The performance of a polyester mooring line is non-linear and its elongation plays a significant role in the dynamic response of an offshore moored structure. However, unlike chain, the tension–elongation relationship and the overall behavior of elastic polyester ropes are complex. In this paper, by applying an enhanced stiffness model of the mooring line, the traditional elastic rod theory has been extended to allow for large elongations. One beneficial feature of the present method is that the tangent stiffness matrix is symmetric; in non-linear formulations the tangent stiffness matrix is often non-symmetric. The static problem was solved by Newton–Raphson iteration, whereas a direct integration method was used for the dynamic problem. The computed mooring line tension was validated against the proprietary OrcaFlex software. Results of mooring line top tension predicated by different elongations are compared and discussed. The present method was then used for a simulation of an offshore floating wind turbine moored with taut lines. From a comparison between linear and non-linear formulations, it is seen that a linear spring model under-estimates the mean position when the turbine is operating, but over-estimates the amplitude of the platform response at low frequencies when the turbine has shut down.