## 1 Introduction  
A hybrid bridge with both cable-stayed and suspension systems is a collaboration of a cable-stayed bridge and a suspension bridge, which, with the optimal design, is able to combine the optimal properties of these two bridge systems [1, 2, 3, 4]. The suspension cable system can guarantee stable and safe erection processes, and meanwhile, the additional stayed cables can increase the stiffness and reduce the beam deformability. Hence, the hybrid system is able to improve the span capacity and structural stability and reduce the pylon height and anchorage volume [5]. The hybrid system has a long history of development for about 170 years, with the first system proposed in 1855 by American bridge designer [PERSON] for the Niagara Falls highway railway bridge [6], which has a main span of 251 m. The first hybrid system is also called the Roebling system where the force is mainly taken by suspenders, while the stay cables are used to increase the stiffness and bearing capacity of the suspension bridge. In 1883, the second hybrid bridge system was proposed by [PERSON], which was named as the Dichinger system [7]. In this system, the cable-stayed bridge part is independent of the suspension bridge part, where the cable-stayed bridge sustains the loads on both sides of the side span and the middle-span, while the suspension bridge takes the load of the middle part of the middle span with a ground-anchored suspension system. With continuous research and development, novel cable-stayed suspension hybrid bridge systems have been proposed, such as the Troitsky system [8], the Lin Tongyan system [9], and the Gimsing system [10]. Meanwhile, the span capacity of the hybrid bridge system has been improved significantly by the design optimization to take full advantage of both cable-stayed and suspension bridges. Currently, the largest-span hybrid bridge is the Third Bosphorus Bridge with a span length of 1408 m [11]. The Xhiounen Rail-cum-Road Bridge, which is currently under construction, has a span of 1488 m [12]. While the hybrid bridge system has a long history of development, only a few cases of applications have been constructed. In addition, its longest span is still far less than that of a suspension bridge, which is the 1915 Canakkale Bridge with a span of 2023 m [13]. The reasons could be the complex force transformation mechanism between various load-carrying components, including the main cables, stayed cables, beam, and tower, where the forces shared by various components are unclear. Meanwhile, a lack of systematic experimental and theoretical analyses, as well as the optimization scheme, could also be another important concern. For the complex hybrid bridge system, the most straightforward method to determine the mechanical static and dynamic behavior is through the finite element method (FEM), which is used to combine the modeling techniques of cable-stayed and suspension bridges. With proper modeling techniques, the deflection and internal forces of the hybrid system can be solved directly [14, 15, 16]. However, for the complex hybrid system with a long span, the FEM involves complex modeling elaboration with a significant amount of input data and complex compiling process, which are all susceptible to computational errors. Only the engineers with a sophisticated modeling background and experience can complete the modeling process. In addition, in the preliminary design stage, a significant amount of design work will focus on the optimization of the geometric configuration of the hybrid system, resulting in varying configurations with different geometric parameters, which is not easily adjusted in the FEM. In order to facilitate the preliminary optimization process of the hybrid system, researchers also make many efforts on the development of an efficient analytical model with simple input parameters based on elastic beam theory and the potential energy method.