# Fatigue Optimization of Structural Parameters for Orthotropic Steel Bridge Decks Using RSM and NSGA-II

Authors: Zhiwen Wang; Shouwang Sun; Youliang Ding
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study presents an optimization flow for structural parameters of orthotropic steel bridge decks (OSDs) based on the fatigue performance and the structural weight. The effects of structural parameters, namely, the thicknesses of the deck plate, the U-rib, the diaphragm, and the longitudinal diaphragm, on equivalent fatigue stress ranges were investigated using surrogate modelling based on the response surface methodology (RSM). Finally, the Pareto optimum front was obtained via the nondominated sorting genetic algorithm II (NSGA-II). The results show that the established surrogate response surface model can predict the local equivalent stress amplitudes well. The deck thickness is the most significant factor for the fatigue performance of the deck to U-rib weld and the U-rib to diaphragm weld. The objectives of minimizing the equivalent stress range and the total structural weight are truly conflicting, and the optimum solutions for structural parameters can be obtained from the Pareto optimum front.


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## Body

## 1. Introduction

In the past few decades, orthotropic steel bridge decks (OSDs) have been widely used as the main load-bearing structure in long-span bridges. It has been proven to have many advantages over the option of truss beams but also has some drawbacks, especially fatigue issues [1, 2]. However, relevant fatigue design methodologies are still insufficient, and the failure of OSD structures is prevalent and has not been properly addressed in recent years [3, 4]. Therefore, it is necessary to provide an optimization method for the usual design parameters to enhance the safety and economy of such structures.Previously, several parametric studies have been conducted to study the optimization of typical OSD structural parameters including the plate deck thickness [5, 6], the cross-beam height [7], and the cutout geometry [8, 9], among others [10–12]. The common feature of these studies is that they all use a single-parameter approach, where only one parameter can be changed each time during the optimization process while keeping the other parameters constant. The obvious drawback of this approach is it ignores the interaction effect between different structural parameters and therefore cannot provide the best design solution [13]. In recent years, some multiparameter approaches have been developed during the optimization of steel bridge deck systems. Zhao and Qian [14] used the optimization design module of commercial finite element software to design structural parameters for steel bridge deck systems with structural weight as the optimization objective. Although such commercial software can consider multiple structural parameters well, the premise is that the functional relationship between the optimization objective and the design parameters is known in advance; however, there is no clear functional equation between the detail fatigue stress and the structural parameters. Zhuang et al. [15] performed the optimization of the OSD system with the objective of extending the fatigue life of weld details, in which the neural network was adopted to build the relationship between the optimization object and the multiple design structural parameters. Although the neural network method successfully took into account the interaction of multiple parameters, its training was complicated and time-consuming and the prediction ability of the trained model needs to be further verified.This paper presents a multiobjective optimization method for typical structural parameters of the OSD system based on the response surface methodology (RSM) [16] and the nondominated sorting genetic algorithm II (NSGA-II) [17]. Compared with the previous studies, this paper thoroughly considers the interactions between multiple structural parameters and uses the RSM-based surrogate modelling to achieve the fast establishment of the optimization objective functions. In contrast with the neural networks, the simple form of the RSM surrogate model helps the convergence of the genetic algorithm. The remainder of this paper is organized as follows: Section 2 presents the materials and methods of this study. Specifically, Sections 2.1 and 2.2 introduce the case bridge and the corresponding finite element modelling process; Section 2.3 describes the numerical experiment design used to obtain samples for fitting RSM surrogate models; and Section 2.4 provides the theoretical background of the RSM and NSGA-II and the formulation of the multiobjective optimization problem. Then, Section 3 details the predictive performance of the RSM surrogate model, the effects of typical structural parameters on the optimization objectives, and the optimal solution results for the multiobjective optimization. Finally, Section 4 collects the concluding remarks.


## 2. Materials and Methods

### 2.1. Physical Bridge Description

The research case object in this paper is the Qingzhou Channel Bridge. As a main component of the Hong Kong-Zhuhai-Macao Bridge, the Qingzhou Channel Bridge is a steel box-girder cable-stayed bridge with double pylons and double cable planes. As shown in Figure1(a), the span layout of the bridge is about 236 + 458 + 236 m. The cross-section dimensions and arrangement of vehicle lanes are shown in Figure 1(b). Generally, the welds under heavy lanes are more prone to fatigue cracking, where a pair of the deck to U-rib and U-rib to diaphragm welds marked in the red box were selected as two typical fatigue details of interest in this study.Figure 1
Details of the Qingzhou Channel Bridge (unit: mm): (a) longitudinal layout of the bridge; (b) cross section of the main girder.
(a)(b)


