Guest lectures and seminars - Page 196

This seminar will be focused on some elementary structural systems such as the cantilever beam. The cantilever is an old problem in structural mechanics already investigated by Galileo (1638) from equilibrium and strength arguments. This structural paradigm will be reconsidered here using buckling, post-buckling and inelastic theory. We will first present some fundamental buckling results for axially loaded columns. This model covers the case of a tree under its own weight or gives an answer to Babel mythology, at least from the stability theory point of view. This in-plane buckling problem in presence of distributed and concentrated axial forces has been recently exactly solved using hypergeometric functions. The post-buckling behavior associated with a nonlinear boundaryvalue problem will be also discussed using some asymptotic and numerical methods. The out-of-plane buckling problem of this cantilever beam will be further investigated. The lateral-torsiona l buckling problem of Prandtl (1899) dealing with the stability boundary of a beam loaded by its own weight and a concentrated force will be also solved. The convexity theorem of Papkovitch and Schaefer (1934) will be shown for these structural problems. The seminar will be concluded by the inelastic analysis of the beam in bending. We will show the need to develop a nonlocal plasticity law to describe the post-failure behavior of a beam in presence of softening. Wood’s paradox (1968) is overcome by using a nonlocal plasticity model. The Galileo problem is then revisited in the light of nonlocal mechanics. Applications of such theoretical studies can be found in the field of civil engineering at the macro scale (reinforced concrete design, timber beams, steel or composite beams…), but also at micro- or nano-scales including for instance nanostructures.

Challamel is Professor of Civil Engineering, University of South Brittany, Lorient, France, and Marie Curie Fellow in solid mechanics (faststoffmekanikk) at UiO (2011/2012).