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Split-type steel box girders are widely used in long-span bridges because of their good wind-resistance performance. In the design stage, a simple finite element model is usually established based on the beam element for wind-resistance design. However, since the irregular cross-beams and diaphragms in the split steel box girder cannot be virtually established, the stiffness of the girder will be underestimated. To improve the accuracy in simulating stiffness of the split-type triple-box steel box girder (STSBG) with the beam element model (BEM), a correction is made to the initial beam element model (IBEM) based on the result of a more refined finite element model. ANSYS is adopted to make a refined model (RM) of a bridge with STSBG as its girder and to calculate its aerostatic responses and dynamic characteristics in 3 typical construction states and 1 finished state. With the reference value, an objective function of the overall residual sum of squares is constructed for the torsion angle of the girder and the frequency of the bridge. Then, the beam element is used for conventional modelling of the bridge, and artificial bee colony (ABC) algorithm is adopted for the optimization and correction of structure parameters of the BEM of the girder. Finally, static and dynamic characteristics of the IBEM and the corrected beam element model (CBEM) are compared with values of the corresponding RM to evaluate the validity of the correction of the model. The results show that the aerostatic responses and dynamic characteristics of the CBEM are close to calculated values of the RM. In more detail, the relative error between the torsion angle of the girder in the middle span of the BEM and the corresponding reference value in the finished state is decreased from +61.71% to +4.94%, and the relative error of torsional fundamental frequency is decreased from −17.43% to +3.66%. According to the calculated value of the RM, ABC algorithm would satisfactorily improve the accuracy in simulating torsional stiffness of the STSBG with the IBEM. This research is expected to provide reference for beam element modelling, which is conducive to accurately simulating torsional stiffness of the STSBG.

It is common that the finite element model of the long-span bridge is built by simulating girder or member based on one-dimensional two-node beam elements. In the construction of such a bridge, closed steel box girder is widely applied by virtue of its light weight, fast construction speed, and good aerodynamic performance. When using beam element to build model for single-box steel girder, single-girder, double-girder, and triple-girder are adopted [

When the span of long-span bridges is increased, the aerodynamic stability of the bridges with closed steel box girders would be impaired. Therefore, a new girder structure, the split steel box girder, with better flutter performance was innovated. Currently, the split steel box girder is divided into the split-type double-box steel box girder (SDSBG) and STSBG. Tsing Lung Bridge in Hong Kong, China, is one of the earliest bridges designed with the SDSBG [

At the same time, many scholars have done a lot of researches on the improvement of torsional stiffness of girder. Bottom bracing systems can be added in curved girder bridge design to improve their dynamic characteristics [

In the wind-resistance design of long-span bridges, it is very important to analyze aerostatic stability and flutter stability. Typical aerostatic instability is the so-called torsional divergence, which may give rise to abrupt structural failure [

At present, studies of the torsional stiffness of STSBG have not yet been reported. Previous studies mainly focused on the influence of changes of parameters on the dynamic characteristics of bridges and some methods to improve the torsional stiffness of structures but failed to fundamentally solve the problem of BEM simulation accuracy of such steel box girders. To simulate the torsional stiffness of such steel box girders more accurately, this paper uses a cable-stayed suspension bridge with STSBG as the engineering background to build a RM to study the torsional stiffness of the girder; meanwhile, it uses this as the reference value to correct the torsional stiffness of the IBEM by using an ABC algorithm. RM can accurately simulate the performance of the girder structure, and ABC algorithm can quickly and accurately correct the stiffness of the girder of IBEM, so that CBEM can replace RM for subsequent simulation calculation.

The torsional stiffness of a beam is codetermined by its elastic modulus and the form of geometric cross sections. Figure

Straight beam with different cross-section forms.

