Third-order shear deformation beam model for flexural waves and free vibration of pipes

A third-order shear deformation beam model is proposed to analyze dynamic behavior of straight hollow cylinders of annular cross-section, in which shear stress vanishes on the inner and outer surfaces of the pipe. Shear deformation, warping, and rotational inertia of cross-section are all considered, and the shear correction factor is not needed. A single governing differential equation is derived for analyzing flexural wave propagation and free vibration of straight pipe-beams. The phase and group speeds of flexural waves propagating in pipes are determined for acoustic and optical modes. The dispersion of flexural waves is analyzed. The frequency equations are obtained explicitly for pipe-beams with ten typical boundary conditions including clamped, pinned, guided, and free ends. The natural frequencies of clamped-free, clamped-clamped, and pinned-pinned pipe-beams are evaluated for the first four vibration modes. A comparison of this paper's numerical results of the natural frequencies with the previous ones is made and turns out the effectiveness of the suggested method. The influences of the pipe's thickness and length on the natural frequencies and mode shapes for a cantilever pipe are presented.