  The linear instability of a beam tensioned by its own weight is considered.
It is shown that for long beams, in the sense of an adequate dimensionless
parameter, the characteristics of the instability caused by a follower force do
not depend on the length. The asymptotic regime significantly differs from that
of short beams: flutter prevails for all types of follower loads, and flutter
is localized at the edge of the beam. An approximate solution using matched
assymptotic expansion is proposed for the case of a semi-infinite beam. Using a
local criterion based on the stability of waves, the characteristics of this
regime as well as its range of application can be well predicted. These results
are finally discussed in relation with cases of flow-induced instabilities of
slender structures.