When analyzing the stress on the torsional cross section of round straight beam, it is necessary to rely on its assumption of plane cross section. However, when analyzing nonround straight beam, the cross section of the beam is no longer flat after torsion. Therefore, the formula for the torsion of round straight beam is not applicable to the torsion of nonround straight beam. In the design of bridge, common rectangular beams are particularly popular. As shown in Figure

Some thin-walled cross-section components are commonly used in the design of bridge, including open thin-walled cross sections and closed thin-walled cross sections. The centerline of the wall thickness of open thin-walled cross sections is an unclosed broken line or curve, such as the T-shape in Figure

As for closed thin-walled cross-section components, the centerline of the wall thickness is a closed broken line or curve, as shown in Figure

The cross section of closed thin-walled box girder is often designed as a single box with multichambers, and the torsional constant of the cross section can be calculated by the following formula [

If there are numerous chambers in the cross section of steel box girder, it is more complicated to solve the shear flow in each chamber. The torsional stiffness of the single-box girder can be calculated by the existing analytical formula. If the calculation of the cross section of complex steel box girders is cumbersome in actual operation, it can also be obtained by reading the cross section with the aid of finite element software.

When the girder is a split steel box girder (as shown in Figure

Figure

Bridge layout (unit in m). (a) Elevation layout of the main bridge. (b) Cross section of the girder. (c) Elevation layout of the bridge tower.

The finite element analysis software ANSYS is adopted to establish the IBEMs. The girder is simulated by BEAM44 element, as shown in Figure

Local construction diagram. (a) IBEM. (b) RM.

The IBEMs of typical construction stages (the common double-cantilever state, the longest double-cantilever state, and the preclosure states) and the finished state are established by compiling command stream, as shown in Figure

Finite element models in different stages. (a) The common double-cantilever state. (b) The longest double-cantilever state. (c) The preclosure states. (d) The finished state.

Meanwhile, the RMs corresponding to the IBEMs are established, as shown in Figure

According to the design drawings, the material properties of the bridge are shown in Table

Material properties.

Structure | Material | ||||
---|---|---|---|---|---|

Girder | 2.679 × | Steel | 7850 | 0.3 | |

Steel | 7850 | 0.3 | |||

Cross-girder | 3.588 × | Steel | 7850 | 0.3 | |

5.194 × | Steel | 7850 | 0.3 | ||

Tower | (5.489–8.684) × | Concrete | 2500 | 0.167 | |

(3.254–3.535) × | Concrete | 2500 | 0.167 | ||

(3.608–4.319) × | Concrete | 2500 | 0.167 | ||

Main cables | 4.230 × | Hot-dip galvanized steel wire | 8400 | 0.3 | |

Stay cables | (8.582–19.204) × | Hot-dip galvanized steel wire | 8400 | 0.3 | |

Suspension cables | 8.851 × | Hot-dip galvanized steel wire | 8400 | 0.3 |

In the finished state, the girder restricts the vertical and transverse displacement at the bridge pier to serve as the bearing, as well as the transverse displacement at the intersection of the tower and girder. The bottom of the bridge tower and the anchor end of the main cable are fixed. There is no constraint in the longitudinal direction of the whole bridge. The girder in each construction state is consolidated at the intersection of the tower and girder to serve as the provisional bearing. Through geometric form-finding analysis, the alignment of main cable and girder under dead load is consistent with the designed alignment, and the subsequent analysis of aerostatic responses and dynamic characteristics is also based on these models. In the coordinate system of the finite element model, the transverse bridge direction is taken as the

In this paper, BEM refers to the finite element model based on the beam element that is established by the traditional modelling method. IBEM refers to the finite element model of the girder based on the BEAM44 element (the girder stiffness has not been corrected). RM refers to the finite element model of the girder based on the SHELL181 element. CBEM refers to the corrected IBEM; that is, the finite element model after stiffness of the girder has been corrected. Modelling differences are shown in Table

Modelling differences.

Structure | BEM | IBEM | RM | CBEM |
---|---|---|---|---|

Girder | BEAM44 | BEAM44 | SHELL181 | BEAM44 |

Tower | BEAM44 | BEAM44 | SOLID185 | BEAM44 |

Cables | LINK10 | LINK10 | LINK10 | LINK10 |

After the BEM and RM are established, to make sure that the two have the same mass is a prerequisite for comparing the static and dynamic characteristics. Figure

Girder quality comparison diagram.

