Two-Dimensional Elasticity Solution for Arbitrarily Supported Axially Functionally Graded Beams

Document Type: Research Paper


Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India


First time, an analytical two-dimensional (2D) elasticity solution for arbitrarily supported axially functionally graded (FG) beam is developed. Linear gradation of the material property along the axis of the beam is considered. Using the strain displacement and constitutive relations, governing partial differential equations (PDEs) is obtained by employing Ressiner mixed variational principle. Then PDEs are reduced to two set of ordinary differential equations (ODEs) by using recently developed extended Kantorovich method. The set of 4n ODEs along the z-direction has constant coefficients. But, the set of 4n nonhomogeneous ODEs along x-direction has variable coefficients which is solved using modified power series method. Efficacy and accuracy of the present methodology are verified thoroughly with existing literature and 2D finite element solution. Effect of axial gradation, boundary conditions and configuration lay-ups are investigated. It is found that axial gradation influence vary with boundary conditions. These benchmark results can be used for assessing 1D beam theories and further present formulation can be extended to develop solutions for 2D micro or Nanobeams.  


[1] Nakamura T., Singh R., Vaddadi P., 2006, Effects of environmental degradation on flexural failure strength of fiber reinforced composites, Experimental Mechanics 46(2): 257-268.
[2] Barbero E., Cosso F., Campo F., 2013, Benchmark solution for degradation of elastic properties due to transverse matrix cracking in laminated composites, Composite Structures 98: 242-252.
[3] Adda-Bedia E., Bouazza M., Tounsi A., Benzair A., Maachou M., 2008, Prediction of stiffness degradation in hygrothermal aged [θm/90n]S composite laminates with transverse cracking, Journal of Materials Processing Technology 199(1): 199-205.
[4] Naebe M., Shirvanimoghaddam K., 2016, Functionally graded materials: A review of fabrication and properties, Applied Materials Today 5: 223-245.
[5] Sankar B., 2001, An elasticity solution for functionally graded beams, Composites Science and Technology 61(5): 689-696.
[6] Ding H., Huang D., Chen W., 2007, Elasticity solutions for plane anisotropic functionally graded beams, International Journal of Solids and Structures 44(1): 176-196.
[7] Zhong Z., Yu T., 2007, Analytical solution of a cantilever functionally graded beam, Composites Science and Technology 67(3): 481-488.
[8] Kapuria S., Bhattacharyya M., Kumar A., 2008, Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation, Composite Structures 82(3): 390-402.
[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] Nguyen T.K., Vo T.P., Thai H.T., 2013, Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory, Composites Part B: Engineering 55: 147-157.
[11] Pradhan K., Chakraverty S., 2014, Effects of different shear deformation theories on free vibration of functionally graded beams, International Journal of Mechanical Sciences 82: 149-160.
[12] Sallai B., Hadji L., Daouadji T.H., Adda B., 2015, Analytical solution for bending analysis of functionally graded beam, Steel and Composite Structures 19(4): 829-841.
[13] Filippi M., Carrera E., Zenkour A., 2015, Static analyses of FGM beams by various theories and finite elements, Composites Part B: Engineering 72: 1-9.
[14] Jing L.I., Ming P.J., Zhang W.P., Fu L.R., Cao Y.P., 2016, Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method, Composite Structures 138: 192-213.
[15] Aldousari S., 2017, Bending analysis of different material distributions of functionally graded beam, Applied Physics A 123(4): 296.
[16] Ghumare S.M., Sayyad A.S., 2017, A new fifth-order shear and normal deformation theory for static bending and elastic buckling of P-FGM beams, Latin American Journal of Solids and Structures 14:1893-1911.
[17] Elishakoff I., Candan S., 2001, Apparently first closed-form solution for vibrating: inhomogeneous beams, International Journal of Solids and Structures 38(19): 3411-3441.
[18] Huang Y., Li X.F., 2010, A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration 329(11): 2291-2303.
