Comparison study between layered and functionally graded composite beams for static deflection and stress analyses

Document Type: Research Paper

Authors

1 The General Directorate of Highways, Ankara, Turkey

2 Civil Engineering, Engineering Fac., Bursa Technical University, Bursa,Turkey

Abstract

The aim of this paper is to compare the static deflections and stress results of layered and functionally graded composite beams under static load. In the comparison study, the results obtained for a cantilever beam under point load. The Timoshenko beam and the Euler-Bernoulli beam theories are used in the beam model. The energy based Ritz method is used for the solution of the problem and algebraic polynomials are used with the trivial functions for the Ritz method. Two different materials are considered as layered and functionally graded distribution in a cantilever beam and their static deflections, stress distributions are compared under a point load at free end of the beam. For two different distributions, the formulations of Ritz method are obtained and solved numerically. In the numerical results, the effects of material distribution parameter, aspect ratio on the static deflections and stress distribution of functionally graded beams are obtained and compared with the results of the layered composite beam. Difference among of beam theories are compared for functionally graded and layered beams. Also, some comparison studies are performed in order to validate the using formulations.

Keywords

[1] Lewandowski R., 1987, Application of the Ritz method to the analysis of non-linear free vibrations of beams, Journal of Sound and Vibration 114(1): 91-101.
[2] Reddy J., 2000, Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering 47(1-3): 663-684.
[3]     Sankar B. V., 2001, An elasticity solution for functionally graded beams, Composites Science and Technology 61(5): 689-696.
[4] Deschilder M.,  Eslami H., Zhao Y., 2006, Nonlinear Static Analysis of a Beam Made of Functionally Graded Material, in: Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, Rhode Island.
[5] Chi S.H., Chung Y.L., 2006, Mechanical behavior of functionally graded material plates under transverse load – Part I: Analysis , International Journal of Solids and Structures  43(13): 3657-3674.
[6] Palanivel S., 2006, Dynamic analysis of laminated composite beams using higher order theories and finite elements, Composite Structures 73(3): 342-353.
[7] Zhong Z., Yu T., 2007, Analytical solution of a cantilever functionally graded beam, Composites Science and Technology 67(3-4): 481-488.
[8] Aydogdu M., Taskin V., 2007, Free vibration analysis of functionally graded beams with simply-supported edges, Materials & Design 36(5): 1651-1656.
[9] Kadoli R., Akhtar K., Ganesan N., 2008, Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modelling 32(12): 2509-2525.
[10] Li X.F., 2008, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams, Journal of Sound and Vibration 318(4-5): 1210-1229.
[11] Benatta M.A., Mechab I., Tounsi A., Bedia E.A.A., 2008, Static analysis of a functionally graded short beams including warping and shear deformation effects, Computational Materials Science 44(2): 765-773.
[12] Şimşek M., 2009, Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method, International Journal of Engineering and Applied Sciences 1(3): 1-11.
[13] Saidi A. R., Rasouli A., Sahraee S., 2009, Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory, Composite Structures 89(1): 110-119.
[14] Amirani M.C., Khalili S.M.R., Nemati N., 2009, Free Vibration Analysis of Sandwich beam with FG Core Using the Element Free Galerkin Method, Composite Structures 90: 373-379.
[15] Kang YA., Li, X F (2009), Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force, International Journal of Non-Linear Mechanics, 44(6): 696-703.
[16] Mohammadi M., Saidi A. R., Jomehzadeh E., 2010, Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates, Applied Composite Materials 17(2): 81-93.
[17] Akbaş ŞD., 2011, Static analysis of a functionally graded beam with edge cracks on elastic foundation, in: Proceedings of the 9th International Fracture Conference, Istanbul, Turkey, 70-80.
