Static Buckling of 2D FG Porous Plates Resting on Elastic ‎Foundation based on Unified Shear Theories‎

Document Type : Research Paper


1 Mechanical Engineering Department, Faculty of Engineering, Jazan University, P. O. Box 45142, Jazan, Kingdom of Saudi Arabia

2 Department of Mechanical Design and Production, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

3 Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

4 Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt‎

5 Mathematical Department, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

6 Mechanical Dept., Faculty of Engineering, King Abdualziz University, Jeddah, Saudi Arabia

7 Mechanical Design and Production Dept., Faculty of Engineering, Zagazig University, Zagazig, Egypt


This article develops a mathematical model to study the static stability of bi-directional functionally graded porous unified plate (BDFGPUP) resting on elastic foundation. The power function distribution is proposed for the gradation of material constituent through thickness and axial directions. Three types of porosity are selected to portray the distribution of voids and cavities through the thickness of the plate. Unified theories of plate are exploited to present the kinematic fields and satisfy the zero-shear strain/stress at the top and bottom surfaces without shear correction factor. Hamilton’s principle is employed to derive the governing equations of motions including the higher terms of force resultants. An efficient numerical method namely differential integral quadrature method (DIQM) is manipulated to discretize the structure spatial domain and transform the coupled variable coefficients partial differential equations to a system of algebraic equations. Problem validation and verification have been proven with previous works for bucking phenomenon. Parametric studies are exemplified to exhibit the significant impacts of kinematic shear relations, gradation indices, porosity type, and boundary conditions on the static stability and buckling loads of BDFGP plate. The proposed model is economical in different applications in nuclear, mechanical, aerospace, naval, dental and medical fields.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Hassan, A.H, Kurgan, N., Can, N., The Relations between the Various Critical Temperatures of Thin‎ FGM Plates, Journal of Applied and Computational Mechanics, 6, 2020, 1404-1419.
[2] Dorduncu, M., Olmus, I., Rabczuk, T., A peridynamic approach for modeling of two dimensional functionally graded plates, Composite Structures, 279, 2022, 114743.
[3] Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F., Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35(1), 2011, 412-425.
[4] Esen, I., Özarpa, C., Eltaher, M.A., Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment, Composite Structures, 261, 2021, 113552.
[5] Shanab, R.A., Attia, M.A., Semi-analytical solutions for static and dynamic responses of bi-directional functionally graded nonuniform nanobeams with surface energy effect, Engineering with Computers, 2020, 1-44. DOI: 10.1007/s00366-020-01205-6
[6] Wu, C.P., Huang, H.Y., A semianalytical finite element method for stress and deformation analyses of bi-directional functionally graded truncated conical shells, Mechanics Based Design of Structures and Machines, 48(4), 2020, 433-458.
[7] Abo-Bakr, H.M., Abo-Bakr, R.M., Mohamed, S.A., Eltaher, M.A., Weight optimization of axially functionally graded microbeams under buckling and vibration behaviors, Mechanics Based Design of Structures and Machines, 2020, 1-22. DOI: 10.1080/15397734.2020.1838298
[8] Abo-Bakr, H.M., Abo-bakr, R.M., Mohamed, S.A., Eltaher, M.A., Multi-objective shape optimization for axially functionally graded microbeams, Composite Structures, 258, 2021, 113370.
[9] Abo-Bakr, R.M., Abo-Bakr, H.M., Mohamed, S.A., Eltaher, M.A., Optimal weight for buckling of FG beam under variable axial load using Pareto optimality, Composite Structures, 258, 2021, 113193.
[10] Abo-bakr, R.M., Shanab, R.A., Attia, M.A., Multi-objective optimization for lightweight design of bi-directional functionally graded beams for maximum frequency and buckling load, Composite Structures, 278, 2021, 114691.
