Nonlinear vibration of edge cracked functionally graded Timoshenko beams

Nonlinear vibration of beams made of functionally graded materials (FGMs) containing an open edge crack is studied in this paper based on Timoshenko beam theory and von Kármán geometric nonlinearity. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through beam thickness. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of cracked FGM beams with different end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, material property gradient, slenderness ratio, and end supports on the nonlinear free vibration characteristics of cracked FGM beams. It is found that unlike isotropic homogeneous beams, both intact and cracked FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of bending–extension coupling in FGM beams.