Geometrically nonlinear analysis of axially functionally graded beams by using finite element method
Document Type : Research Paper
Author
Department of Civil Engineering, Bursa Technical University, Bursa, Turkey
Abstract
The aim of this paper is to investigate geometrically nonlinear static analysis of axially functionally graded cantilever beam subjected to transversal non follower load. The considered problem is solved by finite element method with total Lagrangian kinematic approach. The material properties of the beam vary along the longitudinal direction according to the power law function. The finite element model of the beam is considered in the three dimensional continuum approximation for an eight-node quadratic element. The geometrically nonlinear problem is solved by Newton-Raphson iteration method. In the numerical results, the effects of the material distribution on the geometrically nonlinear static displacements of the axially functionally graded beam are investigated. Also, the differences between of material distributions are investigated in geometrically analysis.
Keywords
[1] Agarwal S, Chakraborty A, Gopalakrishnan S, 2006, Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions, Composite Structures 72:91-104.
[2] Ke LL, Yang J, Kitipornchai S, 2009, Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening, Composite Structures 90:152-160.
[3] Kang YA, Li XF, 2010, Large deflections of a non-linear cantilever functionally graded beam, Journal of Reinforced Plastics and Composites 29:1761-1774.
[4] Su HD, Li SR, Gao Y, 2010, Thermal post-buckling of functionally graded material Timoshenko beams with surface-bonded piezoelectric layers, Chinese Journal of Computational Mechanics 27:1067-1072.
[5] Kocatürk T, Şimşek M, Akbaş ŞD, 2011, Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material, Science and Engineering of Composite Materials 18: 21-34.
[6] Soleimani A, Saadatfar M, 2012, Numerical study of large defection of functionally graded beam with geometry nonlinearity, Advanced Materials Research 403-408: 4226-4230.
[7] Anandrao KS, Gupta RK, Ramchandran P, Rao GV, 2012, Non-linear free vibrations and post- buckling analysis of shear flexible functionally graded beams, Structural Engineering and Mechanics 44:339-361.
[8] Askari H, Younesian D, Saadatnia Z, Esmailzadeh E, 2012, Nonlinear free vibration analysis of the functionally graded beams, Journal of Vibroengineering 14:1233-1245.
[9] Anandrao KS, Gupta RK, Ramchandran P. and Rao GV, 2012, Flexural stress analysis of uniform slender functionally graded material beams using non-linear finite element method, IES Journal Part A: Civil and Structural Engineering 5:231-239.
[10] Machado SP, Piovan MT, 2013, Nonlinear dynamics of rotating box FGM beams using nonlinear normal modes, Thin-Walled Structures 62:158-168.
[11] Akbaş ŞD, 2013, Geometrically nonlinear static analysis of edge cracked Timoshenko beams composed of functionally graded material, Mathematical Problems in Engineering 2013:1-14.
[12] Tung HV, Duc ND, 2014, Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions, Applied Mathematical Modelling 38:2848-2866.
[13] Akbaş ŞD, 2015, Post-Buckling Analysis of Axially Functionally Graded Three-Dimensional Beams, International Journal of Applied Mechanics 7, 1550047, Doi: 10.1142/S1758825115500477.
[14] Nguyen DK, Gan BS, Trinh TH, 2014, Geometrically nonlinear analysis of planar beam and frame structures made of functionally graded material, Structural Engineering and Mechanics 49:727-743.
[15] Mohammadi M., Rastgoo A., 2018, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium. Mechanics of Advanced Materials and Structures, 1-22.
[16] Mohammadi M., Rastgoo A., 2019, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core. Structural Engineering and Mechanics, 69(2), 131.
[17] Mohammadi M., Hosseini M., Shishesaz M., Hadi A., Rastgoo, A., 2019, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads. European Journal of Mechanics-A/Solids, 77, 103793.
[18] Kocatürk T., Akbaş Ş.D., 2011, Post-buckling analysis of Timoshenko beams with various boundary conditions under non-uniform thermal loading. Structural Engineering and Mechanics, 40(3): 347-371.
[19] Akbaş Ş.D., 2015, On post-buckling behavior of edge cracked functionally graded beams under axial loads. International Journal of Structural Stability and Dynamics, 15(04), 1450065.
[20] Akbaş Ş.D., 2017, Post-buckling responses of functionally graded beams with porosities. Steel and Composite Structures, 24(5), 579-589.
