Rotation-free shell formulations were proved to be an effective approach to speed up solving large-scaled problems. It reduces systems' degrees-of-freedom (DOF) and avoids shortages of using rotational DOF, such as singular problem and rotational interpolation. The rotation-free element can be extended for solving geometrically nonlinear problems using a corotational (CR) frame. However, its accuracy may be lost if the approach is used directly. Therefore, a new nonlinear rotation-free shell element is formulated to improve the accuracy of the local bending strain energy using a CR frame. The linear strain for bending is obtained by combining two re-derived elements, while the nonlinear part is deduced with the side rotation concept. Furthermore, a local frame is presented to correct the conventional local CR frame. An explicit tangential stiffness matrix is derived based on plane polar decomposition local frame. Simple elemental rotation tests show that the stiffness matrix and the proposed local frame are both correct. Several numerical examples and the application of drape simulations are given to verify the accuracy of nonlinear behavior of the presented element, and some of the results show that the presented method only requires few elements to obtain an accurate solution to the problem studied.