In order to increase the filling factor of conductor in superconducting coils and improve the mechanical stability of superconducting magnet, the pre-tension is always applied to the conductor during winding the coils. Because the winding pre-tension has a great effect on the quench and degradation performance of superconducting magnet, it is necessary to analyze the mechanical stress caused by the fabrication. First, the winding physical process of conductor is analyzed. Then the theoretical model is developed to calculate the winding stress of superconducting coils based on some reasonable assumptions and approximations. And some formulas used for stress and strain are derived from the theory of elastic mechanics. Two kinds of superconducting coils (one consists of one type of wire, and the other one consists of two types of wires.) are researched according to the model. The effects of winding pre-stress and material anisotropy on radial stress and hoop stress in superconducting coils are also analyzed. On the basis of the analyzed results, one can further research the stress and strain of superconducting coils under the effect of multiphysics and give some theoretical suggestions for the design and construction of superconducting coils.