**2. Bridge Background and Finite Element Model**

The Shawei Zuojiang Bridge is a through-arch bridge with varying cross-section CFST truss ribs. It is designed as a basket-handle arch bridge. It has a main span of 360 m (an effective span of 340 m) and a rise-to-span ratio of 1/4.533. The arch axis follows a catenary curve, and the arch axis coefficient ism=1.55. The two arch ribs are inclined 10 degrees towards the centerline of the bridge, as shown in Figure 1a. The transverse spacing between the arch feet is 38 m. Each arch rib is constructed from a four-tube truss with a varying cross-section. The cross section has a height of 7 m at the arch crown section and 12 m at the arch foot section, and a rib width of 3.2 m. Both upper and lower chord members of the arch rib are made of steel pipes with a diameter of 1200 mm and a wall thickness of 24–32 mm, filled with C60 self-compacting concrete to compensate for shrinkage. The rectangular cross-section is formed by connecting the chord members of the main arch rib with 720 mm-diameter connecting pipes and two vertical web members with a diameter of 610 mm. The chord members of the main arch rib are all made of Q345 steel.

**Figure 1.** Arch rib structure diagram. (**a**) Schematic diagram of arch rib inclination angle; (**b**) arch rib arch crown section; and (**c**) arch rib and arch foot section.

The two arch ribs are connected by "X"-braces. There are 12 "X"-braces arranged along the upper chord and 10 along the lower chord, with the steel pipes of the "X"-braces measuring Φ900 × 18 mm. The cross-sectional diagrams of the main arch rib at the crown and foot are shown in Figures 1b and 1c, respectively, and the overall layout is shown in Figure 2. The cross-section of the bridge deck is shown in Figure 3. The bridge deck has a concrete slab and steel grid beams supporting it. The steel grid beam consists of 7 longitudinal beams and 24 transverse beams. The transverse beam is made of 20-mmthick steel plates and is located under each suspender. The longitudinal beams are I-beams made of Q345 steel. The thickness and the width of the top and bottom flanges of the longitudinal beams are 20 mm and 600 mm, respectively. The thickness of the web plate is 16 mm. The No. 2 and No. 6 beams in Figure 3 have a height of 1924 mm. The No. 1 and No. 7 beams have a height of 1113 mm. The No. 3 and No. 5 beams have a height of 1274 mm. The No. 4 beam has a height of 1400 mm. A C40 concrete slab with a thickness of 140 mm is located at the top of the grid beams. The steel grid beams and concrete slab are connected by a steel plate with a thickness of 100 mm and shear nails. The entire bridge has 24 pairs of suspenders, which are composed of steel strands with a tensile strength of 1860 MPa. This type of through-arch bridge has a clear load path. The dead load and live load are transmitted to the arch ribs through the suspenders and then to the abutment at the arch foot.

**Figure 2.** Elevation of Shawei Zuojiang Bridge.

**Figure 3.** Cross section of the main beam.

MIDAS/CIVIL is software for structural and spatial finite element model building. It was wildly used in bridge construction, temperature, dynamics, and mechanical performance analysis [23,24].

In this study, the finite element software MIDAS/CIVIL was used to implement the spatial finite element modeling of the bridge. The appropriate spatial elements were selected based on the actual stress conditions of each component. The corresponding constraints were applied according to the boundary conditions. The following assumptions were made in the modeling process: (1) all section deformations comply with the crosssectional assumption; (2) no slippage occurs between steel tubes and concrete; and (3) linear elastic theory is adopted and nonlinear effects are not considered.

The arch ribs and the steel grid beams were modeled using beam elements, the bridge slab was modeled using plate elements, and the suspenders were modeled using truss elements. The entire bridge has a total of 3930 nodes and 5714 elements, including 4942 beam elements, 724 plate elements, and 48 truss elements. The abutments were fixed. The suspenders and arch ribs were rigidly connected using elastic connections. The supports at two ends of the main girder were modeled as simply supported. The second-stage load, such as the pavement and parapet, was simulated using uniformly distributed loads applied to the main girder. The types and quantities of elements used in the model are shown in Table 2. The finite element model is shown in Figure 4.

