Document Type : Research Paper

**Author**

Mechanical Engineering Dept., Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Tel: +96653908744, Jeddah, Saudi Arabia

**Abstract**

The present study investigates buckling characteristics of both nonlinear symmetric power and sigmoid functionally graded (FG) beams. The volume fractions of metal and ceramic are assumed to be distributed through a beam thickness by the sigmoid-law distribution (S-FGM), and the symmetric power function (SP-FGM). These functions have smooth variation of properties across the boundary rather than the classical power law distribution which permits gradually variation of stresses at the surface boundary and eliminates delamination. The Voigt model is proposed to homogenize micromechanical properties and to derive the effective material properties. The Euler-Bernoulli beam theory is selected to describe Kinematic relations. A finite element model is exploited to form stiffness and buckling matrices and solve the problem of eignivalue numerically. Numerical results present the effect of material graduations and elasticity ratios on the buckling behavior of FG beams. The proposed model is helpful in stability of mechanical systems manufactured from FGMs.

**Keywords**

- Static Stability
- Buckling
- Functional graded materials
- Symmetric Power-Law
- Sigmoid Function
- Finite element

**Main Subjects**

[1] Eltaher, M. A., Alshorbagy, A. E., Mahmoud, F. F, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams, *Composite Structures*, 99, 2013, 193-201.

[2] Wang, C. M., Wang, C. Y., Reddy, J.N., *Exact solutions for buckling of structural members*, 2005, CRC press.

[3] Reddy, J. N., Analysis of functionally graded plates, *International Journal for Numerical Methods in Engineering*, 47(1-3), 2000, pp. 663-684.

[4] Rastgo, A., Shafie, H., Allahverdizadeh, A., Instability of curved beams made of functionally graded material under thermal loading, *International Journal of Mechanics and Materials in Design*, 2(1), 2005, pp. 117-128.

[5] Alshorbagy, A. E., Eltaher, M. A., Mahmoud, F. F., Free vibration characteristics of a functionally graded beam by finite element method, *Applied Mathematical Modelling*, 35(1), 2011, pp. 412-425.

[6] Kocaturk, T., Akbas, S. D., Post-buckling analysis of Timoshenko beams made of functionally graded material under thermal loading, *Structural Engineering and Mechanics*, 41(6), 2012, pp. 775-789.

[7] Eltaher, M. A., Emam, S. A., Mahmoud, F. F., Static and stability analysis of nonlocal functionally graded nanobeams, *Composite Structures*, 96, 2013, pp. 82-88.

[8] Li, S. R., Batra, R. C., Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams, *Composite Structures*, 95, 2013, pp. 5-9.

[9] Fu, Y., Chen, Y., Zhang, P., Thermal buckling analysis of functionally graded beam with longitudinal crack, *Meccanica*, 48(5), 2013, pp. 1227-1237.

[10] Akbaş, Ş. D., Kocatürk, T., Post-buckling analysis of functionally graded three-dimensional beams under the influence of temperature, *Journal of Thermal Stresses*, 36(12), 2013, pp. 1233-1254.

[11] Kocaturk, T., Akbas, S. D., Thermal post-buckling analysis of functionally graded beams with temperature-dependent physical properties, *Steel and Composite Structures*, 15(5), 2013, pp. 481-505.

[12] Eltaher, M. A., Hamed, M. A., Sadoun, A. M., Mansour, A., Mechanical analysis of higher order gradient nanobeams, *Applied Mathematics and Computation*, 229, 2014, pp. 260-272.

[13] Ebrahimi, F., Salari, E., Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams, *Smart Materials and Structures*, 24(12), 2015, p. 125007.

[14] Fu, Y., Zhong, J., Shao, X., Chen, Y., Thermal postbuckling analysis of functionally graded tubes based on a refined beam model, *International Journal of Mechanical Sciences*, 96, 2015, pp. 58-64.

[15] Ghiasian, S. E., Kiani, Y., Eslami, M. R., Nonlinear thermal dynamic buckling of FGM beams, *European Journal of Mechanics-A/Solids*, 54, 2015, pp. 232-242.

[16] Amara, K., Bouazza, M., Fouad, B., Postbuckling Analysis of Functionally Graded Beams Using Nonlinear Model, *Periodica Polytechnica. Engineering. Mechanical Engineering*, 60(2), 2016, p. 121.

[17] Rezaiee-Pajand, M., Masoodi, A. R., Exact natural frequencies and buckling load of functionally graded material tapered beam-columns considering semi-rigid connections, *Journal of Vibration and Control*, 2016, doi: 1077546316668932.

[18] Kiani, K., Postbuckling scrutiny of highly deformable nanobeams: A novel exact nonlocal-surface energy-based model, *Journal of Physics and Chemistry of Solids*, 110, 2017, pp. 327-343.

[19] Maleki, V. A., Mohammadi, N., Buckling analysis of cracked functionally graded material column with piezoelectric patches, *Smart Materials and Structures*, 26(3), 2017, p. 035031.

[20] Ben-Oumrane, S., Abedlouahed, T., Ismail, M., Mohamed, B. B., Mustapha, M., El Abbas, A. B., A theoretical analysis of flexional bending of Al/Al 2 O 3 S-FGM thick beams, *Computational Materials Science*, 44(4), 2009, pp. 1344-1350.

[21] Chi, S. H., Chung, Y. L., Cracking in sigmoid functionally graded coating*, **International Journal of Structural Stability and Dynamics*, 18, 2002, pp. 41-53.

[22] Mahi, A., 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(8), 2010, pp. 1877-1887.

[23] Fereidoon, A., Mohyeddin, A., Bending analysis of thin functionally graded plates using generalized differential quadrature method, *Archive of Applied Mechanics*, 81(11), 2011, pp. 1523-1539.

