Nonlinear free vibration analysis of non-uniform axially graded beam on variable elastic foundation

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