**Table 2.** Element type and quantity.


To verify the accuracy of the finite element model, the measured vertical displacement of the arch ribs during construction was compared with the finite element values. Reflectors are arranged at the center of the top chord of the arch rib at the key sections (i.e., L/8, 2L/8, 3L/8, 4L/8, 5L/8, 6L/8, and 7L/8). A Leica total station was used to measure the vertical displacement of the arch rib before and after the installation of the bridge deck. The resolution of the total station is 0.1 mm. To avoid the impact of solar radiation on structural deformation, measurements were made in the early morning with similar atmospheric temperatures. Meanwhile, the displacements of ribs induced by bridge deck installation were calculated by the FE model. Two strategies were used during the calculation (i.e., the geometric nonlinearity strategy and the linear elastic strategy). Figure 5a is the onsite construction drawing, and Figure 5b is a comparison between the measured values and calculated values of the vertical displacement at key sections of the arch rib. The deviation is shown in Table 3 as well. In both strategies, the calculated vertical displacement of the

key section of the arch rib matches the measured value. The geometric nonlinearity strategy has slightly better performance than the linear elastic strategy, but it is not significant.

**Figure 4.** Finite element model diagram.

**Figure 5.** (**a**) Construction drawing of bridge deck installation Site. (**b**) Comparison of vertical displacement of key sections of arch rib.

**Table 3.** The deviation between calculated values and measured values of vertical displacement at key sections of arch ribs (mm).


In the table, (1) is the linear elastic finite element value, (2) is the geometrically nonlinear finite element value, and (3) is the measured value.

#### **3. Analysis of Reasonable Value of Arch Rib Inclination Angle**

For the basket-handle through the arch bridge, the inclination angle of the arch rib cannot infinitely increase due to the geometrical limit. The width and height of the bridge deck, the arch rise, and other factors are shown in Figure 6. The relationship between the inclination angle of the arch rib and these parameters can be expressed as follows:

tan *<sup>θ</sup>* <sup>=</sup> *<sup>B</sup>*<sup>1</sup> <sup>−</sup> *<sup>B</sup>*<sup>2</sup>

**Figure 6.** Schematic diagram of arch rib inclination calculation.

The arch rib inclination angle is:

$$\theta = \arctan(\frac{B\_1 - B\_2}{2(f - h)}) \tag{2}$$

<sup>2</sup>(*<sup>f</sup>* <sup>−</sup> *<sup>h</sup>*) (1)

where *θ*—Arch rib inclination angle;

*B*1—The bridge deck width between the arch ribs;

*B*2—The clear distance between arch ribs at the crown;

*f*—Rise of the arch;

*h*—The height from the bridge deck to the center of the arch foot.

When the inclination angle is too large, the arch ribs at the top of the arch will intersect, thereby limiting the increase in the inclination angle. On the other hand, a too-small inclination angle will not be able to take advantage of the benefits of the basket-handle arch. A typical basket-handle arch adjusts the inclination angle of the arch ribs by keeping the bridge deck width constant. For the Shawei Zuojiang Bridge, *B*<sup>1</sup> = 28 m, *f* = 75 m, *h* = 16.2 m. When the inclination angle is zero, *θ* = 0◦, it is a parallel arch. The inclination angle reaches its maximum value when *B*<sup>2</sup> = 0, which can be calculated by the formula.

$$\theta = \arctan(\frac{28 - 0}{2(75 - 16.2)}) = 13.393^{\circ \circ}$$

Therefore, the range of inclination angles for arch ribs is 0~13◦. To analyze the mechanical performance with different rib inclination angles, 0◦, 1◦, 2◦, 3◦, 4◦, 5◦, 6◦, 7◦, 8◦, 9◦, 10◦, 11◦, 12◦, and 13◦ were employed in this study. The natural vibration characteristics, linear elastic stability coefficient, internal force, and displacement of arch ribs under static load were calculated for structures at each selected angle. The optimal range of inclination angle for arch ribs was then determined by considering all the mechanical performances above comprehensively.

