A modified higher-order theory for FG beams

Functionally graded (FG) beams are widely used in many fields. However, the corresponding beam theory is not well established. This paper begins with distinguishing the centroid and the neutral point of cross section. First, the deformation mode is mathematically suggested for axial displacement as a general higher-order form, and then orthogonally decomposed with the help of shear stress free conditions and definitions of generalized displacements (i.e. deflection, rotation and stretch). On this basis, the generalized stresses are defined together with the work conjugated generalized strains, and the decoupled constitutive relations are then derived. Next, the principle of virtual work is proposed for beam problems, and the variationally consistent higher-order theory is established for FG beams, which is as simple as that for a homogeneous beam. Finally, the present theory is demonstrated by typical FG beam problems for both the simply supported case and the clamped case. It is indicated that the analytical solution to the present modified higher-order theory can be regarded as the benchmark of FG beam problems. Furthermore, the relation with the traditional higher-order theory is clarified, which is beneficial to conduct a comparative study on different higher-order beam theories.