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 localised at
the edge of the beam. An approximate solution using matched asymptotic
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.