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The steel bridge deck system, directly subjected to the vehicle load, is an important component to be considered in the optimization design of the bridges. Due to its complex structure, the design parameters are coupled with each other, and many fatigue details in the system result in time-consuming calculation during structure optimization. In view of this, a nonlinear optimization method based on the response surface methodology (RSM) is proposed in this study to simplify the design process and to reduce the amount of calculations during optimization. The optimization design of the steel bridge deck system with two-layer pavement on the top of the steel deck plate is taken as an example, the influence of eight structural parameters is considered. The Box-Behnken design is used to construct a sample space in which the eight structural parameters can be distributed evenly to reduce the calculation workload. The finite element method is used to model the mechanical responses of the steel bridge deck system. From the regression analysis by the RSM, the explicit relationships between the fatigue details and the design parameters can be obtained, based on which the nonlinear optimization design of the bridge deck system is conducted. The influence of constraint functions, objective functions, and optimization algorithms is also analyzed. The method proposed in this study is capable of considering the influence of different structural parameters and different optimization objectives according to the actual needs, which will effectively simplify the optimization design of the steel bridge deck system.

China has constructed hundreds of long-span steel bridges since the 1990s in the last century and accumulated a lot of experience in the design and construction of such bridges. Orthotropic steel box girder is the main structural form of stiffening girders for long-span bridges at present. It has the advantages of light weight, strong ultimate bearing capacity, easy assembly, and construction. However, the related design methods are still inadequate, and the fatigue failure of orthotropic steel bridge deck system is prominent and has not been effectively solved in recent years. It is necessary to investigate the optimization design method of orthotropic steel decks for long-span bridges to improve their safety and economy.

The steel bridge deck system mainly includes the orthotropic steel plate and the pavement on the top of the plate, which directly bears the repeated traffic loads. Due to the complex structure and the characteristics of orthotropic, it is difficult to use the analytical method to guide the optimization design of the steel bridge deck system. Instead, the finite element method is generally used to carry out the related optimization design.

In recent years, the optimization methods of the steel bridge deck system have developed from the single-parameter method to the multiparameter method. In the single-parameter method [

The response surface methodology (RSM) is an effective way to solve the multiparameter optimization problem. The response values are obtained by experiments, and RSM uses multiple regression equations to establish the relationships between the structural parameters and the response values. The optimized structural parameters are finally determined according to the optimization objective. RSM has the advantages of simplifying calculation and predicting the result of the randomly combined parameters. Since being proposed, RSM has been widely used to solve the optimization problems in fields of microorganisms [

Cui et al. [

Due to the mutual coupling effect from different fatigue details of the steel bridge deck system, it is necessary to carry out research on optimization design that multiple fatigue details can be considered to improve the rationality and accuracy of the optimized results. This study proposes a nonlinear optimization method for the design of the steel bridge deck system based on the response surface methodology. A finite element model is developed to analyze the mechanical response of the samples. The explicit relationships between the six fatigue details and the eight structural parameters are obtained through the response surface methodology, based on which the nonlinear optimization design of the bridge deck system is conducted. The influence of constraint functions, objective functions, and the optimization algorithms on the results of nonlinear optimization is analyzed.

Compared to previous research, this study takes into account the influence of steel orthotropic plate and pavement parameters on the structural performance of the steel bridge deck system. Because this study combines RMS and nonlinear optimization, different objectives can be quickly realized based on the objective functions and the constraint functions after the explicit functional relationships between the fatigue details and the structural parameters are obtained by RMS.

Due to the complexity of the steel bridge deck system, the finite element method is normally used to analyze the structural responses such as stress, strain, and deflection. However, the computation workload will increase significantly for optimization problems with multiple objectives, which results in a reduction in design efficiency and is unfavorable to the engineering applications. To improve this situation, this study proposes to use the response surface methodology for the nonlinear optimization design.

The method mainly includes four steps: sample group construction, finite element modeling, function fitting, and nonlinear optimization, as shown in Figure

The process of nonlinear optimization based on response surface methodology.

In view of the complex parameters of the steel bridge deck system and their coupling effects, this study utilizes the response surface methodology (RSM) to carry out the sample group construction of the steel bridge deck system. The explicit relationships between the structural parameters and the response of the steel bridge deck system are obtained from the calculated results of samples by finite element analysis. The multiple quadratic regression equations are normally used in the RSM to obtain the explicit relationships between the response values and the structural parameters, which is a common method for solving multivariable optimization problems to seek the most optimal structural parameters.

Figure

The diagram of response surface representing the explicit relationship between response value and structural parameters.

There are many structural parameters in the steel bridge deck system, and it is challenging to consider the influence of all the structural parameters during the optimization design process. Therefore, it is necessary to firstly determine the major structural parameters and their value ranges that affect the mechanical response of the steel bridge deck system the most. In this study, the structural parameter set is expressed as follows:
^{th} structural parameter of the steel bridge deck system.

Similarly, the response value set is expressed as follows:
^{th} response value.

The explicit functional relationship between the response values and the structural parameters can be expressed as follows:

The major structural parameters can be directly selected if the importance of each one is known before the optimization. If the importance of structural parameters cannot be judged in advance, the Plackett-Burman Design [

Existing studies have shown that for conventional steel bridge deck systems, the eight structural parameters have greater impact on the mechanical response of the system, which include the thickness of the top plate [

The value ranges of the eight structural parameters are summarized based on the survey on the steel bridge deck system in China, as listed in Table

Structural parameters and their value ranges used for optimization design in this study.