[19] Giunta G., Belouettar S., Carrera E., 2010, Analysis of FGM beams by means of classical and advanced theories, Mechanics of Advanced Materials and Structures 17(8): 622-635.
[20] Sarkar K., Ganguli R., 2013, Closed-form solutions for non-uniform Euler-Bernoulli free-free beams, Journal of Sound and Vibration 332(23): 6078-6092.
[21] Sarkar K., Ganguli R., 2014, Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition, Composites Part B: Engineering 58: 361-370.
[22] Li X.F., Kang Y.A., Wu J.X., 2013, Exact frequency equations of free vibration of exponentially functionally graded beams, Applied Acoustics 74(3): 413-420.
[23] Tang A.Y., Wu J.X., Li X.F., Lee K., 2014, Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams, International Journal of Mechanical Sciences 89: 1-11.
[24] Nguyen N., Kim N., Cho I., Phung Q., Lee J., 2014, Static analysis of transversely or axially functionally graded tapered beams, Materials Research Innovations 18: 260-264.
[25] Kukla S., Rychlewska J., 2016, An approach for free vibration analysis of axially graded beams, Journal of Theoretical and Applied Mechanics 54(3): 859-870.
[26] Huang Y., Rong H.W., 2017, Free vibration of axially inhomogeneous beams that are made of functionally graded materials, International Journal of Acoustics & Vibration 22(1): 68-73.
[27] Shahba A., Attarnejad R., Marvi M.T., Hajilar S., 2011, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites Part B: Engineering 42(4): 801-808.
[28] Shahba A., Attarnejad R., Hajilar S., 2013, A mechanical-based solution for axially functionally graded tapered Euler-Bernoulli beams, Mechanics of Advanced Materials and Structures 20(8): 696-707.
[29] Shahba A., Rajasekaran S., 2012, Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials, Applied Mathematical Modelling 36(7): 3094-3111.
[30] Li S., Hu J., Zhai C., Xie L., 2013, A unified method for modeling of axially and/or transversally functionally graded beams with variable cross-section profile, Mechanics Based Design of Structures and Machines 41(2): 168-188.
[31] Arefi M., Rahimi G. H., 2013, Nonlinear analysis of a functionally graded beam with variable thickness, Scientific Research and Essays 8(6): 256-264.
[32] Giunta G., Belouettar S., Ferreira A., 2016, A static analysis of three-dimensional functionally graded beams by hierarchical modelling and a collocation meshless solution method, Acta Mechanica 227(4): 969-991.
[33] Arefi M., Zenkour A. M., 2017, Vibration and bending analysis of a sandwich micro-beam with two integrated piezo-magnetic face-sheets, Composite Structures 159: 479-490.
[34] Arefi M., Zenkour A. M., 2017, Size-dependent vibration and bending analyses of the piezo-magnetic three-layer nano-beams, Applied Physics A 123(3): 202.
[35] ZenkourA. M., Arefi M., Alshehri N. A., 2017, Size-dependent analysis of a sandwich curved nano-beam integrated with piezo-magnetic face-sheets, Results in Physics 7: 2172-2182.
[36] Arefi M., Zenkour A. M., 2016, A simplified shear and normal deformations nonlocal theory for bending of functionally graded piezo-magnetic sandwich nano-beams in magneto-thermo-electric environment, Journal of Sandwich Structures & Materials 18(5): 624-651.
[37] Li X., Li L., Hu Y., Ding Z., Deng W., 2017, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures 165: 250-265.
[38] Sayyad A.S., Ghugal Y.M., 2017, Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures 171: 486-504.
[39] Kapuria S., Kumari P., 2012, Multi-term extended Kantorovich method for three dimensional elasticity solution of laminated plates, Journal of Applied Mechanics 79(6): 061018.
[40] Kumari P., Kapuria S., Rajapakse R., 2014, Three-dimensional extended Kantorovich solution for Levy-type rectangular laminated plates with edge effects, Composite Structures 107: 167-176.
[41] Kumari P., Singh A., Rajapakse R., Kapuria S., 2017, Three-dimensional static analysis of Levy-type functionally graded plate with in-plane stiffness variation, Composite Structures 168: 780-791.
[42] Kapuria S., Dumir P., Jain N., 2004, Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams, Composite Structures 64(3):317-327.
[43] ABAQUS/STANDARD, 2009, User’s Manual, Version: 6.9-1.