This work describes the application of classical Rayleigh method (CRM), modified Rayleigh method (MRM), and ANSYS finite element method (FEM) to calculate the natural frequency of non-homogenous cantilever beam. Two-step cantilever stepped beam was investigated through six studied cases. Each step has different material properties and the same cross section area. Results showed that the combination of materials is useful in order to increase the natural frequencies and reduce the weight of the beam at the same time when the cantilever beam is fixed by the side of the stronger material. There is a good agreement between the CRM and FEM for the region with length larger than half length of beam, on the other hand, there is an excellent agreement between the MRM and FEM for the region with length smaller than half length of beam.

In this paper, a new model of beam was built to study and simulate the buckling behavior of function graded beam. All equations of motion are derived using the principal of the minimum total potential energy and based on Euler-Bernoulli, first and high order shear deformation Timoshenko beam theory. The Navier solution is used for simply supported beam, and exact formulas found for buckling load. The properties of material of FG beam are assumed to change in thickness direction by using the power law formula. The dimensionless critical buckling load is calculated analytically by the FORTRAN program and numerically by ANSYS software. In the beginning, the analytical and numerical results are validated with results available in previous works and it is also has very good agreement in comparison with and some researchers. In the present study, the lower layer of the graded beam is made up of aluminum metal. As for the properties of the rest of the layers, they are calculated based on the modulus ratios studied. The effect of length to thickness ratio, modulus ratio, and power law index on the dimensionless critical buckling load of function graded beam calculating by FORTRAN and ANSYS programs are discussed. The numerical analysis of function graded beam offers accurate results and very close to the analytical solution using Timoshenko Beam theory.

The functionally graded beam is a wide field of research, which attracts great interest today in the field of engineering, science, and medicine society. This type of beam is made from functionally graded material that is characterized by several properties one of them is the high strength to weight ratio. In the current years, this beam has witnessed great developments in the mechanism of its composition and the materials used in its manufacture. This research provides an overview of the properties, types, advantages and challenges, and applications of the functionally graded materials. In addition, this paper review provides a summary of the analysis of bending and buckling that occurs on the functionally graded beam with and without crack effect from (2008-2021) year. Through this review, the following was noted: Firstly, a small number of researchers have worked experimentally, and the properties of a beam in most of the research are gradual towards thickness using the mixing rule. Secondly, the crack has a very severe effect on the behavior of both bending and buckling for the graded beam. This critical review can be considered a milestone in future analyzes of the graded beam and is also beneficial to designers and researchers working in this field.

Euler, Timoshenko and high shear deformation theories to analyze the free vibration of the functionally graded (FG) beam were developed. The mechanical properties of this beam were assumed to differ in thickness direction according to the model of a power-law distribution. The principle of Hamilton was used to find equations of motion. For free vibration, the analytical solution of these equations was presented using the Navier method. The effect of power index, aspect ratio, modulus ratio, and deformation theories on dimensionless frequency were studied numerically by Ansys software and analytically according to different beam theories using the Fortran program. The obtained results from these programs were compared with each other and with some previous research. Results showed an excellent agreement with the previous research. The numerical and analytical results showed that the use of this new FG beam model especially based on first and high shear deformation theories leads to the reduction of dimensionless frequency. It may be concluded that, the including of shear's effect leads to a decrease in the dimensionless frequency. From the modeling and analysis of this model, it is possible to know what is the appropriate design for this FG beam model to reduce the vibration.