Abstract: In this paper, the Poisson locking on the consistent rotation-based formulation (CRBF) of the planar beam element based on the absolute nodal coordinate formulation (ANCF) kinematic description is discussed. First of all, in order to fully understand the locking problem of ANCF/CRBF element, two new ANCF/CRBF planar beams are developed by constraining all of the position-vector gradients of ANCF planar fully-parameterized beam with an orthogonal matrix. Using this orthogonal matrix, a nonlinear velocity transformation matrix is evaluated to write the time derivatives of the ANCF gradients at the nodes in terms of the time derivatives of the rotation parameter. Two new ANCF/CRBF beams have a rigid cross-section and no shear effect. One of the new ANCF/CRBF beams has two position vectors, one rotation angle and one axial extensibility parameter as the nodal coordinates, while the other does not have a longitudinal extensibility parameter. Then, the difference in the Poisson locking problem among the two new ANCF/CRBF beams and traditional ANCF/CRBF beam is discussed. It can be concluded that since the constraints imposed on the position-gradient vectors, ANCF/CRBF beam has an insufficient Poisson effect on the nodes and a sufficient Possion effect in the interior of the element. This inconsistent Poisson effect will lead to a more severe locking problem than ANCF fully-parameterized elements. Furthermore, the locking problem increases with the number of constraints on the nodal gradients, that is ANCF/CRBF new beams have a more serious locking effect compared to traditional ANCF/CRBF beam. Finally, in order to achieve a better convergence of the new elements, two locking alleviation methods, Elastic Center Line (ECL) and Strain Split Method (SSM), are used. The locking problem on the performance of ANCF/CRBF beams is tested using several examples that include static and dynamic examples, in order to identify the scope of applicability of such elements. The numerical results obtained from ANCF/CRBF beams are compared with the ANCF beam, and with the conventional beam implemented in a commercial finite element software LS-DYNA.