Nonlinear elastic analysis of composite beams curved in-plan
Introduction
Composite steel and concrete beams which are curved in-plan are used very widely in highway bridges. In deference to curved steel bridges (e.g. [1]), comparatively few studies have been reported on curved composite beams, and in particular on their numerical modelling. Colville [2], Thevendran et al. [3], [4] and Shanmugam et al. [5], conducted experiments on steel–concrete composite curved beams to investigate the ultimate load behaviour, while Giussani and Mola [6] recently developed an analytical formulation for elastic composite beams curved in-plan by assuming full interaction between the steel girder and the concrete deck. Chang and White [7] discussed the modelling considerations for composite curved steel bridges and illustrated the effects of cross-sectional distortions.
Studies (e.g. [8], [9]) have shown that it is also important to consider geometric nonlinearity in order to accurately predict the response of curved beams, even under service loads. This is because the secondary bending about the minor principal axis and torsion actions may develop and become increasingly profound with an increase of the twist rotations. Bradford et al. [10] showed that during unpropped construction of curved composite bridges, effects of geometric nonlinearity may cause early yielding of the I-girder since I-beam girder acts as a separate individual curved beam when concrete is wet. Thus, time effects are critical for the construction of composite beams. Topkaya et al. [11] conducted experimental and numerical studies to establish the behaviour of composite curved bridges during construction. Liew et al. [12] also showed the effects of geometric nonlinearity on the inelastic behaviour of curved I-beams and proposed simplified equations to evaluate the ultimate strength. The behaviour of curved beams has also been studied by Hall [13], Zureick et al. [14] and Gimena et al. [15], [16], [17].
On the other hand, the behaviour of composite beams is significantly influenced by the flexibility of the shear connection. A beam theory that considers the partial interaction in the longitudinal direction of straight beams was presented by Newmark et al. [18]. Adequate modelling of composite curved beams with flexible shear connectors, however, needs to account for the partial interaction in the radial direction as well as in the tangential direction, because in curved beams radial deflections occur even for vertical loading. Accurate beam models for composite beams curved in-plan accounting for partial interaction and incorporating the important coupling of bending and torsion actions and deflections do not appear to have been reported in the open literature. Beam models always have the advantage of easy structural modelling and easy interpretation of the output results. The objective of this paper is therefore to present a 3D geometrically nonlinear beam finite element that considers the partial interaction in the tangential as well as in the radial directions in the analysis of composite beams curved in-plan. Examples are considered to illustrate the effects of initial curvature, geometric nonlinearity and partial interaction on the behaviour of composite curved beams. The proposed finite element formulation is validated by comparing the numerical results obtained from the formulation with more sophisticated yet complex ABAQUS shell element models and with available experimental results reported in the literature. The beam finite element is shown to provide a very efficient technique for modelling curved composite beams.
Section snippets
Basic assumptions
Fig. 1 shows a composite beam curved in-plan, for which the following assumptions are made:
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The steel girder is a doubly symmetric I-beam that is curved in plan;
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The deck has a rectangular cross-section and has the same initial curvature as the girder in the undeformed configuration;
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Both cross-sections remain rigid throughout the deformation, i.e. no distortion occurs;
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There is no uplifting between the girder and the deck;
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Radius of curvature is constant along the beam;
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The shear connection between
Weak formulation of nonlinear equilibrium equations
The equilibrium equations can be obtained in their weak form by applying the principle of virtual work; hence the sum of the work done by the external forces on the virtual displacements is equal to the internal work done by the stresses on kinematically admissible virtual strains, so that where the first integral is the internal virtual work due to the deformations of the girder and the deck, the second integral is the internal virtual work due
Consistent linearisation of the equilibrium equations
The incremental forms may be obtained by subtracting the virtual work expressions at two neighbouring equilibrium states represented as and , and then linearising the result by omitting the second- and higher-order terms. Using a Taylor expansion, the differences in the virtual work expressions of the two neighbouring configurations can be written as
The first term of the incremental virtual work in Eq. (25) becomes
Nonlinear formulation
A nonlinear finite element formulation is obtained by interpolating the displacement and rotation fields , and by using cubic Hermitian interpolation functions, and the longitudinal deflection and the slip deflections , and by using linear interpolation functions. Thus the nonlinear equilibrium equations in Eq. (12) become discretised as where is the shape function matrix, i.e. . The left-hand side of Eq. (40) is due to the
Composite curved beam of Thenvendran et al.
One of the worked examples (SP4 beam) of Thenvendran et al. [3] is analysed herein for validation purposes. The beam is simply supported at both ends as shown in Fig. 2 (), with one of the ends being a roller support which allows tangential displacements, thus the tangential displacement is restrained only at one end (). The curved span of the beam is and the initial curvature is (). The steel girder is a UB356×171×57 kg/m with an overall height of
Conclusions
A total Lagrangian finite element formulation for the elastic analysis of steel–concrete composite beams that are curved in plan has been developed. The formulation includes the effects of geometric nonlinearity as well as partial interaction in tangential and radial directions, and which hitherto does not seem to have been addressed elsewhere. Comparisons of the numerical results with experimental and ABAQUS shell element results has shown that the developed formulation is efficient and
Acknowledgement
The work in this paper was supported by the Australian Research Council through a Discovery Project awarded to the second author.
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2017, International Journal of Mechanical SciencesCitation Excerpt :It is interesting to note that the number of existing studies on geometric nonlinear analysis of these composite beams is very limited [17,18]. The effect of geometric nonlinear response is incorporated in the finite element models by Ranzi et al. [17], and Erkmen and Bradford [18] for the analysis of composite beams having curved and straight alignments respectively. The authors however have not considered the effect of transverse shear deformation of the beam material layers, as the models are based on EBT.
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