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Here Fundamental Frequency is the main factor used to calculate the vibration of a vehicle when passing across a bridge. With knowledge of the fundamental frequency, bridges can be evaluated and designed in such a manner so as to avoid the critical range of 1.5 Hz to 4.5 Hz, which is occupied by most vehicles. Therefore it is crucial to develop a reliable method to estimate the fundamental frequency of bridges. To overcome the above issue, numerical analysis combined with a theoretical method is applied to estimate the fundamental frequency of multicell box-girder bridges. The effect of span-length, number of boxes and skew angle on the estimation of this factor is discussed. Finally, reliable expressions are proposed to predict the first fundamental frequency of this type of bridge, and the accuracy of the expressions is verified. The results indicate that the fundamental frequency decreases when span length increases, due to development of crack as well as decrease stiffness of girders.

The dynamic effects on a bridge may have a considerable influence on its ultimate limit state behavior by amplifying the maximum stress experienced by each of its members. The dynamic responses of a bridge are influenced by several factors. In order to evaluate these factors for determining the dynamic responses it is necessary to characterize the bridge in term of its fundamental frequency and mode shapes.

Traditionally bridges are designed using static loads, which are increased by the dynamic load allowance (DLA) factor of the bridge (Ashebo et al., 2007a,b; Billing, 1984;Chang and Lee, 1994; Kashif, 1992). Extensive research and development has been carried out to understand the vibration of bridges as a result of natural sources of vibration, and to determine the dynamic allowance factor as a function of the fundamental frequency due to its uniqueness (OHBDC, 1983; Samaan et al., 2007; Sennah et al., 2004; Senthilvasan et al., 2002; Zhang et al., 2003). Meanwhile, in practice heavy trucks establish a quite narrow range of frequencies, 1.5 Hz to 4.5 Hz, therefore it is important to find a reliable method to estimate the fundamental frequency of bridges and design structures in such a way as to avoid this critical range of frequencies (Moghimi and Ronagh, 2008). In addition, due to vibration, the dynamic deflection can cause discomfort to pedestrians using the bridge. It has been known that the human body tends to react more to torsional oscillations than flexural ones.

A large number of studies on free-vibration analysis have been performed on box-girder bridges. Komatsu and Nakai (1970) and Sennah(1998) evaluated the natural vibration responses of straight and curved I- or box-girder bridges using Vlasov's beam theory. Subsequently, Heins and Sahin (1979) applied finite difference methods to solve the differential equations of motion based on Vlasov's thin wall beam theory.

Culver and Oestel (1969) used a close form solution for the equation of motion to determine the fundamental frequencies of horizontal curved beams. Cheung and Cheung (1972)used the finite strip technique to evaluate the free-vibration analysis of straight and curved bridges.

Cantieni (1984) conducted an experimental study to establish a reliable relationship between the fundamental frequency and maximum span length of bridges. It was found that the bracing system significantly affected the fundamental frequency of composite bridges. Finite element analysis was used to extract the dynamic characteristics of bridge-vehicle interaction and establish reliable expressions for predicting the dynamic responses of curved multiple box-girder bridges, but the recommendations are unacceptable in the case of skew bridges.

A number of studies used the finite element method to examine the forced vibration response of instrumented passing vehicles and free-vibration responses of composite cellular box-shaped bridges. Comparison of the studies indicated that the finite element method obtained sufficiently reliable results, compared with other numerical methods (Brownjohn et al., 2008; Fujino et al., 2010; Lin and Yang, 2005; Siringoringo and Fujino, 2012). In order to reduce the significant difference between the estimation of the fundamental frequency obtained from the codes and theoretical methods, Gao et al.(2012) proposed a numerical improved method for straight bridges. However, some limitations still exist for these methods to actually be used in practice. Some enhancements, improvements or further research is required to allow them to be applied in the field (Wang et al., 2013). The results of a study on effects of damage and decay ratio on fundamental frequency by Pandey and Benipal (2011) indicated that that even SDOF bilinear beams have multiple resonance frequencies at which the resonance can take place. However the actual structure is never a SDOF and may have infinite degree of freedom, hence multiple fundamental frequencies depending upon the actual stiffness operating based on cracking pattern.

According to the above, the present studies have mainly concentrated on the free-vibration analysis of straight and curved bridges, and there are no reliable methods to determine the fundamental frequencies of skewed bridges. Therefore, in this study the results of an extensive numerical study on the free-vibration feature of continuous skewed concrete multicell box-girder bridges are evaluated. The prototype bridges are analyzed by using a three-dimensional finite element method. The empirical expressions are established using regression analysis, to determine the fundamental frequencies of such bridges.

The prototype bridge used in this study is highly representative of the majority of concrete skew multicell box-girder bridges. Eighty-five typical bridges with a span length ranging from 30 to 90 m have been designed based on the Canadian Highway Bridge Design Code(CHBDC, 2006). The number of boxes varies from two to six, dependent on the width of the bridge. All selected bridges are two-equal-span continuous, with bridge widths (W) of 9.14 m, 14.00 m and 17.00 m. Since most heavily vehicle frequencies occupy a relatively narrow frequency band in practice, 1.5 To 4.5, it is preferred to design bridge in such a manner as to avoid this critical range, if at all possible.

The preliminary study indicated that the thickness of the deck has an insignificant effect on the dynamic response of multicell box-girder bridges, so constant values of 20cm and 15 cm were selected for the upper and bottom deck thicknesses, respectively. To consider the effect of skewness on the free-vibration response of this type of bridge, the skew angle was ranged from 0 to 60º, which is within the range of applicability introduced by the American Association of State Highway and Transportation Officials' Load Resistance Factor Design(AASHTO, 2008). Table 1 shows the characteristics of the selected bridges, in which the symbols NB and NL stand for the number of boxes and number of loaded lanes, respectively. A typical bridge cross-section is shown in Figure 1. Reinforced concrete intermediate diaphragms are used for all prototype bridges at spacing of 7.5 m (25 ft), alongside the end diaphragms to enhance the stability of the structure under construction loads.

In this study, the prototype bridges are modeled with CSIBRIDGE software V15, using a four node, three-dimensional shell element with six degrees of freedom at each node. The top and bottom shell elements of the webs are integrated with the top and bottom slabs at connection points to improve the compatibility of the deformations (Mohseni and Khalim Rashid, 2013). The bridge modeling was verified by comparing the live load distribution factor (LDF) derived from field testing, and those from the method adopted herein. Boundary conditions are simulated using hinge-bearing at the starting abutment, and roller-bearings for all other supports. Figure 2 shows a finite element model of a 200 m three-box multicell box-girder bridge.

A sensitivity study is first performed to investigate the influence of the key parameters on the fundamental frequency of the prototype bridges. These parameters consisted of the span length, number of lanes loaded, skew angle, and number of boxes.

The effect of span length on the fundamental frequency for three-box bridges with span lengths ranging from 30 m to 90 m is plotted in Figure 4. It can be observed that the fundamental frequency decreases considerably with increasing span length of bridge, within a maximum range of 62.5%. This difference is caused by the influence of other parameters, especially the skew angle. The mode shapes for a three-lane, three-box bridges with a span length of 45 m are plotted in Figure 3. As