Free vibration of nonlocal radially graded nanotubes with cross-sectional warping

This article presents a nonlocal higher-order beam theory to treat the transverse free vibration of functionally graded (FG) nanotubes with radial inhomogeneity considering cross-sectional warping. A novel exponential shape function is constructed to model the cross-sectional warping. The suggested beam model does not need the shear correction coefficient but is capable of handling both size-dependent behavior and fulfilling shear-stress-free conditions on the inner and outer surfaces. A governing equation is derived and the frequency equations of free vibration of FG nanotubes are obtained under typical boundary conditions. The influences of the nonlocal, geometric, and material parameters on the natural frequencies are discussed.