Analyses of a composite functionally graded material beam with a new transverse shear deformation function
Abstract
In the present paper, we offer a higher-order shear deformation theory for bending of functionally graded beam. A new polynomial shear function is used which satisfies the stress-free boundary conditions (exact boundary conditions on the stress) at both, top and bottom surfaces of the beam. Hence, the shear correction factor is not necessary. Additionally, the present theory has strong similarities with Timoshenko beam theory in some concepts such as equations of movement, boundary conditions and stress resultant expressions. The governing equations and boundary conditions are derived from the principle of minimum potential energy. Functionally graded material FGM beams have a smooth variation of material properties due to continuous (unbroken) change in micro structural details. The variation of material properties is along the beam thickness and assumed to follow a power-law of the volume fraction of the constituents. Finite element numerical solutions obtained with the new polynomial shear function are presented and the obtained results are evaluated versus the existing solutions to verify the validity of the present theory. At last, the influences of power law indicator and the new shear deformation polynomial function on the bending of functionally graded beams are explored.References
Alshorbagy A.E., M.A. Eltaher, F.F. Mahmoud (2011) Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling 35(1): 412-425.
Ambartsumyan, S.A. (1958) On a general theory of bending theory of anisotropic plates and shallow shells. Investiia Akad. Nauk SSSR. Proceedings of the USSR Academy of Sciences, No. 5, 69-77.
Aydogdu, M. (2006) Comparison of various shear deformation theories for bending, buckling, and vibration of rectangular symmetric cross-ply plate with simply supported edges. Journal of Composite materials 40(23): 2143-2155.
Aydogdu, M. (2009) A new shear deformation theory for laminated composite plates. Composite structures, 89(1): 94-101.
Aydogdu, M., V. Taskin (2007) Free vibration analysis of functionally graded beams with simply supported edges. Materials and Design 28(5): 1651-1656.
Ben-Oumrane, S., T. Abedlouahed, M. Ismail, B.B. Mohamed, M. Mustapha, & A.B. El Abbas (2009) A theoretical analysis of flexional bending of Al/Al 2 O 3 S-FGM thick beams. Computational Materials Science, 44(4): 1344-1350.
Carrera, E. (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering 10(3):215-296.
Carrera, E., G. Giunta, M. Petrolo (2011) Beam Structures: Classical and Advanced Theories. West Sussex, UK, John Wiley & Sons, Ltd.
Chakraborty, A., S. Gopalakrishnan, J.N. Reddy (2003) A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences 45(3): 519-539.
Delale, F., F. Erdogan (1983) The crack problem for a nonhomo-geneous plane. Journal of Applied Mechanics 50(3): 609-614.
Euler, L. (1744). Methodus inveniendi lineas curvas maximi minimive propietate gaudentes, Additamentum I, De curvis elasticis. Lausanne, Genf: Bousqet & Socios.
Ferreira, A.J.M., C.M.C. Roque, R.M.N. Jorge (2005) Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Computers & structures 83(27): 2225-2237.
Giunta, G., D. Crisafulli, S. Belouettar, E. Carrera (2011) Hierarchical theories for the free vibration analysis of functionally graded beams. Composite Structures 94(1): 68-74.
Giunta, G., S. Belouettar, E. Carrera (2010a) Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures 17(8): 622-635.
Giunta, G., S. Belouettar, E. Carrera (2010b) Analysis of FGM beams by means of a unified formulation. IOP Conference Series: Materials Science and Engineering 10(1): 012073.
Grover, N., D. Maiti & B. Singh (2013) A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates. Composite Structures 95: 667-675.
Kaczkowski, Z. (1968) Plates-statistical calculations. Warsaw: Arkady.
Kadoli, R., K. Akhtar & N. Ganesan (2008) Static analysis of functionally graded beams using higher order shear deformation theory. Applied Mathematical Modelling, 32(12), 2509-2525.
Karama, M., K.S. Afaq & S. Mistou (2003) 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, 40(6): 1525-1546.
