Abstract: For one-dimensional (1D) free boundary problems, such as an elastic string or beam in contact with rigid foundations, it derives all the necessary boundary conditions using the free boundary variation. Then, the problem is transformed into a series of two-point boundary value problems with fixed boundaries, and the constraints at the interface point are reasonably relaxed. Furthermore, the implementation of the adaptive finite element solution based on the EEP (element energy projection) technique is implemented. And then the C iteration based on the secant method is conducted to accurately determine the position of the interface point. Finally, the finite element displacement solution and interface point position can both meet the user-specified error tolerances in the maximum norm. The numerical examples show that the algorithm has the advantages such as being able to determine the interface points, being applicable to higher order elements and solving various problems flexibly.