### 2.2. Finite Element Models

To obtain the stress response at fatigue details of interest, numerical simulation of the Qingzhou Channel Bridge under vehicle loads was performed. Multiscale modelling of the case bridge was conducted based on the submodel technique provided by ABAQUS. The modelling process is illustrated in Figure2 where two scales were taken into account: a global model and two refined submodels. The global model was a semibox girder with five diaphragms along the longitudinal direction. All degrees of freedom at both ends of the model were constrained given the real boundaries in the physical bridge [18]. Due to short influence lines in the transverse direction, the vehicle loading on the right half of the steel girder has a negligible impact on the details in the left half. For this reason, the symmetrical constraint was applied at the center plane of the girder to reduce the simulation burden [19]. Two submodels with the deck to U-rib weld and the U-rib to diaphragm weld were established and connected to the global model using the provided submodel boundary condition. To improve the calculation efficiency, the FEMs were meshed using 4-node doubly curved shell elements. The global mesh sizes of the global model and submodels were set as 100 and 20 mm, respectively, while the welded connections were further refined with 1 mm meshes. The stress response of OSDs under vehicle loading has been extensively studied in [18], and all the results show that the vehicle-induced fatigue stresses are much less than the yield strength of the steel used. Therefore, the linear elastic model was used to simulate the mechanical response of the steel material in ABAQUS. The elastic modulus and Poisson’s ratio of the steel were 2.06 × 105 MPa and 0.3, respectively, according to the used Chinese Steel Q345 (“345” represents the designated yield strength, “q” represents the application to bridges, “D” represents the quality grade corresponding to the required impact toughness at −20°C) in the case bridge.Figure 2
Multiscale model of the steel girder in the Qingzhou Channel Bridge.The FEMs were loaded with the standard fatigue vehicle model III specified in the Chinese standard “Specification for Design of Highway Steel Bridge” [20]. Figure 3 provides the views of the vehicle's longitudinal and transverse sections. It has been pointed out that change of the structural parameters will not affect the global unfavorable loading position [15]. Given the transverse randomness of the vehicles, the most unfavorable transverse loading position was first determined herein using the actual structural parameters to save simulation time and computational power. The most unfavorable transverse loading position is the location where the vehicle travels in the bridge longitudinal direction and obtains the stress history with the maximum equivalent fatigue stress range. To obtain this position, 23 transverse loading cases were set at 150 mm intervals along the bridge in the transverse direction, as shown in Figure 4. Then, at each transverse loading case, the vehicle was moved along the bridge in the longitudinal direction to obtain the fatigue stress history and calculate the corresponding equivalent fatigue stress range. Finally, the transverse loading case with the maximum equivalent fatigue stress range was determined as the most unfavorable transverse loading position. After that, for other samples of the structural parameter combination, the specified vehicle load was directly placed at the derived unfavorable loading positions to obtain and calculate the corresponding equivalent fatigue stress range.Figure 3
Fatigue vehicle model: (a) vehicle cross section; (b) vehicle longitudinal section [12].
(a)(b)Figure 4
Schematic for transverse loading cases.


### 2.3. Design of Numerical Experiments

Much research has shown the significant effect of component thickness on fatigue performance of OSD structures [5, 8, 15, 21]. Therefore, the thicknesses of the deck plate (tdec), U-rib (trib), diaphragm (tdia), and longitudinal diaphragm (tldia) were selected as the design variables. We would like to emphasize that the proposed optimization framework in this study is not limited to the structural parameters mentioned above. The optimal Latin-hypercube design (OLhd) proposed by Park [22] was adopted to generate evenly distributed sample points for the design variables in their domains. In this paper, the domains were determined to be tdec of 10–20 mm, trib of 5–10 mm, tdia of 10–20 mm, and tldia of 10–20 mm.Finally, a total of 71 samples of the structural parameter combination were collected, and corresponding fatigue stress histories at the most unfavorable transverse loading position were obtained using the built FEMs. According to fatigue damage equivalence and the abovementioned Chinese specification, the equivalent stress rangeSeq of different stress histories can be calculated as follows:(1)Seq=∑Si≥ΔσDniSi3ΔσD2+∑Sj<ΔσDnjSj51/5,where ΔσD is the fatigue limit for the weld detail; Si and Sj are the fatigue ranges greater and less than ΔσD, respectively; and ni and nj are the numbers of Si and Sj, respectively. Herein, the parameters ni, nj, Si, and Sj were obtained by the rainflow counting algorithm. The rainflow algorithm is a cycle counting method that stores service measurements in a form suitable for structural fatigue analysis: fatigue life prediction and simulation testing. For more details about the rainflow method, we refer the readers to [23]. Based on the above process, the samples of structural parameters and corresponding equivalent stress ranges at the most unfavorable transverse loading position are listed in Table 1. Note that the deck to U-rib weld and the U-rib to diaphragm weld are denoted as D1 and D2 in this paper.Table 1
Results of the numerical experiment design.
No.Design variablesEquivalent stress rangestdec (mm)trib (mm)tdia (mm)tldia (mm)Seq,D1 (MPa)Seq,D2 (MPa)1117161329.912.22177171412.110.83185131911.110.54127151925.011.85138191421.311.56178191912.210.77117111829.911.48146121118.110.991810181211.09.110127191125.012.211135181721.012.112179151012.39.313167111313.810.01419615159.710.41519817179.710.016135141421.011.517117181929.912.71819712189.69.819187162010.810.6201710162012.49.22119619149.710.922129171925.211.023178201512.210.724136111521.111.125108121736.411.42619611119.79.7271210121525.39.428189111110.98.629158161815.910.730159101816.09.231148121718.310.232116201529.713.2331310151221.59.634146161118.111.535167131013.810.2361410191218.59.937108171636.412.43820818128.79.939139131921.510.140109141336.411.141148201818.311.542165171613.811.3431510141616.19.144149121118.49.445175141212.310.646117131529.911.747166191713.711.648116131229.711.949166141613.710.950129151625.210.651156161915.711.55219714129.69.953116151829.712.5541710131312.48.65520915198.79.156168191113.910.75719618189.710.958178132012.29.959156191315.711.760119201630.111.861166101713.710.46219911179.78.763127121025.011.064189171610.99.665137171621.211.866169101414.08.967159161416.0106820711148.79.369157132015.710.870186181110.810.971148141318.310.5


