The absolute nodal coordinate formulation(ANCF) proposed by Shabana describes the large rotation by introducing slope coordinates as nodal degrees of freedom. For composite structures composed of beams, plates,shells or blocks,the slope coordinates at the nodes of the elements intersecting at the structural joint points are discontinuous, which causes the difficulties in the modeling and analysis of composite structures. In this paper the methods of dealing with the problem of slope discontinuity in the literature were discussed. Firstly, the absolute nodal coordinate method was briefly introduced. Secondly, the slope discontinuity problems in the classical example of folded beam and in the straight beam with stepped section were discussed in detail. Then by numerically calculating the axial strain on the neutral layer of two examples, the constrained function method was compared with the coordinate transformation method in the existing literature. It was pointed out that the coordinate transformation method in current literatures neglects the difference between the slope degree of freedom and the rotation degree of freedom,and thus the correct axial strains cannot be achieved at the joint node with slope discontinuity, which needs a further study.