Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.