Affiliations: Department of Mathematics, Federal University of
Goiás, UFG, Campus of Catalão, 75705-220 Catalão, GO, Brazil | Civil Engineering Department, Pontifical Catholic
University, PUC-Rio, 22453-900, Rio de Janeiro, RJ, Brazil
Note: [] Corresponding author: Prof. Paulo Batista Gonçalves, Civil
Engineering Department, Pontifical Catholic University, PUC-Rio, 22453-900, Rio
de Janeiro, RJ, Brazil. Tel.: +55 21 3114 1188; Fax: +55 21 3114 1195; E-mail:
paulo@civ.puc-rio.br
Abstract: The complexity of the response of a beam resting on a nonlinear
elastic foundation makes the design of this structural element rather
challenging. Particularly because, apparently, there is no algebraic relation
for its load bearing capacity as a function of the problem parameters. Such an
algebraic relation would be desirable for design purposes. Our aim is to obtain
this relation explicitly. Initially, a mathematical model of a flexible beam
resting on a non-linear elastic foundation is presented, and its non-linear
vibrations and instabilities are investigated using several numerical methods.
At a second stage, a parametric study is carried out, using analytical and
semi-analytical perturbation methods. So, the influence of the various physical
and geometrical parameters of the mathematical model on the non-linear response
of the beam is evaluated, in particular, the relation between the natural
frequency and the vibration amplitude and the first period doubling and
saddle-node bifurcations. These two instability phenomena are the two basic
mechanisms associated with the loss of stability of the beam. Finally
Melnikov's method is used to determine an algebraic expression for the boundary
that separates a safe from an unsafe region in the force parameters space. It
is shown that this can be used as a basis for a reliable engineering design
criterion.
Keywords: Beam on elastic foundation, non-linear oscillations, Duffing equation, Melnikov method, Ramberg-Osgood constitutive law