Parameters | Unit | Value range | |
---|---|---|---|

The thickness of the top plate | mm | [12, 20] | |

The thickness of the U-rib | mm | [6, 14] | |

The thickness of the transverse diaphragm plate | mm | [10, 20] | |

The spacing of the transverse diaphragm plate | mm | [2400, 3600] | |

The thickness of the bottom pavement layer | mm | [20, 40] | |

The elastic modulus of the bottom pavement layer | MPa | [4000,17000] | |

The thickness of the top pavement layer | mm | [20, 40] | |

The elastic modulus of the top pavement layer | MPa | [4000,17000] | |

Invariant | The width of the U-rib upper opening | mm | 300 (fixed value) |

Invariant | The height of the U-rib | mm | 300 (fixed value) |

Invariant | The width of the U-rib lower opening | mm | 180 (fixed value) |

Invariant | The center distance of U-ribs | mm | 600 (fixed value) |

Invariant | The height of the transverse diaphragm plate | mm | 700 (fixed value) |

Invariant | The opening form of the transverse diaphragm plate | — | Refer to Eurocode 3 |

Structural parameters of the bridge deck system for some typical long-span steel bridges in China [

Bridge name | The thickness of the top plate | The transverse diaphragm plate | The stiffener | The pavement |
---|---|---|---|---|

Su-Tong Yangtze River Highway Bridge | ≥14 mm | 4 m apart; the opening refers to Japanese specification | U-rib ( |
Double-layer epoxy asphalt (55 mm) |

Yangluo Bridge | 14 mm | 8, 10 mm thick; 3.2 m apart | U-rib ( |
Double-layer epoxy asphalt (60 mm) |

The Second Nanjing Yangtze River Bridge | 14 mm | 10 mm thick; 3.75 m apart | U-rib ( |
One-layer epoxy asphalt (50 mm) |

Jiangyin Yangtze River Bridge | 12 mm | 3.2 m apart | U-rib ( |
Double-layer epoxy asphalt (55 mm) |

Nansha Bridge | 16~18 mm | 3.2 m apart | U-rib ( |
Double-layer epoxy asphalt (65 mm) |

Haicang Bridge | 12 mm | 3.0 m apart | U-rib ( |
Double-layer SMA (65 mm) |

Hong Kong-Zhuhai-Macao Bridge | ≥18 mm | The opening refers to EU specification | U-rib ( |
GMA (lower) + SMA (upper) (68 mm) |

The U-rib parameters are upper opening width (mm) × height (mm) × thickness (mm) × center distance (mm).

Existing research shows that there are multiple fatigue details in the orthotropic steel bridge deck system, which are critical factors controlling the defects of the system [

The locations of the six fatigue details. (a) Joints of orthotropic steel plate members. (b) The locations of

In the process of response surface construction, the design of the sample group will directly affect the accuracy of the explicit relationships to be established and further affect the results of the optimization design. The number of the samples should be neither too small nor too large. A small number of samples is not able to establish the explicit relationships to accurately represent the response in the design space. A large number of samples will significantly increase the optimization workload. In addition, the samples should be evenly distributed within the value ranges to improve the accuracy of the response surface functions which can explicitly describe the relationships between the response values and structural parameters. Therefore, the key to sample design is to determine a suitable number of samples that are evenly distributed in the design space.

At present, the commonly used sample design methods in RSM include the factorial experimental design, central composite design (referred to as “CCD”), Box-Behnken design (referred to as “BBD”), D-optimization design, and Latin square design [

The BBD design method selects the combination of parameters at the mid-points of the edges and the center of the sample space as samples. Each parameter always has 3 levels, that is, the maximum, the minimum, and the median in the value ranges. Figure

The samples in the three-parameter distribution model designed by the Box-Behnken design.

According to the value range of the major structural parameters summarized in Table

The calculated six fatigue details under the most unfavorable loading locations for the 120 samples generated in this study.

No. | ^{-6}) |
|||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 16 | 6 | 15 | 3600 | 20 | 10500 | 40 | 10500 | 24.96 | 35.65 | 32.17 | 1.14 | 66.24 | 0.054 |

2 | 16 | 6 | 10 | 3000 | 20 | 10500 | 30 | 4000 | 44.04 | 55.02 | 44.87 | 1.20 | 184.17 | 0.089 |

3 | 16 | 14 | 20 | 3000 | 30 | 4000 | 20 | 10500 | 44.11 | 28.55 | 23.33 | 0.76 | 117.46 | 0.056 |

4 | 16 | 14 | 15 | 3600 | 20 | 10500 | 20 | 10500 | 44.29 | 37.09 | 24.61 | 1.19 | 151.16 | 0.067 |

5 | 16 | 10 | 10 | 3600 | 30 | 10500 | 40 | 4000 | 31.22 | 49.14 | 29.00 | 1.01 | 128.85 | 0.056 |

6 | 12 | 14 | 10 | 3600 | 30 | 10500 | 30 | 10500 | 34.29 | 47.29 | 18.68 | 1.19 | 107.34 | 0.047 |

7 | 12 | 6 | 10 | 2400 | 30 | 10500 | 30 | 10500 | 29.88 | 48.12 | 33.76 | 1.33 | 87.08 | 0.047 |

8 | 16 | 10 | 10 | 3600 | 40 | 4000 | 30 | 10500 | 31.03 | 46.55 | 27.63 | 0.67 | 67.21 | 0.046 |

