INTRODUCTION AND LITERATURE REVIEW
1.2 LITERATURE REVIEW
1.2.3 Free Vibration of Thin-Walled Box-Girder Bridges
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.
Mukhopadhyay and Sheikh (1995) investigated the large amplitude free flexural vibrations of horizontally curved beams using finite element approach.
Yoon et al. (2005) presented the finite element formulation for free vibration analysis of horizontally curved steel I- girder bridges. Stiffness as well as mass matrices of both the curved and straight beam elements were formulated. Each node of both of them possessed seven degrees of freedom including the warping degree of freedom.
Fam and Turkstra (1975) dealt finite element scheme for static and free vibration analysis of box bridge with orthogonal boundaries and arbitrary combinations of straight and horizontally curved sections.
Lees et al. (1976) calculated the natural frequencies and mode shapes of a number of box beams using the finite element displacement methods. Both the in- plane and transverse motion in the vibrations of box beam were presented.
Tabba and Turkstra (1977) examined the problem of free vibrations of curved thin-walled girders of non-deformable asymmetric cross-section. The general governing differential equations were derived for quadruple coupling between the two flexural, tangential and torsional vibrations. A parametric study was conducted to investigate the effect of relevant parameter on natural frequencies. Eigen functions satisfying the orthogonality condition were given. The solution derived herein for the general case was also shown to cover a variety of special cases of straight and curved girders with doubly symmetric or singly symmetric cross-sections.
Shanmugam and Balendra (1986) described dynamic analysis for free vibration characteristics of multi-cell structures by using the simplified grillage technique.
Comparisons of the results were obtained for structure with different boundary
conditions with those obtained by the finite element method. The effect of the natural frequencies was also investigated.
Noor et al. (1989) developed the simple mixed finite element models for free vibration of thin-walled beams with arbitrary open cross section. The analytical formulation was based on Vlasov’s type thin-walled beam theory, which included the effect of flexural-torsional coupling, the additional effects of transverse shear deformation and rotary inertia. The high accuracy and effectiveness of the mixed models developed were demonstrated by means of numerical examples of thin walled beams with symmetrical and unsymmetrical cross-section.
Snyder and Wilson (1992) presented a closed form solution for the out-of-plan free vibration frequencies of horizontally curved thin-walled beam which was continuous over multiple supports. The bending and twisting of this beam was coupled in the mathematical model used. This coupled bending/twisting behavior caused two free vibration frequencies to be associated with each vibration mode shape.
Kou et al. (1992) presented the free vibration analysis of continuously curved girder bridges, based on the direct stiffness method, which included warping and adopting lumped mass matrices. The results obtained were compared with results based on other theories.
Stavridis and Michaltsos (1999) proposed for the evaluation of the eigen frequencies of a thin-walled beam curved in plan for transverse bending and torsion mode with various boundary conditions. The static differential equations for curved thin-walled beams using established Vlasov theory were appropriately extended and treated through the introduction of four dimensionless geometric quantities.
Yoon et al. (2006) investigated new equation of motion governing dynamic behavior of thin-walled curved beams. Explicit numerical expressions were derived to
predict the complex dynamic behavior of the thin-walled curved beams. Stiffness as well as mass matrices of the curved beam elements for finite element analysis was formulated to allow explicit evaluation of the dynamic behavior. Each node possessed seven degree of freedom including the warping degree of freedom. They included comparisons of the natural frequencies of the thin-walled curved beams from the finite element formulations with those reported by other investigators.
. The literature survey showed that a good deal of effort had been put forward by past researchers on the evaluation of modal parameters of box-girder bridges using different techniques. While some researcher carried out free vibration analysis based on closed form solution, a few conducted finite element analysis without considering torsion and torsional warping, distortional and distortional warping effect in the analysis. Since natural frequencies are important parameters to be considered in design, especially for checking the safety of bridge under dynamic excitation, use of appropriate finite element formulation is necessary to accurately predict the same. It is also desired to use a computationally efficient model which can represent all complex structural actions in a horizontally curved thin walled box girder bridge. In the present study, the thin-walled box beam finite element developed by Zhang and Lyons (1984), which is computationally efficient as well as reasonably representative of thin-walled box girder behaviour have been considered for the free vibration analysis of thin walled box girder bridges. However, as per the records available, this element has not been used for the dynamic analysis. Laboratory vibration testing of thin walled curved box girder bridge model is one of the ways to validate the theoretical model and to accept its applicability for further use. Thus, in the present work, experimental free vibration analysis has also been conducted on a Perspex sheet model of the curved
bridge to validate the natural frequencies and mode shapes of the theoretical bridge model.