An Analytical Study on Thermally Induced Vibration Analysis of FG Beams Using Different HSDTs

Abstract:

Thermo-mechanical vibrations of functionally graded (FG) beams are developed. Governing equations of functionally graded beams are obtained based on higher-order variation of transverse shear strain through the depth of the beam. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion and boundary condition are derived from Hamilton’s principle. Beam is assumed under uniform thermal loading and simply supported boundary condition. Analytical solution is presented, and the obtained results are compared with the existing solutions to verify the validity of the developed theories. Numerical computations are performed for a functionally graded simply supported beam with a gradient index obeying power law and the results are displayed graphically and tabular to show the effects of the gradient index, temperature rise, and geometrical parameters on the fundamental natural frequency of FG beams, indicating that natural frequency is sensitive to the gradient variation of material properties, geometrical parameters, shear deformations and temperature rise.

• Periodical: Applied Mechanics and Materials (Volumes 249-250)
• Pages: 784-791
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.249-250.784
• Online since: December 2012
• Authors: Soraya Mareishi, Mohsen Mohammadi, Mohammad Rafiee
• Keywords: Functionally Graded Beam, Higher Order Shear Deformation Theory, Thermal Condition, Vibration Analytical Solution

References

[1] Sankar BV, Taeng JT. Thermal stresses in functionally graded beams. AIAA J 2002; 40: 1228–32.

DOI: 10.2514/3.15185

[2] Venkataraman S, Sankar BV. Elasticity solution for stresses in a sandwich beam with functionally graded core. AIAA J 2003; 41: 2501–5.

DOI: 10.2514/2.6853

[3] Zhu H, Sankar BV. A combined Fourier series – Galerkin method for the analysis of functionally graded beams. ASME J Appl Mech 2004; 71: 421–3.

DOI: 10.1115/1.1751184

[4] Shi ZF, Chen Y. Functionally graded piezoelectric cantilever beam under load. Arch Appl Mech 2004; 74: 237–47.

[5] Nirmala K, Upadhyay PC, Prucz J, Loyns D. Thermoelastic stresses in composite beams with functionally grade layer. J Reinf Plast Compos 2005; 24: 1965–77.

DOI: 10.1177/0731684405054375

[6] Ching HK, Yen SC. Meslless local Petrov–Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads. Compos Part B – Eng 2005; 36: 223–40.

DOI: 10.1016/j.compositesb.2004.09.007

[7] Ching HK, Yen SC. Transient thermoelastic deformations of 2D functionally graded beams under nonuniformly connective heat supply. Compos Struct 2006; 73: 381–93.

DOI: 10.1016/j.compstruct.2005.02.021

[8] Lu CF, Chen WQ. Free vibration of orthotropic functionally graded beams with various end conditions. Struct Eng Mech 2005; 13: 1430–7.

[9] Wu L, Wang QS, Elishakoff I. Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode. J Sound Vib 2005; 284: 1190–202.

DOI: 10.1016/j.jsv.2004.08.038

[10] Aydogdu M, Taskin V. Free vibration analysis of functionally graded beams with simply supported edges. Mater Des 2007; 28: 1651–6.

DOI: 10.1016/j.matdes.2006.02.007

[11] Yang J, Chen Y. Free vibration and buckling analysis of functionally graded beams with edge cracks. Compos Struct 2008; 83: 48–60.

DOI: 10.1016/j.compstruct.2007.03.006

[12] Kapuria S, Bhattacharyya M, Kumar AN. Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation. Compos Struct 2008; 82: 390–402.

DOI: 10.1016/j.compstruct.2007.01.019

[13] Zhong Z, Yu T. Analytical solution of a cantilever functionally graded beam. Compos Sci Technol 2007; 67: 481–8.

[14] Ying J, Lü CF, Chen WQ. Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Compos Struct 2008; 84: 209–19.

DOI: 10.1016/j.compstruct.2007.07.004

[15] Li XF. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vib 2008; 318: 1210–29.

DOI: 10.1016/j.jsv.2008.04.056

[16] Ke LL, Yang J, Kitipornchai S., An analytical study on the nonlinear vibration of functionally graded beams, Meccanica (2010) 45: 743–752.

DOI: 10.1007/s11012-009-9276-1

[17] Rahmani O, Khalili SMR, Malekzadeh K, Hadavinia H. Free vibration analysis of sandwich structures with a flexible functionally graded syntactic core. Compos Struct 2009. doi: 10. 1016/j. compstruct. 2009. 05. 007.

DOI: 10.1016/j.compstruct.2009.05.007

[18] Li Xian-Fang, Wang Bao-Lin, Han Jie-Cai, A higher-order theory for static and dynamic analyses of functionally graded beams, Arch Appl Mech (2010) 80: 1197–1212.

DOI: 10.1007/s00419-010-0435-6

[19] Reddy JN. A simple higher-order theory for laminated composite plates. Journal of 16Applied Mechanics 1984; 51(4): 745-752.

DOI: 10.1115/1.3167719

[20] Touratier M. An efficient standard plate theory. International Journal of Engineering Science 1991; 29(8): 901-916.

DOI: 10.1016/0020-7225(91)90165-y

[21] Soldatos K. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mechanica 1992; 94(3): 195-220.

DOI: 10.1007/bf01176650

[22] Karama M, Afaq KS, Mistou S. Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. International Journal of Solids and Structures 2003; 40(6): 1525-1546.

DOI: 10.1016/s0020-7683(02)00647-9

[23] Simsek M, Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories.

[24] Mahi A, Adda Bedia E. A, Tounsi A, Mechab I, An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions, Composite Structures, 92(2010) 1877-1887.

DOI: 10.1016/j.compstruct.2010.01.010