[11] Pham, Q.H., Nguyen, P.C., Tran, V.K., Nguyen-Thoi, T., Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium, Defence Technology, 2021. DOI: 10.1016/j.dt.2021.09.006
[12] Attia, M.A., Shanab, R.A., Vibration characteristics of two-dimensional FGM nanobeams with couple stress and surface energy under general boundary conditions, Aerospace Science and Technology, 111, 2021, 106552. 
[13] Shanab, R.A., Attia, M.A., On bending, buckling and free vibration analysis of 2D-FG tapered Timoshenko nanobeams based on modified couple stress and surface energy theories, Waves in Random and Complex Media, 2021, 1-47. DOI: 10.1080/17455030.2021.1884770
[14] Chen, X., Chen, L., Huang, S., Li, M., Li, X., Nonlinear forced vibration of in-plane bi-directional functionally graded materials rectangular plate with global and localized geometrical imperfections, Applied Mathematical Modelling, 93, 2021, 443-466. 
[15] Hendi, A.A., Eltaher, M.A., Mohamed, S.A., Attia, M.A., Abdalla, A.W., Nonlinear thermal vibration of pre/post-buckled two-dimensional FGM tapered microbeams based on a higher order shear deformation theory, Steel and Composite Structures, 41(6), 2021, 787-802.
[16] Ohab-Yazdi, S.M.K., Kadkhodayan, M., Free vibration of bi-directional functionally graded imperfect nanobeams under rotational velocity, Aerospace Science and Technology, 119, 2021, 107210.
[17] Banh, T.T., Luu, N.G., Lieu, Q.X., Lee, J., Kang, J., Lee, D., Multiple bi-directional FGMs topology optimization approach with a preconditioned conjugate gradient multigrid, Steel and Composite Structures, 41(3), 2021, 385-402.
[18] Arasan, U., Venkatachalam, S., Murthy, H., Solution to two-dimensional elastic problems involving functionally graded material in radial co-ordinates, Acta Mechanica, 233, 2022, 343–362.
[19] Fu, P., Zhao, J., Zhang, X., Kang, G., Wang, P., Kan, Q., Elastic shakedown analysis of two-dimensional thermo-elastic rolling/sliding contact for a functionally graded coating/substrate structure with arbitrarily varying thermo-elastic properties, Composite Structures, 280, 2022, 114891.
[20] Brischetto, S., Torre, R., 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions, Applied Sciences, 12(1), 2022, 512. 
[21] Barati, A., Hadi, A., Nejad, M.Z., Noroozi, R., On vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, 50(2), 2022, 468-485.
[22] Whitney, J.M., Pagano, N.J., Shear deformation in heterogeneous anisotropic plates, Journal of Applied Mechanics, 37(4), 1970, 1031-1036.
[23] Reddy, J.N., Free vibration of antisymmetric, angle-ply laminated plates including transverse shear deformation by the finite element method, Journal of Sound and Vibration, 66(4), 1979, 565-576.
[24] Touratier, M., An efficient standard plate theory, International Journal of Engineering Science, 29(8), 1991, 901-916.
[25] Assie, A.E., Mahmoud, F.F., Dynamics of thick hygrothermal viscoelastic composite laminates through finite element method, Structural Engineering and Mechanics, 17(5), 2004, 727-734.
[26] Mantari, J.L., Oktem, A.S., Soares, C.G., Bending response of functionally graded plates by using a new higher order shear deformation theory, Composite Structures, 94(2), 2012, 714-723.
[27] Neves, A.M.A., Ferreira, A.J., Carrera, E., Cinefra, M., Jorge, R.M.N., Soares, C.M.M., Static analysis of functionally graded sandwich plates according to a hyperbolic theory considering Zig-Zag and warping effects, Advances in Engineering Software, 52, 2012, 30-43.
[28] Sedighi, H.M., Shirazi, K.H., Noghrehabadi, A.R., Yildirim, A., Asymptotic investigation of buckled beam nonlinear vibration, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 36(M2), 2012, 107-116.
[29] Sedighi, H.M., Reza, A., Zare, J., The effect of quintic nonlinearity on the investigation of transversely vibrating buckled Euler-Bernoulli beams, Journal of Theoretical and Applied Mechanics, 51, 2013, 959-968.
[30] Mantari, J.L., Soares, C.G., A novel higher-order shear deformation theory with stretching effect for functionally graded plates, Composites Part B: Engineering, 45(1), 2013, 268-281.
[31] Assie, A.E., Kabeel, A.M., Mahmoud, F.F., Optimum design of laminated composite plates under dynamic excitation, Applied Mathematical Modelling, 36(2), 2012, 668-682. 
