  In this paper, the finite free-form beam element is formulated by the
isogeometric approach based on the Timoshenko beam theory to investigate the
free vibration behavior of the beams. The non-uniform rational B-splines
(NURBS) functions which define the geometry of the beam are used as the basis
functions for the finite element analysis. In order to enrich the basis
functions and to increase the accuracy of the solution fields, the h-, p-, and
k-refinement techniques are implemented. The geometry and curvature of the
beams are modelled in a unique way based on NURBS. All the effects of the the
shear deformation, and the rotary inertia are taken into consideration by the
present isogeometric model. Results of the beams for non-dimensional
frequencies are compared with other available results in order to show the
accuracy and efficiency of the present isogeometric approach. From numerical
results, the present element can produce very accurate values of natural
frequencies and the mode shapes due to exact definition of the geometry. With
higher order basis functions, there is no shear locking phenomenon in very thin
beam situations. Finally, the benchmark tests described in this study are
provided as future reference solutions for Timoshenko beam vibration problem.