[21] Akbaş Ş.D., 2018, Thermal post-buckling analysis of a laminated composite beam. Structural Engineering and Mechanics, 67(4), 337-346.
[22] Akbaş Ş.D., 2018, Post-buckling responses of a laminated composite beam. Steel and Composite Structures, 26(6), 733-743.
[23] Akbaş Ş.D., 2018, Post-buckling analysis of a fiber reinforced composite beam with crack Engineering Fracture Mechanics, 212, 70-80.
[24] Akbaş Ş.D., 2019, Hygrothermal post-buckling analysis of laminated composite beams. International Journal of Applied Mechanics 11(01), 1950009.
[25] Wu L., Wangi Q., Elishakoff I., 2005, Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode, Journal of Sound and Vibration, 284, 1190–202.
[26] Huang Y., Li X.F., 2010, A new approach for free vibration of axially functionally graded with non-uniform cross-section, Journal of Sound and Vibration, 329, 2291–303.
[27] Hein H., Feklistova L., 2011, Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets, Engineering Structures, 33(12), 3696-3701.
[28] Alshorbgy A.E., Eltaher M.A. and Mahmoud, F.F., 2011, Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35, 412–25.
[29] Eltaher M.A., Emam S.A., Mahmoud F.F., 2012, Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation, 218(14), 7406-7420.
[30] Shahba A., Attarnejad R., Marvi M.T., Hajilar S., 2011, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites Part B, 42, 801–8.
[31] Shahba A.,and Rajasekaran S, 2012, Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling, 36 (7), 3094-3111.
[32] Akbaş Ş.D. 2017, Vibration and static analysis of functionally graded porous plates. Journal of Applied and Computational Mechanics, 3(3), 199-207.
[33] Akbaş,Ş.D. 2017, Thermal effects on the vibration of functionally graded deep beams with porosity. International Journal of Applied Mechanics, 9(05), 1750076.
[34] Akbaş Ş.D. 2017, Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory. International Journal of Structural Stability and Dynamics, 17(03), 1750033.
[35] Akbaş Ş.D., 2013, Free vibration characteristics of edge cracked functionally graded beams by using finite element method. International Journal of Engineering Trends and Technology, 4(10), 4590-4597.
[36] Akbaş Ş.D., 2015, Free vibration and bending of functionally graded beams resting on elastic foundation. Research on Engineering Structures and Materials, 1(1), 25-37.
[37] Farajpour A., Shahidi A.R., Mohammadi M., Mahzoon M., 2012, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Composite Structures, 94(5), 1605-1615.
[38] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics. Physica E: Low-dimensional Systems and Nanostructures, 44(3), 719-727.
[39] Şimşek M, Kocatürk T, Akbaş Ş.D., 2012, Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load, Composite Structures, 94 (8), 2358-2364.
[40] Huangi Y., Yangi, L.E., Luoi Q.Z., 2013, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites Part B:Engineering, 45 (1), 1493-1498.
[41] Rajasekaran S., 2013, Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature, Applied Mathematical Modelling, 37 (6), 4440-4463.
[42] Rajasekaran S., 2013, Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach, Meccanica, 48(5), 1053-1070.
[43] Akgöz B., Civalek Ö., 2013, Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory, Composite Structures, 98, 314-322.
[44] Nguyen D.K., 2013, Large displacement response of tapered cantilever beams made of axially functionally graded material, Composites Part B: Engineering, 55, 298-305.
[45] Babilio E., 2013, Dynamics of an axially functionally graded beam under axial load, European Physical Journal: Special Topics, 222(7), 1519-1539.
[46] Akbaş Ş.D., 2014, Free vibration of axially functionally graded beams in thermal environment, International Journal of Engineering and Applied Sciences, 6(3), 37-51.
[47] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas R., 2013, Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. Journal of Solid Mechanics, 5(2), 116-132.
[48] Mohammadi, M., Farajpour, A., Goodarzi, M., & Mohammadi, H., 2013, Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundati. Journal of Solid Mechanics, 5(3), 305-323.
[49] Akgöz B., Civalek Ö., 2015, Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134, 294-301.
[50] Akgöz B., Civalek Ö., 2017, Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Composites Part B: Engineering, 129, 77-87.
[51] Akbaş Ş.D., 2019, Forced vibration analysis of functionally graded sandwich deep beams. Coupled Syst. Mech, 8(3), 259-271.