9 | 20 | 14 | 15 | 3000 | 30 | 10500 | 20 | 4000 | 34.12 | 35.56 | 24.65 | 0.87 | 137.83 | 0.050 |

10 | 20 | 6 | 20 | 2400 | 30 | 10500 | 30 | 10500 | 20.41 | 26.52 | 29.64 | 0.95 | 65.39 | 0.038 |

11 | 16 | 6 | 10 | 3000 | 30 | 4000 | 20 | 10500 | 40.91 | 52.51 | 41.71 | 0.83 | 105.33 | 0.072 |

12 | 16 | 10 | 15 | 3000 | 20 | 17000 | 40 | 4000 | 37.10 | 36.92 | 29.93 | 1.32 | 158.17 | 0.061 |

13 | 16 | 10 | 20 | 3600 | 40 | 17000 | 30 | 10500 | 19.47 | 26.31 | 20.96 | 1.17 | 55.52 | 0.034 |

14 | 12 | 6 | 15 | 3000 | 30 | 10500 | 40 | 17000 | 20.78 | 31.66 | 26.98 | 1.16 | 45.25 | 0.034 |

15 | 20 | 6 | 10 | 3600 | 30 | 10500 | 30 | 10500 | 20.18 | 47.25 | 34.12 | 0.94 | 56.98 | 0.048 |

16 | 16 | 6 | 20 | 3000 | 40 | 10500 | 30 | 4000 | 26.99 | 28.97 | 31.66 | 1.11 | 99.48 | 0.054 |

17 | 16 | 14 | 20 | 3000 | 20 | 10500 | 30 | 4000 | 46.70 | 29.45 | 24.90 | 1.08 | 185.91 | 0.066 |

18 | 16 | 6 | 15 | 2400 | 40 | 10500 | 40 | 10500 | 16.86 | 30.46 | 26.94 | 0.94 | 44.38 | 0.028 |

19 | 20 | 10 | 15 | 2400 | 40 | 10500 | 30 | 4000 | 23.31 | 33.36 | 25.86 | 0.85 | 95.71 | 0.037 |

20 | 16 | 6 | 15 | 2400 | 20 | 10500 | 20 | 10500 | 40.66 | 38.00 | 39.73 | 1.33 | 127.73 | 0.070 |

21 | 12 | 10 | 10 | 3000 | 30 | 4000 | 30 | 4000 | 55.28 | 52.65 | 33.18 | 0.94 | 20.77 | 0.083 |

22 | 12 | 10 | 10 | 3000 | 40 | 10500 | 40 | 10500 | 23.53 | 43.56 | 21.55 | 1.01 | 51.79 | 0.031 |

23 | 12 | 10 | 15 | 3600 | 20 | 10500 | 30 | 17000 | 34.38 | 36.08 | 24.31 | 1.47 | 94.85 | 0.054 |

24 | 16 | 10 | 15 | 3000 | 40 | 4000 | 20 | 17000 | 34.47 | 34.43 | 27.31 | 0.72 | 65.81 | 0.045 |

25 | 16 | 10 | 15 | 3000 | 20 | 4000 | 20 | 4000 | 60.99 | 39.74 | 38.24 | 0.75 | 21.28 | 0.096 |

26 | 20 | 10 | 20 | 3000 | 40 | 10500 | 40 | 10500 | 16.42 | 24.61 | 20.80 | 0.77 | 43.83 | 0.026 |

27 | 20 | 14 | 10 | 2400 | 30 | 10500 | 30 | 10500 | 23.34 | 44.92 | 21.58 | 0.83 | 74.29 | 0.030 |

28 | 20 | 10 | 10 | 3000 | 30 | 17000 | 30 | 4000 | 26.25 | 48.85 | 29.83 | 1.12 | 115.01 | 0.047 |

29 | 16 | 10 | 15 | 3000 | 40 | 17000 | 20 | 4000 | 27.82 | 35.25 | 26.31 | 1.35 | 114.85 | 0.047 |

30 | 12 | 10 | 20 | 3000 | 40 | 10500 | 20 | 10500 | 33.19 | 28.25 | 22.86 | 1.25 | 99.60 | 0.047 |

31 | 12 | 14 | 20 | 2400 | 30 | 10500 | 30 | 10500 | 33.46 | 26.49 | 16.61 | 1.20 | 97.38 | 0.038 |

32 | 12 | 6 | 15 | 3000 | 20 | 4000 | 30 | 10500 | 47.63 | 38.67 | 38.30 | 1.19 | 126.74 | 0.079 |

33 | 20 | 14 | 15 | 3000 | 30 | 10500 | 40 | 17000 | 17.50 | 30.27 | 17.61 | 0.79 | 43.10 | 0.023 |

34 | 16 | 14 | 10 | 3000 | 40 | 10500 | 30 | 4000 | 30.08 | 47.08 | 21.93 | 0.98 | 122.04 | 0.043 |

35 | 16 | 6 | 15 | 3600 | 40 | 10500 | 20 | 10500 | 24.96 | 35.65 | 32.17 | 1.13 | 66.24 | 0.054 |

36 | 16 | 14 | 10 | 3000 | 30 | 4000 | 40 | 10500 | 30.66 | 45.23 | 21.21 | 0.68 | 75.85 | 0.037 |

37 | 12 | 10 | 15 | 2400 | 40 | 10500 | 30 | 17000 | 23.72 | 31.23 | 20.04 | 1.07 | 51.18 | 0.028 |

