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Dynamic load allowance (DLA) is a key factor for evaluating the structural condition of bridges; however, insufficient research has been performed regarding the characteristics of DLA in concrete-filled steel tube (CFST) arch bridges. To address this issue, based on an actual CFST arch bridge, the DLA characteristics of bridges are investigated numerically in this study. First, aiming at different structural components, such as the arch rib, main girder, and suspenders, the DLA values obtained at various locations of different structural components are compared in detail, and then the changing regulations of the DLA, considering the influence of different vehicle speeds and various extents of pavement roughness, are summarized and analyzed. Additionally, the relationship between the different DLAs obtained by using the different response indices, that is, displacement, bending moment, and axis force of structure, is investigated. Finally, some conclusions that are significantly beneficial for evaluating or detecting the condition of CFST arch bridges are drawn.

The concrete-filled steel tube (CFST) arch bridge, with its beautiful shape as a landmark building, has been favored by many engineers worldwide. In the CFST structure, the compressive strength of concrete is significantly improved due to the restraint of the steel tube while the stability of the steel tube is improved due to the concrete filling. The mechanical properties of the CFST structure have great advantages. In recent years, plenty of CFST arch bridges have been constructed worldwide, among which hundreds of CFST arch bridges have been built in China during the past 20 years [

The bridge structure bears a variety of loads, the most important of which is the moving vehicle loads. An understanding of the complex interaction between vehicle and bridge vibrations involves nonlinear and dynamic analyses, which have always been a concern in the engineering field. According to numerical analyses, some formulas for dynamic load allowance (DLA) have been suggested for CFST arch bridges [

The dynamic performance of bridge structures under vehicle loads is an important issue in bridge design and operating state assessment, which has attracted the attention of researchers and engineers. The DLA is the regular parameter to express the impact effect of moving vehicles. In recent years, many useful conclusions have been obtained on the coupling vibration of different bridge types, such as metro train-bridge, short-span slab bridge, composite steel bridge, and so forth [

The data show that the DLA plays a vital role in the dynamic performance of a bridge subjected to moving vehicular loads [

To address this issue, based on an actual CFST arch bridge, in this study, the DLA characteristics are investigated numerically. First, aiming at different structural components, the DLA values obtained at various locations of different structural components are compared in detail, and then the changing regulations of the DLA, considering the influence of different vehicle speeds and various extents of pavement roughness, are summarized and analyzed. Additionally, the relationship between different DLAs obtained by using different response indices is investigated. Finally, some conclusions that are beneficial for evaluating or detecting the condition of CFST arch bridges are drawn.

The bridge adopted in this study is the Yilan Mudan River Bridge, which was completed in 1997 and is the first CFST arch bridge in northeastern China (Figure

Photos of the bridge.

The main bridge selected in this paper is shown in Figures

Elevation view of the main bridge (unit: m).

Component drawings (unit: cm): (a) arch rib; (b) longitudinal beam; (c) transverse beam; and (d) details of wind bracing.

The basic information about the materials.

Materials | Young’s modulus (MPa) | Poisson’s ratio | Unit weight (kN/m^{3}) | Components |
---|---|---|---|---|

Concrete C50 | 3.45 × 10^{4} | 0.2 | 26 | Transversal and longitudinal beams, concrete in steel tubes |

Concrete C30 | 3.00 × 10^{4} | 0.2 | 26 | Piers |

Steel 16 Mn | 2.06 × 10^{5} | 0.3 | 78.5 | Arch ribs, wind bracings |

Steel wire | 1.95 × 10^{5} | 0.3 | 78.5 | Suspenders |

The completion of the Yilan Mudan River Bridge has led to the vigorous development of the local economy. Because of this rapid economic development, there has been a surge in traffic flow, and, thus, the vehicle weight is seriously overloaded. Therefore, the bridge is in the long-term overloaded situation, which means that the bridge vibration cannot be ignored.

It should be emphasized that this bridge is very famous in China. For the sake of permeability, the original bridge has no wind bracing, which is also the first CFST arch bridge in China without wind bracing. However, due to the occurrence of a variety of issues, many subsequent reinforcements and transformations have been performed during these years.

For the highway bridge, the moving vehicle load is one of the main loads during its operation. When a vehicle traverses the bridge at a certain speed, the bridge will vibrate under the excitation of the vehicle. In turn, the vibration of the bridge is also an excitation for the vehicle. There are mutual influences between vehicles and bridges, and the resulting vibration is called the vehicle-bridge coupled vibration [

The finite element analysis of this CFST arch bridge is carried out with ANSYS (Figure

Finite element model: (a) spatial grillage model; (b) spatial grillage model with visual shapes; (c) wind bracing; (d) transverse beam.

