  In this paper, we consider an imperfect finite beam lying on a nonlinear
foundation, whose dimensionless stiffness is reduced from $1$ to $k$ as the
beam deflection increases. Periodic equilibrium solutions are found
analytically and are in good agreement with a numerical resolution, suggesting
that localized buckling does not appear for a finite beam. The equilibrium
paths may exhibit a limit point whose existence is related to the imperfection
size and the stiffness parameter $k$ through an explicit condition. The limit
point decreases with the imperfection size while it increases with the
stiffness parameter. We show that the decay/growth rate is sensitive to the
restoring force model. The analytical results on the limit load may be of
particular interest for engineers in structural mechanics
