Multi-objective optimization algorithm for upper stiffened double-deck steel truss girder bridge based on response surface
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摘要: 为优化上加劲双层钢桁梁桥设计参数，在确保结构安全性的同时降低上部结构工程量，提出了一种基于响应面的适用于多目标优化的改进AdaDelta算法。以一座公轨两用桥为依托工程，建立了关于桥梁初始设计参数的多尺度混合单元模型，进行了计算精度验证；基于桥型结构及受力特点，选取上下弦杆高度、桁架高度、加劲竖杆最大长度和桥面板厚度作为待优化结构参数，选定结构跨中活载挠度、加劲弦最大拉应力、桥面板最大弯曲应力、墩顶截面下弦杆最大压应力、腹杆最大稳定压应力和上部结构工程量为目标函数，建立了反映待优化参数与目标函数非线性关系的响应面拟合方程组；采用改进AdaDelta算法对上加劲双层钢桁梁桥的设计参数进行优化，并与PSO及NSGA-Ⅲ的优化结果进行了对比。研究结果表明：响应面法可准确拟合待优化参数与目标响应函数间的数值关系，采用轻量化方程拟合精度达97%以上、变异系数小于1%，在保证计算精度的同时大幅提升了计算效率；提出的改进AdaDelta算法保留了传统AdaDelta算法收敛速度快、优化过程抖动小的特点，引入的满意度与重要性双权重控制准则使优化结果针对性更强、工程适用性更广，多目标优化满意度结果可平均提升6%；设计参数优化后发生了结构内力重分布，结构跨中活载挠度、加劲弦最大拉应力和桥面板最大弯曲应力分别增加了10.37%、4.65%和4.99%，墩顶截面下弦杆最大压应力和腹杆最大稳定压应力分别降低了0.86%和4.34%，上部结构的工程量降低了4.11%，在保证结构安全的前提下大幅降低了工程造价。提出的算法高效且易于操作，可根据具体优化需求灵活调控权重以达成不同目标导向的设计决策，适合推广至其他体系桥梁的优化设计实践中。Abstract: To optimize the design parameters of the upper stiffened double-deck steel truss girder bridge to ensure structural safety and reduce the engineering workload of the superstructure, an improved AdaDelta algorithm based on the response surface method suitable for multi-objective optimization was proposed. Based on a bridge serving both highways and railways, a multi-scale hybrid element model was established for the initial design parameters, and its calculation accuracy was validated. Based on the structural and mechanical characteristics of the bridge, the height of the upper and lower chords, the height of the truss, the maximum length of the stiffened vertical rods, and the thickness of the bridge deck were selected as the structural parameters to be optimized. In addition, the mid-span live load deflection, the maximum tensile stress of stiffened chords, the maximum bending stress of the bridge deck, the maximum compressive stress of lower chords at the top section of piers, the maximum stabilizing compressive stress of web rods, and the engineering workload of the superstructure were selected as the objective functions. A set of response surface fitting equations reflecting the nonlinear relationships between the parameters to be optimized and the objective functions was established. The improved AdaDelta algorithm was applied to optimize the design parameters of the upper stiffened double-deck steel truss girder bridge and compared with the optimization results of PSO and NSGA-Ⅲ. Analysis results show that the response surface method can accurately fit the numerical relationship between the parameters to be optimized and the target response function. The lightweight equation can achieve a fitting accuracy exceeding 97% and a variation coefficient below 1%. This approach significantly enhances computational efficiency while maintaining accuracy. The improved AdaDelta algorithm retains the fast convergence speed and low fluctuation characteristic of the traditional AdaDelta algorithm. By introducing a dual-weight control criterion based on satisfaction and importance, the optimization results become more targeted and applicable for engineering projects. The multi-objective optimization satisfaction results can be improved by an average of 6%. After the optimization of design parameters, the structural internal forces are redistributed. The mid-span live load deflection, the maximum tensile stress of stiffened chords, and the maximum bending stress of the bridge deck increase by 10.37%, 4.65%, and 4.99%, respectively. The maximum compressive stress of the lower chord at the top section of piers and the maximum stabilizing compressive stress of web rods decrease by 0.86% and 4.34%, respectively, and the engineering workload of the superstructure decreases by 4.11%. The project cost is greatly reduced under the premise of ensuring structural safety. The proposed algorithm is efficient and user-friendly. The weights can be flexibly adjusted according to specific optimization requirements to achieve design decisions oriented toward different goals. It is suitable for application in the optimization design practice of bridges in other systems.
