Abstract: For arbitrarily spatial elastic curved rod elements with circular cross-section, a set of displacement functions fully reflecting the rigid body modes is derived using the classical elasticity theory and mathematic theories of the differential geometry and matrix methods. Both natural (curvilinear) and intrinsic (Lagrangian) coordinate systems are used in the derivation. The displacement functions involve all rigid body and constant strain modes of the arbitrarily spatial elastic curved rods. To verify the formulation, two examples are analyzed using the curved rod element based on the displacement functions derived herein for a static situation. Numerical results are well compared with theoretical solutions. The convergence rate of the element based on the proposed displacement functions is better than that of the element in commercial codes. Due to its higher computational efficiency, the proposed curved element may find its practical use in the non-linear analysis of drill-strings confined in various three-dimensional curved wells.