The vibration equation of the bridge structure can be obtained from the structural dynamics, which is shown in the following equation:_{b}} represents the force acting on the bridge, which is caused by the moving vehicles; [_{b}] represents the mass matrix of the bridge, [_{b}] represents the damping matrix of the bridge, and [_{b}] represents the stiffness matrix of the bridge. In addition, {_{b}} is the vibration displacement of the bridge,

To verify the bridge model established by the finite element software ANSYS, the natural frequencies calculated by the model are compared with those tested by the dynamic fielding test. Low-order modes are more important and easily tested. Therefore, the first two transversal bending modes and the first two vertical bending modes are selected. The modal analysis results calculated by the finite element model are shown in Figure

Modal analysis results: (a) first-order transverse bending mode (1^{st} TBM); (b) second-order transverse bending mode (2^{nd} TBM); (c) first-order vertical bending mode (1^{st} VBM); (d) second-order vertical bending mode (2^{nd} VBM).

Dynamic characteristics of the bridge.

Vibration modes | Calculated natural frequencies (Hz) | Tested natural frequencies (Hz) | Relative error (%) |
---|---|---|---|

1^{st} TBM | 0.595 | 0.608 | 2.18 |

2^{nd} TBM | 1.639 | 1.701 | 3.78 |

1^{st} VBM | 1.598 | 1.602 | 0.25 |

2^{nd} VBM | 2.301 | 2.362 | 2.65 |

The largest relative error in Table

The vehicle with multiaxles is adopted and the spatial model is established. Several assumptions are made about the vehicle model. The wheel and the bridge will contact with each other tightly all the time. Only vertical effects between the vehicle and the bridge are considered while longitudinal and transverse effects are ignored. The vehicle body and all wheels are assumed as rigid with corresponding masses while the spring and the damper are linear. The vehicle model can be seen in Figure

Vehicle model.

The dynamic equations of the vehicle can be written as_{V}} is the load vector induced by the bridge, [_{V}], [_{V}], and [_{V}] are, respectively, the mass matrix, damping matrix, and stiffness matrix of the vehicle. Furthermore, {_{V}} is the displacement of the vehicle, the first derivative of the displacement is the vibration velocity, and the second derivative of the displacement is the vibration acceleration.

The mass matrix, displacement vector, and load vector of the vehicle model are, respectively, shown as follows:_{s} is the mass of the vehicle body, _{ti} is the mass of the _{y} and _{x} are, respectively, the moment of inertia of longitudinal swing and transversal swing:_{s} is the vertical vibration degree of the vehicle body, _{ti} is the vertical vibration degree of the _{ti} and _{ti} are, respectively, the stiffness coefficient and damping coefficient of the

The damping matrix of the vehicle model is divided into four parts by taking the boundary line between the vehicle body and the wheel:_{si} and _{si} are, respectively, the stiffness coefficient and damping coefficient of the suspension system of the

Similarly, when calculating the stiffness matrix of the vehicle model,

The pavement roughness is the main excitation of vibration, which plays an important role in the analysis of vehicle-bridge coupled vibration. The pavement roughness model in this paper is expressed as follows [

The pavement roughness is simulated as a steady-state Gaussian random process. Therefore, the inverse Fourier transform of equation (

Pavement roughness: (a) very good (^{−6}); (b) good (^{−6}); (c) average (^{−6}); (d) poor (^{−6}); (e) very poor (^{−6}).

Due to the difficulties in solving nonlinear problems, it is challenging to study the dynamic response of bridges under the load of moving vehicles, especially for bridge engineers. Many tests have shown that the dynamic responses are indeed larger than the static responses. For this reason, the DLA is introduced in the national specifications. The bridge dynamic responses caused by the moving vehicular loads have been investigated. Furthermore, the influencing factors of the DLA have been analyzed, including the parameters of bridge structures, the type of vehicles, the speed of vehicles, pavement roughness, and so forth.

In this study, eight vehicle speeds are adopted, respectively, ranging from 5 m/s to 40 m/s with intervals of 5 m/s. Five different types of pavement roughness are selected, which can be divided into 5 conditions: very good, good, average, poor, and very poor. The pavement roughness has an obvious influence on the dynamic response of the bridge. Even under the same pavement roughness, the corresponding result has a certain degree of randomness. Therefore, in this paper, the simulation for each kind of pavement roughness is run 20 times, and then the average value of the DLAs obtained from these 20 runs is calculated, which shows that the result is sufficiently accurate [

The dynamic responses of different bridge components are analyzed, including the main girder, main rib arch, and suspenders. It can be seen that the bridge is a symmetrical structure, so only half of the components are studied. A total of 15 sections are selected in Figure

Signs of the components and sections.

According to the characteristics of different components, the concerned dynamic responses are not completely consistent. For the main girders, the displacement and bending moment are considered. For the main arch ribs, the displacement, bending moment, and axial force are considered. For the suspenders, only the axial force is considered. Some dynamic responses are demonstrated in Figure

Dynamic responses (

Figure

As the vault section of the half-through arch bridge is high, it is not convenient to arrange the sensors. In particular, the traditional displacement sensor needs to find a fixed point, which is obviously not suitable for arch rib testing. Noncontact displacement sensors can be adopted, but they are so expensive. Therefore, considering that the main girder is more convenient for arranging the sensors, we hope to find the relationship between the DLA of the arch ribs and that of the main girders.