[52] Akbaş Ş.D., 2018, Nonlinear thermal displacements of laminated composite beams. Coupled Syst. Mech, 7(6), 691-705.
[53] Akbaş Ş.D., 2017, Forced vibration analysis of functionally graded nanobeams. International Journal of Applied Mechanics, 9(07), 1750100.
[54] Akbaş Ş.D., 2015, Wave propagation of a functionally graded beam in thermal environments. Steel and Composite Structures, 19(6), 1421-1447.
[55] Akbaş, Ş.D., 2016, Wave propagation in edge cracked functionally graded beams under impact force. Journal of Vibration and Control, 22(10), 2443-2457.
[56] Akbaş Ş.D., 2018, Forced vibration analysis of cracked functionally graded microbeams. Advances in Nano Research, 6(1), 39.
[57] Akbaş Ş.D., 2018), Investigation on free and forced vibration of a bi-material composite beam. Journal of Polytechnic-Politeknik Dergisi, 21(1), 65-73.
[58] Akbaş Ş. D., 2020, Dynamic responses of laminated beams under a moving load in thermal environment. Steel and Composite Structures, 35(6), 729-737.
[59] Ghayesh M.H., 2018, Mechanics of tapered AFG shear-deformable microbeams, Microsystem Technologies, 24(4), 2018,1743-1754.
[60] Liu P., Lin K., Liu H., Qin R., 2016, Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock and Vibration, 5891030.
[61] Çalim F.F., 2016, Transient analysis of axially functionally graded Timoshenko beams with variable cross-section, Composites Part B: Engineering, 98, 472-483.
[62] Alimoradzadeh M., Salehi M., Esfarjani S.M., 2019, Nonlinear dynamic response of an axially functionally graded (AFG) beam resting on nonlinear elastic foundation subjected to moving load, Nonlinear Engineering, 8(1), 250-260.
[63] Uzun, B., Yaylı, M. Ö., Deliktaş, B., “Free vibration of FG nanobeam using a finite-element method”, Micro & Nano Letters, 15(1), 2020, 35-40.
[64] Cao, D., Gao, Y., Free vibration of non-uniform axially functionally graded beams using the asymptotic development method, Applied Mathematics and Mechanics, 40(1), 2019, 85-96.
[65] Barati, A., Adeli, MM., Hadi, A., Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field, International Journal of Applied Mechanics, 12(2), 2020, 2050021.
[66] Sharma, P., Singh, R., Hussain, M., On modal analysis of axially functionally graded material beam under hygrothermal effect, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 234(5), 2020, 1085-1101.
[2] Ke LL, Yang J, Kitipornchai S, 2009, Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening, Composite Structures 90:152-160.
[3] Kang YA, Li XF, 2010, Large deflections of a non-linear cantilever functionally graded beam, Journal of Reinforced Plastics and Composites 29:1761-1774.
[4] Su HD, Li SR, Gao Y, 2010, Thermal post-buckling of functionally graded material Timoshenko beams with surface-bonded piezoelectric layers, Chinese Journal of Computational Mechanics 27:1067-1072.
[5] Kocatürk T, Şimşek M, Akbaş ŞD, 2011, Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material, Science and Engineering of Composite Materials 18: 21-34.
[6] Soleimani A, Saadatfar M, 2012, Numerical study of large defection of functionally graded beam with geometry nonlinearity, Advanced Materials Research 403-408: 4226-4230.
[7] Anandrao KS, Gupta RK, Ramchandran P, Rao GV, 2012, Non-linear free vibrations and post- buckling analysis of shear flexible functionally graded beams, Structural Engineering and Mechanics 44:339-361.
[8] Askari H, Younesian D, Saadatnia Z, Esmailzadeh E, 2012, Nonlinear free vibration analysis of the functionally graded beams, Journal of Vibroengineering 14:1233-1245.
[9] Anandrao KS, Gupta RK, Ramchandran P. and Rao GV, 2012, Flexural stress analysis of uniform slender functionally graded material beams using non-linear finite element method, IES Journal Part A: Civil and Structural Engineering 5:231-239.
[10] Machado SP, Piovan MT, 2013, Nonlinear dynamics of rotating box FGM beams using nonlinear normal modes, Thin-Walled Structures 62:158-168.
[11] Akbaş ŞD, 2013, Geometrically nonlinear static analysis of edge cracked Timoshenko beams composed of functionally graded material, Mathematical Problems in Engineering 2013:1-14.