Karama, M., K.S. Afaq & S. Mistou (2009) A new theory for laminated composite plates. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials Design and Applications, 223(2): 53-62.
Levinson, M. (1980) An accurate, simple theory of the statics and dynamics of elastic plates. Mechanics Research Communications, 7(6): 343-350.
Levy, M. (1877) Mémoire sur la théorie des plaques élastiques planes. Journal de mathématiques pures et appliquées : 219-306.
Li, X.F. (2008) A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. Journal of Sound and vibration, 318(4): 1210-1229.
Li, X.F., B.L. Wang & J.C. Han (2010) A higher-order theory for static and dynamic analyses of functionally graded beams. Archive of Applied Mechanics, 80(10): 1197-1212.
Mantari, J. L., A.S. Oktem & C.G Soares (2011) Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Composite structures, 94(1): 37-49.
Mantari, J.L. & C.G. Soares (2012) Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory. Composite Structures, 94(8): 2640-2656.
Mantari, J.L. A.S. Oktem & , C.G. Soares (2012a) A new higher order shear deformation theory for sandwich and composite laminated plates. Composites Part B: Engineering, 43(3): 1489-1499.
Mantari, J.L., A.S. Oktem & C.G. Soares (2012b) A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. International Journal of Solids and Structures, 49(1): 43-53.
Murthy, M.V.V. (1981) An improved transverse shear deformation theory for laminated anisotropic plates. NASA Technical Paper 1903, 37pp.
Panc, V. (1975) Theories of elastic plates (Vol. 2). Springer Science & Business Media.
Reddy, J.N. (1984) A simple higher-order theory for laminated composite plates. Journal of applied mechanics, 51(4): 745-752.
Reissner, E. (1975) On transverse bending of plates, including the effect of transverse shear deformation. International Journal of Solids and Structures, 11(5): 569-573.
Sahoo, R., & B.N. Singh (2013) A new shear deformation theory for the static analysis of laminated composite and sandwich plates. International Journal of Mechanical Sciences, 75: 324-336.
Savoia, M., & N. Tullini (1996) Beam theory for strongly orthotropic materials. International journal of solids and structures, 33(17): 2459-2484.
Şimşek, M. (2009) Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method. Int J Eng Appl Sci, 1(3): 1-11.
Şimşek, M. (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Engineering and Design 240(4): 697-705.
Sina, S.A., Navazi, H.M., & H. Haddadpour (2009) An analytical method for free vibration analysis of functionally graded beams. Materials & Design 30(3): 741-747.
Soldatos, K.P. (1992) A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mechanica 94(3-4): 195-220.
Stein, M. (1986) Nonlinear theory for plates and shells including the effects of transverse shearing. AIAA journal 24(9): 1537-1544.
Timoshenko, S. P. (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41: 744-746.
Timoshenko, S. P. (1922). On the transverse vibrations of bars of uniform cross-section. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43: 125-131.
Touratier, M. (1991) An efficient standard plate theory. International journal of engineering science 29(8): 901-916.
Wang, C. M., , J.N. Reddy & K.H. Lee (Eds.) (2000) Shear deformable beams and plates: Relationships with classical solutions. Elsevier.
Wattanasakulpong, N., B.G. Prusty & D.W. Kelly (2011) Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. International Journal of Mechanical Sciences 53(9): 734-743.
Wei, D., Y. Liu & Z. Xiang (2012). An analytical method for free vibration analysis of functionally graded beams with edge cracks. Journal of Sound and Vibration 331(7), 1686-1700.
Yang, J. & Y. Chen (2008) Free vibration and buckling analyses of functionally graded beams with edge cracks. Composite Structures 83(1): 48-60.
Yesilce, Y. & H.H. Catal (2011) Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method. Archive of Applied Mechanics 81(2): 199-213.
Yesilce, Y. & S. Catal (2009) Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method. Struct. Eng. Mech 31(4): 453-476.
Yesilce, Y. (2010) Effect of axial force on the free vibration of Reddy-Bickford multi-span beam carrying multiple spring-mass systems. Journal of Vibration and Control 16(1): 11-32.
