Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements

Document Type: Research Paper


1 Production Engineering and mechanical Design Dept, Mansoura University Al Mansurah, Egypt

2 Mechanical Engineering Dept, King Abdulaziz University, Jeddah, Saudi Arabia AND Mechanical Design and Production Eng. Dept., Zagazig University, Al Zagazig, Egypt


This study presents the effect of porosity on mechanical behaviors of a power distribution functionally graded beam. The Euler-Bernoulli beam is assumed to describe the kinematic relations and constitutive equations. Because of technical problems, particle size shapes and micro-voids are created during the fabrication which should be taken into consideration. Two porosity models are proposed. The first one describes properties in the explicit form as linear functions of the porosity parameter. The second is a modified model which presents porosity and Young’s modulus in an implicit form where the density is assumed as a function of the porosity parameter and Young’s modulus as a ratio of mass with porosity to the mass without porosity. The modified proposed model is more applicable than the first model. The finite element model is developed to solve the problem by using the MATLAB software. Numerical results are presented to show the effects of porosity on mechanical behaviors of functionally graded beams.


Main Subjects

[1] Eltaher, M.A., Khairy, A., Sadoun, A.M., Omar, F.A. Static and buckling analysis of functionally graded Timoshenko nanobeams, Applied Mathematics and Computation, 229, 2014, pp. 283-295.

[2] Zhu, J., Lai, Z., Yin, Z., Jeon, J., Lee, S. Fabrication of ZrO 2–NiCr functionally graded material by powder metallurgy, Materials Chemistry and Physics, 68(1), 2001, pp. 130-135.

[3] Aqida, S.N., Ghazali, M.I., Hashim, J. Effects of porosity on mechanical properties of metal matrix composite: an overview, Jurnal Teknologi, 40, 2004, pp. 17-32.

[4] Kim, H.S., Yang, Y., Koh, J.T., Lee, K.K., Lee, D.J., Lee, K.M., Park, S.W. Fabrication and characterization of functionally graded nano‐micro porous titanium surface by anodizing, Journal of Biomedical Materials Research Part B: Applied Biomaterials, 88B, 2009, pp. 427-435.

[5] Wattanasakulpong, N., Prusty, B.G., Kelly, D.W., Hoffman, M. Free vibration analysis of layered functionally graded beams with experimental validation, Materials & Design, 36, 2012, pp. 182-190.

[6] Ji, S., Gu, Q., Xia, B. Porosity dependence of mechanical properties of solid materials, Journal of Materials Science, 41, 2006, pp. 1757-1768.

[7] Wattanasakulpong, N., Ungbhakorn, V. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities, Aerospace Science and Technology, 32, 2014, pp. 111-120.

[8] Wattanasakulpong, N., Chaikittiratana, A. Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory, Chebyshev collocation method, Meccanica, 50(5), 2015, pp. 1331-1342.

[9] Atmane, H.A., Tounsi, A., Bernard, F. Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations, International Journal of Mechanics and Materials in Design, 13, 2015, pp. 71-84.

[10] Ebrahimi, F., Mokhtari, M. Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37, 2015, pp. 1435-1444.

[11] Ebrahimi, F., Jafari, A. A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities, Mechanics of Advanced Materials and Structures, 23, 2016, pp. 1-13.

[12] Ebrahimi, F., Ghasemi, F., Salari, E. Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities, Meccanica, 51, 2016, pp. 223-249.

[13] Shafiei, N., Mousavi, A., Ghadiri, M. On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams, International Journal of Engineering Science, 106, 2016, pp. 42-56.

[14] Ebrahimi, F., Barati, M.R. Size-dependent vibration analysis of viscoelastic nanocrystalline silicon nanobeams with porosities based on a higher order refined beam theory, Composite Structures, 166, 2017, pp. 256-267.

[15] Magnucki, K., Stasiewicz, P. Elastic buckling of a porous beam, Journal of Theoretical and Applied Mechanics, 42(4), 2004, pp. 859-868.

[16] Jabbari, M., Mojahedin, A., Joubaneh, E.F. Thermal Buckling Analysis of Circular Plates Made of Piezoelectric and Saturated Porous Functionally Graded Material Layers, Journal of Engineering Mechanics, 141(4), 2015, pp. 1-12.

[17] Xue, L., Dui, G., Liu, B., Xin, L. A phenomenological constitutive model for functionally graded porous shape memory alloy, International Journal of Engineering Science, 78, 2014, pp. 103-113.

[18] Chen, D., Yang, J., Kitipornchai, S. Elastic buckling and static bending of shear deformable functionally graded porous beam, Composite Structures, 133, 2015, pp. 54-61.

[19] Kitipornchai, S., Chen, D., Yang, J., Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets, Materials & Design, 116, 2017, pp. 656-665.

[20] 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, pp. 829-839.

[21] Ebrahimi, F., Zia, M., Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities, Acta Astronautica, 116, 2015, pp. 117-125.

[22] Sarkar, B.K., Mukherjee, M. K., Natarajan, A. A modification of the rule of mixture in estimating strengths of a composite, Materialwissenschaft und Werkstofftechnik, 13(8), 1982, pp. 269-273.

[23] Bert, C.W. Prediction of elastic moduli of solids with oriented porosity, Journal of Materials Science, 20(6), 1985, pp. 2220-2224.

[24] Hardin, R.A., Beckermann, C. Effect of porosity on the stiffness of cast steel, Metallurgical and Materials Transactions A, 38(12), 2007, pp. 2992-3006.

[25] Wachtman, J.B., Cannon, W.R., Matthewson, M.J. Mechanical properties of ceramics, John Wiley & Sons, New York, 2009.

[26] Sabree, I., Gough, J.E., Derby, B. Mechanical properties of porous ceramic scaffolds: influence of internal dimensions, Ceramics International, 41(7), 2015, pp. 8425-8432.

[27] Zok, F.W., Levi, C.G. Mechanical properties of porous-matrix ceramic composites, Advanced Engineering Materials, 3(1-2), 2001, pp. 15-23.

[28] Revel, G.M. Measurement of the apparent density of green ceramic tiles by a non-contact ultrasonic method, Experimental Mechanics, 47(5), 2007, pp. 637-648.

[29] 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.

[30] 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.

[31] Eltaher, M.A., El-Borgi, S., Reddy, J.N. Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs, Composite Structures, 153, 2016, pp. 902-913.