# An Optimization Algorithm for the Design of an Irregularly-Shaped Bridge Based on the Orthogonal Test and Analytic Hierarchy Process

^{*}

## Abstract

**:**

## 1. Introduction

^{3}= 27, while only nine orthogonal tests can analyze the sensitivity of factors and determine the preferred levels. As shown in Figure 2, nine test points regularly distributed throughout the comprehensive test points, which shows a strong representation and is able to better reflect the basic information in the optimal selection space. The optimum condition determined by part tests has the same trends as that obtained by comprehensive tests.

## 2. Comprehensive Weight Calculation Model Based on AHP and Orthogonal Test

#### 2.1. Orthogonal Analysis

#### 2.2. Influence Weight Calculation Based on AHP

#### 2.2.1. Weight Vector of Test Index Layer

#### 2.2.2. Weights of Factors and Levels in the Orthogonal Test

#### 2.2.3. Comprehensive Weight

## 3. Structure Model of a Typical Irregularly-Shaped Bridge

## 4. Construction of the Index System and the Determination of Factors and Levels

## 5. Sensitivity Analysis and Design Optimization of Parameters

_{i}(i = 1, 2, 3) represents the corresponding index value at each parameter level and R denotes the difference between the largest and the smallest values of k

_{i}. It is apparent that the parameter with the largest R value has the most significant influence on the test index.

