Full length articleLateral-torsional buckling of fixed circular arches having a thin-walled section under a central concentrated load
Introduction
The elastic lateral-torsional buckling of pin-ended circular arches that are subjected to nominal in-plane uniform compression or bending moment has been studied extensively, and analytical solutions for the buckling loads have been obtained by a number of researchers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Although the lateral-torsional buckling of arches is more complicated than the flexural or torsional buckling of columns and the lateral-torsional buckling of beams due to couplings between the lateral and torsional buckling deformations, a trivial prebuckling stress state was assumed in classical lateral-torsional buckling analyses for such arches, which makes the prebuckling analysis relatively simple [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. In classical analyses, uniform compression in a circular arch is produced by a uniform radial load, while uniform bending is produced by applying equal and opposite bending moment to both ends of a simply supported arch. However, Pi et al. [14], [15], [16], [17], [18] found that the stress state in arches subjected to a uniform radial load q is non-trivial. Although the uniform radial load produces a uniform compressive force in a circular arch, its magnitude in shallow arches is much smaller than the trivial value N=qR (R is the radius of the arch) and the bending moment in shallow arches is substantial. When a circular arch is subjected to a central concentrated load, the axial compressive force and bending moment in the arch produced by the load are non-uniform with much more complicated distribution patterns, which makes its lateral-torsional buckling analysis difficult [14], [15], [16], [17], [18]. Pi et al. [14], [15], [16], [17], [18] investigated the lateral-torsional buckling of out-of-plane pin-ended circular arches that are subjected to a uniform radial load or a central concentrated load and derived an analytical solution for the lateral-torsional buckling load. However, in many cases of engineering practice, both ends of arches are out-of-plane fixed. Lateral-torsional buckling of fixed circular arches under nominal axial compression has been studied in [19], [20], [21], [22] by assuming that prebuckling stress state is trivial. Fixed arches under a central concentrated load are subjected to combined axial compressive and bending actions and so the prebuckling stress state is expected to be quite complicated, which have to be considered in the lateral-torsional buckling analysis of fixed arches. In addition, the lateral-torsional buckling mode shape of fixed arches under a central concentrated load is much complicated than that of out-of-plane pin-ended arches. Hence, it is difficult to obtain analytical solutions for the lateral-torsional buckling load of fixed arches under a central concentrated load and recourse to numerical methods such as the finite element methods may be made to compute their lateral-torsional buckling loads [23], [24], [25], [26], [27]. Analytical solutions for the elastic lateral-torsional buckling load of an arch having out-of-plane fixed boundary condition under central concentrated load and comprehensive investigation of the corresponding lateral-torsional buckling behaviour do not appear to be reported in the literature.
The purpose of this paper is to investigate the elastic lateral-torsional buckling behaviour of fixed circular arches having a thin-walled cross-section under an in-plane central concentrated load. Accurate prebuckling analyses are conducted to determine the distributions of prebuckling axial compressive and bending actions. The correct lateral-torsional buckling mode shape is explored. With the accurate axial compressive and bending actions and the correct lateral-torsional buckling mode shape, the analytical solution for the elastic lateral-torsional buckling load of fixed circular arches is derived. The effects of the in-plane boundary conditions, load height, cross-section and slenderness ratio on the lateral-torsional buckling of fixed arches are also investigated. All analytical solutions are verified by independent finite element results. This paper provides structural researchers and designers with a deep insight and useful analytical solutions for lateral-torsional buckling of circular arches.
Section snippets
Prebuckling analysis
The fixed circular arch investigated in this paper is shown in Fig. 1, where R is the radius, S the length, L the span, H the rise, and 2α the included angle of the arch. The lateral, radial and tangential displacements of the arch axis in the direction x, y, and z are denoted by u(φ), v(φ), and w(φ), and the twist torsion of the cross-section by θ(φ), where φ is the angular coordinate.
Before tackling the lateral-torsional buckling analysis of a fixed circular arch, it is important that
Lateral-torsional buckling
During lateral-torsional buckling, the deformations of an arch bifurcate from a primary in-plane equilibrium configuration to a lateral-torsional buckled configuration under the constant central load Q. At the infinitesimally close buckling equilibrium point, the central load Q, the prebuckling stresses, the axial compressive force N, the bending moment M and the in-plane radial and axial displacements and remain unchanged in association with the lateral buckling displacement and twist
Comparisons with finite element results
To verify the analytical solution of Eq. (51) for the lateral–torsional buckling load, the results obtained from Eq. (51) are compared with the results of ANSYS [28] for arches with different in-plane boundary condition, included angles and cross-sections. The elastic lateral-torsional buckling loads were obtained by the eigenvalue analysis provided by ANSYS [28]. In the FE analyses, the rectangular box section and I-section and the material properties used in Section 3 are adopted; arches were
Conclusions
This paper has investigated the elastic lateral–torsional buckling of out-of-plane fixed circular arches having a thin-walled section under a central concentrated load. Accurate prebuckling analyses were carried out to derive the analytical solutions for the in-plane axial compressive force and bending moment of in-plane fixed and pin-ended arches. Analytical solutions of the buckling load for out-of-plane fixed arches having in-plane fixed and pin-ended boundary conditions were derived using
Acknowledgements
The research of this paper was financially supported by the National Natural Science Foundation of China (No. 51578166) and by a Yangcheng Research Fellowship (No. 1201541551) awarded to the first author by Technology Planning Project of Guangdong Province (No. 2016B050501004).
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