beams. It will be shown herein that the problem arises from a misconception that the immovable supporting points are located at the mid-plane when the supporting points are actually at the neutral plane when basing the formulation on the neutral plane. The positioning of the immovable simple supports at two different planes leads to the difference in results. However, in the case of movable simple supports, the mid-plane formulation furnishes the same vibration solutions as the neutral plane formulation, even though the supports are located on different planes. Both formulations furnish the same frequency results for clamped ends. The conclusion is that there is nothing wrong with using the mid-plane formulation for FGM beams. In fact, by using the neutral plane formulation, it would be difficult to solve FGM beams with a constraint on the longitudinal displacement at the midplane.