[24] Duc, N. D., Cong, P. H., Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation, *Journal of Vibration and Control*, 21(4), 2015, pp. 637-646.

[25] Lee, C. Y., Kim, J. H., Thermal post-buckling and snap-through instabilities of FGM panels in hypersonic flows, *Aerospace Science and Technology*, 30(1), 2013, pp. 175-182.

[26] Jung, W. Y., Han, S. C., Analysis of sigmoid functionally graded material (S-FGM) nanoscale plates using the nonlocal elasticity theory, *Mathematical Problems in Engineering*, 2013, Article ID 476131, 10p.

[27] Akbaş, Ş. D., On post-buckling behavior of edge cracked functionally graded beams under axial loads, *International Journal of Structural Stability and Dynamics*, 15(4), 2015, p. 1450065.

[28] Akbaş, Ş. D., Post-buckling analysis of axially functionally graded three-dimensional beams, *International Journal of Applied Mechanics*, 7(3), 2015, p. 1550047.

[29] Ebrahimi, F., Salari, E., Analytical modeling of dynamic behavior of piezo-thermo-electrically affected sigmoid and power-law graded nanoscale beams, *Applied Physics A*, 122(9), 2016, p. 793.

[30] Hamed, M. A., Eltaher, M. A., Sadoun, A. M., Almitani, K. H., Free vibration of symmetric and sigmoid functionally graded nanobeams, *Applied Physics A*, 122(9), 2016, p. 829.

[31] Swaminathan, K., Sangeetha, D. M., Thermal analysis of FGM plates–A critical review of various modeling techniques and solution methods, *Composite Structures*, 160, 2017, pp. 43-60.

[32] Yahia, S. A., Atmane, H. A., Houari, M. S. A., Tounsi, A., Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories, *Structural Engineering and Mechanics*, 53(6), 2015, pp. 1143-1165.

[33] Atmane, H. A., Tounsi, A., Bernard, F., Mahmoud, S. R., A computational shear displacement model for vibrational analysis of functionally graded beams with porosities, *Steel and Composite Structures*, 19(2), 2015, pp. 369-384.

[34] Attia, A., Tounsi, A., Bedia, E. A., Mahmoud, S. R., Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories, *Steel and Composite Structures*, 18(1), 2015, pp. 187-212.

[35] Beldjelili, Y., Tounsi, A., Mahmoud, S. R., Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory, *Smart Structures and Systems*, 18(4), 2016, pp. 755-786.

[36] Bouderba, B., Houari, M. S. A., Tounsi, A., Mahmoud, S. R., Thermal stability of functionally graded sandwich plates using a simple shear deformation theory, *Structural Engineering and Mechanics*, 58(3), 2016, pp. 397-422.

[37] Bousahla, A. A., Benyoucef, S., Tounsi, A., Mahmoud, S. R., On thermal stability of plates with functionally graded coefficient of thermal expansion, *Structural Engineering and Mechanics*, 60(2), 2016, pp. 313-335.

[38] Boukhari, A., Atmane, H. A., Tounsi, A., Adda, B., Mahmoud, S. R., An efficient shear deformation theory for wave propagation of functionally graded material plates, *Structural Engineering and Mechanics*, 57(5), 2016, pp. 837-859.

[39] Bellifa, H., Benrahou, K. H., Hadji, L., Houari, M. S. A., Tounsi, A., Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position, *Journal of the Brazilian Society of Mechanical Sciences and Engineering*, 38(1), 2016, pp. 265-275.

[40] Houari, M. S. A., Tounsi, A., Bessaim, A., Mahmoud, S. R., A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates, *Steel and Composite Structures*, 22(2), 2016, pp. 257-276.

[41] Chikh, A., Tounsi, A., Hebali, H., Mahmoud, S. R., Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT, *Smart Structures and Systems*, 19(3), 2017, pp. 289-297.

[42] Besseghier, A, Houari, M.S.A, Tounsi , A., Hassan, S., Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory, *Smart Structures and Systems*, 19 (6), 2017, pp. 601-614.

[43] Bellifa, H., Benrahou, K. H., Bousahla, A. A., Tounsi, A., Mahmoud, S. R., A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams, *Structural Engineering and Mechanics*, 62(6), 2017, pp. 695-702.

[44] Mori, T., Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, *Acta Metallurgica*, 21(5), 1973, pp. 571-574.

[45] Tomota, Y., Kuroki, K., Mori, T., Tamura, I., Tensile deformation of two-ductile-phase alloys: Flow curves of α-γ Fe-Cr-Ni alloys, *Materials Science and Engineering*, 24(1), 1976, pp. 85-94.

[46] Li, S. R., Su, H. D., Cheng, C. J., Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment, *Applied Mathematics and Mechanics*, 30, 2009, pp. 969-982.

[47] Komijani, M., Esfahani, S. E., Reddy, J. N., Liu, Y. P., Eslami, M. R., Nonlinear thermal stability and vibration of pre/post-buckled temperature-and microstructure-dependent functionally graded beams resting on elastic foundation, *Composite Structures*, 112, 2014, pp. 292-307.

[48] Eltaher, M. A., Abdelrahman, A. A., Al-Nabawy, A., Khater, M., Mansour, A., Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position, *Applied Mathematics and Computation*, 235, 2014, pp. 512-529.

[49] Reddy, J. N., *An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer*, fluid mechanics, and solid mechanics, 2014, Oxford University Press.

[50] Delale, F., Erdogan, F., The crack problem for a nonhomogeneous plane, *Journal of Applied Mechanics*, 50(3), 1983, pp. 609-614.

April 2018

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