New hybrid approach for free vibration and stability analyses of axially functionally graded Euler-Bernoulli beams with variable cross-section resting on uniform Winkler-Pasternak foundation

AbstractIn this study, free vibration behavior of a multilayered symmetric sandwich beam made of functionally graded materials (FGMs) with variable cross section resting on variable Winkler elastic foundation are investigated. The elasticity and density of the functionally graded (FG) sandwich beam vary through the thickness according to the power law. This law is related to mixture rules and laminate theory. In order to provide this, a 50-layered beam is considered. Each layer is isotropic and homogeneous, although the volume fractions of the constituents of each layer are different. Furthermore, the width of the beam varies exponentially along the length of the beam, and also the beam is resting on an elastic foundation whose coefficient is variable along the length of the beam. The natural frequencies are computed for conventional boundary conditions of the FG sandwich beam using a theoretical procedure. The effects of material, geometric, elastic foundation indexes and slenderness ratio on natural frequencies and mode shapes of the beam are also computed and discussed. Finally, the results obtained are compared with a finite-element-based commercial program, ANSYS®, and found to be consistent with each other.

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Flexural Free Vibrations of Multistep Nonuniform Beams

Mathematical Problems in Engineering ◽

10.1155/2016/7314280 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12

Author(s):

Guojin Tan ◽

Wensheng Wang ◽

Yubo Jiao

Keyword(s):

Differential Equation ◽

Free Vibration ◽

Cross Section ◽

Order Differential Equation ◽

Variable Cross Section ◽

Free Vibrations ◽

Variable Cross ◽

Exact Approach ◽

Constant Coefficients ◽

One Step

This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as I(x)=α11+βxr+4 and m(x)=α21+βxr was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.

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