The RM has a better performance in simulating the mechanical properties of the bridge girder, but the IBEM may not be able to simulate it accurately. After the comparison of the mass in the two models is made, the modal analysis of the IBEM and the RM is to be carried out. The finite element software ANSYS provides many modal analysis methods. This paper adopts the Block Lanczos method, which is applied to deal with the mass matrix and stiffness matrix in large symmetrical structures and has fast convergence. Fundamental frequency of both models is comparatively shown in Figure

Fundamental frequency difference of the finished state. And

According to Figure

Although more accurate results can be obtained by simulating the mechanical properties of bridges based on a RM, due to the laborious modelling workload, long computing time, and high requirements for computer configuration, it is difficult to limit the cost in engineering design. Moreover, the number of meshes of the model in this paper exceeds 5 million, so it is hard to employ the refined bridge model at the current stage for a dynamic analysis. A more realistic approach is still to model the bridge by using the BEM, and to use simulation results of the RM as the reference value to correct the IBEM to achieve equivalence between the two in terms of static and dynamic properties and then analyze dynamic behavior of the CBEM to achieve accuracy and efficiency when simulating.

There are significant differences in mechanical properties between the IBEM and the RM. The IBEM should be corrected by the model correction method, which, in essence, is to optimize related parameters. The optimization method adopted in this paper is the artificial bee colony (ABC) algorithm, which is a new intelligent optimization algorithm produced by bees’ nectar gathering behavior in nature. At present, it has been successfully applied in engineering optimization [

There are three types of bees: employed, onlooker, and scouts. Each species plays a different role in the optimization process. The employed bees stay above the nectar source and keep the neighboring nectar source in mind. Onlooker bees get information from employed bees and make a resource choice to collect the nectar. In addition, the scouts are responsible for calculation. The algorithm consists of three steps:

First, the employed bees are sent to scamper for resources, and the nectar amount here is the employed bees, and the nectar amount is calculated.

Secondly, the onlooker bees make a resource choice in accordance with the information they took from finding new nectar resources.

Finally, one of the employed bees is nominated randomly as a scout bee, and it is sent to the sources to find new sources [

Half of the bees in the colony are appointed as employed, and the rest as onlooker bees in this optimization algorithm. Therefore, the number of employed bees is equal to the number of nectar sources, and the food source in the algorithm refers to the possible solutions of the problem to be optimized. The amount of nectar from a source means the quality of the source. Less nectar means poor optimization effect, while more nectar means better optimization effect. Synthesize the above-mentioned steps and continue the cycle until the best nectar source is found.

Figure

Model correction flowchart.

According to the 0° three-component coefficient in the finished state and the construction states obtained from the segmental model wind tunnel test, the aerostatic moment is calculated through formula (

Thus, it is calculated that the loading moment of each section is −5383882.73

Then, it is necessary to select the correction parameters of the BEM and specify the correction range of the parameters. As the relative errors in the mechanical properties of the IBEM and the RM are caused by the diaphragms and the cross-beams,

Under the strong wind, the girder of the bridge would undergo elastic torsion or flexural deformation [

In terms of the aerostatic responses of the bridge in this paper at the design wind speed, the calculation results of the IBEM/CBEM and the RM are shown in Figure

Comparison diagram of torsional angle of the girder in different states. (a) The common double-cantilever state. (b) The maximum double-cantilever state. (c) The preclosure states. (d) The finished state. And

The calculation results show that the torsion angle of the girder of the IBEM is quite different from the reference value, and its torsional stiffness simulation is distorted, which fails to accurately simulate the torsional stiffness of the girder. The torsion angle of the girder of the CBEM is closer to that of the RM, and the CBEM can quickly and accurately simulate the aerostatic responses of the girder, thereby improving the efficiency of related projects.

Dynamic characteristics of bridges include natural frequency and mode shape, which are the basis of calculating the critical wind speeds of aerostatic instability and flutter [

Transverse bending fundamental frequency and torsion fundamental frequency of the IBEM/CBEM and the RM are comparatively shown in Figure

Fundamental frequency comparison in different stages.

The finite element models in this paper are established according to the design drawings and relevant parameters and are also under constraints as per the actual structure. The basic frequency is calculated by ANSYS. Jiangyin Yangtze River Bridge [

The relative error between the torsional fundamental frequency and the corresponding reference value gradually augments with the increase in the length of girder of the IBEM, which indicates that if the IBEM is directly employed to simulate the construction stages, the deviation between the predicted torsional stiffness of the girder and the real situation may become increasingly larger.

