Vibration analysis of functionally graded beams using a higher-order shear deformable beam model with rational shear stress distribution

Abstract This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams. In the element model proposed, the stress equilibrium equation is employed to derive the rational distribution of transverse shear stress. The beam element formulation oriented with vibration analysis is derived on the basis of the variational principle with mixed energy. To visualise the effectiveness and applicability of the proposed mixed beam element model in vibration analysis, numerical examples have been studied. According to the numerical results, in comparison with the classical beam element model and the beam element models based on the conventional first-order or higher-order shear deformation theory, the vibration frequencies obtained using the mixed beam element model are significantly more satisfactory. Moreover, the comparative study demonstrates that, for vibration analysis of FG sandwich beams, the rational distribution of transverse shear stress is of decisive significance to achieve high-precision solutions. With the mixed beam element used, the load–frequency relationship of FG sandwich beams is discussed. As indicated by the results, in addition to the axial force , the bending moment exerts a significant difference upon the vibration frequencies of the FG beams.