A co-rotational beam element for geometrically nonlinear analysis of plane frames is presented. On the base of the shallow arch expression for local strain, the element is formulated by using exact polynomials to interpolate the transversal displacement and rotation. Using the formulated element, the critical load and equilibrium path are computed with the aid of the bracketing procedure and the arc-length control method, respectively. Numerical examples show that the proposed element is capable to give accurate results with a smaller number of elements comparing to the elements previously used in the examples. The effect of the nonlinear term used in the local strain expression on the numerical results is also investigated and highlighted.