[18] Akbaş ŞD., 2013, Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material, Mathematical Problems in Engineering, doi: 10.1155/2013/871815.
[19] Akbaş ŞD., 2018, Forced vibration analysis of cracked functionally graded microbeams, Advances in Nano Research 6(1): 39-55.
[20] Akbaş ŞD., 2018, Bending of a cracked functionally graded nanobeam, Advances in Nano Research 6(3): 219-242.
[21] Akbaş ŞD., 2019, Post-buckling analysis of a fiber reinforced composite beam with crack, Engineering Fracture Mechanics 212:  70-80.
[22] Akbaş ŞD., 2019, Nonlinear behavior of fiber reinforced cracked composite beams, Steel and Composite Structures 30(4): 327-336.
[23] 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.
[24] Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39(1): 23-27.
[25] Mohammadi M., Farajpour A., Goodarzi M., Mohammadi H., 2013, Temperature effect on vibration analysis of annular graphene sheet embedded on Visco-Pasternak foundation, Journal of Solid Mechanics: 5 (3), 305-323.
[26] Pradhan K.K., Chakraverty S., 2013, Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh – Ritz method, Composites Part B: Engineering 51: 175-184.
[27] Su H., Banerjee J.R., Cheung C.W., 2013, Dynamic stiffness formulation and free vibration analysis of functionally graded beams , Composite Structures 106: 854-862.
[28]  Li S., Batra R., 2013 Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler – Bernoulli beams, Composite Structures 95: 5-9.
[29] 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.
[30] Asemi S.R., Mohammadi M., Farajpour A., 2014, A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures: 11 (9), 1515-1540.
[31] Safarabadi M., Mohammadi M., Farajpour A., Goodarzi M.,  2015, Effect of surface energy on the vibration analysis of rotating nanobeam, Journal of Solid Mechanics: 7 (3), 299-311.
[32] Vo T., Thai T., Nguyen T. K., Inam F., Lee J., 2015, A quasi-3D theory for vibration and buckling of functionally graded sandwich beams, Composite Structures 119: 1–12.
[33]   Avcar M., 2016, Free Vibration of Non-Homogeneous Beam Subjected to Axial Force Resting on Pasternak Foundation, Journal of Polytechnic 19(4): 507-512.
[34] Akbaş ŞD., 2019, Nonlinear static analysis of laminated composite beams under hygro-thermal effect, Structural Engineering and Mechanics 72(4): 433-441.
[35] Akbaş ŞD., 2017, Nonlinear static analysis of functionally graded porous beams under thermal effect, Coupled Systems Mechanics 6(4): 399-415.
[36] Akbaş ŞD., 2015, Wave propagation of a functionally graded beam in thermal environments, Steel and Composite Structures 19(6): 1421-1447.
[37] Akbaş ŞD., 2018, Nonlinear thermal displacements of laminated composite beams, Coupled systems mechanics 7(6): 691-705.
[38] Wattanasakulpong N., Mao Q., 2015, Dynamic response of Timoshenko functionally graded beams with classical and non-classical boundary conditions using Chebyshev collocation method, Composite Structures 119: 346-354.
[39] Nguyen T. K., Nguyen N. D., Vo T., Thai T., 2016, Trigonometric-series solution for analysis of laminated composite beams, Composite Structures 160: 142-151.
[40] Özütok A., Madenci E., 2017, Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method, International Journal of Mechanical Sciences 130: 234-243.
[41] Civalek Ö. 2017, Vibration of laminated composite panels and curved plates with different types of FGM composite constituent, Composites Part B: Engineering 122: 89-108.
[42] 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, doi: 10.1080/15376494.2018.1525453.
[43]  Ghasemi AR., Mohandes M., 2018, Comparison between the frequencies of FML and composite cylindrical shells using beam modal function model, Journal of Computational Applied Mechanics 50(2): 239-245.
[44] Karamanlı A., 2018, Bending analysis of composite and sandwich beams using Ritz method, Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19(1): 10-23.
[45] Mahmoudi A., Benyoucef S., Tounsi A., Benachour A., Bedia EAA. 2018, On the effect of the micromechanical models on the free vibration of rectangular FGM plate resting on elastic foundation, Earthquakes and Structures 14(2): 117-128.
