Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories.

*(English)*Zbl 07193059Summary: This study investigates the nonlinear free vibration of functionally graded material (FGM) beams by different shear deformation theories. The volume fractions of the material constituents and effective material properties are assumed to be changing in the thickness direction according to the power-law form. The von Kármán geometric nonlinearity has been considered in the formulation. The Ritz method and Lagrange equation are adopted to yield the discrete formulations. A direct numerical integration method for the motion equation in matrix form is developed to solve the nonlinear frequencies of FGM beams. Comparing with the global concordant deformation assumption (GCDA), a new deformation assumption named as local concordant deformation assumption (LCDA) is proposed in this study. The LCDA fits with the real deformation of the vibrating beam better, thus more accurate results of the nonlinear frequency can be expected. In numerical results, the comparison study of the GCDA and LCDA is carried out. In addition, the effects of power-law index, slenderness ratio and maximum deflection for different shear deformation theories and boundary conditions on the nonlinear frequency of the beam are discussed.

##### Keywords:

functionally graded material (FGM) beam; nonlinear vibration; different shear deformation beam theories; global concordant deformation assumption (GCDA); local concordant deformation assumption (LCDA)
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\textit{K. Xie} et al., Appl. Math. Modelling 77, Part 2, 1860--1880 (2020; Zbl 07193059)

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