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Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory

Year 2018, Volume: 19 Issue: 2, 375 - 406, 30.06.2018

Armağan Karamanlı

https://doi.org/10.18038/aubtda.361095
Cited By: 5

Abstract

This study presents the free
vibration and buckling behavior of two directional (2D) functionally graded
beams (FGBs) under arbitrary boundary conditions (BCs) for the first time. A
four-known shear and normal deformation (Quasi-3D) theory where the axial and
transverse displacements are assumed to be cubic and parabolic variation
through the beam depth is employed based on the framework of the Ritz
formulation. The equations of motion are derived from Lagrange’s equations. The
developed formulation is validated by solving a homogeneous beam problem and
considering different aspect ratios and boundary conditions. The obtained
numerical results in terms of dimensionless fundamental frequencies and
dimensionless first critical buckling loads are compared with the results from
previous studies for convergence studies. The material properties of the
studied problems are assumed to vary along both longitudinal and thickness
directions according to the power-law distribution. The axial, bending, shear
and normal displacements are expressed in polynomial forms with the auxiliary
functions which are necessary to satisfy the boundary conditions. The effects
of shear deformation, thickness stretching, material distribution, aspect
ratios and boundary conditions on the free vibration frequencies and critical
buckling loads of the 2D-FGBs are investigated.