[32] Yu, T., Yin, S., Bui, T.Q., Xia, S., Tanaka, S., Hirose, S., NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method, Thin-Walled Structures, 101, 2016, 141-156.
[33] Nguyen, H.N., Hong, T.T., Vinh, P.V., Quang, N.D., Thom, D.V., A refined simple first-order shear deformation theory for static bending and free vibration analysis of advanced composite plates, Materials, 12(15), 2019, 2385.
[34] Vu, T.V., Nguyen, H.T., Nguyen-Van, H., Nguyen, T.P., Curiel-Sosa, J.L., A refined quasi-3D logarithmic shear deformation theory-based effective meshfree method for analysis of functionally graded plates resting on the elastic foundation, Engineering Analysis with Boundary Elements, 131, 2021, 174-193.
[35] Raissi, H., Stress analysis in adhesive layers of a five-layer circular sandwich plate subjected to temperature gradient based on layerwise theory, Mechanics Based Design of Structures and Machines, 2020, 1-27. DOI: 10.1080/15397734.2020.1776619
[36] Khiloun, M., Bousahla, A.A., Kaci, A., Bessaim, A., Tounsi, A., Mahmoud, S.R., Analytical modeling of bending and vibration of thick advanced composite plates using a four-variable quasi 3D HSDT, Engineering with Computers, 36(3), 2020, 807-821.
[37] Fallahi, N., Viglietti, A., Carrera, E., Pagani, A., Zappino, E., Effect of fiber orientation path on the buckling, free vibration, and static analyses of variable angle tow panels, Facta Universitatis. Series: Mechanical Engineering, 18(2), 2020, 165-188.
[38] Fallahi, N., GA optimization of variable angle tow composites in buckling and free vibration analysis through layerwise theory, Aerospace, 8(12), 2021, 376.
[39] Arshid, E., Khorasani, M., Soleimani-Javid, Z., Amir, S., Tounsi, A., Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory, Engineering with Computers, 2021, 1-22. DOI: 10.1007/s00366-021-01382-y
[40] Pham, Q.H., Tran, V.K., Tran, T.T., Nguyen-Thoi, T., Nguyen, P.C., A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation, Case Studies in Thermal Engineering, 26, 2021, 101170.
[41] Pavlovic, A., Fragassa, C., Minak, G., Buckling analysis of telescopic boom: theoretical and numerical verification of sliding pads, Tehnički Vjesnik, 24(3), 2017, 729-735. 
[42] Daikh, A.A., Houari, M.S.A., Eltaher, M.A., A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates, Composite Structures, 262, 2021, 113347.
[43] Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Chakraverty, S., Eltaher, M.A., Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates, Engineering with Computers, 2021, 1-22. DOI: 10.1007/s00366-021-01413-8
[44] Li, M., Soares, C.G., Yan, R., Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple quasi-3D HSDT, Composite Structures, 264, 2021, 113643.
[45] Van Vinh, P., Deflections, stresses and free vibration analysis of bi-functionally graded sandwich plates resting on Pasternak’s elastic foundations via a hybrid quasi-3D theory, Mechanics Based Design of Structures and Machines, 2021, 1-32. DOI: 10.1080/15397734.2021.1894948
[46] Sadgui, A., Tati, A., A novel trigonometric shear deformation theory for the buckling and free vibration analysis of functionally graded plates, Mechanics of Advanced Materials and Structures, 2021, 1-16. DOI: 10.1080/15376494.2021.1983679
[47] Li, M., Yan, R., Xu, L., Soares, C.G., A general framework of higher-order shear deformation theories with a novel unified plate model for composite laminated and FGM plates, Composite Structures, 261, 2021, 113560.
[48] Tran, T.T., Nguyen, P.C., Pham, Q.H., Vibration analysis of FGM plates in thermal environment resting on elastic foundation using ES-MITC3 element and prediction of ANN, Case Studies in Thermal Engineering, 24, 2021, 100852.
[49] Sofiyev, A.H., Dikmen, F., Buckling Analysis of Functionally Graded Shells under Mixed‎ Boundary‎ Conditions Subjected to Uniform Lateral Pressure, Journal of Applied and Computational Mechanics, 7(1), 2021, 345-354.
[50] Civalek, Ö., Avcar, M., Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method, Engineering with Computers, 38, 2022, 489–521.
