## 1 Introduction  
A suspension bridge is mainly composed of a main cable, pylon, stiffening beam and suspension cable. Because of its good mechanical performance, strong spanning ability and beautiful outline, it is widely used. In 1883, the Brooklyn Bridge built in the US was regarded as the first modern suspension bridge [1]. In 1915, German engineers built the Cologne--Deutz Bridge on the Rhine River, with a main span of 184.5 m, which became the first long-span, self-anchored suspension bridge in the world [2]. Since then, self-anchored suspension bridges have gradually diversified in terms of span, cable arrangement and types of stiffening beams. Landmark bridges, such as the Konohara Bridge, the Yongjong Grand Bridge and Auckland Bay New Bridge have been successively built [3; 4; 5]. Multi-tower self-anchored suspension bridge combines the characteristics of single-tower and double-tower self-anchored suspension bridges and ground-anchored suspension bridges. Multi-tower self-anchored suspension bridges have better adaptability to terrain and site conditions, more flexible layout and more beautiful linear appearance. In China, the Fuzhou Luozhou Bridge, built in 2013, became the first multi-tower self-anchored suspension bridge in the world [6]. In 2016, the span of the Yinchuan Binhe Yellow River Bridge, a multi-tower, self-anchored suspension bridge, exceeded 200 m [7]. In 2022, the Fenghuang Yellow River Bridge was built in Jinan, China, with a span of 70 m + 168 m + 2 \(\times\) 428 m + 168 m + 70 m, making it the largest three-tower, self-anchored suspension bridge in the world today [8]. In recent decades, scholars have conducted extensive research on the analytical calculation theory of self-anchored suspension bridges and obtained many significant results. [PERSON] and [PERSON]. [2] compare the analysis theory of suspension bridges with finite element analysis, and think that the deflection theory can be used for preliminary approximate analysis of self-anchored suspension bridge. [PERSON] et al. [3] analyzed the empty cable state and completed cable state of the main cable of self-anchored suspension bridges and the nonlinear form-finding calculation method proposed by them has higher accuracy than the previous calculation and analysis. [PERSON] et al. [9] considered the influence of the axial compressive strain energy of stiffened beam and obtained the basic differential equation of deflection theory of self-anchored suspension bridges. [PERSON] et al. [10] established an analytical algorithm and calculation program of deflection theory of self-anchored suspension bridge with single tower and two towers by force method. Based on the deflection theory, [PERSON] [11] deduced the practical formula of internal force and deformation of stiffening girder of self-anchored suspension bridge and established the calculation equation based on the deflection theory. Based on the energy principle, [PERSON] and [PERSON] [12] deduced the basic equations suitable for common self-anchored suspension bridges with two towers and put forward a simplified calculation method. For the two-tower self-anchored suspension bridge whose main cable is a three-dimensional curve, [PERSON] et al. [13] put forward the coordinate iteration method to find the shape. An analytical algorithm by [PERSON] et al. [14] put forward based on the segmented catenary theory to obtain the reasonable range of the stiffness of the center tower of the ground-anchored three-tower suspension bridge with unequal main spans. Most of the above analytical methods of multi-tower self-anchored suspension bridges adopt deflection theory or calculation methods based on deflection theory. From the analysis results, it is appropriate to use deflection theory to approximate the self-anchored suspension bridge. However, the related studies are only aimed at the common single-tower or two-tower self-anchored suspension bridges, and the longitudinal stiffness of the main tower is not considered when the deflection theory is used for analysis.