38 | 16 | 10 | 15 | 3000 | 20 | 17000 | 20 | 17000 | 34.93 | 36.85 | 28.99 | 1.52 | 106.95 | 0.056 |

39 | 16 | 14 | 15 | 2400 | 30 | 17000 | 30 | 4000 | 32.27 | 34.07 | 21.64 | 1.32 | 125.92 | 0.043 |

40 | 16 | 10 | 20 | 3600 | 20 | 4000 | 30 | 10500 | 40.41 | 30.20 | 28.67 | 0.88 | 118.29 | 0.065 |

41 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

42 | 16 | 10 | 20 | 2400 | 20 | 17000 | 30 | 10500 | 30.97 | 27.84 | 25.94 | 1.38 | 103.24 | 0.045 |

43 | 16 | 14 | 15 | 3600 | 30 | 4000 | 30 | 4000 | 44.87 | 36.48 | 24.60 | 0.68 | 178.33 | 0.067 |

44 | 16 | 10 | 10 | 2400 | 30 | 10500 | 20 | 4000 | 40.98 | 51.04 | 33.44 | 1.19 | 157.81 | 0.061 |

45 | 20 | 10 | 15 | 2400 | 20 | 10500 | 30 | 17000 | 23.53 | 33.31 | 25.59 | 1.01 | 66.64 | 0.034 |

46 | 16 | 10 | 10 | 2400 | 20 | 4000 | 30 | 10500 | 39.44 | 49.52 | 31.59 | 0.88 | 105.56 | 0.053 |

47 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

48 | 16 | 10 | 20 | 3600 | 30 | 10500 | 20 | 4000 | 42.04 | 31.32 | 30.24 | 1.20 | 187.97 | 0.077 |

49 | 12 | 10 | 10 | 3000 | 20 | 10500 | 20 | 10500 | 53.19 | 53.70 | 33.29 | 1.62 | 174.64 | 0.083 |

50 | 16 | 10 | 20 | 2400 | 40 | 4000 | 30 | 10500 | 30.63 | 26.26 | 24.71 | 0.67 | 71.93 | 0.038 |

51 | 16 | 14 | 10 | 3000 | 20 | 10500 | 30 | 17000 | 29.99 | 46.96 | 21.66 | 1.16 | 85.13 | 0.040 |

52 | 20 | 10 | 10 | 3000 | 30 | 4000 | 30 | 17000 | 25.63 | 45.89 | 27.84 | 0.64 | 51.82 | 0.037 |

53 | 20 | 10 | 20 | 3000 | 30 | 17000 | 30 | 17000 | 17.72 | 25.75 | 21.93 | 1.08 | 50.13 | 0.029 |

54 | 20 | 10 | 20 | 3000 | 20 | 10500 | 20 | 10500 | 34.40 | 29.81 | 29.41 | 0.96 | 111.30 | 0.057 |

55 | 20 | 10 | 10 | 3000 | 40 | 10500 | 20 | 10500 | 22.77 | 46.57 | 27.30 | 0.88 | 73.52 | 0.038 |

56 | 12 | 10 | 15 | 2400 | 20 | 10500 | 30 | 4000 | 55.88 | 37.75 | 32.12 | 1.52 | 208.45 | 0.080 |

57 | 12 | 10 | 15 | 2400 | 30 | 17000 | 40 | 10500 | 23.79 | 32.15 | 20.69 | 1.36 | 68.85 | 0.031 |

58 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

59 | 16 | 6 | 15 | 2400 | 30 | 17000 | 30 | 17000 | 18.28 | 31.97 | 28.73 | 1.33 | 49.45 | 0.032 |

60 | 16 | 14 | 15 | 2400 | 20 | 10500 | 40 | 10500 | 28.04 | 32.79 | 19.58 | 1.02 | 85.90 | 0.034 |

61 | 20 | 6 | 15 | 3000 | 30 | 10500 | 40 | 4000 | 23.58 | 35.49 | 33.55 | 0.90 | 92.28 | 0.051 |

62 | 12 | 10 | 20 | 3000 | 30 | 4000 | 30 | 17000 | 38.69 | 27.78 | 23.99 | 0.95 | 75.11 | 0.047 |

63 | 12 | 6 | 15 | 3000 | 30 | 10500 | 20 | 4000 | 49.73 | 40.43 | 40.88 | 1.63 | 209.18 | 0.096 |

64 | 20 | 10 | 10 | 3000 | 20 | 10500 | 40 | 10500 | 22.77 | 46.57 | 27.30 | 0.88 | 73.52 | 0.038 |

65 | 20 | 10 | 15 | 2400 | 30 | 4000 | 40 | 10500 | 23.56 | 32.18 | 25.05 | 0.58 | 59.75 | 0.032 |

66 | 16 | 10 | 20 | 2400 | 30 | 10500 | 20 | 17000 | 29.54 | 27.18 | 24.82 | 1.14 | 79.62 | 0.040 |

67 | 20 | 10 | 15 | 2400 | 30 | 17000 | 20 | 10500 | 24.45 | 34.02 | 26.54 | 1.16 | 84.46 | 0.038 |

68 | 12 | 10 | 15 | 3600 | 30 | 4000 | 40 | 10500 | 36.06 | 34.49 | 23.97 | 0.85 | 77.80 | 0.049 |

69 | 20 | 10 | 15 | 3600 | 20 | 10500 | 30 | 4000 | 35.52 | 38.02 | 31.40 | 0.85 | 155.48 | 0.068 |

