# Dynamic buckling of slender variable section members to self-weight

## Palavras-chave:

dynamic stability, self-weight, Rayleigh’s Method, optimization## Resumo

We present a study of the dynamic buckling of variable section very slender members due to their self-weight. Modern materials and powerful new analysis methods are leading to the design of very slender aerospace structures that may be prone to instability issues. Elastic stability of such structural systems is a problem inside the

scope of Non-Linear Dynamics Analysis Methods. An indicator of instability is when the structure's free vibration frequency tends to zero. Two factors affect these frequency results. First the stiffness, composed of elastic stiffness, always positive and non-zero, that diminishes rapidly with height, and the geometric stiffness, negative for compressive forces, whose absolute value grows as the structure gets taller and heavier. Second, the mass, that also grows with the height of the structures, and is always positive. To access this behavior, we present a one-

degree-of-freedom mathematical model of a cantilever vertical member via Rayleigh's Method, adopting a cubic polynomial as shape function. Closed form formulas are obtained for elastic and geometric equivalent stiffness, as

well as for equivalent mass, dependent on the member length and transverse section variable characteristics. For some adopted numerical geometric and material properties we use an optimization algorithm to maximize the member length. Check of the formulation is possible for the prismatic member case.