#### 5.1. Result Analysis for a Single Index

#### 5.1.1. Influence of Parameters on Maximum Stress

#### 5.1.2. Influence of Parameters on the Stress Variation Coefficient

#### 5.1.3. Influence of Parameters on the Torsion Vibration Fundamental Frequency

#### 5.2. Result Analysis for the Mechanical Properties of an Irregularly-Shaped Bridge

#### 5.2.1. Parameter Optimization Based on the Overall Balance Method

#### 5.2.2. Parameter Optimization Based on the Comprehensive Weight Analysis Method

## 6. Conclusions

- (1)
- Bifurcation diaphragm stiffness and box girder height are the main factors that affect the maximum stress and stress variation coefficient relative to the ramp radius and supporting condition. The increasing bifurcation diaphragm thickness decreases the maximum stress and stress variation coefficient. As box girder height increases, the maximum stress decreases and uneven stress distribution becomes serious.
- (2)
- The supporting condition plays the most important role in the influence on torsion vibration fundamental frequency. Double torsional supports set at the end of an irregularly-shaped bridge can improve the ability of the irregularly-shaped bridge to resist torsional vibration, which is more beneficial to driving safety.
- (3)
- Based on the influence weights of factors to the mechanical properties of an irregularly-shaped bridge, the influence order of parameters is: supporting condition > bifurcation diaphragm stiffness > box girder height > ramp radius. According to the influence weights of levels in each factor, the best parameters for an irregularly-shaped bridge design are obtained when the ramp radius, bifurcation diaphragm thickness, box girder height, and supporting conditions are R = 50 m, D = 2.0 m, H = 2.0 m, and ①, respectively.
- (4)
- Comparative analysis with test results of optimal parameters obtained from the overall balance method reveals that the comprehensive weight analysis method is superior to the overall balance method. The comprehensive weight analysis method possesses more favorable accuracy and validity in the sensitivity analysis and design optimization of parameters for the mechanical properties of irregularly-shaped bridges.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Lu, P.Z.; Xie, X.; Shao, C.Y. Experimental study and numerical analysis of a composite bridge structure. Constr. Build. Mater.
**2012**, 30, 695–705. [Google Scholar] [CrossRef] - Tayşi, N.; Özakça, M. Free vibration analysis and shape optimization of box-girder bridges in straight and curved planform. Eng. Struct.
**2002**, 24, 625–637. [Google Scholar] [CrossRef] - Lu, P.Z.; Zhang, J.P.; Zhao, R.D. Analysis and study on model experiment of typical Y-shape bridge. Eng. Mech.
**2008**, 27, 139–144. [Google Scholar] - Foti, D.; Ivorra, S.; Bru, D.; Dimaggio, G. Dynamic identification of a pedestrian bridge using OMA: Previous and post-reinforcing. In Proceedings of the Eleventh International Conference on Computational Structures Technology, Dubrovnik, Croatia, 4–7 September 2012.
- Ivorra, S.; Foti, D.; Bru, D.; Baeza, F.J. Dynamic behavior of a pedestrian bridge in Alicante (Spain). J. Perform. Constr. Facil.
**2015**, 29. 04014132:1–10. [Google Scholar] [CrossRef] - Kim, K.; Yoo, C.H. Effects of external bracing on horizontally curved box girder bridges during construction. Eng. Struct.
**2006**, 28, 1650–1657. [Google Scholar] [CrossRef] - Park, N.H.; Choi, Y.J.; Kang, Y.J. Spacing of intermediate diaphragms in horizontally curved steel box girder bridges. Finite Elements Anal. Des.
**2005**, 41, 925–943. [Google Scholar] [CrossRef] - Kim, N.; Shin, D.K. Torsional analysis of thin-walled composite beams with single-and double-celled sections. Eng. Struct.
**2009**, 31, 1509–1521. [Google Scholar] [CrossRef] - Dowell, R.K. Closed-form moment solution for continuous beams and bridge structures. Eng. Struct.
**2009**, 31, 1880–1887. [Google Scholar] [CrossRef] - Dowell, R.K.; Johnson, T.P. Closed-form shear flow solution for box–girder bridges under torsion. Eng. Struct.
**2012**, 34, 383–390. [Google Scholar] [CrossRef] - Wu, Y.P.; Zhu, Y.L.; Lai, Y.M.; Pan, W.D. Analysis of shear lag and shear deformation effects in laminated composite box beams under bending loads. Compos. Struct.
**2002**, 55, 147–156. [Google Scholar] - Lin, Z.B.; Zhao, J. Least-work solutions of flange normal stresses in thin-walled flexural members with high-order polynomial. Eng. Struct.
**2011**, 33, 2754–2761. [Google Scholar] [CrossRef] - Bazant, Z.; Nimeiri, M.El. Stiffness method for curved box girders at initial stress. J. Struct. Div.
**1974**, 100, 2071–2090. [Google Scholar] - Razaqpur, A.G.; Li, H.B. Thin-walled multicell box-girder finite element. J. Struct. Eng.
**1991**, 117, 2953–2971. [Google Scholar] [CrossRef] - Yoon, K.Y.; Park, N.H.; Choi, Y.J.; Kang, Y.J. Natural frequencies of thin-walled curved beams. Finite Elements Anal. Des.
**2006**, 42, 1176–1186. [Google Scholar] [CrossRef] - Sapountzakis, E.J.; Mokos, V.G. Dynamic analysis of 3-D beam elements including warping and shear deformation effects. Int. J. Solids Struct.
**2006**, 43, 6707–6726. [Google Scholar] [CrossRef] - Zhou, S.J. Finite beam element considering shear-lag effect in box girder. J. Eng. Mech.
**2010**, 136, 1115–1122. [Google Scholar] [CrossRef] - Huang, H.Y.; Zhang, J.P. Parameters analysis for mechanical properties of Y-shaped bridge. J. Guangzhou Univ. (Nat. Sci. Ed.)
**2003**, 2, 472–476. [Google Scholar] - Hinton, E.; Rao, N.V.R. Analysis and shape optimization of variable thickness prismatic folded plates and curved shells—Part 2: Shape optimization. Thin-Walled Struct.
**1993**, 17, 161–183. [Google Scholar] [CrossRef] - Özakça, M.; Hinton, E.; Rao, N.V.R. Free vibration analysis and shape optimization of prismatic folded plates and shells with circular curved planform. Int. J. Numer. Methods Eng.
**1994**, 37, 1713–1739. [Google Scholar] [CrossRef] - Fang, K.T.; Ma, C.X. Orthogonal and Uniform Design of Experiments; Science Press: Beijing, China, 2001. [Google Scholar]
- Liang, R.J. Orthogonal test design for optimization of the extraction of polysaccharides from Phascolosoma esulenta and evaluation of its immunity activity. Carbohydr. Polym.
**2008**, 73, 558–563. [Google Scholar] - Sun, Y.X.; Li, Y.J.; Li, M.Q.; Tong, H.B.; Yang, X.D.; Liu, J.C. Optimization of extraction technology of the Anemone raddeana polysaccharides (ARP) by orthogonal test design and evaluation of its anti-tumor activity. Carbohydr. Polym.
**2009**, 75, 575–579. [Google Scholar] [CrossRef] - Wei, L.K.; Huang, X.X.; Huang, Z.Z.; Zhou, Z.G. Orthogonal test design for optimization of lipid accumulation and lipid property in Nannochloropsis oculata for biodiesel production. Bioresour. Technol.
**2013**, 147, 534–538. [Google Scholar] [CrossRef] [PubMed] - Bai, Y.; Gao, H.M.; Wu, L.; Ma, Z.H.; Cao, N. Influence of plasma-MIG welding parameters on aluminum weld porosity by orthogonal test. Trans. Nonferrous Met. Soc. China
**2010**, 20, 1392–1396. [Google Scholar] [CrossRef] - Zhang, Q.; Leung, Y.W. An orthogonal genetic algorithm for multimedia multicast routing. IEEE Trans. Evolut. Comput.
**1999**, 3, 53–62. [Google Scholar] [CrossRef] - Liu, Z.X.; Huang, R.H.; Tian, A.M. Test Design and Data Processing; Chemical Industry Press: Beijing, China, 2005. [Google Scholar]
- Satty, T.L.; Vargas, L.G. Prediction, Projection and Forecasting; Kluwer Academic Publishers: Boston, MA, USA, 1991. [Google Scholar]
- Code for design of the municipal bridge; CJJ 11–2011; China Architecture & Building Press: Beijing, China, 2011. (In Chinese)