Ziou, H., H. Guenfoud, & M. Guenfoud (2016) Numerical modelling of a Timoshenko FGM beam using the finite element method. International Journal of Structural Engineering 7(3): 239-261.
Ambartsumyan, S.A. (1958) On a general theory of bending theory of anisotropic plates and shallow shells. Investiia Akad. Nauk SSSR. Proceedings of the USSR Academy of Sciences, No. 5, 69-77.
Aydogdu, M. (2006) Comparison of various shear deformation theories for bending, buckling, and vibration of rectangular symmetric cross-ply plate with simply supported edges. Journal of Composite materials 40(23): 2143-2155.
Aydogdu, M. (2009) A new shear deformation theory for laminated composite plates. Composite structures, 89(1): 94-101.
Aydogdu, M., V. Taskin (2007) Free vibration analysis of functionally graded beams with simply supported edges. Materials and Design 28(5): 1651-1656.
Ben-Oumrane, S., T. Abedlouahed, M. Ismail, B.B. Mohamed, M. Mustapha, & A.B. El Abbas (2009) A theoretical analysis of flexional bending of Al/Al 2 O 3 S-FGM thick beams. Computational Materials Science, 44(4): 1344-1350.
Carrera, E. (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering 10(3):215-296.
Carrera, E., G. Giunta, M. Petrolo (2011) Beam Structures: Classical and Advanced Theories. West Sussex, UK, John Wiley & Sons, Ltd.
Chakraborty, A., S. Gopalakrishnan, J.N. Reddy (2003) A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences 45(3): 519-539.
Delale, F., F. Erdogan (1983) The crack problem for a nonhomo-geneous plane. Journal of Applied Mechanics 50(3): 609-614.
Euler, L. (1744). Methodus inveniendi lineas curvas maximi minimive propietate gaudentes, Additamentum I, De curvis elasticis. Lausanne, Genf: Bousqet & Socios.
Ferreira, A.J.M., C.M.C. Roque, R.M.N. Jorge (2005) Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Computers & structures 83(27): 2225-2237.
Giunta, G., D. Crisafulli, S. Belouettar, E. Carrera (2011) Hierarchical theories for the free vibration analysis of functionally graded beams. Composite Structures 94(1): 68-74.
Giunta, G., S. Belouettar, E. Carrera (2010a) Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures 17(8): 622-635.
Giunta, G., S. Belouettar, E. Carrera (2010b) Analysis of FGM beams by means of a unified formulation. IOP Conference Series: Materials Science and Engineering 10(1): 012073.
Grover, N., D. Maiti & B. Singh (2013) A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates. Composite Structures 95: 667-675.
Kaczkowski, Z. (1968) Plates-statistical calculations. Warsaw: Arkady.
Kadoli, R., K. Akhtar & N. Ganesan (2008) Static analysis of functionally graded beams using higher order shear deformation theory. Applied Mathematical Modelling, 32(12), 2509-2525.
Karama, M., K.S. Afaq & S. Mistou (2003) 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, 40(6): 1525-1546.
Karama, M., K.S. Afaq & S. Mistou (2009) A new theory for laminated composite plates. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials Design and Applications, 223(2): 53-62.
Levinson, M. (1980) An accurate, simple theory of the statics and dynamics of elastic plates. Mechanics Research Communications, 7(6): 343-350.
Levy, M. (1877) Mémoire sur la théorie des plaques élastiques planes. Journal de mathématiques pures et appliquées : 219-306.
Li, X.F. (2008) A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. Journal of Sound and vibration, 318(4): 1210-1229.
Li, X.F., B.L. Wang & J.C. Han (2010) A higher-order theory for static and dynamic analyses of functionally graded beams. Archive of Applied Mechanics, 80(10): 1197-1212.
Mantari, J. L., A.S. Oktem & C.G Soares (2011) Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Composite structures, 94(1): 37-49.
Mantari, J.L. & C.G. Soares (2012) Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory. Composite Structures, 94(8): 2640-2656.