Two sections, the vault section (

DLAs for the displacement of different components with various pavement roughnesses: (a) very good; (b) good; (c) average; (d) poor; (e) very poor.

In Figure

Therefore, in bridge evaluation or detection, for a new bridge, the deck condition is always good enough, and only the DLA of the main girder is tested, which can represent the maximum value of the measured DLAs. However, if the deck condition is poor in old bridges, then the DLA of the arch rib must be tested and will be much larger.

For half-through arch bridges, the sensor arrangement on the main girder is obviously more convenient than that on the arch rib. Therefore, in bridge detection and evaluation, sensors are always arranged on the main girder to test the dynamic response and DLA.

The static analysis results show that the static response of section _{dynamic} denotes the vertical displacement of the main girder caused by the dynamic loads of the vehicles, and _{stat} denotes the vertical displacement of the main girder caused by the static vehicle loads:_{dynamic} denotes the bending moment of the main girder caused by the dynamic loads of the vehicles, and _{stat} is the bending moment of the main girder caused by the static vehicle loads.

DLA for the midspan section of the main girder

In Figure

The results above show that the sensitive speeds are 20 m/s and 30 m/s. Therefore, taking sections G1 to G6 as the research points when the vehicle passes the bridge at these two speeds under the condition of various bridge deck roughnesses, the DLA of the displacement and the DLA of the bending moment are calculated (Figure

DLAs in different locations of the main girder: (a) DLA of the displacement (

In Figure

In the traditional testing techniques, the displacement test requires a fixed point while the strain test does not. For large-span bridges, especially crossing rivers, lakes, and seas or the piers that are high, the fixed point of the displacement sensor is difficult to find. Therefore, bridge engineers hope to replace displacement test results with strain test results. However, according to the above analysis, these two tests are not completely consistent. To this end, the ratio of these two DLAs is introduced and defined as follows:

Taking the midspan section

DLAs for the different responses of the main girder.

In Figure

The arch rib is the framework of the main arch ring, and it is the important component of the arch bridge. The vault section and the foot section are two critical sections of the arch bridge. The vault section is adopted as an example in this study to analyze the DLA of section displacement, bending moment, and axial force.

The DLAs of the vault section (

DLA for the vault section of the arch rib

In Figure

To explore the relationship between the DLAs of different responses, in addition to the DLA (

The calculation results of the DLAs of the arch rib are demonstrated in Figure

DLAs for the different responses of the arch rib: (a) DLA M/D and (b) DLA N/D.

In Figure

Suspenders are the main components combined from the arch rib to the main girder. The stress characteristic of the suspender is the axial force.

The shortest suspender and the longest suspender are selected in this paper. The other parameters are consistent with the above analysis. The results are demonstrated in Figure

DLA of the suspender: (a) DLA of the shortest suspender (S1); (b) DLA of the longest suspender (S7).

In Figure

As the DLA almost increases with increasing vehicle travelling speed, two travelling speeds, 5 m/s and 40 m/s, are selected. The results of the DLAs in the different suspenders are shown in Figure

DLAs of the different suspenders: (a)

When the deck condition is not worse than the average, the DLA of each suspender exhibits little difference. When the rough condition of the deck is poor or very poor, the DLA of the shortest suspender is different from that of other suspenders, and the DLA of the shortest suspender is obviously greater than other suspenders with increasing vehicle travelling speed.

The dynamic response of the shortest suspender is more obvious with the deterioration of the deck condition and increasing vehicle travelling speed, which may be one of the most important reasons why the shortest suspender of all arch bridges can be easily damaged.

Several conclusions can be drawn based on the investigations of this study, which are described as follows:

The DLA of the arch rib is greater than that of the main girder under very poor deck conditions, which is contrary to the conclusion when the deck is in good condition. If the deck condition is poor in old bridges, the DLA of the arch rib must be tested, which will be much larger. For the main girders and arch ribs, the DLA (

The dynamic responses of the main girders and arch ribs are quite sensitive to the vehicle locations, especially their maximum values. In contrast, the dynamic responses of the suspenders are not evidently changed. However, the cycle number of suspenders is more obvious than that of the main girders and arch ribs.

The pavement roughness has a great impact on the DLAs of the displacement, bending moment, and axial force. All of these characteristics increase significantly as the pavement condition worsens; however, these responses do not necessarily show an increasing trend with the increase in travelling speed of vehicles.

For the main girders, both the DLA (

For the suspenders, the DLA almost increases with increasing vehicle travelling speed. The dynamic response of the shortest suspender is more obvious with the deterioration of the deck condition and an increase in the vehicle travelling speed, which may be one of the most important reasons why the shortest suspender of all arch bridges can be easily damaged.

The data used to support the findings of this study are included within the article.

The authors declare no conflicts of interest.

The first author was the main contributor to this work. The other authors have contributed equally to this work.

This research was financially supported by the National Natural Science Foundation of China (Grant no. 51778194), the China Postdoctoral Science Foundation (Grant no. 2017M621282), and the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2019056).