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.

In this study, the free vibration characteristics of a functionally graded (FG) shear-deformable Timoshenko beam were investigated both analytically and numerically. The work is notable for its significant contribution to the literature, particularly in addressing analytically challenging problems related to complex FGM structures using advanced computer-aided finite element methods. For the analytical approach, the governing equations and associated boundary conditions were derived using Hamilton’s principle of minimum potential energy. These equations were then solved using the Navier solution method to determine the natural frequencies of the beam. In the numerical analysis, a 3D FG beam model was developed in the ABAQUS finite element software (2023, Dassault Systèmes, Providence, RI, USA)using the second-order hexahedral (HEX20/C3D20) and 1D three-node quadratic beam (B32) elements. The material gradation was defined layer-by-layer along the thickness direction in accordance with the rule of mixtures. Modal analysis was subsequently performed to extract the natural frequency values. The results show a high level of agreement between the analytical and numerical solutions. and were consistent with previously published studies in the literature.

Based on the unified shear deformation theory, this paper presents the free vibration analysis of linear, isotropic, homogeneous beams resting on a Winkler elastic foundation. Unlike existing transfer matrix formulations that operate on four-variable state-space systems, the present work establishes a six-variable state-space framework in which an arbitrary shear strain function is incorporated into the formulation solely through integral coefficients, enabling the unified treatment of seven distinct shear deformation models without theory-specific rederivation. The governing equations and boundary conditions are derived using energy principles. Natural frequencies are determined using the initial value method via both exact transfer matrix (ETM) and approximate transfer matrix (ATM). Since the analytical derivation of the ETM involves a sixth-order characteristic polynomial with prohibitively complex expressions, an ATM is constructed using a three-term truncated matricant series expansion, thereby eliminating the need for eigenvalue extraction. A correction methodology based on the chain property is proposed to ensure convergence to the exact solution. The formulation reduces the frequency determinant from a $$6 \times 6$$ 6 × 6 system to a $$3 \times 3$$ 3 × 3 system for each boundary condition. Systematic comparisons across different shear deformation theories, boundary conditions, slenderness ratios, and foundation stiffness values confirm the accuracy and reliability of the proposed approach, with ATM results converging to the ETM reference within less than 0.25% relative error for all examined configurations upon moderate interval refinement. These findings establish the proposed formulation as a unified, highly accurate, and versatile framework that enables consistent vibration analysis of shear-deformable beams without requiring theory-specific rederivation.