#### *3.1. Analysis of the Influence of Natural Vibration Characteristics*

To evaluate the natural frequencies and modes of vibration of the bridge under different arch rib inclinations, the structural self-weight and second stage loads (i.e., the

weight of the bridge pavement and the parapet) were considered as the loads. As a result, low-frequency modes contain the majority of the energy, while high-order modes have a negligible impact on the structure's vibration. For the parallel arch (with an inclination angle of 0◦), the first six mode shapes and their natural frequencies are shown in Table 4. The first three mode shapes are shown in Figures 7–9.


**Table 4.** Natural frequency and mode direction of the first six modes.

**Figure 7.** Symmetrical transverse bending of the main arch.

**Figure 8.** Antisymmetric vertical bending of the main beam and arch rib.

**Figure 9.** Antisymmetric transverse bending of the main arch.

According to the analysis in Table 4 and Figures 7–9, the first six modal frequencies of the bridge have a maximum value of 0.8586, indicating that it is a flexible structure. The first two modes were dominated by lateral bending and vertical bending. Torsion

mode was ranked as the sixth mode. It indicated that the structure has a relatively weak lateral stiffness, moderate vertical stiffness, and a relatively strong torsional stiffness. To investigate the modal frequency variation with respect to different inclination angles, the modal frequencies under different inclination angles were compared with those of the parallel arch. The percentage difference of each modal frequency with respect to the parallel arch model is shown in Figure 10.

**Figure 10.** Natural frequency difference of different arch rib angles and parallel arches.

According to Figure 10, the changing rate varied with different modes while increasing the inclination angle. For the first, third, and sixth modes, the curve shows a pattern of increasing first and then decreasing. The natural frequency of the first, third, and sixth modes reached a maximum value at 8◦ to 10◦, indicating the optimal inclination angle for transverse and torsional stiffness improvement. For the second mode, the natural frequency gradually decreases with an increase in the inclination angle. However, the decrease was small—less than 2% at the inclination angle of 8◦ to 10◦. It indicated that the increase in the inclination angle of the arch rib reduced the vertical stiffness of the arch rib, but the effect was small.

#### *3.2. Linear Elastic Stability Analysis*

The linear elastic stability coefficient (λ) is an important indicator for evaluating the stability and safety of structures. The definition of λ can be found in Cao et al. [9], Xu et al. [19], and Wang et al. [25]. Due to the complexity of the structure, it is difficult to obtain its elastic instability limit load using analytical methods. However, using finiteelement numerical analysis methods can yield reliable results [25]. Midas Civil employed the subspace iteration method to calculate λ and the buckling shape of each buckling mode by inputting the load conditions, number of buckling modes, and convergence conditions of structural buckling. The change in inclination angle leads to a change in buckling capacity. Under certain loads, the linear elastic stability coefficient λ increases with the increase in buckling capacity. Therefore, λ can be used as an indicator of structural stability. The dead load was employed as the load condition herein because it is the dominant load on the linear stability of the structure. In practice, only the first buckling mode is important because it has the smallest linear elastic stability coefficient. The shape and λ of the first buckling mode were calculated for both in-plane and out-of-plane buckling. The λ buckling

shape of the parallel arch ribs is shown in Table 5. The corresponding buckling shape diagrams are shown in Figures 11 and 12.