### 2.4. Surrogate Modelling and Optimization

#### 2.4.1. Response Surface Methodology

During the optimization process, two main optimization objectives are considered, namely, maximizations of the equivalent stress range and the structural total weight. Among them, the equivalent stress range was correlated with the structural parameters via the RSM based on the data provided in Table1.RSM constructs a regression model between the optimization objective and the design variables via mathematical methods. The relationship between the objective valuey and the design variables x1, x2, …, xn (n is the number of the design variables) can be described as follows:(2)y=fx1,x2,…,xn+ε,where f denotes an approximation function and ε denotes the residual error between the predicted value and the real value. The approximation function commonly comprises a lower-order degree polynomial, and when a quadratic polynomial is used, the predicted value y’ can be written as follows:(3)y^=b0+∑i=1nbixi+∑i=1nbiixi2+∑i=1n−1∑j=i+1nbijxixj,where xi and xj are the design variables and b0, bi, bii, and bij are polynomial regression coefficients.


#### 2.4.2. Nondominated Sorting Genetic Algorithm II

Herein, the optimum structural parameters were determined via NSGA-II to minimize the local equivalent stress ranges and the total structural weight. The basic idea of NSGA-II is as follows: First, the initial population of sizeN is randomly generated, and the initial offspring generation can be obtained via the basic operations of selection, crossover, and mutation. Then, from the second generation, the parent generation is merged with the offspring generation for fast nondominated sorting, and the crowing distance is calculated in each nondominated layer. The new parent generation is formed by selecting the appropriate individuals according to the nondominance relationship and the crowding distance of individuals. Finally, the new offspring generation is generated by the basic operations and so on until the end of the algorithm. The corresponding flowchart of NSGA-II is shown in Figure 5.Figure 5
Procedures of NSGA-II.


#### 2.4.3. Optimization Problem Formulation

In this study, optimum structural parameters can be optimized with two conflicting objectives associated with the minimization of both the equivalent stress ranges and the total structural weight. This optimization problem can be formulated as follows:(4)Findx=tdec,trib,tdia,tldia,MinFx=Seq,D1,Seq,D2,Wt,s.t.10≤tdec≤20mm,5≤trib≤10mm,10≤tdia≤20mm,10≤tldia≤20mm,where x is the vector of design variables, which consists of component thicknesses of tdec, trib, tdia, and tldia; F(x) is the vector of optimization objective functions; Seq,D1 is the equivalent stress range at the deck to U-rib weld detail; Seq,D2 is the equivalent stress range at the U-rib to diaphragm weld detail; and Wt is the total structural weight, which can be calculated as a product of the steel volume, and its density of 7.85 g/cm3.The NSGA-II complied in a Python file was used to obtain the Pareto optimum solutions for the representative structural parameters based on the trade-off between the conflicting objectives. Crossover and mutation operations were conducted with respective probabilities of 90% and 20%. After 200 generations with 100 populations, the Pareto optimal solution set was found. An overall flowchart for the optimization of structural parameters is provided in Figure6.Figure 6
Overall flowchart for optimization of structural parameters.