[46]   Moradi A., Yaghootian A., Jalalvand M., Ghanbarzadeh A. 2018, Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation, Journal of Computational Applied Mechanics, 49(2): 395-407.
[47]  Zargaripoor A., Daneshmehr A., Isaac Hosseini I., Rajabpoor A. (2018), Free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory using finite element method, Journal of Computational Applied Mechanics, 49(1); 86-101.
[48]    Sayyad A.S., Ghugal Y.M, 2018, Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams, Mechanics of Advanced Composite Structures 5: 13-24.
[49] Chen D., Yang J., Kitipornchai S., 2018, Buckling and bending analyses of a novel functionally graded porous plate using Chebyshev-Ritz method, Archives of Civil and Mechanical Engineering 19 (1): 157-170.
[50] Hadji L., Zouatnia N., Bernard F. 2019, An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models, Structural Engineering and Mechanics 69(2): 231-241.
[51] Zouatnia N., Hadji L., Kassoul A. 2017, An analytical solution for bending and vibration responses of functionally graded beams with porosities, Wind and Structures 25(4): 329-342.
[52] Yayli MÖ., 2019, Free vibration analysis of a rotationally restrained (FG) nanotube, Microsystem Technologies, 25(10); 3723-3734.
[53] Akbaş ŞD., 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.
[54] Akbaş ŞD., 2014, Free vibration of axially functionally graded beams in thermal environment, International Journal of Engineering and Applied Sciences, 6(3): 37-51.
[55] Akbaş ŞD., 2015, Free vibration and bending of functionally graded beams resting on elastic foundation, Research on Engineering Structures and Materials, 1(1): 25-37.
[56] Akbaş ŞD., 2018, Investigation of static and vibration behaviors of a functionally graded orthotropic beam, Journal of Balikesir University Institute of Science and Technology, 20(1): 69-82.
[57] Akbaş ŞD., 2018, Investigation on free and forced vibration of a bi-material composite beam, Journal of Polytechnic, 21(1): 65-73.
[58] Akbaş ŞD., 2019, Forced vibration analysis of functionally graded sandwich deep beams. Coupled Systems Mechanics, 8(3): 259-271.
[59] Akbaş ŞD., 2017, Stability of a non-homogenous porous plate by using generalized differantial quadrature method, International Journal of Engineering and Applied Sciences, 9(2): 147-155.
[60] Akbaş ŞD., 2018, Post-buckling responses of a laminated composite beam, Steel and Composite Structures, 26(6): 733-743.
[61] Akbaş ŞD., 2018, Geometrically nonlinear analysis of a laminated composite beam, Structural Engineering and Mechanics, 66(1): 27-36.
[62] Akbaş ŞD., 2018, Geometrically nonlinear analysis of functionally graded porous beams, Wind and Structures, 27(1): 59-70.
[63] Akbaş ŞD., 2019, Hygro-thermal nonlinear analysis of a functionally graded beam, Journal of Applied and Computational Mechanics, 5(2): 477-485.
[64] Akbaş ŞD., 2019, Hygro-thermal post-buckling analysis of a functionally graded beam, Coupled Systems Mechanics, 8(5): 459-471.
[65] Seyyed Nosrati A., Parvizi A., Afzal SA., Alimirzaloo V. 2019, Elasto-plastic solution for thick-walled spherical vessels with an inner FGM layer, Journal of Computational Applied Mechanics, 50(1): 1-13.
[66] Yüksel YZ., Akbaş ŞD., 2019, Buckling Analysis of a Fiber Reinforced Laminated Composite Plate with Porosity, Journal of Computational Applied Mechanics, 50(2): 375-380.
[67] Barati A., Adeli MM., Hadi A., 2019, Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field, International Journal of Applied Mechanics, 12(2): 2050021.
[68] Khoram MM., Hosseini M., Hadi A., Shishehsaz, M., 2019, subjected to mechanical and magnetic forces and resting on Winkler-Pasternak foundation, International Journal of Applied Mechanics, https://doi.org/10.1142/S1758825120500933.
 [69] 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,
      doi: 10.1016/j.euromechsol.2019.05.008
[70] 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): 131-143.

Volume 51, Issue 2
December 2020
Pages 294-301
  • Receive Date: 20 January 2020
  • Revise Date: 31 January 2020
  • Accept Date: 02 February 2020