Abstract: The explicit expressions of the shape functions and the related stiffness, geometrical stiffness and mass matrices for the partially orthogonal hierarchal finite element of arbitrary high-order for beams are all presented in the closed form in this paper. These expressions are used to evaluate the natural frequencies and the corresponding internal forces of a uniform cantilever beam and a rotating beam with segment-by-segment continuously varying properties, respectively. The results show that the present technique is superior in computational efficiency, accuracy of results, programming, feasibility of use, and so on.