An Efficient Finite Element Formulation Based on Deformation Approach for Bending of Functionally Graded Beams

Document Type : Research Paper


1 National Centre of Integrated Studies and research on Building Engineering (CNERIB), Algeria

2 LGCH Laboratory, University of Guelma, Algeria



Finite element formulations based generally on classical beam theories such as Euler-Bernoulli or Timoshenko. Sometimes, these two formulations could be problematic expressed in terms of restrictions of Euler-Bernoulli beam theory, in case of thicker beams due to non-consideration of transverse shear; phenomenon that is known as shear locking characterized the Timoshenko beam theory, in case of thin beams; problem of slow of convergence in regards to the element of Timoshenko beam. In responding to this problematic, a new beam finite element model is developed to study the static bending of functionally graded beams. The originality of this model lies in the use of a deformation approach with the consideration of a central node positioned in the middle of the beam. The degrees of freedom of this node are subsequently eliminated by the method of static condensation. In addition, this model is suitable for all linear structures regardless of L/h ratio. Functionally graded material beams have a smooth variation of material properties due to continuous change in micro structural details. The mechanical properties of the beam are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. A simply supported beam subjected to uniform load for different length-to-thickness ratio has been chosen in the analysis. Finite element solutions obtained with the new finite element model are presented, and the obtained results are evaluated with the existing solutions to verify the validity of the present model. 


[1] Chakraborty A., Gopalakrishnan S., Reddy J.N., 2003, A new beam finite element for the analysis of functionally graded materials, International Journal Mechanic Science 45(3): 519-539.
[2] Kadoli R., Akhtar K., Ganesan N., 2008, Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modeling 32(12): 2509-2525.
[3] Kapuria S., Bhattacharyya M., Kumar A.N., 2008, Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation, Composite Structures 82(3): 390-402.
[4] Pindera M-J., Dunn P., 1995, An evaluation of coupled microstructural approach for the analysis of functionally graded composites via the finite element method, National Aeronautics and Space Administration NASA, Contractor Report 195455.
[5] Ziou H., Guenfoud H., Guenfoud M., 2016, Numerical modelling of a Timoshenko FGM beam using the finite element method, International Journal of Structural Engineering 7(3): 239-261.
[6] Nguyen D.K., Gan B.S., 2014, Large deflections of tapered functionally graded beams subjected to end forces, Applied Mathematical Modeling 38(11-12): 3054-3066.
[7] Nguyen D.K., 2013, Large displacement response of tapered cantilever beams made of axially functionally graded material, Composites Part B: Engineering 55: 298-305.
[8] Kutiš V., Murin J., Belak R., Paulech J., 2011, Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials, Computers and Structures 89(11): 1192-1205.
[9] Murin J., Kutiš V., 2002, 3D-beam element with continuous variation of the cross-sectional area, Computers and Structures 80(3–4): 329-352.
[10] Lazreg H., Ait Amar Meziane M., Abdelhak Z., Hassaine Daouadji T., Adda Bedia E.A., 2016, Static and dynamic behavior of FGM plate using a new first shear deformation plate theory, Structural Engineering & Mechanics 57(1): 127-140.
[11] Rezaiee-Pajand M., Masoodi A.R., Mokhtari M., 2018, Static analysis of functionally graded non-prismatic sandwich beams, Advances in Computational Design 3(2): 165-190.
[12] Benferhat R., Daouadji T.H., Adim B., 2016, A novel higher order shear deformation theory based on the neutral surface concept of FGM plate under transverse load, Advances in Materials Research 5(2): 107-120.
[13] Lazreg H., Nafissa Z., Fabrice B., 2019, An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models, Structural Engineering & Mechanics 69(2): 231-241.
[14] Guenfoud H., Himeur M., Ziou H., Guenfoud M., 2018, The use of the strain approach to develop a new consistent triangular thin flat shell finite element with drilling rotation, Structural Engineering & Mechanics 68(4): 385-398.
[15] Alshorbagy A.E., Eltaher M.A., Mahmoud F.F., 2011, Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling 35: 412-425.
[16] Eltaher M.A., Samir A.E.,Mahmoud F.F., 2012, Free vibration analysis of functionally graded size-dependent nano-beams, Applied Mathematics and Computation 218: 7406-7420.
[17] Eltaher M.A., Samir A.E., Mahmoud F.F., 2013, Static and stability analysis of nonlocal functionally graded nano-beams, Composite Structures 96: 82-88.
[18] Mahmoud F.F., Eltaher M.A, Alshorbagy A.E., Meletis E.I., 2012, Static analysis of nano-beams including surface effects by nonlocal finite element, Journal of Mechanical Science and Technology 26(11): 3555-3563.
[19] Hamed M.A., Eltaher M.A., Sadoun A.M., Almitani K.H., 2016, Free vibration of symmetric and sigmoid of functionally graded nano-beams, Applied Physics A 122(9): 829.
[20] Li X.F., Wang B.L., Han J.C., 2010, A higher-order theory for static and dynamic analyses of functionally graded beams, Archive of Applied Mechanics 80(10): 1197-1212.