## 2.1. Physical Bridge Description

The research case object in this paper is the Qingzhou Channel Bridge. As a main component of the Hong Kong-Zhuhai-Macao Bridge, the Qingzhou Channel Bridge is a steel box-girder cable-stayed bridge with double pylons and double cable planes. As shown in Figure1(a), the span layout of the bridge is about 236 + 458 + 236 m. The cross-section dimensions and arrangement of vehicle lanes are shown in Figure 1(b). Generally, the welds under heavy lanes are more prone to fatigue cracking, where a pair of the deck to U-rib and U-rib to diaphragm welds marked in the red box were selected as two typical fatigue details of interest in this study.Figure 1
Details of the Qingzhou Channel Bridge (unit: mm): (a) longitudinal layout of the bridge; (b) cross section of the main girder.
(a)(b)


## 2.2. Finite Element Models

To obtain the stress response at fatigue details of interest, numerical simulation of the Qingzhou Channel Bridge under vehicle loads was performed. Multiscale modelling of the case bridge was conducted based on the submodel technique provided by ABAQUS. The modelling process is illustrated in Figure2 where two scales were taken into account: a global model and two refined submodels. The global model was a semibox girder with five diaphragms along the longitudinal direction. All degrees of freedom at both ends of the model were constrained given the real boundaries in the physical bridge [18]. Due to short influence lines in the transverse direction, the vehicle loading on the right half of the steel girder has a negligible impact on the details in the left half. For this reason, the symmetrical constraint was applied at the center plane of the girder to reduce the simulation burden [19]. Two submodels with the deck to U-rib weld and the U-rib to diaphragm weld were established and connected to the global model using the provided submodel boundary condition. To improve the calculation efficiency, the FEMs were meshed using 4-node doubly curved shell elements. The global mesh sizes of the global model and submodels were set as 100 and 20 mm, respectively, while the welded connections were further refined with 1 mm meshes. The stress response of OSDs under vehicle loading has been extensively studied in [18], and all the results show that the vehicle-induced fatigue stresses are much less than the yield strength of the steel used. Therefore, the linear elastic model was used to simulate the mechanical response of the steel material in ABAQUS. The elastic modulus and Poisson’s ratio of the steel were 2.06 × 105 MPa and 0.3, respectively, according to the used Chinese Steel Q345 (“345” represents the designated yield strength, “q” represents the application to bridges, “D” represents the quality grade corresponding to the required impact toughness at −20°C) in the case bridge.Figure 2
Multiscale model of the steel girder in the Qingzhou Channel Bridge.The FEMs were loaded with the standard fatigue vehicle model III specified in the Chinese standard “Specification for Design of Highway Steel Bridge” [20]. Figure 3 provides the views of the vehicle's longitudinal and transverse sections. It has been pointed out that change of the structural parameters will not affect the global unfavorable loading position [15]. Given the transverse randomness of the vehicles, the most unfavorable transverse loading position was first determined herein using the actual structural parameters to save simulation time and computational power. The most unfavorable transverse loading position is the location where the vehicle travels in the bridge longitudinal direction and obtains the stress history with the maximum equivalent fatigue stress range. To obtain this position, 23 transverse loading cases were set at 150 mm intervals along the bridge in the transverse direction, as shown in Figure 4. Then, at each transverse loading case, the vehicle was moved along the bridge in the longitudinal direction to obtain the fatigue stress history and calculate the corresponding equivalent fatigue stress range. Finally, the transverse loading case with the maximum equivalent fatigue stress range was determined as the most unfavorable transverse loading position. After that, for other samples of the structural parameter combination, the specified vehicle load was directly placed at the derived unfavorable loading positions to obtain and calculate the corresponding equivalent fatigue stress range.Figure 3
Fatigue vehicle model: (a) vehicle cross section; (b) vehicle longitudinal section [12].
(a)(b)Figure 4
Schematic for transverse loading cases.


