## 1 Introduction  
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; 2]. 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 0(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 0(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; 6; 7; 8; 9]. However, the main girder and ramp mutually contact and restrict each other at the croft 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. [PERSON] 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. [PERSON] 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. [PERSON] 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].