Nonlinear free vibration of elastic bars possessing initial imperfections is studied. This generalizes the problem of a perfect bar studied previously by Woinowsky-Krieger and Burgreen. For the vibration frequency the differential equation is solved exactly in terms of complete elliptic integrals. Numerical results demonstrate that the initial imperfections usually reduce the frequency of nonlinear free vibration of elastic bars subjected to moderate compression, but there are exceptions with the opposite effect. Such an effect of incerease in frequency was observed previously in unstiffened and cylindrical shells by Singer et al. The exact solution is contrasted with timewise single-term and two-term Galerkin approximations. It turns out that while the single-term treatment fails to show the increase, the two-term treatment brings it out with attendant close agreement with the exact solution. © 1985 Springer-Veralg.