With the development of transportation infrastructure, urban road network density is increasing, and road intersections are worsening. Urban overpasses are an effective way to solve this problem. In the design and planning of urban overpasses, curved bridges can make road alignment smooth and beautiful. They also reduce the effect on ground transportation and play an importation role in relieving urban traffic congestion [1
]. Considering restrictions of topography, space, and other factors, irregular crotches with variable width, slope, and curvature are usually adopted to connect main bridges and ramps. Irregularly-shaped bridges, such as the one shown in Figure 1
a, become the main structural form of urban overpass. However, a number of urban overpasses with irregular crotches, such as the one shown in Figure 1
b, have been destroyed while in operation in China [3
]. It indicates that design methods and techniques are not mature in practice. Research on the modal characteristics of irregularly-shaped bridges can provide references for irregularly-shaped bridge design, and gradually improve the safety operations of these structures.
The cross-section of an irregularly-shaped bridge mainly adopts thin-walled box girders [4
]. It increases the complexity of analyzing the modal characteristics of irregularly-shaped bridges due to the combinational effect of bending and torsion in thin-walled box girders. A number of studies have been conducted on the calculation principles and analysis methods for the modal characteristics of curved thin-walled box girder bridges [5
]. However, the main girder and ramp mutually contact and restrict each other at the crotch of an irregularly-shaped bridge. Its dynamic characteristics present a complicated coupling effect with bending and torsion, which are different from the conventional curved structure.
Finite element analysis provides an effective way to analyze the modal characteristics of this structure. Yoon et al. [10
] proposed a curved beam element with seven degrees of freedom at each node, and put forward the formula of element stiffness and mass matrices to analyze the modal characteristics of thin-walled curved beams. Sapountzakis et al. [11
] dispersed the curved beam by straight line elements—considering warping and shear deformation—to calculate the structural modal characteristics, which verified the accuracy of the curved beam model to replace broken lines. Lu et al. [4
] analyzed the modal characteristics of irregularly-shaped bridges using shell elements. A comparative analysis with a grillage method is performed to prove the accuracy of a proposed method. However, the complex structure is divided into a large number of units with multiple degrees of freedom, which would seriously affect the computational efficiency and accuracy of the finite element method. Especially, when the bridge’s parameters change, the finite element model reconstruction increases the analysis’ complexity. A substructure method is widely used because it can minimize the size of matrices, and reduces the expense of computational time. The basic idea of the substructure method is to decompose the complex structure into several substructures according to structural features. In addition, analysis on each respective substructure is performed. Overall modal characteristics can be obtained by comprehensively assembling all the substructures according to boundary conditions [12
]. If the local design parameter is changed in the structural model, only the modal characteristics of the corresponding substructure should be recalculated. It could effectively improve operational efficiency and achieve overall analysis of the complex structure.
As an important component of substructure methods, the dynamic substructure method can take advantage of the modal characteristics of each subsystem with a simple calculation process to finally obtain the modal characteristics of the whole structure. Since the 1960s, the dynamic substructure method has been developing rapidly [13
]. Especially in recent years, this method has been increasingly applied in the field of Bridges. Biondi et al. [14
] presented a substructure approach for analyzing the dynamic response of the train–rails–bridge system. It regards the train, rail, and bridge deck as three substructures, and simultaneously computes the dynamic responses of the train, rail, and bridge to analyze vehicle–bridge dynamic interaction. Li et al. [15
] proposed the damage identification strategy based on the dynamic substructure method, which divided the complex structure into several substructures, and identified the structural damage based on the dynamic response of the substructures under moving load excitation. Kong et al. [12
] demonstrated a new substructure approach to analyzing the vehicle-induced vibration of long-span hybrid cable-stayed bridges. It divided the bridge into many substructures with a reasonable length and condensed the substructures model in detail with refined mesh into super elements. It used the dynamic substructure mode synthesis method to analyze the vehicle-induced dynamic response under deterministic flows.
Based on different solutions to the problem, dynamic substructure methods can be divided into the component mode synthesis (CMS) method, the interface displacement synthesis method, the migration substructure method, and the super element method. CMS is widely used as the most mature theory among them [16
]. Based on the differences between some treatments to the interface, CMS can be divided into the fixed interface component mode synthesis method, the free interface component mode synthesis method, and the hybrid component mode synthesis method. The free interface component mode synthesis method proposed by Hou has been developing rapidly and is widely used [19
]. It does not contain the interface node displacement coordinates in the comprehensive equation, which is convenient to combine and verify with the experimental mode technique. The double coordinate free-interface mode synthesis method possesses outstanding advantages in structural modal analysis with high calculation accuracy.
The free-interface mode synthesis method is gradually applied to the modal analysis of bridges. However, there is very little investigation on the modal characteristics of irregularly-shaped bridges based on this method. In this paper, the double coordinate free-interface mode synthesis method is adopted to analyze the modal characteristics of an irregularly-shaped bridge. Ramp curve radius, crotches diaphragm beam stiffness, cross-section feature, and bearing conditions are selected as feature parameters. Their influence on the modal characteristics of an overall irregularly-shaped bridge is analyzed. The analysis can provide a reference for design and can improve the safety standard. However, there is a long distance from the parameters’ impact on the modal characteristics of the irregularly-shaped bridge to the impact on dynamic behaviours. The critical analysis on the corresponding structural dynamic response beyond modal analysis will be a matter for future studies. Additionally, the parameters’ impact on the dynamic responses of irregularly-shaped bridges considering nonlinear behaviours is also important [23
]. Pushover analysis on the irregularly-shaped bridge will be conducted in future studies to enrich the conclusion [25
In this paper, a novel double coordinate free-interface mode synthesis-based method is proposed for analyzing the modal characteristics of irregularly-shaped bridges. Taking the numerical model of a typical irregularly-shaped structure as an example, the effects of four design parameters including ramp radius, crotch diaphragm stiffness, cross-section features, and bearing condition on modal characteristics are demonstrated and the following conclusions can be drawn.
First, a comparative analysis with the traditional finite element method reveals that the accuracy of the double coordinates free-interface mode synthesis method is favorable and is suitable for analyzing the modal characteristics of irregularly-shaped bridges.
Second, the effect of ramp radius on the modal characteristics of the irregularly-shaped bridge is not obvious. It has a relatively significant effect on modal frequencies such as the third and fifth modes in which ramp vertical bending vibration is the main vibration form. The capability to resist lateral bending of the ramp continues to weaken with the increasing of the ramp radius.
Third, the effect of crotch diaphragm stiffness on modal frequencies and low-order mode shapes is negligible. The changes of crotch diaphragm stiffness can weaken the third and fourth vertical vibration of the irregularly-shaped bridge to a certain extent.
Fourth, the effect of the cross-section stiffness of the whole bridge on modal frequencies is obvious; modal frequencies increase as the stiffness increases. However, this effect is negligible on mode shapes. As for the change of the cross-section stiffness of substructures, substructures 2 and 3 have greater impact on the modal characteristics of the whole bridge. Additionally, the results also indicate that mode shapes have nothing to do with the whole cross-section stiffness but do influence its relative distribution.
Fifth, different bearing conditions will affect structural vibration. Unreasonable conditions can lead to a decrease in the capability to resist lateral bending vibration. The double fixed bearing condition is set at three ends, and the bifurcation position of the bridge can strengthen the lateral bending stiffness, which is more satisfactory.