[12] Tung HV, Duc ND, 2014, Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions, Applied Mathematical Modelling 38:2848-2866.
[13] Akbaş ŞD, 2015, Post-Buckling Analysis of Axially Functionally Graded Three-Dimensional Beams, International Journal of Applied Mechanics 7, 1550047, Doi: 10.1142/S1758825115500477.
[14] Nguyen DK, Gan BS, Trinh TH, 2014, Geometrically nonlinear analysis of planar beam and frame structures made of functionally graded material, Structural Engineering and Mechanics 49:727-743.
[15] Mohammadi M., Rastgoo A., 2018, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium. Mechanics of Advanced Materials and Structures, 1-22.
[16] Mohammadi M., Rastgoo A., 2019, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core. Structural Engineering and Mechanics, 69(2), 131.
[17] Mohammadi M., Hosseini M., Shishesaz M., Hadi A., Rastgoo, A., 2019, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads. European Journal of Mechanics-A/Solids, 77, 103793.
[18] Kocatürk T., Akbaş Ş.D., 2011, Post-buckling analysis of Timoshenko beams with various boundary conditions under non-uniform thermal loading. Structural Engineering and Mechanics, 40(3): 347-371.
[19] Akbaş Ş.D., 2015, On post-buckling behavior of edge cracked functionally graded beams under axial loads. International Journal of Structural Stability and Dynamics, 15(04), 1450065.
[20] Akbaş Ş.D., 2017, Post-buckling responses of functionally graded beams with porosities. Steel and Composite Structures, 24(5), 579-589.
[21] Akbaş Ş.D., 2018, Thermal post-buckling analysis of a laminated composite beam. Structural Engineering and Mechanics, 67(4), 337-346.
[22] Akbaş Ş.D., 2018, Post-buckling responses of a laminated composite beam. Steel and Composite Structures, 26(6), 733-743.
[23] Akbaş Ş.D., 2018, Post-buckling analysis of a fiber reinforced composite beam with crack Engineering Fracture Mechanics, 212, 70-80.
[24] Akbaş Ş.D., 2019, Hygrothermal post-buckling analysis of laminated composite beams. International Journal of Applied Mechanics 11(01), 1950009.
[25] Wu L., Wangi Q., Elishakoff I., 2005, Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode, Journal of Sound and Vibration, 284, 1190–202.
[26] Huang Y., Li X.F., 2010, A new approach for free vibration of axially functionally graded with non-uniform cross-section, Journal of Sound and Vibration, 329, 2291–303.
[27] Hein H., Feklistova L., 2011, Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets, Engineering Structures, 33(12), 3696-3701.
[28] Alshorbgy A.E., Eltaher M.A. and Mahmoud, F.F., 2011, Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35, 412–25.
[29] Eltaher M.A., Emam S.A., Mahmoud F.F., 2012, Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation, 218(14), 7406-7420.
[30] Shahba A., Attarnejad R., Marvi M.T., Hajilar S., 2011, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites Part B, 42, 801–8.
[31] Shahba A.,and Rajasekaran S, 2012, Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling, 36 (7), 3094-3111.
[32] Akbaş Ş.D. 2017, Vibration and static analysis of functionally graded porous plates. Journal of Applied and Computational Mechanics, 3(3), 199-207.
[33] Akbaş,Ş.D. 2017, Thermal effects on the vibration of functionally graded deep beams with porosity. International Journal of Applied Mechanics, 9(05), 1750076.
[34] Akbaş Ş.D. 2017, Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory. International Journal of Structural Stability and Dynamics, 17(03), 1750033.
[35] Akbaş Ş.D., 2013, Free vibration characteristics of edge cracked functionally graded beams by using finite element method. International Journal of Engineering Trends and Technology, 4(10), 4590-4597.
[36] Akbaş Ş.D., 2015, Free vibration and bending of functionally graded beams resting on elastic foundation. Research on Engineering Structures and Materials, 1(1), 25-37.
[37] Farajpour A., Shahidi A.R., Mohammadi M., Mahzoon M., 2012, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Composite Structures, 94(5), 1605-1615.
[38] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics. Physica E: Low-dimensional Systems and Nanostructures, 44(3), 719-727.
[39] Şimşek M, Kocatürk T, Akbaş Ş.D., 2012, Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load, Composite Structures, 94 (8), 2358-2364.
