A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams

• Published in: Journal of sound and vibration Vol. 318; no. 4; pp. 1210 - 1229
• Main Author: Li, X.-F.
• Format: Journal Article
• Language: English
• Published: London Elsevier Ltd 23-12-2008; Elsevier
• Subjects: Beams (radiation); Dynamic tests; Dynamics; Exact sciences and technology; Functionally gradient materials; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematical models; Physics; Shear deformation; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Timoshenko beams; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Stress analysis; Stratified material; Deformation modulus; Free vibration; Timoshenko beam; Bending wave; Simple support; Bernoulli Euler model; Elastic wave; Modeling; Cut off frequency; Natural frequency; Functionnally graded material; Frequency spectrum; Inertia; Vibrational mode; Power law
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2008.04.056

This paper presents a new unified approach for analyzing the static and dynamic behaviors of functionally graded beams (FGB) with the rotary inertia and shear deformation included. As two special cases, the Euler–Bernoulli and Rayleigh beam theories can be analytically reduced from the Timoshenko beam theory. All material properties are arbitrary functions along the beam thickness. A single fourth-order governing partial differential equation is derived and all physical quantities can be expressed in terms of the solution of the resulting equation. The static result of deflection and stress distribution is presented for a cantilever FGB. Furthermore, two branches of flexural waves propagating in FGB are obtained with different wave speeds. The higher wave speed disappears when the effects of neither the rotary inertia nor shear deformation are considered. Free vibration of an FGB is analyzed and the frequency equation is given. The natural frequencies and mode shapes of a simply supported beam are obtained for frequencies lower than, equal to and higher than the cut-off frequency. Numerical results are presented for an FGB with the power-law gradient and a laminated beam. The second frequency spectrum is found to exist when frequencies exceed the cut-off frequency. In addition, double frequencies may occur for certain specified geometry of the beam. Previous results for a homogeneous Timoshenko beam can be recovered from the present only letting the material properties be constant. The suggested method is also applicable to layered Timoshenko beams.