70 | 16 | 10 | 15 | 3000 | 20 | 4000 | 40 | 17000 | 28.43 | 33.57 | 25.30 | 0.90 | 62.59 | 0.039 |

71 | 12 | 14 | 15 | 3000 | 30 | 10500 | 20 | 17000 | 37.38 | 34.72 | 18.44 | 1.32 | 100.64 | 0.046 |

72 | 16 | 10 | 10 | 3600 | 20 | 17000 | 30 | 10500 | 31.33 | 49.92 | 29.39 | 1.36 | 113.78 | 0.057 |

73 | 20 | 14 | 15 | 3000 | 20 | 4000 | 30 | 10500 | 33.10 | 34.65 | 23.60 | 0.65 | 92.98 | 0.044 |

74 | 12 | 10 | 15 | 3600 | 40 | 10500 | 30 | 4000 | 34.89 | 36.23 | 24.62 | 1.24 | 137.07 | 0.058 |

75 | 16 | 10 | 10 | 2400 | 30 | 10500 | 40 | 17000 | 18.92 | 42.57 | 22.60 | 0.94 | 44.53 | 0.025 |

76 | 12 | 6 | 20 | 3600 | 30 | 10500 | 30 | 10500 | 30.34 | 29.93 | 30.16 | 1.35 | 74.59 | 0.059 |

77 | 16 | 6 | 15 | 3600 | 30 | 17000 | 30 | 4000 | 29.75 | 38.08 | 35.57 | 1.50 | 119.11 | 0.071 |

78 | 16 | 10 | 20 | 2400 | 30 | 10500 | 40 | 4000 | 30.70 | 27.42 | 25.71 | 1.02 | 120.26 | 0.045 |

79 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

80 | 16 | 10 | 10 | 2400 | 40 | 17000 | 30 | 10500 | 19.08 | 43.83 | 23.39 | 1.15 | 59.43 | 0.028 |

81 | 16 | 6 | 20 | 3000 | 30 | 4000 | 40 | 10500 | 27.18 | 27.51 | 30.56 | 0.75 | 55.04 | 0.045 |

82 | 16 | 6 | 10 | 3000 | 30 | 17000 | 40 | 10500 | 18.06 | 44.70 | 31.04 | 1.24 | 50.53 | 0.037 |

83 | 16 | 14 | 20 | 3000 | 40 | 10500 | 30 | 17000 | 21.56 | 24.77 | 16.34 | 0.89 | 50.01 | 0.026 |

84 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

85 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

86 | 12 | 10 | 20 | 3000 | 30 | 17000 | 30 | 4000 | 38.52 | 29.76 | 25.53 | 1.70 | 162.55 | 0.062 |

87 | 12 | 14 | 15 | 3000 | 20 | 17000 | 30 | 10500 | 38.85 | 35.55 | 19.33 | 1.59 | 131.46 | 0.052 |

88 | 16 | 14 | 10 | 3000 | 30 | 17000 | 20 | 10500 | 31.10 | 48.01 | 22.56 | 1.32 | 108.93 | 0.044 |

89 | 16 | 6 | 10 | 3000 | 40 | 10500 | 30 | 17000 | 17.63 | 43.08 | 29.81 | 0.98 | 34.05 | 0.032 |

90 | 20 | 10 | 15 | 3600 | 30 | 17000 | 40 | 10500 | 16.99 | 32.05 | 22.64 | 1.01 | 51.86 | 0.032 |

91 | 12 | 10 | 15 | 3600 | 30 | 17000 | 20 | 10500 | 35.63 | 37.08 | 25.40 | 1.69 | 127.24 | 0.061 |

92 | 16 | 10 | 15 | 3000 | 40 | 17000 | 40 | 17000 | 14.21 | 29.08 | 19.14 | 1.01 | 30.04 | 0.021 |

93 | 20 | 10 | 20 | 3000 | 30 | 4000 | 30 | 4000 | 34.14 | 29.33 | 29.23 | 0.56 | 131.64 | 0.763 |

94 | 16 | 14 | 20 | 3000 | 30 | 17000 | 40 | 10500 | 21.78 | 25.49 | 16.82 | 1.10 | 68.05 | 0.029 |

95 | 12 | 10 | 20 | 3000 | 20 | 10500 | 40 | 10500 | 33.19 | 28.25 | 22.86 | 1.27 | 99.60 | 0.047 |

96 | 16 | 6 | 20 | 3000 | 20 | 10500 | 30 | 17000 | 26.73 | 28.82 | 31.37 | 1.31 | 68.29 | 0.050 |

97 | 12 | 10 | 15 | 2400 | 30 | 4000 | 20 | 10500 | 52.37 | 36.42 | 29.63 | 1.03 | 129.93 | 0.066 |

98 | 20 | 14 | 20 | 3600 | 30 | 10500 | 30 | 10500 | 23.93 | 26.90 | 19.47 | 0.84 | 79.88 | 0.037 |

99 | 16 | 6 | 15 | 3600 | 30 | 4000 | 30 | 17000 | 28.80 | 34.95 | 32.98 | 0.83 | 43.32 | 0.051 |

100 | 16 | 14 | 15 | 3600 | 40 | 10500 | 40 | 10500 | 20.76 | 30.81 | 16.66 | 0.83 | 51.39 | 0.028 |

101 | 16 | 14 | 15 | 2400 | 40 | 10500 | 20 | 10500 | 28.04 | 32.79 | 19.58 | 1.00 | 85.90 | 0.034 |