Keywords

2D Functionally Graded Beam Ritz Method Quasi-3D Theory Vibration

References

  • [1] Chakraborty, A., Gopalakrishnan, S., Reddy, J.N. A new beam finite element for the analysis of functionally graded materials. Int J Mech Sci. 2003; 45(3) 519–539.
  • [2] Li, X.F. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vib. 2008; 318 (4–5) 1210–1229.
  • [3] Li, S.R., Batra, R.C. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Compos Struct. 2013; 95 5–9.
  • [4] Nguyen, T.K., Vo, T.P., Thai, H.T. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Composites B. 2013; 55 147–57.
  • [5] Pradhan , K., Chakraverty, S. Free vibration of euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites B. 2013; 51 175–84.
  • [6] Sina, S., Navazi, H., Haddadpour, H. An analytical method for free vibration analysis of functionally graded beams, Mater Des. 2009; 30(3) 741–7.
  • [7] Aydogdu, M., Taskin, V. Free vibration analysis of functionally graded beams with simply supported edges. Mater Des. 2007; 28 1651–1656.
  • [8] Kapuria, S., Bhattacharyya, M., Kumar, A.N. Bending and free vibration response of layered functionally graded beams A theoretical model and its experimental validation. Compos Struct. 2008; 82(3) 390–402.
  • [9] Kadoli, R., Akhtar, K., Ganesan, N. Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model. 2008; 32 (12): 2509–2525.
  • [10] Benatta, M.A., Mechab, I., Tounsi, A., Bedia, E.A.A. Static analysis of functionally graded short beams including warping and shear deformation effects. Comput Mater Sci. 2008; 44(2): 765–773.
  • [11] Ben-Oumrane, S., Abedlouahed, T., Ismail, M., Mohamed, B.B., Mustapha, M., Abbas, A.B.E.. A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams. Comput Mater Sci. 2009; 44(4): 1344–1350.
  • [12] Li, X.F, Wang, B.L., Han, J.C. A higher-order theory for static and dynamic analyses of functionally graded beams. Arch Appl Mech. 2010; 80: 1197–1212.
  • [13] Zenkour, A.M., Allam, M.N.M., Sobhy, M. Bending analysis of fg viscoelastic sandwich beams with elastic cores resting on Pasternaks elastic foundations. Acta Mech. 2010; 212: 233–252.
  • [14] Simsek, M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl Eng Des. 2010; 240(4): 697–705.
  • [15] Thai, H.T., Vo, T.P. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int J Mech Sci. 2012; 62 (1): 57–66.
  • [16] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica. 2014; 49(1): 155–68.
  • [17] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A., Lee, J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct. 2014; 64: 12–22.
  • [18] Nguyen, T.K., Nguyen, T.T.P., Vo, T.P., Thai, H.T. Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Composites B. 2015; 76: 273–85.
  • [19] Nguyen, T.K., Nguyen, B.D.. A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams. J Sandwich Struct Mater. 2015; 17(6): 613–31.
  • [20] Carrera, E., Giunta, G., Petrolo, M. Beam structures classical and advanced theories. John Wiley & Sons; 2011.
  • [21] Giunta, G., Belouettar, S., Carrera, E. Analysis of FGM beams by means of classical and advanced theories. Mech Adv Mater Struct. 2010; 17(8): 622–35.
  • [22] Mashat, D.S., Carrera, E., Zenkour, A.M., Khateeb, S.A.A., Filippi, M. Free vibration of FGM layered beams by various theories and finite elements. Composites B. 2014; 59: 269–78.
  • [23] Filippi, M., Carrera, E., Zenkour, A.M. Static analysis of FGM beams by various theories and finite elements. Composites B. 2015; 72: 1–9.
  • [24] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F., Lee, J. A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos Struct. 2015; 119: 1–12.
  • [25] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F., Lee, J. Static behaviour of functionally graded sandwich beams using a quasi-3D theory, Composites B. 2015; 68: 59–74.
  • [26] Mantari, J.L., Yarasca, J. A simple and accurate generalized shear deformation theory for beams, Compos Struct. 2015; 134: 593–601.
  • [27] Mantari, J.L. A refined theory with stretching effect for the dynamics analysis of advanced composites on elastic foundation. Mech Mater. 2015; 86: 31–43.
  • [28] Mantari, J.L.. Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells, Composites B. 2015; 83: 142–52.
  • [29] Nguyen, T.K., Vo, T.P., Nguyen, B.D., Lee J. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Compos. Struct. 2016; 156: 238-252.
  • [30] A. Frikha, A. Hajlaoui, M. Wali, F. Dammak. A new higher order C0 mixed beam element for FGM beams analysis, Composites B. 2016; 106: 181-9.
  • [31] Nemat-Alla, M. Reduction of thermal stresses by developing two-dimensional functionally graded materials. Int. Journal of Solids and Structures. 2003; 40: 7339–7356.
  • [32] Goupee, A.J., Vel, S.S. Optimization of natural frequencies of bidirectional functionally graded beams. Struct. Multidisc. Optim. 2006; 32: 473–484.
  • [33] Lü, C.F., Chen, W.Q., Xu, R.Q., Lim, C.W. Semi-analytical elasticity solutions for bidirectional functionally graded beams. Int. Journal of Solids and Structures. 2008; 45: 258–275.
  • [34] Zhao, L., Chen, W.Q., Lü, C.F. Symplectic elasticity for two-directional functionally graded materials. Mech. Mater. 2012; 54: 32–42.
  • [35] Simsek, M. Bi-Directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Compos. Struct. 2015; 141: 968–978.
  • [36] Simsek, M. Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions. Compos. Struct. 2016; 149: 304–314.
  • [37] Karamanli, A. Elastostatic analysis of two-directional functionally graded beams using various beam theories and Symmetric Smoothed Particle Hydrodynamics method. Compos. Struct. 2017; 160: 653-669.
  • [38] Nazargah, M. Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach. Aerospace Science and Technology. 2015; 45: 154-164.
  • [39] Pydah, A., Batra, R.C. Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams. Compos. Struct. 2017; 172: 45-60.
  • [40] Karamanli, A. Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory. Compos. Struct. 2017; 174: 70-86.
  • [41] Zafarmand, H., Hassani, B. Analysis of two-dimensional functionally graded rotating thick disks with variable thickness. Act Mech. 2014; 225: 453-464.
  • [42] Nguyen D.K., Nguyen Q.H., Tran, T.T., Bui, V.T. Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Act Mech. 2017; 228: 141-155.

