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Abstract: This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses. [Copyright &y& Elsevier]

Abstract: Arches are often supported elastically by other structural members. This paper investigates the in-plane nonlinear elastic behaviour and stability of elastically supported shallow circular arches that are subjected to a radial load uniformly distributed around the arch axis. Analytical solutions for the nonlinear behaviour and for the nonlinear buckling load are obtained for shallow arches with equal or unequal elastic supports. It is found that the flexibility of the elastic supports and the shallowness of the arch play important roles in the nonlinear structural response of the arch. The limiting shallownesses that distinguish between the buckling modes are obtained and the relationship of the limiting shallowness with the flexibility of the elastic supports is established, and the critical flexibility of the elastic radial supports is derived. An arch with equal elastic radial supports whose flexibility is larger than the critical value becomes an elastically supported beam curved in elevation, while an arch with one rigid and one elastic radial support whose flexibility is larger than the critical value still behaves as an arch when its shallowness is higher than a limiting shallowness. Comparisons with finite element results demonstrate that the analytical solutions and the values of the critical flexibility of the elastic supports and the limiting shallowness of the arch are valid. [Copyright &y& Elsevier]

Abstract: An arch is often connected with other structural members and may be supported by elastic foundations that provide elastic-like restraints to the arch. These elastic restraints participate in the structural response of the arch, and may significantly influence its structural behaviour. A curved-beam element for the structural analysis of arches should therefore consider the effects of these elastic restraints. It is known that the elasto-plastic properties of materials also affect the behaviour of a structure significantly, and so the nonlinearities of the materials need to be included in the curved element for elasto-plastic analysis of an arch. An elastic curved-beam element for the in-plane nonlinear analysis of arches is extended here to an elasto-plastic curved-beam element by formulating the elastic restraints and the material nonlinearities into the element. Geometric nonlinearities in the element are based on an accurate rotation matrix of the two dimensional special orthogonal group that satisfies the desired orthogonality and unimodular conditions. Comparisons of results from the curved element with test results and with analytical solutions demonstrate that the curved-beam element can provide accurate predictions for elastic behaviour, and acceptablly accurate predictions for the elasto-plastic behaviour of elastically restrained arches. [Copyright &y& Elsevier]

Abstract: An arch with an open thin-walled section that is subjected to a radial load uniformly distributed around the arch axis may suddenly buckle out of its plane of loading and fail in a flexural-torsional buckling mode. The classical flexural-torsional buckling load for an arch with an open thin-walled section under a uniform radial load has been obtained by a number of researchers, based on the consideration that the uniform radial load produces a uniform axial compressive force without in-plane bending prior to the occurrence of flexural-torsional buckling. This assumption is correct for deep arches. However, the uniform radial load may produce substantial bending actions in shallow arches prior to flexural-torsional buckling, and so the classical buckling analysis based on the assumption of uniform axial compression may produce incorrect flexural-torsional buckling loads for shallow arches. This paper investigates the flexural-torsional buckling of shallow arches with an open thin-walled section that are subjected to a radial load uniformly distributed around the arch axis. It is found that shallow arches under a uniform radial load are subjected to combined in-plane compressive and bending actions prior to flexural-torsional buckling, and that using the classical buckling solution for circular arches under uniform compression produces incorrect buckling loads for shallow arches. A rational finite element model is developed for the flexural-torsional buckling and postbuckling analysis of shallow arches with an open thin-walled section, which allows the buckling loads to be obtained correctly. [Copyright &y& Elsevier]

No abstract received. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops a closed-form solution for the elastic flexural–torsional buckling of a laterally fixed steel circular arch that is subjected to uniform bending. The formulation makes recourse to an orthogonal rotations matrix in order to derive the finite strains and the energy equation for buckling. Whilst the flexural–torsional buckling of pinned arches has received some degree of treatment in the literature, it appears that the flexural–torsional buckling of fixed arches has not been reported openly. The buckling solutions are very different from those of pinned arches, and significantly they illustrate that the concept of an ‘effective length’ approach, that is often useful for determining the buckling response of straight beams or columns, is not applicable for the flexural–torsional buckling of arches. The ramifications of this observation are that design procedures for the out-of-plane buckling of pinned arches cannot be used as a basis for design procedures for fixed arches. [Copyright &y& Elsevier]

