Thermal Buckling Analysis of Functionally Graded Euler-Bernoulli Beams with Temperature-dependent Properties

Document Type : Research Paper


1 Department of Mechanical Engineering, Faculty, Chinese Culture University, Taipei, Taiwan

2 Department of Mechanical Engineering, Faculty, Lunghwa University of Science and Technology, Guishan Shiang 33306, Taiwan


Thermal buckling behavior of functionally graded Euler-Bernoulli beams in thermal conditions is investigated analytically. The beam with material and thermal properties dependent on the temperature and position is considered. Based on the transformed-section method, the functionally graded beam is considered as an equivalent homogeneous Euler-Bernoulli beam with an effective bending rigidity under an eccentric thermal load. Then, the thermal elastic buckling equation associated with the bending deflection about the neutral axis is established. The easily usable closed-form solutions for the critical thermal buckling temperature of functionally graded beams under uniform and non-linear temperature rise are obtained and used to calculate the thermal buckling temperature. Some results are evaluated and compared with those by other investigators to validate the accuracy of the presented method. The effects of material compositions, temperature-dependent material properties, slenderness ratios and restraint conditions on thermal buckling behaviors are discussed. It is believed that the proposed model provides engineers and designers an easy and useful method to investigate the effects of various parameters affecting the thermal buckling characteristics of functionally graded beams.