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Key words:
- bridge engineering /
- steel truss girder /
- optimization algorithm /
- response surface /
- light weight /
- hybrid modeling
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图 9 上加劲弦杆设计荷载最大轴力
Figure 9. Maximum axial forces of upper stiffened chord under designed load
图 10 加劲竖杆设计荷载最大轴力
Figure 10. Maximum axial forces of stiffened vertical rod under designed load
表 1 杆件编号与位置对应关系
Table 1. Corresponding relationship between number and position of rods
杆件类型 杆件位置 W1墩顶 第1跨跨中 W2墩顶 第2跨跨中 W3墩顶 第3跨跨中 W4墩顶 第4跨跨中 W5墩顶 中跨1/4截面 中跨跨中 上弦杆 A00A01 A05A06 A10A11 A15A16
A16A17A21A22 A26A27
A27A28A32A33 A39A40 A46A47 A53A54 A58A59 下弦杆 E00E01 E04E05
E05E06E09E10
E10E11E15E16 E20E21
E21E22E26E27 E31E32
E32E33E38E39
E39E40E45E46
E46E47E52E53
E53E54E58E59 斜杆 A01E00 A05E05
A06E05A10E10
A11E10A16E15
A16E16A21E21
A22E21A27E26
A27E27A32E32
A33E32A39E39
A40E39A46E46
A47E46A53E53
A54E53A59E58 上加劲弦杆 A35S38~S55A58 加劲竖杆 S38A38~S55A55 
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表 2 待优化参数设置
Table 2. Parameters to be optimized
编号 待优化参数 优化范围/% 水平 低水平 初始值 高水平 A 上下弦杆高度/mm ±20 1 280 1 600 1 920 B 桁架高度/mm ±10 10 800 12 000 13 200 C 加劲竖杆最大长度/mm ±10 28 800 32 000 35 200 D 桥面板厚度/mm ±10 14.4 16.0 17.6
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表 3 目标函数设置
Table 3. Objective functions
编号 目标函数 R1 跨中活载挠度/mm R2 加劲弦最大拉应力/MPa R3 桥面板最大弯曲应力/MPa R4 墩顶截面下弦杆最大压应力/MPa R5 腹杆最大稳定压应力/MPa R6 上部结构工程量/t
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表 4 设计试验工况
Table 4. Test conditions of design
工况号 待优化参数取值比例/% 工况号 待优化参数取值比例/% A B C D A B C D 1 100 100 100 80 16 120 90 90 90 2 100 100 100 100 17 80 110 110 90 3 120 110 110 90 18 80 90 110 90 4 120 90 110 90 19 80 90 90 110 5 80 110 90 110 20 100 100 100 100 6 100 100 100 120 21 100 100 100 100 7 120 90 90 110 22 60 100 100 100 8 100 100 120 100 23 100 80 100 100 9 80 110 110 110 24 100 120 100 100 10 100 100 100 100 25 120 90 110 110 11 100 100 100 100 26 100 100 100 100 12 100 100 80 100 27 120 110 90 90 13 80 110 90 90 28 120 110 90 110 14 140 100 100 100 29 80 90 90 90 15 80 90 110 110 30 120 110 110 110
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表 5 计算概率评估
Table 5. Evaluation of probability calculation
