05.41: A unified catastrophe theory approach for the in-plane buckling of steel arches under point gravitational loading

The problem of the buckling response of various types of steel arches has been the subject of a vast number of earlier as well as recent papers and chapters of books. Namely, the majority of all these works aimed – via analytical and/or computational approaches – to establish the nature of the postbuckling response of the foregoing structures. It has been found over the years that two types of loss of in-plane stability are dominant: (a) snap-through, a characteristic of shallow (flat) arches (low height to span ratio), in a symmetric or asymmetric mode (limit point or bifurcation), and (b) sideways buckling (deep arches). In the latter situation, the axial inextensibility (being a justified simplifying assumption) becomes dominant and in the case of radial load and circular arch configuration the whole analysis is rather easy and convenient, using equilibrium considerations. On the other hand, low arches (with unavoidable shortening of their center line) can be thought of as curved beams; hence their buckling analysis may be performed from the viewpoint of energy. However, there not seems to be a unified approach overall, especially since the geometry of arches may be significantly varying, from circular to elliptical, parabolic, sinusoidal and catenary, and hence the height at the crown and the span are not the only parameters this geometry is dependent on. Moreover, if one considers a point gravitational loading acting on the arch, a case scarcely reported, then the whole analysis is multiparametric, since it involves (a) the value of the load and its position, (b) the cross-sectional and material characteristics and (c) the geometrical configuration parameters. This work aims to provide a unified approach for the foregoing problem based on the Theory of Catastrophes. The total potential energy function involving all the parameters is firstly established, and furthermore all restrictions related to boundary conditions, geometry, material and cross-sectional properties. Using an approximate 4th order polynomial postbuckling shape, it was found that the problem at hand is governed by a singularity that strongly resembles the butterfly. The work is ongoing and will hopefully reveal interesting future results for structural design purposes. © Ernst Sohn Verlag für Architektur und technische Wissenschaften GmbH Co. KG, Berlin.