The elastic flexural–torsional buckling behaviour of arches with a central elastic torsional restraint has been reported elsewhere by the authors, and it was found that a central elastic torsional restraint restricts the buckling shape of an elastic arch and increases its elastic buckling resistance. However, both the inelastic buckling and strength of arches with a central elastic torsional restraint have hitherto not been investigated. It is not known whether the threshold stiffness for elastic buckling can be applied to arches which buckle inelastically, nor is it known how to determine the strength of steel arches with a central elastic torsional restraint. This paper modifies a finite element model for the nonlinear inelastic flexural–torsional analysis of steel I-section arches by including the effects of elastic restraints, and uses it to investigate the influence of central elastic torsional restraints on the inelastic flexural–torsional buckling and strength of steel I-section arches. It is found that a central elastic torsional restraint increases the strength of steel arches, but that the increase in strength decreases as the modified slenderness of the arches decreases. The threshold value of the stiffness of a central elastic torsional restraint at which the inelastic strength of an arch is equal to that of the corresponding arch with a rigid restraint is related to both the modified slenderness and included angle of the arch. For an arch with a low modified slenderness and with a small included angle which buckles inelastically, the threshold restraint stiffness is much smaller than that for an arch which buckles elastically. Design formulae for the strengths of steel I-section arches in uniform bending and in uniform compression with a central elastic torsional restraint are proposed. Comparisons with finite element results show that the proposed formulae provide good predictions for the strength of thin-walled steel I-section arches with a central elastic torsional restraint. [Copyright &y& Elsevier]

This paper is concerned with the in-plane elastic stability of arches with a symmetric cross section and subjected to a central concentrated load. The classical methods of predicting elastic buckling loads consider bifurcation from a prebuckling equilibrium path to an orthogonal buckling path. The prebuckling equilibrium path of an arch involves both axial and transverse deformations and so the arch is subjected to both axial compression and bending in the prebuckling stage. In addition, the prebuckling behavior of an arch may become nonlinear. The bending and nonlinearity are not considered in prebuckling analysis of classical methods. A virtual work formulation is used to establish both the nonlinear equilibrium conditions and the buckling equilibrium equations for shallow arches. Analytical solutions for antisymmetric bifurcation buckling and symmetric snap-through buckling loads of shallow arches subjected to this loading regime are obtained. Approximations for the symmetric buckling load of shallow arches and nonshallow fixed arches and for the antisymmetric buckling load of nonshallow pin-ended arches, and criteria that delineate shallow and nonshallow arches are proposed. Comparisons with finite element results demonstrate that the solutions and approximations are accurate. It is found that the existence of antisymmetric bifurcation buckling loads is not a sufficient condition for antisymmetric bifurcation buckling to take place. [ABSTRACT FROM AUTHOR]

Arches are often connected to other members which influence the structural behaviour of the arch. These members induce restraining actions during flexural-torsional buckling which restrict the buckled shapes of the arch and may significantly influence its buckling response. This paper uses an energy method to study the elastic flexural-torsional buckling of continuously restrained arches of doubly symmetric open thin-walled cross-section in uniform bending and in uniform axial compression. Closed form solutions for the flexural-torsional buckling moment of restrained arches in uniform bending and for the flexural-torsional buckling load of restrained arches in uniform axial compression are obtained. It is found that the elastic continuous restraints are more effective in increasing the buckling resistance for arches with a large included angle than for arches with a small included angle. The first buckling mode may not correspond to the lowest buckling moment or load for arches restrained by minor axis rotational or lateral-translational restraints. It is also found that the number of half sine waves corresponding to the lowest buckling moment or load increases with the stiffness of the restraints. For restrained arches in uniform axial compression, when the restraining stiffness exceeds a limiting value, the buckling mode may change from flexural-torsional to torsional. [Copyright &y& Elsevier]