## 2.3. Design of Numerical Experiments

Much research has shown the significant effect of component thickness on fatigue performance of OSD structures [5, 8, 15, 21]. Therefore, the thicknesses of the deck plate (tdec), U-rib (trib), diaphragm (tdia), and longitudinal diaphragm (tldia) were selected as the design variables. We would like to emphasize that the proposed optimization framework in this study is not limited to the structural parameters mentioned above. The optimal Latin-hypercube design (OLhd) proposed by Park [22] was adopted to generate evenly distributed sample points for the design variables in their domains. In this paper, the domains were determined to be tdec of 10–20 mm, trib of 5–10 mm, tdia of 10–20 mm, and tldia of 10–20 mm.Finally, a total of 71 samples of the structural parameter combination were collected, and corresponding fatigue stress histories at the most unfavorable transverse loading position were obtained using the built FEMs. According to fatigue damage equivalence and the abovementioned Chinese specification, the equivalent stress rangeSeq of different stress histories can be calculated as follows:(1)Seq=∑Si≥ΔσDniSi3ΔσD2+∑Sj<ΔσDnjSj51/5,where ΔσD is the fatigue limit for the weld detail; Si and Sj are the fatigue ranges greater and less than ΔσD, respectively; and ni and nj are the numbers of Si and Sj, respectively. Herein, the parameters ni, nj, Si, and Sj were obtained by the rainflow counting algorithm. The rainflow algorithm is a cycle counting method that stores service measurements in a form suitable for structural fatigue analysis: fatigue life prediction and simulation testing. For more details about the rainflow method, we refer the readers to [23]. Based on the above process, the samples of structural parameters and corresponding equivalent stress ranges at the most unfavorable transverse loading position are listed in Table 1. Note that the deck to U-rib weld and the U-rib to diaphragm weld are denoted as D1 and D2 in this paper.Table 1
Results of the numerical experiment design.
No.Design variablesEquivalent stress rangestdec (mm)trib (mm)tdia (mm)tldia (mm)Seq,D1 (MPa)Seq,D2 (MPa)1117161329.912.22177171412.110.83185131911.110.54127151925.011.85138191421.311.56178191912.210.77117111829.911.48146121118.110.991810181211.09.110127191125.012.211135181721.012.112179151012.39.313167111313.810.01419615159.710.41519817179.710.016135141421.011.517117181929.912.71819712189.69.819187162010.810.6201710162012.49.22119619149.710.922129171925.211.023178201512.210.724136111521.111.125108121736.411.42619611119.79.7271210121525.39.428189111110.98.629158161815.910.730159101816.09.231148121718.310.232116201529.713.2331310151221.59.634146161118.111.535167131013.810.2361410191218.59.937108171636.412.43820818128.79.939139131921.510.140109141336.411.141148201818.311.542165171613.811.3431510141616.19.144149121118.49.445175141212.310.646117131529.911.747166191713.711.648116131229.711.949166141613.710.950129151625.210.651156161915.711.55219714129.69.953116151829.712.5541710131312.48.65520915198.79.156168191113.910.75719618189.710.958178132012.29.959156191315.711.760119201630.111.861166101713.710.46219911179.78.763127121025.011.064189171610.99.665137171621.211.866169101414.08.967159161416.0106820711148.79.369157132015.710.870186181110.810.971148141318.310.5


## 2.4. Surrogate Modelling and Optimization

### 2.4.1. Response Surface Methodology

During the optimization process, two main optimization objectives are considered, namely, maximizations of the equivalent stress range and the structural total weight. Among them, the equivalent stress range was correlated with the structural parameters via the RSM based on the data provided in Table1.RSM constructs a regression model between the optimization objective and the design variables via mathematical methods. The relationship between the objective valuey and the design variables x1, x2, …, xn (n is the number of the design variables) can be described as follows:(2)y=fx1,x2,…,xn+ε,where f denotes an approximation function and ε denotes the residual error between the predicted value and the real value. The approximation function commonly comprises a lower-order degree polynomial, and when a quadratic polynomial is used, the predicted value y’ can be written as follows:(3)y^=b0+∑i=1nbixi+∑i=1nbiixi2+∑i=1n−1∑j=i+1nbijxixj,where xi and xj are the design variables and b0, bi, bii, and bij are polynomial regression coefficients.


### 2.4.2. Nondominated Sorting Genetic Algorithm II

Herein, the optimum structural parameters were determined via NSGA-II to minimize the local equivalent stress ranges and the total structural weight. The basic idea of NSGA-II is as follows: First, the initial population of sizeN is randomly generated, and the initial offspring generation can be obtained via the basic operations of selection, crossover, and mutation. Then, from the second generation, the parent generation is merged with the offspring generation for fast nondominated sorting, and the crowing distance is calculated in each nondominated layer. The new parent generation is formed by selecting the appropriate individuals according to the nondominance relationship and the crowding distance of individuals. Finally, the new offspring generation is generated by the basic operations and so on until the end of the algorithm. The corresponding flowchart of NSGA-II is shown in Figure 5.Figure 5
Procedures of NSGA-II.


### 2.4.3. Optimization Problem Formulation

In this study, optimum structural parameters can be optimized with two conflicting objectives associated with the minimization of both the equivalent stress ranges and the total structural weight. This optimization problem can be formulated as follows:(4)Findx=tdec,trib,tdia,tldia,MinFx=Seq,D1,Seq,D2,Wt,s.t.10≤tdec≤20mm,5≤trib≤10mm,10≤tdia≤20mm,10≤tldia≤20mm,where x is the vector of design variables, which consists of component thicknesses of tdec, trib, tdia, and tldia; F(x) is the vector of optimization objective functions; Seq,D1 is the equivalent stress range at the deck to U-rib weld detail; Seq,D2 is the equivalent stress range at the U-rib to diaphragm weld detail; and Wt is the total structural weight, which can be calculated as a product of the steel volume, and its density of 7.85 g/cm3.The NSGA-II complied in a Python file was used to obtain the Pareto optimum solutions for the representative structural parameters based on the trade-off between the conflicting objectives. Crossover and mutation operations were conducted with respective probabilities of 90% and 20%. After 200 generations with 100 populations, the Pareto optimal solution set was found. An overall flowchart for the optimization of structural parameters is provided in Figure6.Figure 6
Overall flowchart for optimization of structural parameters.