Year 2018, Volume: 19 Issue: 2, 375 - 406, 30.06.2018

Armağan Karamanlı

https://doi.org/10.18038/aubtda.361095
Cited By: 5

Abstract

References

  • [1] Chakraborty, A., Gopalakrishnan, S., Reddy, J.N. A new beam finite element for the analysis of functionally graded materials. Int J Mech Sci. 2003; 45(3) 519–539.
  • [2] Li, X.F. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vib. 2008; 318 (4–5) 1210–1229.
  • [3] Li, S.R., Batra, R.C. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Compos Struct. 2013; 95 5–9.
  • [4] Nguyen, T.K., Vo, T.P., Thai, H.T. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Composites B. 2013; 55 147–57.
  • [5] Pradhan , K., Chakraverty, S. Free vibration of euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites B. 2013; 51 175–84.
  • [6] Sina, S., Navazi, H., Haddadpour, H. An analytical method for free vibration analysis of functionally graded beams, Mater Des. 2009; 30(3) 741–7.
  • [7] Aydogdu, M., Taskin, V. Free vibration analysis of functionally graded beams with simply supported edges. Mater Des. 2007; 28 1651–1656.
  • [8] Kapuria, S., Bhattacharyya, M., Kumar, A.N. Bending and free vibration response of layered functionally graded beams A theoretical model and its experimental validation. Compos Struct. 2008; 82(3) 390–402.
  • [9] Kadoli, R., Akhtar, K., Ganesan, N. Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model. 2008; 32 (12): 2509–2525.
  • [10] Benatta, M.A., Mechab, I., Tounsi, A., Bedia, E.A.A. Static analysis of functionally graded short beams including warping and shear deformation effects. Comput Mater Sci. 2008; 44(2): 765–773.
  • [11] Ben-Oumrane, S., Abedlouahed, T., Ismail, M., Mohamed, B.B., Mustapha, M., Abbas, A.B.E.. A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams. Comput Mater Sci. 2009; 44(4): 1344–1350.
  • [12] Li, X.F, Wang, B.L., Han, J.C. A higher-order theory for static and dynamic analyses of functionally graded beams. Arch Appl Mech. 2010; 80: 1197–1212.
  • [13] Zenkour, A.M., Allam, M.N.M., Sobhy, M. Bending analysis of fg viscoelastic sandwich beams with elastic cores resting on Pasternaks elastic foundations. Acta Mech. 2010; 212: 233–252.
  • [14] Simsek, M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl Eng Des. 2010; 240(4): 697–705.
  • [15] Thai, H.T., Vo, T.P. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int J Mech Sci. 2012; 62 (1): 57–66.
  • [16] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica. 2014; 49(1): 155–68.
  • [17] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A., Lee, J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct. 2014; 64: 12–22.
  • [18] Nguyen, T.K., Nguyen, T.T.P., Vo, T.P., Thai, H.T. Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Composites B. 2015; 76: 273–85.
  • [19] Nguyen, T.K., Nguyen, B.D.. A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams. J Sandwich Struct Mater. 2015; 17(6): 613–31.
  • [20] Carrera, E., Giunta, G., Petrolo, M. Beam structures classical and advanced theories. John Wiley & Sons; 2011.
  • [21] Giunta, G., Belouettar, S., Carrera, E. Analysis of FGM beams by means of classical and advanced theories. Mech Adv Mater Struct. 2010; 17(8): 622–35.
  • [22] Mashat, D.S., Carrera, E., Zenkour, A.M., Khateeb, S.A.A., Filippi, M. Free vibration of FGM layered beams by various theories and finite elements. Composites B. 2014; 59: 269–78.
  • [23] Filippi, M., Carrera, E., Zenkour, A.M. Static analysis of FGM beams by various theories and finite elements. Composites B. 2015; 72: 1–9.
  • [24] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F., Lee, J. A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos Struct. 2015; 119: 1–12.
  • [25] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F., Lee, J. Static behaviour of functionally graded sandwich beams using a quasi-3D theory, Composites B. 2015; 68: 59–74.
