Abstract: The catenary problem is a very classical mechanical problem. There are many methods to derive the catenary equation from static equilibrium methods to variational principles. This paper establishes and solves the catenary equation based on the discrete static equilibrium model. Furthermore, this paper establishes a discrete static equilibrium model with torsion spring, then derives and solves the beam displacement equation. In the end, the beam displacement equations with different parameters are solved to obtain the gradual solution, and the existence of various equilibrium states is analyzed.