## 2.4.1. Response Surface Methodology

During the optimization process, two main optimization objectives are considered, namely, maximizations of the equivalent stress range and the structural total weight. Among them, the equivalent stress range was correlated with the structural parameters via the RSM based on the data provided in Table1.RSM constructs a regression model between the optimization objective and the design variables via mathematical methods. The relationship between the objective valuey and the design variables x1, x2, …, xn (n is the number of the design variables) can be described as follows:(2)y=fx1,x2,…,xn+ε,where f denotes an approximation function and ε denotes the residual error between the predicted value and the real value. The approximation function commonly comprises a lower-order degree polynomial, and when a quadratic polynomial is used, the predicted value y’ can be written as follows:(3)y^=b0+∑i=1nbixi+∑i=1nbiixi2+∑i=1n−1∑j=i+1nbijxixj,where xi and xj are the design variables and b0, bi, bii, and bij are polynomial regression coefficients.


## 2.4.2. Nondominated Sorting Genetic Algorithm II

Herein, the optimum structural parameters were determined via NSGA-II to minimize the local equivalent stress ranges and the total structural weight. The basic idea of NSGA-II is as follows: First, the initial population of sizeN is randomly generated, and the initial offspring generation can be obtained via the basic operations of selection, crossover, and mutation. Then, from the second generation, the parent generation is merged with the offspring generation for fast nondominated sorting, and the crowing distance is calculated in each nondominated layer. The new parent generation is formed by selecting the appropriate individuals according to the nondominance relationship and the crowding distance of individuals. Finally, the new offspring generation is generated by the basic operations and so on until the end of the algorithm. The corresponding flowchart of NSGA-II is shown in Figure 5.Figure 5
Procedures of NSGA-II.


## 2.4.3. Optimization Problem Formulation

In this study, optimum structural parameters can be optimized with two conflicting objectives associated with the minimization of both the equivalent stress ranges and the total structural weight. This optimization problem can be formulated as follows:(4)Findx=tdec,trib,tdia,tldia,MinFx=Seq,D1,Seq,D2,Wt,s.t.10≤tdec≤20mm,5≤trib≤10mm,10≤tdia≤20mm,10≤tldia≤20mm,where x is the vector of design variables, which consists of component thicknesses of tdec, trib, tdia, and tldia; F(x) is the vector of optimization objective functions; Seq,D1 is the equivalent stress range at the deck to U-rib weld detail; Seq,D2 is the equivalent stress range at the U-rib to diaphragm weld detail; and Wt is the total structural weight, which can be calculated as a product of the steel volume, and its density of 7.85 g/cm3.The NSGA-II complied in a Python file was used to obtain the Pareto optimum solutions for the representative structural parameters based on the trade-off between the conflicting objectives. Crossover and mutation operations were conducted with respective probabilities of 90% and 20%. After 200 generations with 100 populations, the Pareto optimal solution set was found. An overall flowchart for the optimization of structural parameters is provided in Figure6.Figure 6
Overall flowchart for optimization of structural parameters.


## 3. Results and Discussion

### 3.1. Stress Characteristics under Vehicle Loads

Under the case of actual structural parameters (tdec = 6 mm, trib = 10 mm, tdia = 12 mm, and tldia = 14 mm), the equivalent stress amplitudes at the two typical weld details under different transverse loading cases can be seen in Figure 7. The stress characteristics can be analyzed by comparing them with each other at loading cases 1–23. It can be observed from Figure 7 that the equivalent stress ranges at the deck to U-rib weld (D1) and the U-rib to diaphragm weld (D2) reached their maximum values in loading cases 11 and 8, respectively. In other words, loading case 11 was the most unfavorable transverse loading position for the deck to U-rib weld, and the most unfavorable transverse loading position for the U-rib to diaphragm weld was loading case 8. Therefore, loading cases 11 and 8 were selected to obtain the fatigue stress histories at the deck to U-rib weld and the U-rib to diaphragm weld, respectively, under different cases of combination of structural parameters.Figure 7
Equivalent stress amplitudes under different loading cases.


### 3.2. Performance of RSM Surrogate Modelling

Based on OLhD, a total of 71 samples were generated, as shown in Table1. Among them, 50 samples were used for regression of the response surface model and the others were left for evaluating the performance of the established surrogate model. In the present study, a cubic polynomial function was selected as the form of RSM and the corresponding model coefficients were obtained via least-square regression. The performance of the response surface model was evaluated by comparing the predicted values with the actual values in Table 1, as shown in Figure 8. It can be observed that the predicted values are consistent with the actual values well, demonstrating the validity of the built response surface models.Figure 8
Comparison of the predicted and actual equivalent stress ranges.