[51] Hamed, M.A., Abo-Bakr, R.M., Mohamed, S.A., Eltaher, M.A., Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core, Engineering with Computers, 36(4), 2020, 1929-1946.
[52] Abdollahi, M., Saidi, A.R., Bahaadini, R., Aeroelastic analysis of symmetric and non-symmetric trapezoidal honeycomb sandwich plates with FG porous face sheets, Aerospace Science and Technology, 119, 2021, 107211.
[53] Sobhy, M., Zenkour, A.M., Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory, Composite Structures, 220, 2019, 289-303.
[54] Coskun, S., Kim, J., Toutanji, H., Bending, free vibration, and buckling analysis of functionally graded porous micro-plates using a general third-order plate theory, Journal of Composites Science, 3(1), 2019, 15.
[55] Kim, J., Zur, K.K., Reddy, J.N., Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates, Composite Structures, 209, 2019, 879-888.
[56] Babaei, M., Asemi, K., Kiarasi, F., Static response and free-vibration analysis of a functionally graded annular elliptical sector plate made of saturated porous material based on 3D finite element method, Mechanics Based Design of Structures and Machines, 2020, 1-25. DOI: 10.1080/15397734.2020.1864401
[57] Esmaeilzadeh, M., Golmakani, M.E., Sadeghian, M., A nonlocal strain gradient model for nonlinear dynamic behavior of bi-directional functionally graded porous nanoplates on elastic foundations, Mechanics Based Design of Structures and Machines, 2020, 1-20. DOI: 10.1080/15397734.2020.1845965
[58] Cho, I.H., Wave energy dissipation by a floating horizontal porous plate in oblique incident waves, Wave Motion, 105, 2021, 102765.
[59] Hieu, D.V., Chan, D.Q., Sedighi, H.M., Nonlinear bending, buckling and vibration of functionally graded nonlocal strain gradient nanobeams resting on an elastic foundation, Journal of Mechanics of Materials and Structures, 16(3), 2021, 327-346.
[60] Katiyar, V., Gupta, A., Vibration response of a geometrically discontinuous bi-directional functionally graded plate resting on elastic foundations in thermal environment with initial imperfections, Mechanics Based Design of Structures and Machines, 2021, 1-29. DOI: 10.1080/15397734.2021.1929313
[61] Akbas, S.D., Bashiri, A.H., Assie, A.E., Eltaher, M.A., Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support, Journal of Vibration and Control, 27(13-14), 2021, 1644-1655.  
[62] Akbas, S.D., Fageehi, Y.A., Assie, A.E., Eltaher, M.A., Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load, Engineering with Computers, 38, 2022, 365–377.
[63] Sah, S.K., Ghosh, A., Influence of porosity distribution on free vibration and buckling analysis of multi-directional functionally graded sandwich plates, Composite Structures, 279, 2022, 114795.
[64] Karama, M., Afaq, K.S., Mistou, S., Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids and Structures, 40(6), 2003, 1525-1546.
[65] Thai, H.T., Kim, S.E., Closed-form solution for buckling analysis of thick functionally graded plates on elastic foundation, International Journal of Mechanical Sciences, 75, 2013, 34-44.
[66] Taibi, F.Z., Benyoucef, S., Tounsi, A., Bachir Bouiadjra, R., Adda Bedia, E.A., Mahmoud, S.R., A simple shear deformation theory for thermo-mechanical behaviour of functionally graded sandwich plates on elastic foundations, Journal of Sandwich Structures & Materials, 17(2), 2015, 99-129.
[67] Bellman, R., Kashef, B.G., Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics, 10(1), 1972, 40-52.
[68] Shu, C., Differential quadrature and its application in engineering, Springer Science & Business Media, 2012.
[69] Zong, Z., Advanced differential quadrature methods, Chapman and Hall/CRC, 2009.
[70] Mohamed, S.A., A fractional differential quadrature method for fractional differential equations and fractional eigenvalue problems, Mathematical Methods in the Applied Sciences, 2020, DOI: 10.1002/mma.6753
[71] Mohamed, S.A., Mohamed, N.A., Abo‐Hashem, S.I., A novel differential‐integral quadrature method for the solution of nonlinear integro‐differential equations, Mathematical Methods in the Applied Sciences, 44(18), 2021, 13945-13967.
[72] Thai, H.T., Choi, D.H., An efficient and simple refined theory for buckling analysis of functionally graded plates, Applied Mathematical Modelling, 36(3), 2012, 1008-1022.