102 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

103 | 20 | 6 | 15 | 3000 | 30 | 10500 | 20 | 17000 | 22.74 | 35.01 | 32.79 | 1.02 | 58.03 | 0.046 |

104 | 16 | 14 | 15 | 3600 | 30 | 17000 | 30 | 17000 | 22.02 | 32.19 | 17.59 | 1.16 | 62.38 | 0.032 |

105 | 16 | 6 | 20 | 3000 | 30 | 17000 | 20 | 10500 | 28.04 | 29.71 | 32.69 | 1.52 | 93.19 | 0.057 |

106 | 16 | 14 | 15 | 2400 | 30 | 4000 | 30 | 17000 | 32.04 | 32.36 | 20.34 | 0.75 | 65.52 | 0.035 |

107 | 20 | 10 | 15 | 3600 | 30 | 4000 | 20 | 10500 | 33.64 | 36.67 | 29.70 | 0.61 | 94.01 | 0.057 |

108 | 12 | 14 | 15 | 3000 | 40 | 4000 | 30 | 10500 | 39.83 | 33.46 | 18.47 | 0.78 | 90.83 | 0.045 |

109 | 12 | 10 | 10 | 3000 | 30 | 17000 | 30 | 17000 | 24.79 | 45.71 | 23.01 | 1.42 | 63.42 | 0.036 |

110 | 20 | 6 | 15 | 3000 | 40 | 4000 | 30 | 10500 | 23.09 | 33.63 | 32.08 | 0.58 | 49.86 | 0.042 |

111 | 16 | 6 | 15 | 2400 | 30 | 4000 | 30 | 4000 | 41.02 | 37.29 | 39.27 | 0.76 | 151.74 | 0.069 |

112 | 16 | 10 | 10 | 3600 | 30 | 10500 | 20 | 17000 | 29.82 | 48.49 | 28.07 | 1.14 | 83.19 | 0.050 |

113 | 12 | 6 | 15 | 3000 | 40 | 17000 | 30 | 10500 | 20.89 | 32.82 | 27.94 | 1.45 | 52.81 | 0.038 |

114 | 16 | 10 | 15 | 3000 | 40 | 4000 | 40 | 4000 | 33.59 | 34.85 | 27.55 | 0.63 | 113.24 | 0.050 |

115 | 20 | 6 | 15 | 3000 | 20 | 17000 | 30 | 10500 | 24.01 | 36.08 | 34.14 | 1.23 | 80.28 | 0.052 |

116 | 20 | 14 | 15 | 3000 | 40 | 17000 | 30 | 10500 | 17.68 | 31.08 | 18.11 | 0.96 | 57.93 | 0.026 |

117 | 12 | 14 | 15 | 3000 | 30 | 10500 | 40 | 4000 | 39.20 | 35.02 | 19.12 | 1.20 | 152.03 | 0.053 |

118 | 16 | 10 | 15 | 3000 | 30 | 10500 | 30 | 10500 | 27.63 | 34.45 | 25.45 | 1.04 | 87.11 | 0.042 |

119 | 20 | 10 | 15 | 3600 | 40 | 10500 | 30 | 17000 | 16.65 | 30.98 | 21.91 | 0.81 | 36.04 | 0.028 |

120 | 16 | 10 | 20 | 3600 | 30 | 10500 | 40 | 17000 | 19.29 | 25.46 | 20.38 | 0.95 | 39.24 | 0.030 |

Based on the above generated sample group, finite element analysis is conducted to calculate the mechanical responses in terms of the six fatigue details (

To obtain the mechanical responses of the steel bridge deck system under the traffic loads, a finite element model is established by using the ABAQUS software, as shown in Figure

(a) FE model of bridge deck system. (b) The opening type of the transverse diaphragm plate (unit : mm).

The finite element model of the second system of the steel box girder bridge established in this study is composed of four transverse diaphragms in the longitudinal direction and seven U-shaped stiffeners in the transverse direction. Existing research shows that this type of model can better reflect the mechanical responses of the steel bridge deck system [

The steel orthotropic plate was meshed with S4 and S3 shell elements, and the pavement layer was meshed with C3D8 solid elements. The mesh size in this study is set as 10 mm. According to the results of the trial calculation, the mesh size can reduce the calculation workload and maintain the accuracy of the calculation results. The calculation is simulated by a finite element with static implicit scheme.

The finite element model established in this study requires the material mechanical parameters as inputs. The steel parameters are selected according to the provisions of the “Specifications for Design of Highway Steel Bridge (JTG D64-2015)” [

The boundary conditions used in this study are as follows. The bottom of the diaphragms is fixed, and the two sides of the diaphragms are symmetrical about the center line in the transverse direction. There is no displacement between the top steel plate and the pavement layer in the horizontal direction. The

As mentioned earlier, the mechanical response of the orthotropic steel bridge deck system has local effects. According to “Specifications for Design of Highway Steel Bridge (JTG D64-2015)” [

Illustration of finding the unfavorable loading position. (a) The double-wheel load model (unit : mm). (b) The load application area on the steel bridge deck system. (c) The transverse distribution of the double-wheel load.

