Stretchability of the stretchable and flexible electronics involves the post-buckling behaviors of internal connectors that are designed into various shapes of curved beams. The linear displacement–curvature relation is often used in the existing post-buckling analyses. Koiter pointed out that the post-buckling analysis needs to account for curvature up to the fourth power of displacements. A systematic method is established for the accurate post-buckling analysis of curved beams in this paper. It is shown that the nonlinear terms in curvature should be retained, which is consistent with Koiter's post-buckling theory. The stretchability and strain of the curved beams under different loads can be accurately obtained with this method.