Keywords: arches, curved beam element, elastic restraints, elasto-plastic, geometric, integrated curved element, material, membrane locking, nonlinearity.

This paper presents a new integrated and membrane-locking free curved-beam element for the in-plane large deformation analysis of arches. When the lateral and torsional deformations of an arch are fully restrained, the arch may buckle in an in-plane bifurcation mode or in a snap-through mode under in-plane loading. After buckling, the deformations of the arch increase rapidly and become very large. Hence, in order to predict postbuckling behaviour correctly, the effects of large deformations have to be considered. However, in the conventional formulations of curved-beam elements, the nonlinear strains under in-plane loading consist of nonlinear membrane strains and linear bending strains. The higher-order bending strain components produced by the higher-order curvature terms do not appear to have been considered hitherto. Because of this, some significant terms in the strains due to large deformations may be lost and the large postbuckling deformations cannot be predicted correctly. In addition, the linear bending strains in the conventional formulation introduce membrane deformation to resist the bending deformation. Because the membrane stiffness is far greater than the bending stiffness in a slender arch, this leads to excessive stiffness in bending, i.e. membrane-locking. In the present curved beam element, the higher-order curvature components due to bending are included and the additional higher-order curvature terms due the interaction of the axial deformations with the deformed curvatures are also included in the formulation. These higher order terms of deformed curvature cancel the effects of the membrane stiffness on the bending deformation and so eliminate the membrane locking problem in the conventional formulation. It is well-known that when an arch is subjected to in-plane loading, the axial compression is the major primary action in the arch. In order to produce axial compression, the supports of arches are usually fixed or pin-ended. However, in practice, arches may be supported by elastic foundations or by other structural members which provide elastic types of restraints to the arches. The elastically restrained actions of the other elements of the structure on an arch can be replaced by equivalent springs and the arch can be considered to be restrained by the elastic springs. In many cases, by knowing the structural configuration connecting to the arch, the stiffness of the corresponding elastic springs (or spring constant) can be estimated accurately. The elastic restraints participate in the structural behaviour of the arch and may influence significantly the structural behaviour of the arch. These elastic restraints may be continuous or discrete. The curved-beam element, therefore, should consider the effects of these elastic restraints. In addition to elastic buckling and postbuckling, steel, concrete or concrete-filled tubular section arches may buckle elasto-plastically. Hence, the nonlinearity of the material needs to be included in the curved-beam element as well. The effects of geometric and material nonlinearities, residual stresses, initial geometric imperfections, and various types of elastic restraints are integrated in the formulation of the curved beam element.

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