  This paper is a theoretical and numerical study of the uniform growth of a
repeating sinusoidal imperfection in the line of a strut on a nonlinear elastic
Winkler type foundation. The imperfection is introduced by considering an
initially deformed shape which is a sine function with an half wavelength. The
restoring force is either a bi-linear or an exponential profile. Periodic
solutions of the equilibrium problem are found using three different
approaches: a semi-analytical method, an explicit solution of a Galerkin method
and a direct numerical resolution. These methods are found in very good
agreement and show the existence of a maximum imperfection size which leads to
a limit point in the equilibrium curve of the system. The existence of this
limit point is very important since it governs the appearance of localization
phenomena.
  Using the Galerkin method, we then establish an exact formula for this
maximum imperfection size and we show that it does not depend on the choice of
the restoring force. We also show that this method provides a better estimate
with respect to previous publications. The decrease of the maximum compressive
force supported by the beam as a function of the imperfection magnitude is also
determined. We show that the leading term of the development has a different
exponent than in subcritical buckling of elastic systems, and that the exponent
values depend on the choice of the restoring force.