[40] Huangi Y., Yangi, L.E., Luoi Q.Z., 2013, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites Part B:Engineering, 45 (1), 1493-1498.
[41] Rajasekaran S., 2013, Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature, Applied Mathematical Modelling, 37 (6), 4440-4463.
[42] Rajasekaran S., 2013, Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach, Meccanica, 48(5), 1053-1070.
[43] Akgöz B., Civalek Ö., 2013, Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory, Composite Structures, 98, 314-322.
[44] Nguyen D.K., 2013, Large displacement response of tapered cantilever beams made of axially functionally graded material, Composites Part B: Engineering, 55, 298-305.
[45] Babilio E., 2013, Dynamics of an axially functionally graded beam under axial load, European Physical Journal: Special Topics, 222(7), 1519-1539.
[46] Akbaş Ş.D., 2014, Free vibration of axially functionally graded beams in thermal environment, International Journal of Engineering and Applied Sciences, 6(3), 37-51.
[47] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas R., 2013, Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. Journal of Solid Mechanics, 5(2), 116-132.
[48] Mohammadi, M., Farajpour, A., Goodarzi, M., & Mohammadi, H., 2013, Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundati. Journal of Solid Mechanics, 5(3), 305-323.
[49] Akgöz B., Civalek Ö., 2015, Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134, 294-301.
[50] Akgöz B., Civalek Ö., 2017, Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Composites Part B: Engineering, 129, 77-87.
[51] Akbaş Ş.D., 2019, Forced vibration analysis of functionally graded sandwich deep beams. Coupled Syst. Mech, 8(3), 259-271.
[52] Akbaş Ş.D., 2018, Nonlinear thermal displacements of laminated composite beams. Coupled Syst. Mech, 7(6), 691-705.
[53] Akbaş Ş.D., 2017, Forced vibration analysis of functionally graded nanobeams. International Journal of Applied Mechanics, 9(07), 1750100.
[54] Akbaş Ş.D., 2015, Wave propagation of a functionally graded beam in thermal environments. Steel and Composite Structures, 19(6), 1421-1447.
[55] Akbaş, Ş.D., 2016, Wave propagation in edge cracked functionally graded beams under impact force. Journal of Vibration and Control, 22(10), 2443-2457.
[56] Akbaş Ş.D., 2018, Forced vibration analysis of cracked functionally graded microbeams. Advances in Nano Research, 6(1), 39.
[57] Akbaş Ş.D., 2018), Investigation on free and forced vibration of a bi-material composite beam. Journal of Polytechnic-Politeknik Dergisi, 21(1), 65-73.
[58] Akbaş Ş. D., 2020, Dynamic responses of laminated beams under a moving load in thermal environment. Steel and Composite Structures, 35(6), 729-737.
[59] Ghayesh M.H., 2018, Mechanics of tapered AFG shear-deformable microbeams, Microsystem Technologies, 24(4), 2018,1743-1754.
[60] Liu P., Lin K., Liu H., Qin R., 2016, Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock and Vibration, 5891030.
[61] Çalim F.F., 2016, Transient analysis of axially functionally graded Timoshenko beams with variable cross-section, Composites Part B: Engineering, 98, 472-483.
[62] Alimoradzadeh M., Salehi M., Esfarjani S.M., 2019, Nonlinear dynamic response of an axially functionally graded (AFG) beam resting on nonlinear elastic foundation subjected to moving load, Nonlinear Engineering, 8(1), 250-260.
[63] Uzun, B., Yaylı, M. Ö., Deliktaş, B., “Free vibration of FG nanobeam using a finite-element method”, Micro & Nano Letters, 15(1), 2020, 35-40.
[64] Cao, D., Gao, Y., Free vibration of non-uniform axially functionally graded beams using the asymptotic development method, Applied Mathematics and Mechanics, 40(1), 2019, 85-96.
[65] Barati, A., Adeli, MM., Hadi, A., Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field, International Journal of Applied Mechanics, 12(2), 2020, 2050021.
[66] Sharma, P., Singh, R., Hussain, M., On modal analysis of axially functionally graded material beam under hygrothermal effect, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 234(5), 2020, 1085-1101.
)
Volume 51, Issue 2
December 2020Pages 411-416
- Receive Date: 28 August 2020
- Revise Date: 02 September 2020
- Accept Date: 03 September 2020