Keywords: arches, buckling, central, concentrated, flexural-torsional, load, postbuckling.

This paper is concerned with the elastic flexural-torsional buckling and postbuckling of circular arches that are subjected to a central concentrated load. The elastic buckling of circular arches that are subjected to uniform bending and uniform compression has been studied extensively [1,2,3,4,5], and closed form solutions have been obtained for their buckling loads [1,2,3]. However, the elastic flexural-torsional buckling and postbuckling of arches that are subjected to a central concentrated load have received little attention, and research into this issue of instability is presented in this paper.

Because an arch subjected to a central concentrated load experiences combined axial compressive and bending actions, it may be thought that the elastic flexural-torsional buckling behaviour of the arch is similar to that of an arch in uniform bending or in uniform compression. This is correct in some cases. However, the axial compressive and bending actions produced within an arch subjected to a central concentrated load are related to its included angle and slenderness. When the slenderness of an arch is constant, the values of the axial compressive actions produced by the central concentrated load in a shallow arch (with a small included angle) may be higher than those in a deep arch (with a large included angle). When the included angle of a shallow arch is constant, the values of the axial compressive actions may be higher in a slender arch than in a stocky arch. The large axial compressive actions combined with the bending actions may cause premature elastic flexural-torsional buckling. For fixed arches, the deformed profile of an arch prior to buckling is related to its included angle and slenderness. The number of the inflexion points along a deep arch (with a large included angle) may be more than that along a shallow arch (with a small included angle), so that the effective length of a deep arch may be smaller than that of a shallow arch, which will lead to a higher buckling load for a deep fixed arch than that for a shallow fixed arch.

Finite element models based on a classical buckling theory and eigenvalue analysis are often used to compute the elastic flexural-torsional buckling load of structures both in research and practice. However, a postbuckling analysis cannot be carried out by these types of finite element models. This paper uses a rational finite element model for the 3D elastic large deformation analysis of arches, developed by the authors, to investigate the elastic flexural-torsional buckling and postbuckling behaviour of circular arches that are subjected to a central concentrated load.

It is found that the buckling load of a simply supported arch decreases as its included angle increases, but the slenderness of a simply supported arch has little effect on its buckling behaviour. For shallow pin-ended arches, the elastic flexural-torsional buckling loads are reduced significantly by the large axial compressions developed in the arch prior to buckling. As the included angle increases, the buckling load of the pin-ended arches decreases significantly until a minimum value of the buckling load is reached, and then increases. The increase of the buckling load stops at a certain value of the included angle, and thereafter the buckling load steadily decreases with the increase of the included angle. For stocky pin-ended arches, the buckling load decreases steadily with an increase of the included angle. The slenderness of slender and moderate pin-ended arches has a significant effect on their buckling behaviour while the slenderness of a deep pin-ended arch has a very small effect on its buckling behaviour.

The increase in the postbuckling strength of simply supported arches is negligible or very small. There is a substantial postbuckling response for shallow pin-ended arches due to a postbuckling relaxation of the axial compression. The slenderness of fixed arches has a significant effect on their buckling behaviour. The large axial compression developed in shallow fixed arches reduces the elastic flexural-torsional buckling loads significantly. Shallow fixed arches also have a substantial postbuckling response due to the postbuckling relaxation of the axial compression and the moment redistribution. For slender fixed arches with moderate or large included angles, four inflexion points can be developed in their deformed profile, which reduce their effective length and lead to a significant increase of their flexural-torsional buckling load. After buckling, moment redistribution takes place which increases the moments at the supports and decreases the moment at mid-span, thereby increasing the postbuckling strength. For stocky fixed arches, however, only two inflexion points can be developed in their deformed profile prior to buckling and the buckling load decreases slightly as the included angle increases.

1

Vlasov, V. Z. Thin walled elastic beams. 2d Ed. National Science Foundation, Washington, D.C., 1961

2

Papangelis, J.P. and Trahair, N.S. "Flexural-torsional buckling of arches", Journal of Structural Engineering, ASCE 113(4), 889-906, 1987. doi:10.1061/(ASCE)0733-9445(1987)113:4(889)

3

Pi, Y.-L., Papangelis, J. P. and Trahair, N.S. "Prebuckling deformations and flexural-torsional buckling of arches", Journal of Structural Engineering, ASCE 121(9), 1313-1322, 1995. doi:10.1061/(ASCE)0733-9445(1995)121:9(1313)

4

Yang, Y-B., Kuo, S-R. and Cherng, Y.D. "Curved beam elements for non-linear analysis", Journal of Engineering Mechanics, ASCE. 1989, 115(4), 840-855. doi:10.1061/(ASCE)0733-9399(1989)115:4(840)

5

Rajasekaran, S. and Padmanabhan, S. "Equations of curved beams", Journal of Engineering Mechanics, ASCE 115(5), 1094-1111, 1989. doi:10.1061/(ASCE)0733-9399(1989)115:5(1094)

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