Due to anisotropy and complex nature of the steel bridge deck structure, multiple fatigue details exist, such as

The most unfavorable loading position of each fatigue detail can be identified by the trial calculation through load traversal. To reduce the calculation workload, one case with structural parameters as follows was carried out first to identify the most unfavorable loading position for each fatigue details. The thickness of the top plate is 14 mm. For U-rib, the thickness is 8 mm, the width of the upper opening is 300 mm, the width of the lower opening is 180 mm, the height is 300 mm, and the center distance between the two adjacent U-ribs is 600 mm. For the diaphragm, the thickness is 10 mm and the center distance between the two adjacent diaphragms is 3200 mm. The pavement includes two layers of epoxy asphalt mixture. The thickness of each layer is 30 mm, the elastic modulus of the pavement materials is 17000 MPa, and the Poisson ratio of the pavement materials is 0.35.

Considering the symmetry of the steel bridge deck structure, the longitudinal range of the loading area is between the second diaphragm and its mid-span, and the transverse range is between the two adjacent U-rib centerlines (Figure

During the traversal of the double-wheel load, the most unfavorable loading position where the maximum stress, strain, or deflection are achieved for each fatigue detail can be determined from the finite element analysis, which will be detailed in Section

The fatigue cracking at the joint between the U-rib and the top plate is mainly caused by the excessive stress amplitude (

The maximum

(a) The location of

Similarly, the fatigue cracking at the diaphragm opening is mainly caused by the excessive stress amplitude at that location (Figure

(a) The location of

The relationship between

(a) The location of

Similar to the calculations of the previous two stress amplitudes, the center point of the double-wheel load is traversed and loaded in the gray area in Figure

The variation of

Considering the shear resistance between the steel plate and the pavement layer, it is necessary to emphasize the shear stress at the bottom of pavement layer in the transverse direction (

(a) The location of

To prevent longitudinal fatigue cracking at the top pavement layer, the tensile strain at the top pavement in the transverse direction (

(a) The location of

In particular, when the center point of the double-wheel load is located at mid-span between the two diaphragm plates and 50 mm away from the U-rib centerline,

To prevent too much deflection of the top plate (

(a) The location of

The relationship between the local deflection of the top plate and the transverse position of the load at the mid-span is plotted in Figure

According to the calculated results shown in Figures

The most unfavorable loading locations and the most unfavorable stress or deflection locations for the six fatigue details.

Fatigue details | The most unfavorable stress or deflection locations | The most unfavorable loading locations | ||
---|---|---|---|---|

Longitudinal distance from the transverse diaphragm plate | Transverse distance from the centerline of the U-rib | Longitudinal distance from the transverse diaphragm plate | Transverse distance from the centerline of the U-rib | |

About 0.09 times of diaphragm plates spacing | 0.5 times the upper opening width of the U-rib | About 0.09 times of diaphragm plates spacing (tensile) |
0.5 times the upper opening width of the U-rib (tensile) | |

Near the transverse diaphragm plate | — | About 0.09 times of diaphragm plates spacing | 0.33 times the upper opening width of the U-rib | |

Near the transverse diaphragm plate | 0.5 times the upper opening width of the U-rib | Near the transverse diaphragm plate | At the centerline of the U-rib | |

Near the transverse diaphragm plate | 0.5 times the upper opening width of the U-rib | Near the transverse diaphragm plate | 0.33 times the upper opening width of the U-rib | |

Mid-span | 1.5 times the upper opening width of the U-rib | Mid-span | 0.17 times the upper opening width of the U-rib | |

Mid-span | At the centerline of the U-rib | Mid-span | At the centerline of the U-rib |

The response surface methodology (RSM) is based on the experimental or numerical results of the sample group to find the explicit relationships between the response values and the structural parameters. At present, the commonly used response surface functions include elementary functions such as polynomial functions, exponential functions, and logarithmic functions. Given that the multivariable quadratic polynomial has simple expressions and can reflect the coupling relationship between structural parameters, this study will use the quadratic polynomial (as shown in Equation (^{th} and ^{th} structural parameters such as

The regressed response surface functions for the six fatigue details are shown in Equations (

The stress amplitude at the welding joint between the top plate and the U-rib in the transverse direction (

The stress amplitude at the opening of the diaphragm plate in the height direction (

The stress amplitude at the inner side of stiffener in the oblique rib direction (

The shear stress at the bottom pavement layer in the transverse direction (

The tensile strain at the top pavement layer in the transverse direction (^{-6}): (predicted

The local deflection of the top plate (

It is seen from the response surface functions that the structural parameters have different degrees of influence on different response values. For

To ensure the applicability of the response surface functions, the correlation between the response surface function and the structural parameters of the initial sample group (shown in Table

The normal residual plot is used to show the relationship between the cumulative frequency distribution of the sample results and the cumulative probability distribution of the theoretical normal distribution. If the distribution of each point is approximate to a straight line, the normal distribution assumption of the sample results is acceptable and the response surface functions obtained by RMS are acceptable. The normal residual plots of all six fatigue details are shown as Figure

The normal residual plots of six fatigue details.

After the explicit functional relationships between the fatigue details and the structural parameters have been obtained, the optimization can be carried out according to the objectives and constraints, which is detailed in Section

The design of the steel bridge deck system can be based on the requirements of both safety and the mass of the system. To balance the safety and the mass, nonlinear optimization is used for design, which is capable of solving the optimization problem with several nonlinear objective functions or constraint functions. The expression of nonlinear optimization is shown in Equation (

Different constraints and optimization objectives will give different optimized results for nonlinear optimization problems. This study will provide both the single-objective optimization and the multiobjective optimization, as detailed follows.

The constraint condition of the single-objective optimization problem mainly considers the safety issue. Moreover, it needs to satisfy the constraints of the structural parameter ranges. According to the relevant provisions “Specifications for Design of Highway Steel Bridge (JTG D64-2015)” [

The single-objective function mainly considers the mass of the unit area materials. The different single optimization objectives and their objective functions are set as follows.