### 3.3. Effect of Structural Parameters on Equivalent Stress Range

Figures9 and 10 show the normalized equivalent stress ranges at the deck to U-rib weld and the U-rib to diaphragm weld, according to the obtained response surface surrogate models. Each subplot shows the interaction effects of two structural parameters while keeping the other two structural parameters at their minimum values of the corresponding domains.Figure 9
Response surfaces of equivalent stress range at D1: (a)Seq-tdec, trib; (b) Seq-tdec, tdia; (c) Seq-tdec, tldia; (d) Seq-trib, tdia; (e) Seq-trib, tldia; (f) Seq-tdia, tldia.
(a)(b)(c)(d)(e)(f)Figure 10
Response surfaces of equivalent stress range at D2: (a)Seq-tdec, trib; (b) Seq-tdec, tdia; (c) Seq-tdec, tldia; (d) Seq-trib, tdia; (e) Seq-trib, tldia; (f) Seq-tdia, tldia.
(a)(b)(c)(d)(e)(f)For the deck to U-rib weld detail, it can be clearly observed from Figure9 that the deck thickness tdec was the most significant factor to the local equivalent stress range Seq,D1. Seq,D1 decreased significantly with the increase in tdec, which is in good agreement with the simulation results reported by Cao et al. [5, 24]. This is because a thicker deck plate leads to a higher bending stiffness, thus reducing the local out-of-plane deformation as well as the detail fatigue stress amplitudes. However, the influence of thicknesses of U-rib trib, of diaphragm tdia, and of longitudinal diaphragm tldia on the local fatigue stress range was minor.From Figure10, a similar influence trend of the deck thickness tdec on the equivalent stress range Seq,D2 can be observed. The increase in tdec resulted in a decrease in Seq,D2. Besides, the Seq,D2 also decreased with the increase in the thickness of U-rib trib. This is possible owing to the fact that the reinforced U-rib stiffness due to increased thickness would lower the action effect of bending and torsion at the U-rib to diaphragm weld joint. As can be seen from the tdia-tldia subplot of Figure 10, the thicknesses of the transverse and longitudinal diaphragms have reverse effects on the local fatigue stress amplitude when compared with the thicknesses of the deck and U-rib. Seq,D2 increased with the increase in tdia or tldia.


### 3.4. Multiobjective Optimization Results

Using the NSGA-II, the Pareto optimum solutions forSeq,D1, Seq,D2, and Wt were obtained, as illustrated in Figure 11. From the distribution of the Pareto front, it can be concluded that the objectives of minimizing both Seq and Wt are truly conflicting. The total weight of the OSD is bound to increase when reducing the local stress ranges by adjusting the component thicknesses. Since Seq,D1 and Seq,D2 both decreased significantly with the increase in the deck thickness, the objectives associated with minimization of Seq,D1 and Seq,D2 are consistent with each other. Table 2 lists the values of design variables and optimization objectives of the five representative solutions marked by yellow stars in Figure 11.Figure 11
Pareto optimum front forSeq,D1, Seq,D2, and Wt.Table 2
Values of design variables and optimization objectives of representative solutions.
SolutionsDesign variablesOptimization objectivestdec (mm)trib (mm)tdia (mm)tldia (mm)Seq,D1 (MPa)Seq,D2 (MPa)Wt (ton)A129111025.49.755.5B19811109.79.067.1C1010111036.49.853.8D155121015.610.652.9E20610108.59.464.0


## 3.1. Stress Characteristics under Vehicle Loads

Under the case of actual structural parameters (tdec = 6 mm, trib = 10 mm, tdia = 12 mm, and tldia = 14 mm), the equivalent stress amplitudes at the two typical weld details under different transverse loading cases can be seen in Figure 7. The stress characteristics can be analyzed by comparing them with each other at loading cases 1–23. It can be observed from Figure 7 that the equivalent stress ranges at the deck to U-rib weld (D1) and the U-rib to diaphragm weld (D2) reached their maximum values in loading cases 11 and 8, respectively. In other words, loading case 11 was the most unfavorable transverse loading position for the deck to U-rib weld, and the most unfavorable transverse loading position for the U-rib to diaphragm weld was loading case 8. Therefore, loading cases 11 and 8 were selected to obtain the fatigue stress histories at the deck to U-rib weld and the U-rib to diaphragm weld, respectively, under different cases of combination of structural parameters.Figure 7
Equivalent stress amplitudes under different loading cases.


