Dynamic Performance and Anti-earthquake Analysis of Cable-stayed Arch Bridge

The strength, stiffness, and stability check calculations and the effect of earthquakes should be considered in the design of cable-stayed arch bridges with collaborative systems. This study aims to investigate the dynamic performance and structural response of cable-stayed arch bridges under seismic action. The space analysis model is enhanced of the Xiang Feng River Bridge using finite element software Midas Civil, whose lower foundation considers the effects of piles and soil. Firstly the vibration period, vibration frequency, and modal characteristics are computed, thus the dynamic performance is summarized of the bridge. Then, a proper seismic wave is selected according to engineering conditions and in terms of three orthogonal directions: inputting the adjusted El Centro seismic wave, considering Rayleigh damping, and calculating via the Newmark method. Furthermore, a time-history response analysis under the action of one-dimensional and multidimensional earthquake is performed. Lastly, the results of the response analysis is compared and the behavior characteristics of arch bridge is summarized under seismic action. The results show that the transverse stability problem of bridges is prominent and should be the focus of antiearthquake fortification, the inclined cable tower of this bridge is not conducive to the earthquake resistance of the structure in comparison with the vertical cable tower. and the influence of horizontal and vertical earthquake actions should be considered in antiearthquake designs.


Introduction
The major load-bearing parts of cable-stayed bridges are the cable stays, bridge towers, and stiffening beams [1]. In the early 20th century, cable-stayed bridges were developed rapidly due to the development, improvement, and production of high-strength, high-elastic-mold steel wires and its anchorage systems and the improvement of orthotropic steel bridge decks [2]. Currently, the cable-stayed bridge with largest span is the Russky Island Bridge built in 2012; as shown in Figure 1, its center span measures 1104 m, its longest stay cable is 483 m, and the cantilever length of its main beam is 852 m [3]. However, with the increase of the span of cable-stayed bridges, the stability of the cantilever section of the stiffening beam before closing is difficult to guarantee. As the axial force in the stiffening beam is remarkable increased, the proportion of its own weight and the height of the tower are also increased, and the sag effect of the cable stay becomes evident.  As one of the basic forms of bridges, arch bridges have a history of more than 3,000 years [4]. With the continuous innovation of arch bridge building materials, construction technology, and design theory, arch bridges have achieved major breakthroughs in its span and structural form, from stone arch bridges in BC to concrete and simple steel arch bridges in the 19th century and then to truss and concrete-filled steel-tube arch bridges in the 20th century. In 2009, the Chaotianmen Yangtze River Bridge opened to traffic in China. It is the arch bridge with the largest main span in the world. Its mid-through continuous steel truss arch bridge structure is adopted in the main bridge (190 + 552 + 190) m. The main span is installed with a buckle tower to assist the full extension arm from the arch to the beam, and the bridge is integrated in the span. The finished bridge is shown in Figure 2 [5]. With the increase of the span, the traditional arch bridge increased in its own weight, making cable construction difficult. Concrete-filled steel-tube arch bridges are prone to problems, such as corrosion and concrete hollowing in the tubes. Meanwhile, steel arch bridges are characterized by high cost, high maintenance, and stability problems.
Cable-stayed and arch bridges are long-span bridges that are commonly used worldwide. However, the further development of bridge spans is restricted by their respective shortcomings. As shown in Table 1, many cooperative systems of cable-stayed or arched bridges have been proposed by the engineering community due to the increasing demand for bridge aesthetics. These new types of bridges can maximize the respective advantages of cable-stayed and arch bridges, complementing each other's strengths and weaknesses. (1) The spanning capacity of the structure is increased, and its own rigidity is improved. (2) The shape is improved aesthetically. (3) The stability of the bridge structure is increased. (4) The safety of the bridge during the construction and operation phases is enhanced. (5) The height of the bridge tower is reduced. (6) The tension of the stay cable or suspension rod is decreased. (7) Coordination performance is great, and internal force distribution is uniform to reduce local stress. (8) Structural performance and economic benefits are enhanced. The main span is 120 m. The main beam is made of steel truss, and the cable on one side is anchored on the concrete tunnel.
Although cable-stayed or arched bridges appeared early, their development has been slow. People remain skeptical about bridges with collaborative systems due to the complex mechanical characteristics of its structure, the immature related design theory, the imperfect construction management, and some bridge accidents on the cooperative systems that have been completed and opened to traffic. However, the development of bridges in related collaboration systems has played a crucial role in achieving breakthroughs for bridges [6]. Cable-stayed arch bridges are a new type of bridge with a composite system that emerged at the beginning of this century. Theoretical studies on cable-stayed arch bridges are still in its infancy. Pascal Klein introduced the structure and construction process of Malaysia's Jambatan God Shawjala Bridge [7] in detail. With the Liancheng Bridge as the background, Yang Xiangzhan, H. J. Kang, Luo Shidong, Tu Yangzhi, Jiang Hua, Wang Meizhi, etc. performed analysis and verification of seismic performance, dynamic performance experiment analysis, local stress analysis, cable force optimization analysis, parameter analysis, and temperature gradient effect analysis of the box girder section of cable-stayed arch bridges [8][9][10][11][12][13]. Sun Quansheng et al. analyzed the static and construction stress characteristics of the Xiangfeng River Bridge under construction in Dalian [14]. In this study, the dynamic characteristics, the structural response under seismic action, and the seismic measures of the Xiangfeng River Bridge were investigated. The results of this study can provide a reference for the seismic design of similar bridges and promote the development of cable-stayed arch bridges. With the development of society and economy, the increasing demand for bridge aesthetics will promote the construction of collaborative bridges with beautiful and unique shapes, good structural performance, and considerable economic benefits.

