The postbuckling shape of a slender elastic rod clamped at both ends and subject to an axial compressive force is considered. The critical value \(P_c\) of the magnitude of the axial force is assumed to cause the rod to bend so far out as to come into contact with itself for the case of the first buckling mode. A perturbation analysis for the shape function in terms of the expansion parameter \(P - P_c = \varepsilon >0\) is developed. The dependence of the location of the contact point on the parameter \(\varepsilon\) has the effect of introducing nonuniformity into the usual asymptotic expansion. These difficulties are over come by use of a nonlinear transformation of the independent variable which freezes the contact point.