Table

Comparison of optimized structural parameters based on different optimization objectives for the single-objective optimization.

Objective | ||||||||
---|---|---|---|---|---|---|---|---|

1: minimize pavement thickness | 20 | 8 | 20 | 3471 | 20 | 8066 | 22 | 17000 |

2: minimize the steel mass per unit area | 12 | 6 | 14 | 3600 | 40 | 17000 | 40 | 17000 |

3: minimize the steel and pavement mass per unit area | 12 | 6 | 19 | 3600 | 20 | 8479 | 35 | 17000 |

4: minimize the steel and pavement mass per unit area (enhance the constraints by _{1} ≥ 14 mm, _{2} ≥ 8 mm, _{3} ≥ 12 mm, _{4} ≤ 3200 mm, and the others are the same as the constraints list below) |
14 | 8 | 18 | 3200 | 20 | 7579 | 34 | 17000 |

Constraints:

For the optimization of Objective 1, the pavement thickness (

For the optimization of Objective 2, the thickness of the steel deck plate, the U-rib, and the diaphragm (

For the optimization of Objective 3, since the steel density (about 7.90 t/m^{3}) is much higher than that of the asphalt concrete (about 2.45 t/m^{3}), the optimization algorithm tends to reduce steel consumption to minimize the total amount of steel and pavement materials. The thickness and the elastic modulus of the top pavement layer (

The multiobjective optimization problem only constrains the value range of the structural parameters, while taking the structural safety (values of fatigue details, as calculated from Equations (

For the safety objectives, different fatigue details have different limits and their values need to be normalized first for combination. The weights for different objective functions and the normalization methods are summarized in Table

Weight and normalization of different optimization functions for the multiobjective optimization.

Optimization function | ||||||
---|---|---|---|---|---|---|

Weight | ||||||

Normalization |

For the multiobjective optimization composed of the six optimization objects shown in Table

Comparison of optimized structural parameters under different weights.

No. | Weight |
||||||||
---|---|---|---|---|---|---|---|---|---|

1 | (1 : 1 : 1 : 1 : 1 : 1) | 12 | 6 | 20 | 3600 | 40 | 17000 | 40 | 17000 |

2 | (1 : 1 : 1 : 1 : 1 : 10) | 12 | 6 | 19 | 3600 | 20 | 13572 | 35 | 17000 |

3 | (10 : 10 : 10 : 10 : 10 : 1) | 12 | 14 | 20 | 2400 | 40 | 17000 | 40 | 10032 |

4 | (10 : 10 : 10:1 : 1 : 1) | 12 | 14 | 20 | 2400 | 40 | 17000 | 40 | 17000 |

5 | (1:1:1:1 : 10 : 10:1) | 20 | 6 | 20 | 3600 | 40 | 17000 | 40 | 17000 |

Table

Current nonlinear optimization algorithms suitable for computer operation include interior-point, sqp, and active-set [

The comparison between the number of iterations and the optimized results of the three optimization algorithms is shown in Table

Comparison of optimized results by different algorithms.

Algorithm | Initial value | Iteration times | Optimized parameter combination | Optimized results (kg/m^{2}) |
---|---|---|---|---|

Interior-point | 97 | (12,6,18,3200,20,8134,36,17000) |
327.1 | |

Interior-point | 80 | (12,6,18,3200,20,8134,36,17000) |
327.1 | |

Interior-point | 48 | (12,6,18,3200,20,8134,36,17000) |
327.1 | |

sqp | 46 | (12,6,18,3200,20,8134,36,17000) |
327.1 | |

sqp | 39 | (12,6,18,3200,20,8134,36,17000) |
327.1 | |

sqp | 8 | (12,6,16,3200,40,17000,30,17000) |
355.5 | |

Active-set | 34 | (12,6,18,3200,20,8134,36,17000) |
327.1 | |

Active-set | 88 | (12,6,18,3200,34,8366,29,12880) |
344.7 | |

Active-set | 9 | (12,6,16,3200,40,17000,30,17000) |
355.5 |

This study proposes a response surface methodology- (RSM-) based nonlinear method for optimizing the steel bridge deck system to simplify the design process and reduce the calculation workload. The optimization method proposed is first to generate a sample space, within which the samples can be evenly distributed by using the Box-Behnken design to improve the accuracy of the response surface functions. The FE method is used to analyze the mechanical responses (fatigue details) of the sample groups. The regression analysis based on RSM is then conducted to obtain the explicit relationships between the six fatigue details and the eight design parameters of the steel bridge deck system. Finally, the nonlinear optimization design of the system is performed. Five constraint functions were selected in this study in terms of the limit stress or strain referring to the relevant codes. Considering the mass and the safety of the steel bridge deck system, six objectives with assigned weights are taken into account to obtain the optimized result.

In summary, three conclusions can be drawn from this study:

From the calculated results by FE analysis,

It is found that the thickness of pavement on the steel deck can be reduced by increasing the thickness of the steel plate or increasing the elastic modulus of the pavement materials. Because the density of steel is much larger than that of the asphalt pavement materials, increasing the thickness or the elastic modulus of the pavement is an effective method if both the safety and the mass of the steel bridge deck system are considered

The optimized results by different nonlinear optimization algorithms are affected by the initial value of the iteration. The interior-point algorithm is less sensitive to the initial value and can achieve a stable optimization design results of the steel bridge deck system.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.