Project Background
The Xiangfeng River Bridge, which is located in the Wolong Bay Business District of Dalian, is a bridge that crosses the Xiangfeng River under the East Huanghai Road and alleviates traffic stress and styling.
This bridge, which has two spans measuring 40 m + 90 m=130 m, features a cable-stayed bridge without backstays and a special-shaped arch bridge. The bridge deck is a variable-width structure with a full deck width of 39.0-43.0 m. The main beam is a PC cast-in-place box beam. The height of the beam in the middle span is 2.7 m and is changed to 3.8 m within a range of 21 m on the left and right sides of the pier top. The tower column is a reinforced concrete structure, with a horizontal inclination angle of 56°, and the tower height above the bridge deck is 59.5 m. The bottom of the tower is consolidated with the main beam, arch foot, and main pier. The cable is a finished cable; that is, the cable system is formed by the entire strand extrusion of the steel, with a total of eight cables that are arranged in 8.5 m intervals. The arch rib is a special-shaped arch with a steel box structure and divided into left and right pieces. The steel box is hollow, and the concrete is poured only near the arch foot. The rise-span ratio of the arch axis is 1/3, in which the rise height is 28 m, and the span is 84 m, showing a quadratic parabola and straight line. The main pier is a wall pier, which is consolidated with the main beam, tower, and arch. The auxiliary pier is a rectangular column pier, and the foundation is a pile cap foundation. The design reference period of the bridge structure is 100 years, and the seismic fortification intensity is VII. The overall layout of the bridge is shown in Figure 3.

Theoretical Calculation and Analysis Method
The problem of the dynamic response of a multidegree of freedom system can be concluded mathematically as the initial value problem of a second-order ordinary differential equation system, as follows [15,16]: Equations (1) and (2) show that the damping and stiffness matrices of the structure are time functions. The structure takes different values for different stress statuses, which can be solved using the step-by-step integration method. The equation of motion via the step-by-step integration method is solved as follows: First, the duration of the seismic action is discretized into 1 t , 2 t … n t finite time nodes. Equation (1) can be satisfied at each discrete time node; that is, the value of the structural displacement, velocity, and acceleration at each time node under seismic action can be solved. Assuming the structure's seismic response, such as displacement, velocity, acceleration, meets a certain relationship within the time period t ∆ , t ∆ must be selected to ensure the accuracy and stability of the analysis calculation.

Establishment of Finite Element Model
In this study, the finite element analysis software Midas Civil 2015 is used to analyze the structure of the bridge. The space beam grillage method is used to establish the finite element analysis model of full bridges in proportion to the actual position of each member. The units are divided in accordance with the principle of satisfying calculation accuracy and calculation convenience. In the model, the truss unit is used for the boom and the stay cable, and the elastic modulus of the truss unit is modified in accordance with actual material parameters. The beam units are used for the main beam, cable tower, arch ribs, and lower components. The rigid connection is used for the connection of the boom and the cable stay unit to the main beam, cable tower, or arch rib unit.
The grillage system of the main beam is composed of different types of longitudinal beams and cross beams. The main span box girder is divided into three longitudinal beams in accordance with the cross section, and the side span is divided into seven longitudinal beams. Virtual longitudinal beams are provided on both sides of the bridge deck for the convenience of loading; beams are set in accordance with actual beam and diaphragm positions, and a certain number of virtual beams are set for the side span.
The top of the main pier is consolidated with the main beam, skewbacks, and bottom of the tower. Moreover, the elastic connection between the top of the auxiliary pier and the crossbeam is used to simulate the support in accordance with the stiffness equivalent method. In accordance with the engineering geological conditions, the equivalent soil spring stiffness is calculated in different depths, and the elastic support of the nodes is set at each element node of the base components to simulate the boundary conditions. According to the above modeling principles, the full bridge structure is divided into 2619 units. The finite element model of the space is shown in Figure 4. On the basis of the basic theory of structural dynamic characteristics, a modal analysis of the Xiangfeng River Bridge is conducted with the assist of finite element software. The first 50 orders of natural frequency, period, and vibration shape are calculated using the subspace iteration method. The characteristics of the vibration mode is described, and a period change graph of the natural vibration is drawn to master the basic dynamic performance of the Xiangfeng River Bridge.