试验设计项 标准差 方差膨胀因子 设计评估计算概率/% 0.5标准差 1.0标准差 2.0标准差 A 0.20 1.00 20.9 63.0 99.5 B 0.20 1.00 20.9 63.0 99.5 C 0.20 1.00 20.9 63.0 99.5 D 0.20 1.00 20.9 63.0 99.5 AB 0.25 1.00 15.5 46.5 96.2 AC 0.25 1.00 15.5 46.5 96.2 AD 0.25 1.00 15.5 46.5 96.2 BC 0.25 1.00 15.5 46.5 96.2 BD 0.25 1.00 15.5 46.5 96.2 CD 0.25 1.00 15.5 46.5 96.2 A2 0.19 1.05 68.7 99.8 99.9 B2 0.19 1.05 68.7 99.8 99.9 C2 0.19 1.05 68.7 99.8 99.9 D2 0.19 1.05 68.7 99.8 99.9
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表 6 目标函数限值
Table 6. Optimization target limit
编号 目标函数 限值Rilim R1 跨中活载挠度/mm 272.7 R2 加劲弦最大拉应力/MPa 312 R3 桥面板最大弯曲应力/MPa 220 R4 墩顶截面下弦杆最大压应力/MPa 370.5 R5 腹杆最大稳定压应力/MPa 247.1
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表 7 优化程度控制
Table 7. Optimization degree control
编号 目标函数 满意度权重hi 重要性权重wi R1 跨中活载挠度 0.2 1 R2 加劲弦最大拉应力 0.2 3 R3 桥面板最大弯曲应力 0.5 1 R4 墩顶截面下弦杆最大压应力 0.5 3 R5 腹杆最大稳定压应力 0.5 3 R6 上部结构工程量 5.0 5
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表 8 目标函数优化结果
Table 8. Objective function optimization results
算法 优化方案 参数优化值 优化系数/% 满意度函数值 A/mm B/mm C/mm D/mm R1 R2 R3 R4 R5 R6 改进AdaDelta 1 1 280.0 11 805.6 29 017.6 14.4 11.55 4.96 4.99 -0.80 -4.46 -4.13 0.698 2 1 280.0 11 768.4 29 382.4 14.4 11.08 4.87 5.04 -0.81 -4.40 -4.12 0.698 3 1 280.0 11 773.2 29 436.8 14.4 10.94 4.79 4.98 -0.84 -4.40 -4.12 0.698 4 1 280.0 11 828.4 28 812.8 14.4 11.80 5.00 4.96 -0.81 -4.49 -4.13 0.698 5 1 280.0 11 737.2 29 849.6 14.4 10.37 4.65 4.99 -0.86 -4.34 -4.11 0.698 PSO 6 1 280.0 11 041.3 33 241.2 14.4 8.98 5.87 8.39 0.41 -3.44 -4.42 0.662 7 1 280.0 11 127.5 33 271.5 14.4 8.23 5.28 7.72 0.08 -3.52 -4.34 0.673 8 1 280.0 11 227.1 30 479.2 14.4 13.22 7.34 8.37 0.61 -3.61 -4.54 0.667 9 1 280.0 11 021.7 33 574.6 14.4 8.47 5.66 8.39 0.37 -3.42 -4.41 0.662 10 1 280.0 11 036.6 32 258.8 14.4 11.04 6.87 8.89 0.75 -3.43 -4.53 0.648 NSGA-Ⅲ 11 1 282.8 11 061.8 33 966.3 14.5 7.21 5.01 7.85 0.10 -3.36 -4.17 0.652 12 1 282.8 11 892.1 30 989.3 14.5 6.67 2.64 3.23 -1.84 -4.12 -3.68 0.670 13 1 282.8 12 174.9 30 989.3 14.5 4.59 1.00 1.32 -2.70 -4.38 -3.41 0.658 14 1 282.8 12 447.1 29 616.5 14.5 5.24 0.58 0.25 -3.09 -4.63 -3.28 0.653 15 1 282.8 12 365.9 29 437.8 14.5 6.18 1.17 0.88 -2.81 -4.56 -3.38 0.658
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