Free vibration of axially functionally graded tapered micro-beams considering uncertain properties

Document Type : Research Article

Authors

Civil Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

Experimental observations reveal that the classical continuum theory cannot accurately describe the mechanical behavior of micro/nano scale structures. In fact, the size-effect will arise when the order of structure dimensions is the same as the material characteristic length. The current work presents free vibration and stability of axially functionally graded (AFG) tapered micro-beams with random properties. The size-dependent behavior of the micro-structure is modeled by the modified couple stress theory. The mathematical formulations are developed based on Euler-Bernoulli beam model and von Kármán geometric nonlinearity. The minimum total potential energy principle is employed to obtain governing differential equations and the corresponding boundary conditions. The governing equations are solved by Galerkin method. Due to the complexity in the fabrication process of FGMs, their mechanical and structural properties may vary significantly from sample to sample. Hence, achieving the desired FGMs specification is almost impossible and they are not deterministic, inherently. To incorporate uncertainties in the mathematical model of this study, a First-Order Second-Moment (FOSM) technique is applied to estimate the reliability index of the micro-structure, stochastically. Finally, numerical examples are presented for both deterministic and reliability analysis to show the effects of geometry, length scale parameter, material distribution, and axial load on the natural frequency of vibration and the reliability index of the AFG tapered micro-beam. It can be concluded that by increasing the coefficient of variation (COV) of random variables, the reliability index will decrease. Indeed, by enhancing the length scale parameter, a higher natural frequency of vibration is expected.

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