  • [26] Mantari, J.L., Yarasca, J. A simple and accurate generalized shear deformation theory for beams, Compos Struct. 2015; 134: 593–601.
  • [27] Mantari, J.L. A refined theory with stretching effect for the dynamics analysis of advanced composites on elastic foundation. Mech Mater. 2015; 86: 31–43.
  • [28] Mantari, J.L.. Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells, Composites B. 2015; 83: 142–52.
  • [29] Nguyen, T.K., Vo, T.P., Nguyen, B.D., Lee J. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Compos. Struct. 2016; 156: 238-252.
  • [30] A. Frikha, A. Hajlaoui, M. Wali, F. Dammak. A new higher order C0 mixed beam element for FGM beams analysis, Composites B. 2016; 106: 181-9.
  • [31] Nemat-Alla, M. Reduction of thermal stresses by developing two-dimensional functionally graded materials. Int. Journal of Solids and Structures. 2003; 40: 7339–7356.
  • [32] Goupee, A.J., Vel, S.S. Optimization of natural frequencies of bidirectional functionally graded beams. Struct. Multidisc. Optim. 2006; 32: 473–484.
  • [33] Lü, C.F., Chen, W.Q., Xu, R.Q., Lim, C.W. Semi-analytical elasticity solutions for bidirectional functionally graded beams. Int. Journal of Solids and Structures. 2008; 45: 258–275.
  • [34] Zhao, L., Chen, W.Q., Lü, C.F. Symplectic elasticity for two-directional functionally graded materials. Mech. Mater. 2012; 54: 32–42.
  • [35] Simsek, M. Bi-Directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Compos. Struct. 2015; 141: 968–978.
  • [36] Simsek, M. Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions. Compos. Struct. 2016; 149: 304–314.
  • [37] Karamanli, A. Elastostatic analysis of two-directional functionally graded beams using various beam theories and Symmetric Smoothed Particle Hydrodynamics method. Compos. Struct. 2017; 160: 653-669.
  • [38] Nazargah, M. Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach. Aerospace Science and Technology. 2015; 45: 154-164.
  • [39] Pydah, A., Batra, R.C. Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams. Compos. Struct. 2017; 172: 45-60.
  • [40] Karamanli, A. Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory. Compos. Struct. 2017; 174: 70-86.
  • [41] Zafarmand, H., Hassani, B. Analysis of two-dimensional functionally graded rotating thick disks with variable thickness. Act Mech. 2014; 225: 453-464.
  • [42] Nguyen D.K., Nguyen Q.H., Tran, T.T., Bui, V.T. Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Act Mech. 2017; 228: 141-155.
There are 42 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Armağan Karamanlı 0000-0003-3990-6515

Publication Date June 30, 2018
Published in Issue Year 2018 Volume: 19 Issue: 2

Cite

APA Karamanlı, A. (2018). Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering, 19(2), 375-406. https://doi.org/10.18038/aubtda.361095
AMA Karamanlı A. Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory. AUJST-A. June 2018;19(2):375-406. doi:10.18038/aubtda.361095
Chicago Karamanlı, Armağan. “Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19, no. 2 (June 2018): 375-406. https://doi.org/10.18038/aubtda.361095.
EndNote Karamanlı A (June 1, 2018) Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19 2 375–406.
IEEE A. Karamanlı, “Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory”, AUJST-A, vol. 19, no. 2, pp. 375–406, 2018, doi: 10.18038/aubtda.361095.
ISNAD Karamanlı, Armağan. “Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19/2 (June 2018), 375-406. https://doi.org/10.18038/aubtda.361095.
JAMA Karamanlı A. Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory. AUJST-A. 2018;19:375–406.
MLA Karamanlı, Armağan. “Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering, vol. 19, no. 2, 2018, pp. 375-06, doi:10.18038/aubtda.361095.
Vancouver Karamanlı A. Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory. AUJST-A. 2018;19(2):375-406.

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