Influence of an elastic end support on the dynamic stability of Beck's column with multiple weak sections

Recent investigations have shown that the presence of weak cross sections can deeply modify flutter and divergence instability of a cantilever beam-column. Position, intensity of weakness and degree of non-conservativeness not only can alter the value of the critical load of the healthy column but can also produce a modification of the type of instability. In this paper, issues involved in the influence of an elastic end support on flutter and buckling instability of the Beck's column in presence of an arbitrary number of weak sections is investigated. In the literature, a numerical study, restricted to the case of a single weak section only, has been presented. The study here proposed has been motivated by an extension to the case of multiple weak sections by means of a model that does not require continuity conditions to be enforced. On the other hand, the latter extension has led to a surprising contradiction of the results previously divulgated for the propped Beck's column with a single weak section. The exact solution, in terms of mode shapes and characteristic eigen-value equation of the weakened propped cantilever has been obtained in an explicit suitable form, through the use of generalised functions. The extensive numerical applications aim at the investigation of the effect of different debilitation scenarios in the flutter and buckling instability of the propped cantilever. In particular, the results relative to the case of single-weak section propped column, already investigated in the literature, are discussed in this work and they are shown that do not match those obtained by other authors. The latter incongruence is duly highlighted and discussed to infer the specific motivation.