Forced vibration of non-uniform axially functionally graded (AFG) Timoshenko beam on elastic foundation is performed under harmonic excitation. A linear elastic foundation is considered with three different classical boundary conditions. AFG materials are an advanced class of materials that have potential for application in various engineering fields. In the present work, variation of material properties along the longitudinal axis of the beam are considered according to power-law forms. Five values of material gradation parameter provides different functional variation and their effect on the frequency response of the system is studied. The present approximate method is displacement based and Von-Karman type of geometric nonlinearity is considered with rotational component to incorporate transverse shear. Hamilton’s principle is used to derive nonlinear set of governing equation and Broyden method is implemented to solve the nonlinear equations numerically. The results are successfully validated with previously published article. Frequency vs. amplitude curve corresponding to different combinations of system parameters are presented and are capable of serving as benchmark results. A separate free vibration analysis is undertaken to include backbone curves with the frequency response curves in the non-dimensional plane.

The present work investigates the nonlinear free vibration of an axially functionally graded (AFG) beam supported on the variable foundation. The beam geometry is non-uniform, with a linear cross-section variation along the length. The beam material is graded along the axial direction following the power-law relation. A Winkler type of variable elastic foundation is taken, in which variation of stiffness is considered along the length of the foundation. Geometrical nonlinearity produced by the beam's large-amplitude deflection is also considered. To attain the desired objectives, the problem is divided into two parts. The static problem is solved first, then a subsequent free vibration analysis is executed on the statically deformed beam configuration. The governing differential equations of the system are derived using suitable energy methods. A numerical technique of direct substitution with relaxation is utilized to obtain the solution of the derived nonlinear differential equations. A suitable validation study is presented to ensure the appropriateness of the present methodology. Benchmark results are also presented by means of natural frequency, backbone curve, and mode shape plot to investigate the influences of elastic foundation, material gradation, and non-uniform geometry on nonlinear vibration.

This article presents geometrically nonlinear forced vibration analysis of an axially functionally graded (AFG) non-uniform beam resting on an elastic foundation. The mathematical formulation is displacement based and derivation of governing equations is accomplished following Hamilton's principle. The foundation has been mathematically incorporated into the analysis as a set of linear springs. According to the basic assumption of the present method force equilibrium condition is satisfied at a maximum excitation amplitude value. Thus, the dynamic problem is equivalently represented as a static one, which is solved by following a numerical implementation of the Broyden method. It is a method that utilizes the Jacobian matrix and subsequent correction of the initial Jacobian to solve a system of nonlinear equations. The large amplitude dynamic behaviour of the system in terms of non-dimensional frequency response curves is validated against established results and new results are furnished for a parabolic tapered AFG beam on a linear elastic foundation.

The analysis of beams resting on elastic foundations has attracted significant attention in structural and mechanical engineering due to its extensive applications in different domains. Beams on elastic foundations interact continuously with deformable media. To represent this interaction mathematically, researchers have developed a variety of single-parameter, two-parameter and three-parameter models as well as continuous models. Among the structural elements employed to investigate these models, functionally graded material (FGM) beams have emerged as an attractive platform due to their continuously varying material properties and extensive use in modern engineering applications. Over the past two decades, extensive research has been conducted on the behavior of graded beams resting on elastic foundations using various beam theories, foundation models and analytical as well as numerical solution techniques. The present paper puts forward a detailed review of the existing literature concerning static bending, buckling, free and forced vibration analyses of FG beams supported by elastic foundations. Various material gradation schemes, including power law, exponential, sigmoid, porous and carbon nanotube-reinforced FGMs, are discussed. The review further examines the application of different classical foundation models, such as the Winkler, Pasternak, Kerr and Vlasov models. These classical models may be linear or nonlinear depending on the nature of the foundation medium. A comparative evaluation of the available studies is provided to identify current trends and limitations. Finally, potential future research directions involving multi-physics coupling, advanced foundation modelling, machine-learning-assisted prediction and micro/nanoscale applications are highlighted. The review aims to serve as a valuable reference for researchers and engineers working in the field of FG beams interacting with elastic foundations.