Shear deformation mathematical modeling of functional graded clamped beams under bending

In this paper, high order bending theories are used to develop an analytical model considering shear strains in displacement fields which have not been taken into account by other theories for functionally graded material (FGM) symmetric beams under bending. A new polynomial shear function satisfied the stress-free boundary conditions, is used in this investigation.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. 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 for the beam case with clamped ends represents the originality of this research work presented in this paper. Results in terms of displacement fields including rotation of the section, deformations, and stresses, predicted from the proposed model and compared with those of simply supported end beams found in the literature, are presented. This contribution helps to understand the FGM beams and the use of the proposed mathematical model.