**1. Introduction**

In mountainous regions of China, arch bridges are widely used due to their advantages of high stiffness, high bearing capacity, good seismic performance, and low cost [1,2]. The number of existing concrete-filled steel tube (CFST) arch bridges in China has exceeded 400, and their spans continue to break records. However, there are few CFST basket-handle arch bridges with spans above 300 m [3–5]. Therefore, the research on the large-span basket-handle arch bridge is relatively limited, especially on the mechanical performance affected by inclination angle [1,6].

As the span of steel-concrete composite arch bridges increases, the problem of lateral stability becomes more prominent. The basket-handle arch not only has an elegant appearance but also has good lateral stability. Table 1 shows the statistics of some highway CFST basket-handle arch bridges. The inclination angle of the arch ribs is between 4.6◦ and 13◦. Existing literature [7–9] has shown that if an improper inclination angle of the basket-handle arch ribs is used, the lateral stiffness decreases. In addition, for truss structures, changes in structural form may lead to a redistribution of member axial forces, thereby affecting the safety of the structure. Scholars have proposed methods to estimate the axial forces on the members of the structure. Therefore, when determining the optimal

**Citation:** Liu, Z.; Wu, Y.; Wang, C.; Fan, Y.; Luo, C.; Wang, S. Research on Optimal Arch Rib Inclination of Large Span Highway CFST through Arch Bridge. *Buildings* **2023**, *13*, 1415. https://doi.org/10.3390/ buildings13061415

Academic Editors: Simon X. Yang, Elena Ferretti, Jingzhou Xin, Yan Jiang and Bo Wu

Received: 25 April 2023 Revised: 23 May 2023 Accepted: 29 May 2023 Published: 30 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

inclination angle of the arch ribs, the effects on the mechanical performance, such as axial forces and bending moments of the structure, need to be considered [10,11]. Many scholars have studied the effects of inclination angle on the structural mechanical performance of CFST arch bridges. Yang et al. [12] studied a 220-m railway CFST arch bridge with varying inclination angles. It was found that when the inclination angle ranges from 0◦ to 10◦, the modal frequencies of lateral and torsional vibrations of the arch ribs increase with inclination. However, the vertical and longitudinal vibration frequencies remain almost constant. Yun et al. [13] used the finite element analysis method to study the influence of the arch ribs rise-to-span ratio and inclination angle on the structural vibration mode and stability. A mid-supporting steel-concrete composite arch bridge was employed in this study. It was found that the inclination angle of the arch ribs should not exceed 10◦. Wang et al. [14] conducted a self-vibration characteristic analysis of a 240-m railway CFST arch bridge. In this study, the lateral vibration frequency was found to be more sensitive than the other two directions when the inclination angle varied from 7.5◦ to 9.5◦. Wei et al. [15] conducted a parameter analysis of several existing CFST arch bridges and built a CFST arch bridge with a span of 105 m. The study showed that the linear elastic stability safety factor gradually increased as the inclination angle of the arch ribs increased within 0–15◦ and reached an optimal state at 9◦. Zeng et al. [16] studied the structural stability of a steel box arch rib bridge with varying arch rib inclination angles from 0◦ to 12◦ using the finite element analysis method. The results showed that the out-of-plane stability safety factor increased first and then decreased with the increase in inclination angle. The out-of-plane stability safety factor reached its peak at 10◦. However, the in-plane stability safety factor decreased with the increase in the inclination angle. Therefore, it is necessary to comprehensively analyze and determine the optimal inclination angle. Zhao et al. [17] studied the variation of internal forces at key positions of the arch ribs when the inclination angles were 0◦, 7◦, and 15◦. A flying-swallow-shaped CFST cable-stayed arch bridge with a main span of 360 m was employed in this study. The results showed that the internal forces of the arch ribs increased with the increase in the inclination angle, especially the bending moment at the crown. Huang et al. [18] examined the variation of internal forces and vertical displacement of a 260-m CFST arch bridge when the inclination angle of the arch rib ranged from 0◦ to 12◦. The results showed that with an increase in the inclination angle of the arch rib, the bending moment and horizontal thrust force at the abutments decreased significantly, reaching a minimum of 8◦. The increase in inclination angle beyond 8◦ did not result in a significant change in the internal forces or vertical displacement. Xu et al. [19] used a railway CFST arch bridge as an example to investigate the influence of the arch rib inclination angle on the seismic performance of the structure. The study found that an inclination angle of 3.5~4◦ not only reduced the displacement and axial force of the arch rib but also avoided the problem of excessive growth of tensile stress in the concrete of the arch rib, which could result in inadequate strength. Wang et al. [20] determined the optimal inclination angle of the arch rib by considering the effects of the rise-to-span ratio, width-to-span ratio, and number of transverse braces. A centrally supported dumbbellshaped steel-concrete composite basket-handle arch bridge was employed in this study. The study found that the optimal inclination angle of the arch rib was negatively correlated with the rise-to-span ratio and not significantly correlated with the number of transverse braces. Ji et al. [21] investigated the variation patterns of the linear elastic and ultimate bearing capacities of a large-span railway steel-concrete composite lever arch bridge with a tube-shaped structure. The study found that the structural stability of the arch bridge would increase first and then decrease as the arch rib inclination angle increased. The optimal value of the inclination angle was determined in this paper as well. Pan et al. [22] conducted a study on the influence of the inclination angle on the lateral stability of a tied arch bridge using the finite element method. Through parameter analysis, the study obtained an approximate expression for the reasonable inclination angle of the double-rib arch. The studies of the aforementioned scholars have achieved rich results. However, the majority of studies were focused on railway bridges. Related studies on highway bridges

are relatively few, especially for bridges with spans larger than 300 m. Furthermore, the evaluation is not comprehensive enough. The relationship between multiple factors of the structure and the arch rib inclination angle needs to be systematically studied. Further research is required to comprehensively determine the reasonable range of inclination angles for the arch rib.


**Table 1.** Partial basket-handle arch bridges were built in China.

The objective of this paper is to study the influence of the arch rib inclination angle on the large span highway CFST basket-handle arch bridge (i.e., 300 m level). The mechanical performance of the structure under different rib inclination angles was investigated. A reasonable range of arch rib inclination angles for large-span CFST arch bridges was proposed to enhance their safety and provide a reference for the design of similar bridges in the future. Based on the world's largest CFST basket-handle arch bridge with a span of 360 m—the Shaowei Zuojiang Extra-large Bridge—this paper employs Midas Civil to establish a full-bridge finite element model. Firstly, the accuracy of the finite element was verified by using the measured vertical displacement values of the main arch during the installation of the main beam and bridge pavement. Then, different arch rib inclination angles were simulated in the FE model. The structural vibration characteristics, linear elastic stability, internal forces, and displacement under static loads with different inclination angles were analyzed. The relationship between the arch rib inclination angles and the structural mechanical performance was systematically studied.