## 3.2. Performance of RSM Surrogate Modelling

Based on OLhD, a total of 71 samples were generated, as shown in Table1. Among them, 50 samples were used for regression of the response surface model and the others were left for evaluating the performance of the established surrogate model. In the present study, a cubic polynomial function was selected as the form of RSM and the corresponding model coefficients were obtained via least-square regression. The performance of the response surface model was evaluated by comparing the predicted values with the actual values in Table 1, as shown in Figure 8. It can be observed that the predicted values are consistent with the actual values well, demonstrating the validity of the built response surface models.Figure 8
Comparison of the predicted and actual equivalent stress ranges.


## 3.3. Effect of Structural Parameters on Equivalent Stress Range

Figures9 and 10 show the normalized equivalent stress ranges at the deck to U-rib weld and the U-rib to diaphragm weld, according to the obtained response surface surrogate models. Each subplot shows the interaction effects of two structural parameters while keeping the other two structural parameters at their minimum values of the corresponding domains.Figure 9
Response surfaces of equivalent stress range at D1: (a)Seq-tdec, trib; (b) Seq-tdec, tdia; (c) Seq-tdec, tldia; (d) Seq-trib, tdia; (e) Seq-trib, tldia; (f) Seq-tdia, tldia.
(a)(b)(c)(d)(e)(f)Figure 10
Response surfaces of equivalent stress range at D2: (a)Seq-tdec, trib; (b) Seq-tdec, tdia; (c) Seq-tdec, tldia; (d) Seq-trib, tdia; (e) Seq-trib, tldia; (f) Seq-tdia, tldia.
(a)(b)(c)(d)(e)(f)For the deck to U-rib weld detail, it can be clearly observed from Figure9 that the deck thickness tdec was the most significant factor to the local equivalent stress range Seq,D1. Seq,D1 decreased significantly with the increase in tdec, which is in good agreement with the simulation results reported by Cao et al. [5, 24]. This is because a thicker deck plate leads to a higher bending stiffness, thus reducing the local out-of-plane deformation as well as the detail fatigue stress amplitudes. However, the influence of thicknesses of U-rib trib, of diaphragm tdia, and of longitudinal diaphragm tldia on the local fatigue stress range was minor.From Figure10, a similar influence trend of the deck thickness tdec on the equivalent stress range Seq,D2 can be observed. The increase in tdec resulted in a decrease in Seq,D2. Besides, the Seq,D2 also decreased with the increase in the thickness of U-rib trib. This is possible owing to the fact that the reinforced U-rib stiffness due to increased thickness would lower the action effect of bending and torsion at the U-rib to diaphragm weld joint. As can be seen from the tdia-tldia subplot of Figure 10, the thicknesses of the transverse and longitudinal diaphragms have reverse effects on the local fatigue stress amplitude when compared with the thicknesses of the deck and U-rib. Seq,D2 increased with the increase in tdia or tldia.


## 3.4. Multiobjective Optimization Results

Using the NSGA-II, the Pareto optimum solutions forSeq,D1, Seq,D2, and Wt were obtained, as illustrated in Figure 11. From the distribution of the Pareto front, it can be concluded that the objectives of minimizing both Seq and Wt are truly conflicting. The total weight of the OSD is bound to increase when reducing the local stress ranges by adjusting the component thicknesses. Since Seq,D1 and Seq,D2 both decreased significantly with the increase in the deck thickness, the objectives associated with minimization of Seq,D1 and Seq,D2 are consistent with each other. Table 2 lists the values of design variables and optimization objectives of the five representative solutions marked by yellow stars in Figure 11.Figure 11
Pareto optimum front forSeq,D1, Seq,D2, and Wt.Table 2
Values of design variables and optimization objectives of representative solutions.
SolutionsDesign variablesOptimization objectivestdec (mm)trib (mm)tdia (mm)tldia (mm)Seq,D1 (MPa)Seq,D2 (MPa)Wt (ton)A129111025.49.755.5B19811109.79.067.1C1010111036.49.853.8D155121015.610.652.9E20610108.59.464.0


## 4. Conclusion

This study provides the basic flow for fatigue optimization of structural parameters of OSDs considering the conflicting criteria. In the case of multiple design variables, surrogate modelling can be used. The results indicated that the built response surface model based on RSM can accurately predict the equivalent stress ranges at the weld details of interest. Among the several typical structural parameters, the deck thickness has a significant impact on the local stress of the deck to U-rib weld and the U-rib to diaphragm weld. Besides, the distribution of the Pareto solution set showed that the objectives of minimizing the equivalent stress component and the total structural weight are truly conflicting and the optimum solutions for structural parameters in such case can be obtained from the Pareto front. Unfortunately, due to the finical and material limitations, the corresponding verification was not provided in this study, which should be emphasized in future work.



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*Source: 4179898-2022-04-15.xml*