Seismic Wave Selection and Input
In seismic engineering, seismic waves are considered elastic waves; that is, particles vibrate in the form of waves when an earthquake occurs and propagate in different directions along the epicenter. Seismic waves are mainly divided into surface and body waves due to different propagation media. The seismic waves can be reflected at the junction of different soil layers in the inhomogeneous medium to meet the continuous deformation condition and stress balance at the interface. Therefore, surface waves are formed. In an ideal infinite homogeneous medium, seismic waves are not reflected; thus, body waves are formed. In accordance with the rotation or distortion, surface waves can be divided into P waves (longitudinal waves) and S waves (transverse waves. [17,18] The soil of the bridge site is type II site soil, that is, medium and soft site soil. The spectral characteristic range of the input seismic wave is determined in accordance with the period of the site characteristic, and the El Centro wave is selected as the input seismic wave for structural time history analysis. Afterward, the maximum value of acceleration response spectrum is designed in accordance with the E2 earthquake action levels to determine the peak horizontal acceleration PGA. In accordance with the PGA, the adjustment coefficient of the peak value of the input seismic wave is calculated. Five to 10 times of the basic structure period is selected because the duration of the seismic wave in the calculation is 20 s. The adjustment data of the El Centro wave are shown in Table 4. The input seismic waves in all directions after adjustment are shown in Figure 6.

Time-history Response Under One-dimensional Earthquake Conditions
The stiffness of the bridge structure varies in all directions, and responses under the action of single-and multi-direction earthquakes differ. First, the time-history response at each control point of the structure under the action of a one-dimensional earthquake is analyzed. By comparing the response spectrum analysis, six displacement response control points are selected for time-history analysis. The points are as follows: main tower top (P1), arch top (P2), main pier top (P3), 2# midspan (P4), 2# span beam end (P5), and 2# pier top (P6). The calculation results are shown in Figures 7-9.
Three orthogonal directions are separately considered in the one-dimensional seismic wave input, which is divided into three working conditions, namely, Working condition 1: input in the transverse direction, Working condition 2: input in the direction along the bridge, Working condition 3: input in the vertical direction. Analysis of the above figures reveals the following: 1) Under working condition 1, the displacement response value in the longitudinal and vertical structures is small when responses to the displacement in the transverse direction occur in the structure.
2) Under working conditions 2 and 3, the deformation of the structure leads to vertical (or along the bridge) displacement when the structure is displaced along the bridge (or vertically). Therefore, the displacement response of the structure along the bridge direction and the vertical direction occurs, and a structural displacement response in the transverse direction is slight. 3) With circumstances (1) and (2) combined, the displacement response of the structure mainly occurs under seismic action in the transverse direction, and the structural deformation is small. Structural deformation occurs in addition to the displacement response and structural deformation under seismic action in the direction along the bridge and the vertical direction. In consideration of the ductile design concept and the principle of energy dissipation, the seismic resistance of the transverse bridge is weak.

Time-history Response Under Multidimensional Earthquake Conditions
Results of the time-history response of the structure under multidimensional earthquake conditions at each control point is shown in Figures 10-11. Horizontal seismic input and that from three orthogonal directions are simultaneously considered in the multidimensional seismic wave input, which is divided into two working conditions, namely, working condition 4: input in the transverse direction + direction along the bridge and working condition 5: input in the transverse direction + direction along the bridge + vertical direction.

Comparative Analysis of the Time-history Response to One-dimensional and Multidimensional Earthquakes
The response internal forces of the main stress positions of the bridge tower, arch rib, and main girder under the five earthquake load conditions are compared and analyzed. The maximum internal force value is shown in Table 5.
A summary table of the maximum values of internal force at key positions of the bridge structure under different earthquake loading conditions is shown below.

Conclusions
The strength, stiffness, and stability check calculations and the effect of earthquakes should be considered in the design of cable-stayed arch bridges with collaborative systems. In this study, a cable-stayed arch bridge is modeled and analyzed using Midas Civil finite element analysis software. Analyses of the dynamic characteristics and structural response of the structure under seismic action are also conducted. The following conclusions are drawn: 1. The low fundamental frequency of bridges with the cooperative systems indicates that the overall rigidity of the structure is small, the spectrum is scattered, the low-order frequencies are dense, and the changes are uniform. The overall mass space distribution of the structure is similar to that of the cable-stayed bridge without backstays, indicating that the dynamic characteristics of the bridge are similar to those of the cable-stayed bridge without backstays. The seismic analysis of the bridge focuses on the analysis of the cable tower, main span, main beam, and arch ribs. The ratio of in-plane fundamental frequency to out-of-plane fundamental frequency is 1.8124/0.5038, indicating that in-plane stiffness is considerably greater than out-of-plane stiffness. This result illustrates that the transverse stability problem of bridges is prominent and should be the focus of antiearthquake fortification. 2. The time-history response analysis reveals that the structural displacement response value under the transverse bridge action is the largest under the earthquake in the single direction. Compared with the horizontal and three-directional earthquake actions, fluctuations occur in the transverse direction and that along the bridge at each control point. The analysis of the control point at the top of the cable tower indicates that the inclined cable tower of this bridge is not conducive to the earthquake resistance of the structure in comparison with the vertical cable tower. The results of the analysis of the internal force response of the structure show that the bridge structural system is complicated, and the influence of horizontal and vertical earthquake actions should be considered in antiearthquake designs.