**Table 5.** First-order stability coefficient and instability mode of a parallel arch.


**Figure 11.** Out-of-plane buckling shape of a parallel arch plane.

According to Table 5 and Figures 11 and 12, the first buckling modes of the parallel arch are the anti-symmetric transverse bending and the anti-symmetric vertical bending for out-of-plane buckling and in-plane buckling, respectively. The linear elastic stability coefficient of the out-of-plane buckling is smaller than that of the in-plane buckling for the parallel arch. It indicated that the lateral stiffness of the arch rib is weaker than the vertical stiffness of the parallel arch. The percentage difference of λ with respect to the parallel arch under different arch rib inclination angles is shown in Figure 13.

According to Figure 13, when the inclination angle of the arch rib ranges from 0◦ to 13◦, the λ for the out-of-plane and in-plane directions showed different changing patterns with the increase of the inclination angle. The λ of out-of-plane increased at the beginning and then decreased with the increase in inclination angle. It reached its peak at an inclination angle of 9◦ with a 20.2% improvement and then sharply decreased after the inclination angle exceeded 10◦. The λ of in-plane mode showed a negative correlation with the inclination angle. The decrease was small (3%) when the inclination angle was less than 10◦. However, the λ of in-plane mode decreases significantly when the inclination angle exceeds 10◦. The results indicated that the optimal inclination angle of the arch rib is between 8◦ and 10◦. It effectively improves the out-of-plane stability of the structure with an insignificant decrease in in-plane stability. Therefore, it is recommended to use an appropriate inclination angle of 8◦ to 10◦ for the arch rib while ensuring the optimal first-order stability safety factors for both the out-of-plane and in-plane directions.

**Figure 13.** Percentage difference of the linear elastic stability coefficient with respect to parallel arch.

In order to consider the impact of geometric nonlinearity on the stability of the structure, the λ of the parallel arch was calculated using a geometric nonlinear model. The results are shown in Table 6.

**Table 6.** Stability comparison.


From Table 6, it can be observed that, considering the geometric nonlinearity of the structure, the λ is smaller with respect to that of the linear elastic model. However, the difference is not significant, with 1.5% for out-of-plane mode and 5.86% for in-plane mode. Therefore, the geometric nonlinearity of the structure is no longer considered.

#### *3.3. Static Performance Analysis*

Deformation and internal forces are macroscopic manifestations of the stress state of bridges [26,27], and the internal forces of an arch bridge are mainly affected by the dead load. As the span increases, the proportion of internal forces generated by the dead load becomes larger. Therefore, the analysis of the effect of dead loads on the internal forces of arch bridges is essential [28,29]. The dead load-induced internal forces and displacements of the arch rib were calculated under different inclination angles. As a result, the arch rib is a symmetrical structure, and the arch foot, L/8, 2L/8, 3L/8, and 4L/8 (arch top) were selected as the key sections for displacement analysis. The key sections for internal force selection were the arch foot, L/4, and 4L/8 (arch top). For a parallel arch, the vertical displacements of the arch rib at the arch foot (L/8, 2L/8, 3L/8, and 4L/8) are 0, −16.5 mm, −83 mm, −162.8 mm, and −208.8 mm, respectively. Figure 14 shows the difference in displacement for different arch rib inclination angles with respect to a parallel arch.

**Figure 14.** Vertical displacement difference with respect to parallel arch at key sections.

According to Figure 14, with the increase in the inclination angle, the vertical displacements of the L/8 and 2L/8 sections show a continuous increase. When the inclination angle is less than 10◦, the vertical displacement changes slowly, but it increases rapidly in later stages. Vertical displacement at L/8 shows the largest percentage increase of 10% at the inclination angle of 13◦. However, the vertical displacement value of the L/8 section is relatively small. The 3L/8 and 4L/8 sections show a trend of first decreasing and then increasing. The vertical displacement percentage difference reached its smallest for both 3L/8 and 4L/8 sections when the inclination angle was 8◦. It indicated that when the inclination angle of the arch rib is 8◦, the vertical displacement of the 3L/8 section and the 4L/8 section of the arch rib is the smallest, and the percentage difference is less than 5%. As far as the vertical displacement of the arch rib is concerned, the influence of the inclination angle on the vertical displacement of the structure is not significant.

