Abstract: Conventional finite element methods, together with the absolute nodal coordinate formulation (ANCF), are subject to inherent limitations in the accurate geometric representation of circular-arc structures. For key components with circular-arc geometric features that are commonly encountered in engineering fields such as aerospace and robotics, including ring brackets, curved guide rails, and springs, it is typically necessary to employ a multi-segment discretization, whereby the underlying circular arc is approximated piecewise by multiple elements. However, such a multi-segment approximation not only degrades the geometric consistency of flexible system models and introduces non-negligible deviations in the predicted dynamic responses, but also substantially increases the element count and the overall computational cost. To address the above issues, this study proposes a direct global-coordinate definition method for a standard circular arc element, according to its geometric characteristics, based on the rational absolute nodal coordinate formulation (RANCF). Inter-element continuity constraints and the corresponding control strategy between adjacent elements are further formulated. By explicitly introducing the unit normal vector \boldsymbol\nu associated with the plane in which each circular-arc element is embedded, together with the scalar magnitude λ of the element gradient vector, the proposed formulation provides a clear and unambiguous geometric characterization of the element, thereby guaranteeing the uniqueness of the circular-arc element definition. In addition, these two geometric descriptors allow the continuity constraints at inter-element junctions to be expressed in a more direct manner, which substantially simplifies the practical procedure for enforcing and controlling continuity at connection points. As a result, further enhances the overall geometric consistency, and controllability of the modeling process. Furthermore, a curved-beam pendulum model and a flexible circular-arch model are employed as representative examples to comparatively investigate the dynamic responses and solution efficiency under different continuity conditions (C¹ and C²). The simulation results demonstrate that the C² curvature-continuous model exhibits a pronounced advantage in the efficiency and numerical stability of dynamic simulations, thereby providing a practical modeling and analysis approach for engineering simulations of flexible components with circular-arc geometric features.