Mantari, J.L. A.S. Oktem & , C.G. Soares (2012a) A new higher order shear deformation theory for sandwich and composite laminated plates. Composites Part B: Engineering, 43(3): 1489-1499.
Mantari, J.L., A.S. Oktem & C.G. Soares (2012b) A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. International Journal of Solids and Structures, 49(1): 43-53.
Murthy, M.V.V. (1981) An improved transverse shear deformation theory for laminated anisotropic plates. NASA Technical Paper 1903, 37pp.
Panc, V. (1975) Theories of elastic plates (Vol. 2). Springer Science & Business Media.
Reddy, J.N. (1984) A simple higher-order theory for laminated composite plates. Journal of applied mechanics, 51(4): 745-752.
Reissner, E. (1975) On transverse bending of plates, including the effect of transverse shear deformation. International Journal of Solids and Structures, 11(5): 569-573.
Sahoo, R., & B.N. Singh (2013) A new shear deformation theory for the static analysis of laminated composite and sandwich plates. International Journal of Mechanical Sciences, 75: 324-336.
Savoia, M., & N. Tullini (1996) Beam theory for strongly orthotropic materials. International journal of solids and structures, 33(17): 2459-2484.
Şimşek, M. (2009) Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method. Int J Eng Appl Sci, 1(3): 1-11.
Şimşek, M. (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Engineering and Design 240(4): 697-705.
Sina, S.A., Navazi, H.M., & H. Haddadpour (2009) An analytical method for free vibration analysis of functionally graded beams. Materials & Design 30(3): 741-747.
Soldatos, K.P. (1992) A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mechanica 94(3-4): 195-220.
Stein, M. (1986) Nonlinear theory for plates and shells including the effects of transverse shearing. AIAA journal 24(9): 1537-1544.
Timoshenko, S. P. (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41: 744-746.
Timoshenko, S. P. (1922). On the transverse vibrations of bars of uniform cross-section. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43: 125-131.
Touratier, M. (1991) An efficient standard plate theory. International journal of engineering science 29(8): 901-916.
Wang, C. M., , J.N. Reddy & K.H. Lee (Eds.) (2000) Shear deformable beams and plates: Relationships with classical solutions. Elsevier.
Wattanasakulpong, N., B.G. Prusty & D.W. Kelly (2011) Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. International Journal of Mechanical Sciences 53(9): 734-743.
Wei, D., Y. Liu & Z. Xiang (2012). An analytical method for free vibration analysis of functionally graded beams with edge cracks. Journal of Sound and Vibration 331(7), 1686-1700.
Yang, J. & Y. Chen (2008) Free vibration and buckling analyses of functionally graded beams with edge cracks. Composite Structures 83(1): 48-60.
Yesilce, Y. & H.H. Catal (2011) Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method. Archive of Applied Mechanics 81(2): 199-213.
Yesilce, Y. & S. Catal (2009) Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method. Struct. Eng. Mech 31(4): 453-476.
Yesilce, Y. (2010) Effect of axial force on the free vibration of Reddy-Bickford multi-span beam carrying multiple spring-mass systems. Journal of Vibration and Control 16(1): 11-32.
Ziou, H., H. Guenfoud, & M. Guenfoud (2016) Numerical modelling of a Timoshenko FGM beam using the finite element method. International Journal of Structural Engineering 7(3): 239-261.
Published
2016-12-22
How to Cite
GUENFOUD, Hamza et al.
Analyses of a composite functionally graded material beam with a new transverse shear deformation function.
Journal of Applied Engineering Science & Technology, [S.l.], v. 2, n. 2, p. 105-113, dec. 2016.
ISSN 2571-9815.
Available at: <https://revues.univ-biskra.dz/index.php/jaest/article/view/1898>. Date accessed: 19 nov. 2024.
Issue
Section
Section C: Geotechnical and Civil Engineering
Keywords
Functionally Graded Material; Power-law; Finite Element Method; Timoshenko’s beam; Shear function
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