For the parallel arch, the internal forces of the key sections of the arch ribs are shown in Table 7. The axial force unit in the table is kN and the moment unit is kN·m. The internal force differences of different inclinations with respect to a parallel arch are shown in Figure 15.


**Table 7.** Internal force values of key sections of parallel arches.

From Table 7 and Figure 15, the axial force of each key section of the arch rib shows a different trend with the increase of the inclination angle. The axial forces at the upper chord arch top section and the lower chord 2L/8 section decrease with the inclination angle when the inclination angle is less than 10◦ and then increase. The axial force of the other key sections of the arch rib shows an increasing trend, but the percentage difference is less than 5%, and the change in axial force is relatively small. The bending moment of different sections also shows different trends. The bending moment of the upper chord arch foot section increases with the arch rib inclination angle at first and then decreases. It reached a peak when the inclination angle was 10◦ with a 30% larger parallel arch. The effects of inclination angle on bending moments at other key sections are not significant.

**Figure 15.** Internal force difference of arch ribs at different arch rib angles with respect to parallel arch. (**a**) Axial force and (**b**) bending moment.

#### **4. Conclusions**

Based on the world's largest CFST basket-handle arch bridge, this paper conducted a study on the effect of changes in arch rib inclination angle on the mechanical performance of large CFST basket-handle arch bridges. Different arch rib inclination angles were analyzed in terms of structural natural frequency, linear elastic stability, internal forces, and displacement under static load. The relationship between mechanical performance and arch rib inclination angle was systematically studied, and the following main conclusions were obtained:

(1) According to the analysis of the vibration characteristics, it is recommended that the reasonable range of the rib inclination angle be 8◦ to 10◦, with the optimal angle being 9◦. At this angle, not only does the second-order natural frequency of the structure change slightly, but the first, third, and sixth-order natural frequencies are also increased, which can effectively improve the rib's lateral and torsional stiffness.

(2) Based on the analysis of the structural elastic stability coefficient, it is suggested that the reasonable range of the arch rib inclination angle is 8~10◦, and the optimal angle is 9◦. This not only ensures the best out-of-plane stability of the arch rib, with a stability improvement of 20.2% compared to a parallel arch but also ensures that the decrease in in-plane stability is within 3%.

(3) Based on the analysis of static performance, the variation of rib inclination angle has little effect on rib displacement and internal force. When the rib inclination angle varies within the range of 0◦ to 13◦, the difference ratio of rib displacement is within 5%. The bending moment difference ratio of the upper chord foot section reaches 30% at an inclination angle of 10◦, but the bending moment value is relatively small. Considering the relationship between rib displacement and internal force with rib inclination angle, it is suggested that the rib inclination angle be between 9◦ and 10◦.

(4) Based on the above analysis of the structural mechanics performance under different rib inclination angles, it is recommended that the optimal range of inclination angle for the arch rib should be 8–10◦ for the 300-m-class CFST arch bridge.

In the future, further research will be conducted by considering material nonlinearity, geometric nonlinearity, and material nonlinear coupling effects. Additionally, CFST arch bridges with different span lengths will be investigated as well.

**Author Contributions:** Z.L. and Y.W., methodology and finite element analysis; C.W. and Y.F., drawing and translation; C.L. and S.W., review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** The support from the Guangxi Key R&D Plan (Guike AB22036007), the National Natural Science Foundation Youth Fund Project (51608080), the Chongqing Natural Science Foundation General Project (CSTC2021jcyj-msxm2491), The Science and Technology Innovation Project of the Chongqing Education Commission for the Construction of the Chengdu Chongqing Double City Economic Circle (KJCXZD2020032), and the Special Key Project for Technological Innovation and Application Development in Chongqing (CSTB2022TIAD-KPX0205) are greatly acknowledged.

**Data Availability Statement:** Data is available on request due to restrictions.

**Conflicts of Interest:** The authors declare no conflict of interest.
