  We deal with the as yet unresolved exponential stability problem for Beck's
Problem on a metric star graph with three identical edges. The edges are
stretched Euler--Bernoulli beams which are simply supported with respect to the
outer vertices. At the inner vertex we have viscoelastic damping acting on the
slopes of the edges. We carry out a complete spectral analysis of the system
operator associated with the abstract spectral problem in Hilbert space. Within
this framework it is shown that the eigenvectors have the property of forming a
Riesz (i.e.\ an unconditional) basis, which makes it possible to directly
deduce the exponential stability of the corresponding $C_0$-semigroup using
spectral information for the system operator alone. A physically interesting
conclusion is that the particular choice of vertex conditions ensures the
exponential stability even when the elasticity acting on the slopes of the
edges is absent.