Main Subjects

[1] Yang, J., Chen, Y, Free vibration and buckling analyses of functionally graded beams with edge cracks, Composite Structures, 83, 2008, 48-60.
[2] 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: Part B, 55, 2013, 147–157.
[3] Li, S.R., Batra, R.C., Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams, Composite Structures, 95, 2013, 5-9.
[4] Li, S.R., Wang, X., Wan, Z., Classical and homogenized expressions for buckling solutions of functionally graded material Levinson beams, Acta Mechanica Solida Sinica, 28, 2015, 592-604.
[5] Aydogdu, M., Semi-inverse method for vibration and buckling of axially functionally graded beams, Journal of Reinforced Plastics and Composites, 27, 2008, 683-691.
[6] Shahba, A., Attarnejad, R., Marvi, M.T., Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classic and non-classical boundary conditions, Composites: Part B, 42, 2011, 801-808.
[7] Shahba, A., Rajasekaran, S., Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials, Applied Mathematical Modelling, 36, 2012, 3094-3111.
[8] Rychlewska, J., Buckling analysis of axially functionally graded beams, Journal of Applied Mathematics and Computational Mechanics, 13, 2014, 103-108.
[9] Torki, M.E., Reddy, J.N., Buckling of functionally-graded beams with partially delaminated piezoelectric layers, International Journal of Structural Stability and Dynamics, 16, 2016, 1450104 (25 pages).
[10] Shvartsman, B., Majak, J., Numerical method for stability analysis of functionally graded beams on elastic foundation, Applied Mathematical Modelling, 40, 2016, 3713–3719.
[11] Huang, Y., Zhang, M., Rong, H.W., Buckling analysis of axially functionally graded and non-uniform beams based on Timoshenko theory, Acta Mechanica Solida Sinica, 29, 2016, 200-207.
[12] 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, Composite Structures, 156, 2016, 238-252.
[13] Thai, C.H., Ferreira, A.J.M., Abdel Wahab, M., Nguyen-Xuan, H., A moving Kriging meshfree method with naturally stabilized nodal integration for analysis of functionally graded material sandwich plates, Acta Mechanica, 229, 2018, 2997-3023.
[14] Kiani, Y., Eslami, M.R., Thermal buckling analysis of functionally graded materials beams, International Journal of Mechanics and Materials in Design, 6, 2010, 229-238.
[15] Wattanasakulpong, N., Gangadhara Prusty, B., Kelly, D.W., Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams, International Journal of Mechanical Sciences, 53, 2011, 734-743.
[16] Kiani, Y., Taheri, S., Eslami, M.R., Thermal buckling of piezoelectric functionally graded material beams, Journal of Thermal Stresses, 34, 2011, 835-850.
[17] Kiani, Y., Rezaei, M., Taheri, S., Eslami, M.R., Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams, International Journal of Mechanics and Materials in Design, 7, 2011, 185-197.
[18] Fallah, A., Aghdam, M.M., Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation, Composites: Part B, 43, 2012, 1523–1530.
[19] Fu, Y., Chen, Y., Zhang, P., Thermal buckling analysis of functionally graded beam with longitudinal crack, Meccanica, 48, 2013, 1227–1237.
[20] Anandrao, K.S., Gupta, R.K., Ramchandran, P., Rao, G.V., Thermal buckling and free vibration analysis of heated functionally graded material beams, Defense Science Journal, 63, 2013, 315-322.
[21] Kiani, Y., Eslami, M.R., Thermalmechanical buckling of temperature-dependent FGM beams, Latin American Journal of Solids and Structures, 10, 2013, 223-245.
[22] Esfahani, S.E., Kiani, Y., Eslami, M.R., Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations, International Journal of Mechanical Sciences, 69, 2013, 10-20.
[23] Ghiasian, S.E., Kiani, Y., Eslami, M.R., Nonlinear thermal dynamic buckling of FGM beams, European Journal of Mechanics -A/Solids, 54, 2015, 232-242.
[24] Sun, Y., Li, S.R., Batra, R.C., Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation, Journal of Thermal Stresses, 39, 2016, 11-26.
[25] Trinh, L.C., Vo, T.P., Thai, H.T., Nguyen, T.K., An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads, Composites: Part B, 100, 2016, 152-163.
[26] Shenas, A.G., Malekzadeh, P., Ziaee, S., Thermoelastic buckling analysis of pre-twisted functionally graded beams with temperature-dependent material properties, Acta Astronautica, 133, 2017, 1-13.
[27] Nguyen, T.K., Nguyen, B.D., Vo, T.P., Thai, H.T., Hygro-thermal effects on vibration and thermal buckling behaviours of functionally graded beams, Composite Structures, 176, 2017, 1050-1060.
[28] Hosseini, M., Farhatnia, F., Oveissi, S. Functionally graded Timoshenko beams with elastically-restrained edge supports: thermal buckling analysis via Stokes’ transformation technique, Research on Engineering Structures & Materials, 4, 2018, 103-125.
[29] Majumdar, A., Das, D., A study on thermal buckling load of clamped functionally graded beams under linear and nonlinear thermal gradient across thickness, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 232, 2018, 769-784.
[30] Liu, Y., Su, S., Huang, H., Liang, Y., Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane, Composites: Part B, 168, 2019, 236-242.
[31] Thanh, C.L., Tran, L.V., Bui, T.Q., Nguyen, H.X., Abdel-Wahab, M., Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates, Composite Structures, 221, 2019, 110838.
[32] Zhang, J., Chen, L., Lv, Y., Elastoplastic thermal buckling of functionally graded material beams, Composite Structures, 224, 2019, 111014.
[33] Ugural, A.C., Mechanical Design: An Integrated Approach, McGrow-Hill Company, Singapore, 2004.
[34] Chen, W.R., Chang, H., Closed-form solutions for free vibration frequencies of functionally graded Euler-Bernoulli beams, Mechanics of Composite Materials, 53, 2017, 79-98.
[35] Chen, W.R., Chang, H., Vibration analysis of functionally graded Timoshenko beams, International Journal of Structural Stability and Dynamics, 18, 2018, 1850007 (24 pages).
[36] Touloukian, Y.S., Thermophysical properties of high temperature solids materials, MacMillan, New York, 1967.
[37] Reddy, J.N., Chin, C.D., Thermomechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses, 21, 1998, 593–626.
[38] Shen, H., Functionally graded materials, Non-linear analysis of plates and shells, CRC press, New York, 2009.