Isogeometric vibration analysis of free-form Timoshenko curved beams

In this paper, the finite free-form curved beam element is formulated by the isogeometric approach based on the Timoshenko Rcurved beam theory to investigate the free vibration behavior of the curved beams with arbitrary curvature. The non-uniform rational B-splines (NURBS) functions which define the geometry of the curved 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 free-form curved beams are modelled in a unique way based on NURBS. The gap between the free vibration analysis of the curved beams with constant curvature and those with variable curvature is eliminated. All the effects of the axis extensibility, the shear deformation, and the rotary inertia are taken into consideration by the present isogeometric model. Results of the parabolic and elliptic curved beams for non-dimensional frequencies are compared with other available results in order to show the accuracy and efficiency of the present isogeometric approach. Furthermore, the free vibration analysis of the elliptic thick rings is presented. Particularly, the Tschirnhausen's cubic curved beam is considered to study the dynamic behavior as an example of free-form curved beams.