Localization in the buckling or in the vibration of a two-span weakened column

This paper is devoted to buckling or dynamic analysis of two-connected beams with a variation of the bending connection and minor perturbations of the length of the beams. The point of reduced bending stiffness represented by a rotational spring may result from a crack. This rotational spring can also be associated with semi-rigid connection in the field of steel or composite structures for instance. The buckling or the dynamics of this two-span weakened column appears to exhibit strong localization for small values of flexibility of the rotational spring. The buckled mode shapes indicate a strong confinement of the buckling to a fraction of the column. A quantitative criterion of localization has been established and has been correlated to well known phenomena such as the curve veering effect or close eigenvalues. Such a result is quite encouraging as localization is strongly associated with the values of the rotational spring. When considering the open crack analogy, localization only appears for a severely damaged beam. It can then be understood as an indicator of the damage level of the global structure.