To study the geometrically nonlinear internal force and displacement calculation and the in-plane stability of circular arch structures, the analytical displacement shape function is established based on the geometrically nonlinear static analysis model of a circular arch. The analytical geometrically nonlinear element for circular arches is constructed by using energy method and potential energy variational principle. The element formulas are obtained. The element model in this paper is theoretically degenerated without the axial deformation. Then the analytical geometrically nonlinear element for circular arches without compression is obtained. The element was used in the in-plane stability analysis of circular arches. The results show that the critical load coefficient of the in-plane bifurcation instability obtained by this element is consistent with the results of classical Timoshenko model, Dinnik model and static method model. The relative error is 0%. The critical load of in-plane bifurcation instability is also in accordance with that of in-plane extreme point instability. The element presented in this paper is of analytical type, high accuracy and good applicability. It can be used for geometrically nonlinear displacement and internal force analysis and in-plane stability analysis of circular arches under any load cases.