An investigation on the shear deformation under bending conditions of cantilever FGM beams using a new polynomial shear function


  • Aissa Boussouar
  • Attia Bachiri
  • Bachir Taallah
  • Ali Zaidi



high order theories, mathematical model, cantilever beams, new polynomial shear function, displacements fields, shear stress


In this paper, a static analysis to establish a mathematical model using high order bending theories considering shear strains in displacement fields which have not been taken into account by other theories for functionally graded material (FGM) cantilever beams under bending. A new polynomial shear function is used in this investigation, when satisfied the stress-free boundary conditions. These theories do not require a shear correction factor and consider a hyperbolic shape function. Material properties are assumed to vary in the thickness direction, a simple power-law distribution in terms of volume fractions of constituents is considered. Illustrative cases, a cantilever FGM beam subjected to a concentrated shear force at the free end, and also as a cantilever FGM beam with uniformly distributed load are presented the originality of this research work, in this investigation. The mathematical model is established by differential equations which are derived by the principle of virtual work. Equilibrium equations and boundary conditions are introduced. The solution model is based on a variation approach (integrals) to predict the field component of displacements and the basic constitutive laws. The solution of the analytical model is presented. The results in terms of displacement fields including rotation of the section, and shear stresses, predicted from    the proposed model, are presented.


Sankar,.B.V. (2001). An elasticity solution for functionally graded beams. Composite science technologie Journal. 61 , 689–696.


Mahi. A,AddaBedia. E.A., Tounsi.A., Mechab,I.(2010). An analytical method for temperature dependent free vibration analysis of functionally graded beams with general boundary conditions. Composite structures. 92, 1877–1887.

AkbasS., D. (2018). Large deflection analysis of a fiber reinforced composite beam. Steel and Composite Structures. 27, 567-576.‎


Aydogdu, M., V. Taskin. (2007). Free vibration analysis of functionally graded beams with simply supported edges. Materials and Design.28, 1651-1656.

Benatta, M.A., Mechab, I., Tounsi, A., AddaBedia, E.A. (2008). Static analysis of functionally graded short beams including warping and shear deformation effects.J Computational Materials. 44, 765-773.


Benatta, M.A., Tounsi, A., Mechab, I., Bouiadjra, M.B. (2009). Mathematical solution for bending of short hybrid composite beams with variable fibers spacing.Applied Mathematics and Computation. 212, 337-348.


Bouremana M., Houari M.S.A., Tounsi A., Kaci A., Bedia E.A.A. (2013). A new first shear deformation beam theory based on neutral surface position for functionally graded beams. Steel and Composite Structures. 15, 467-479.DOI: 10.12989/scs.2013.15.5.467

Ghugal Y. M., Sharma, R. (2009). A Hyperbolic Shear Deformation Theory for Flexure and Vibration of Thick Isotropic Beams. International Journal of Computational Methods. 6, 585-604.DOI:10.1142/S0219876209002017

Ghugal Y. M. and Sharma, R., (2011).A Refined Shear Deformation Theory for Flexure of Thick Beams, Latin American Journal of Solids and Structures. 8,183-193.

DahakeA. G., Yuwaraj M. Ghugal, (2013). Flexure of Thick Simply Supported Beam Using Trigonometric Shear Deformation Theory. International Journal of Scientific and Research Publications. 51, 1-7.


Gandhe G. R., P. M. Pancake, D. H. Tupe, P. G. Taur(2018). Higher Order Computational Method for Static Flexural Analysis of thick beam. Procedia Manufacturing. 20, 493–498.

Guenfoud, H., Ziou, H., Himeur, M., Guenfoud, H. (2016). Analyses of a composite functionally graded material beam with a new transverse shear deformation function. Journal applied engineering science technologie,2, 105-113.

HuangY., Li X.F.A. (2010). New approach for freevibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration. 329, 2291–2303.DOI: 10.1016/j.jsv.2009.12.029

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, 1-11.

Ziou, H., Guenfoud., M., Guenfoud, H.(2021). A Simple Higher-order Shear Deformation Theory for Static Bending Analysis of Functionally Graded Beams. Jordan Journal of Civil Engineering. 15, 209-224.

Zhong, Z.,Yu, T.(2007). Analytical solution of cantilever functionally graded beams. Composites science technology Journal. 67, 481–488.


Reddy. J.N.(1984). A simple higher order theory for laminated composite plates.Journal of Applied Mechanics. 51, 745-752.


Reddy, J.N. (2000). Analysis of functionally graded plates.International journal fornumerical methods in engineering.47, 663-84.


Zaoui, F. Z., Hanifi, A., Younsi, A., Meradjah, M, Tounsi., A., Ouinas. D. (2017). Free vibration analysis of functionally graded beams using a higher-order shear deformation theory. Mathematical Modelling of Engineering Problems. 4, 7-12.DOI: 10.18280/mmep.040102




How to Cite

Boussouar, A., Bachiri, A., Taallah, B., & Zaidi, A. (2024). An investigation on the shear deformation under bending conditions of cantilever FGM beams using a new polynomial shear function . STUDIES IN ENGINEERING AND EXACT SCIENCES, 5(1), 223–241.