The frequencies of the mode shape at the first 30 orders of the RM in the finished state are in the range of 0.1 Hz–0.38 Hz. The maximum relative error between the IBEM and the corresponding mode of the RM is up to 20.14%. The fundamental frequency of the CBEM is very close to the calculated values of the RM. The relative errors of the other orders are reduced, and the maximum is only 11.86%, which means that the CBEM has high calculation accuracy. The CBEM can analyze in a fast and accurate manner in many aspects, such as buffeting responses prediction, seismic responses analysis, and parameter sensitivity study.

Based on the RM of three typical construction states and one finished state, 4 groups of corrected parameters are obtained as shown in Table

Correction parameters in different states.

Different states | Correction parameters | |
---|---|---|

The common double-cantilever state | 1 | 2.8 |

The longest double-cantilever state | 1 | 2.8 |

The preclosure states | 1 | 2.8 |

The finished state | 10 | 2.8 |

Based on reference value of the RM in the finished state, the correction parameter of transverse bending fundamental frequency of the IBEM in the finished state is 10. It should be noted that the triple-girder model established in this paper is fundamentally different from the triple-girder model established in reference [

The analysis of all models in this paper is carried out on the same computing platform with a 3.6 GHz CPU and a 96 GB RAM. IBEM/CBEM in the finished stage includes 2809 nodes and 4159 elements. It takes only 5 seconds to calculate dead load of the bridge in the geometric nonlinearity and the subsequent dynamic characteristics of the bridge at the first 30 orders. RM in the finished state includes 5,029,138 nodes and 5,953,325 elements. The computing time would decrease with the increase in the capacity of RAM. More details are given in Figure

The RM computing time variations.

This paper mainly studies the differences of torsional stiffness of STSBG under different modelling methods, and ABC algorithm is employed to correct IBEM parameters. The static/dynamic characteristics of IBEM, CBEN, and RM are compared to evaluate the effectiveness of the model correction. The following conclusions can be drawn based on the obtained results:

For the bridge with a STSBG as the girder, the IBEM established directly by the conventional method may be distorted when simulating the torsional stiffness of the structure. The relative error between the torsional fundamental frequency of the IBEM and the corresponding reference value in the finished state is −17.43%, and the relative error of transverse bending fundamental frequency is −20.14%.

The torsion angle of the girder of the IBEM is quite different from the corresponding reference values. The relative errors between the calculated values of IBEM and the corresponding reference values are +17.40%, +17.67%, +61.28%, and +61.71% in the common double-cantilever state, the longest double-cantilever state, the preclosure states, and the finished state, respectively.

Based on the objective function constructed by calculation values in the RM, the optimal correction parameters can be found out quickly through the ABC algorithm, which improves the simulation accuracy of the torsional stiffness and transverse bending stiffness of the IBEM in the finished state to a greater extent, and the subsequent calculation can be conducted fast and more accurately by application. The CBEM, as a substitute for RM, can make a faster and more accurate calculation for the following steps.

The correction parameters of the torsional stiffness of the girder of the IBEMs in the typical construction stages and the finished state are the same, namely, 2.8. In view of this girder structure, a common double-cantilever RM with a few meshes is only needed to be modelled for rapid correction of the torsional stiffness of IBEM in the finished state, which takes only about two hours for computing. In this way, the fast and accurate calculation of torsional fundamental frequency can be achieved, and the bending-torsion frequency ratio can be calculated correctly, thus analyzing the flutter stability more accurately in the bridge engineering design.

The number of meshes in the RM established in this paper is more than 5 million, and its computing efficiency is mainly determined by the hardware configuration of the computer. Through calculation and analysis, when the frequency of the CPU is given, the higher capacity a RAM has, the more efficient it will be. To improve work efficiency, the capacity of RAM of the computer can be appropriately increased according to needs.

It is noted that the stiffness of the girder also has a significant influence on bridges’ critical wind speeds of aerostatic instability and flutter. Further efforts are still needed on this topic by the authors.

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

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the support for the research work jointly provided by the National Natural Science Foundation of China (no. 51808470) and the Scientific Research Project of Education Department of Hunan Province (no. 18K012).