INTRODUCTION AND LITERATURE REVIEW
1.2 LITERATURE REVIEW
1.2.2 Finite Element Method of Analysis of Thin-Walled Box-Girder Bridges Different numerical methods of analyses of thin-walled box bridge include
sections in their analysis, the procedures developed based on closed-form solutions of Vlasov’s (1961) theory considered adequately all such important components like torsion, torsional warping, distortional and distortional warping.
1.2.2 Finite Element Method of Analysis of Thin-Walled Box-Girder Bridges
technique for the computer solution of complex problems in engineering. Nowadays a logical alternative for modeling curved box girder bridges is to combine finite element technique with thin-walled beam theory to develop one dimensional thin-walled box beam elements. Several investigators have combined thin-walled beam theory of Vlasov (1961) and finite element technique to develop a thin-walled box beam element for elastic analysis of straight and curved box bridges.
Fam and Turkstra (1975) introduced finite element scheme analysis of box bridge with orthogonal boundaries and arbitrary combinations of straight and horizontally curved sections. A variety of special purpose elements were developed to suit the behavioral characteristics of thin box sections.
Mikkolo and Paavola (1980) conducted a finite element analysis for rectangular single-cell box-girder with side cantilevers. Shape functions are represented by cubic polynomials in each element as in the finite element solution of beam problems. It was observed that the known displacement functions describing the deformation modes of the cross-section must be chosen in advance for each type of cross-section. Thus, difficulties exist in extending the method for more complicated or more general types of cross section. It was observed that the method presented by them was applicable only in the case of single box cell.
Gunnlaugsson and Pedersen (1982) developed a finite element formulation considering seven degrees of freedom at each node for beam with thin-walled cross- sections. They presented calculation of stresses and deformation of beam with different types thin-walled cross-sections.
Zhang and Lyons (1984) developed a thin-walled box beam finite element, applicable for the analysis of box girder bridges. They considered only one distortional mode and the effective width concept was used to include shear lag effects. In
contrast to conventional beam formulation, warping and distortional effects which were essential for the analysis of box beam formulation were included in the thin- walled beam element formulation. They included three extra additional degrees of freedom for thin-walled box beam finite element of curved bridge. These additional degrees of freedom were the rate of change of twisting angle, distortional angle of the cross-section and the rate of change of distortional angle. The thin-walled box beam element could be used effectively for static analysis of single or multi-cell box beams curved in space and subjected to general loading conditions.
Boswell and Zhang (1985) presented the results of experimental investigation of the behavior of four types of thin-walled box beam and compared the results with those obtained from the specially developed thin-walled box-beam finite element theoretical analysis. The behaviors of the individual models were studied, with particular attention being given to the torsion and distortion of the box-section, the cross-sectional distributions of the longitudinal and transverse bending stresses and the deflection.
Hsu et al. (1990) developed a more exact horizontally curved beam finite element in which the true warping degree of freedom conforms to the bi-moment (warping). The variational method was used to formulate the stiffness matrix in an explicit form.
Shanmugam and Balendra (1991) described an experimental and theoretical study of the behavior of multi-cell structures curved in plan. They made perspex models of multi-cell box-girders that were curved in plan. The results of eight tests on two perspex models of different span/radius ratios subjected to different loading conditions were presented. The experimental results were compared with theoretical
values obtained by employing the finite element method and good agreements between the results were demonstrated.
Razaqpur and Li (1994) used Vlasov’s thin-walled beam theory combined with special shear lag warping function to derive a box beam finite element for curved thin- walled box-girder bridge with exact shape functions that could be used to analyze single and multi-cell box girders.
Paavola (1992) developed a numerical model for analyzing thin-walled girders.
The theory was based on the conventional idea of Vlasov developed originally for torsional problems of thin-walled girders with closed cross-sections. This idea was combined with the finite element method. The effect of torsion, distortion, torsional and distortional warping were included in the analysis.
Kim and Kim (1999) proposed a new C0 –continuous displacement-based box beam finite element for straight box bridges. Direct kinematic variables representing torsion, warping and distortion were used for both static and dynamic analysis.
Kim and Kim (2002) formulated a one-dimensional beam theory for the analysis of thin-walled curved box beams under torsion and out-of-plane bending. In addition to the conventional three kinematic variables, two additional variables representing warping and distortional deformations of a beam cross section were included in the present theory
Yaping et al. (2002) presented a finite curved beam element method of analyzing of curved thin-walled box-beam bridge based on the energy principle. The analysis considered both shear lag and warping torsion effects.
Park et al. (2005) proposed the expanded method for exact distortional behavior of multi-cell box-girder subjected to eccentric loadings. This method decomposes the eccentric loading into flexural, torsional and distortional forces by
using the force equilibrium. Based on the method, a thin-walled box beam finite element, which could be applied to practical distortional analysis of straight multi-cell box girder bridges, was also developed. The box beam element possessed nine degree of freedom per node to consider each separate behavior of multi-cell box girder. The validation of the box beam element was demonstrated through a series of comparative studies using a conventional shell element proposed by other researchers.
Desantiago and Mohammadi (2005) presented a simple three dimensional finite-element analysis for a series of single span horizontally curved bridges. The analyses were carried out using a typical truck load and also the dead load of the bridge as the primary force on bridges. In each analysis, the behavior of bridges was investigated, and the major internal forces developed in members were determined.
Specifically, an increase in the bending moment and the existence of a torsional moment were observed for the cases, where the horizontal angle of curvature was large.
Thus, it has been observed that a large number of finite elements were developed by different researchers for the analysis of box girder bridges. While a few of them were not completely robust in the representation of all the complex actions of a curved box girder bridge, the three noded 1D beam element by Zhang and Lyons (1984) with nine degrees of freedom at each node was observed to be very well representative of the structural actions and computationally efficient. While, shell elements were considered for the accurate analysis of such curved box-girder bridge, the total number of degrees of freedom with shell elements would be much higher in comparison to the 1D idealization considered by Zhang and Lyons (1984). The main advantage of such idealization in the analysis would be to reduce unnecessary computational cost such as during preliminary design stages, where a full three-
dimensional analysis would be unnecessary, the effect on computational time requirement will be more pronounced when forced vibration studies (e.g. vehicle induced vibration) are carried out. Thus it is observed that while accurate estimation of responses are obtained using the 1D beam element, the idealization leads to a computationally less expensive procedure in comparison to the analysis using standard finite elements (e.g. shell). In view of this, it has been proposed to introduce 1D beam element by Zhang and Lyons (1984) in the present study.