Presented herein is the free vibration analysis of functionally graded beams (FGMs) via higher-order shear deformation theory and an artificial neural network method (ANN). The transverse displacement (w) is expressed as bending (w_b) and shear (w_s) components to define the deformation of the beam. The higher-order variation of the transverse shear strains is accounted for through the thickness direction of the FGM beam, and satisfies boundary conditions. The governing equations are derived with the help of Hamilton’s principle. Non-dimensional frequencies are obtained using Navier’s solution. To validate and enrich the proposed research, an artificial neural network method (ANN) was developed in order to predict the dimensionless frequencies. Material properties and previous studies were used to generate the ANN dataset. The obtained frequency values from the analytical solution and ANN method were compared and discussed with respect to the mean error. In conclusion, the solutions were demonstrated for various deformation theories, and all of the results were thereupon tabularized and visualized using 2D and 3D plots.

In this study, the slope deflection method was presented for structures made of small-scaled axially functionally graded beams with a variable cross section within the scope of nonlocal elasticity theory. The small-scale effect between individual atoms cannot be neglected when the structures are small in size. Therefore, the theory of nonlocal elasticity is used throughout. The stiffness coefficients and fixed-end moments are calculated using the method of initial values. With this method, the solution of the differential equation system is reduced to the solution of the linear equation system. The given transfer matrix is unique and the problem can be easily solved for any end condition and loading. In this problem, double integrals occur in terms of the transfer matrix. However, this form is not suitable for numerical calculations. With the help of Cauchy’s repeated integration formula, the transfer matrix is given in terms of single integrals. The analytical or numerical calculation of single integrals is easier than the numerical or analytical calculation of double integrals. It is demonstrated that the nonlocal effect plays an important role in the fixed-end moments of small-scaled beams.