**Figure 6.**Node selection and normal stress distribution in the top plate of the bifurcation diaphragm. (

**a**) Node selected in the top plate; and (

**b**) the normal stress distribution of selected nodes.

Test No. | Factor | Index Value | ||
---|---|---|---|---|

B_{1} | B_{2} | B_{3} | ||

1 | 1 | 1 | 1 | y_{1} |

2 | 1 | 2 | 2 | y_{2} |

3 | 2 | 1 | 2 | y_{3} |

4 | 2 | 2 | 1 | y_{4} |

K_{jl} | K_{11} = y_{1} + y_{2} | K_{21} = y_{1} + y_{3} | K_{31} = y_{1} + y_{4} | - |

K_{12} = y_{3} + y_{4} | K_{22} = y_{2} + y_{4} | K_{32} = y_{2} + y_{3} | - | |

m_{j} | 2 | 2 | 2 | - |

k_{jl} | k_{11} = K_{11}/m_{1} | k_{21} = K_{21}/m_{2} | k_{31} = K_{31}/m_{3} | - |

k_{12} = K_{12}/m_{1} | k_{21} = K_{22}/m_{2} | k_{31} = K_{32}/m_{3} | - | |

R_{j} | R_{1} | R_{2} | R_{3} | - |

Index | A_{1} | A_{2} | ··· | A_{n} |
---|---|---|---|---|

A_{1} | ${a}_{11}$ | ${a}_{12}$ | ··· | ${a}_{1n}$ |

A_{2} | ${a}_{21}$ | ${a}_{22}$ | ··· | ${a}_{2n}$ |

··· | ··· | ··· | ··· | ··· |

A_{n} | ${a}_{n1}$ | ${a}_{n2}$ |

n | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

RI | 0 | 0 | 0.58 | 0.90 | 1.12 |

Target Layer | Mechanical Property of the Irregularly-Shaped Bridge | ||
---|---|---|---|

Index layer | ${\sigma}_{\text{max}}$ | $\lambda $ | ${f}_{d}$ |

Weight | 0.25 | 0.25 | 0.50 |

Support Position | Support Condition | ||
---|---|---|---|

① | ② | ③ | |

C1 | Double movable support | Double movable support | Single vertical support |

C2 | Double fixed support | Double fixed support | Double fixed support |

C3 | Single vertical support | Single vertical support | Single vertical support |

C4 | Double movable support | Double movable support | Single vertical support |

C5 | Single vertical support | Vertical eccentric support | Single vertical support |

C6 | Double movable support | Double movable support | Single vertical support |

Level | Factor | |||
---|---|---|---|---|

Ramp Radius R (m) | Diaphragm Thickness D (m) | Box Girder Height H (m) | Supporting Condition U | |

1 | 30 | 1.2 | 1.6 | ① |

2 | 40 | 1.6 | 1.8 | ② |

3 | 50 | 2.0 | 2.0 | ③ |

Test No. | Factor | Test index | |||||
---|---|---|---|---|---|---|---|

R (m) | D (m) | H (m) | U | ${\mathit{\sigma}}_{\mathbf{\text{max}}}$ (MPa) | $\mathit{\lambda}$ | ${\mathit{f}}_{\mathit{d}}$ (Hz) | |

1 | 30 | 1.2 | 1.6 | ① | 2.95 | 1.76 | 7.7341 |

2 | 30 | 1.6 | 1.8 | ② | 2.36 | 1.66 | 7.8733 |

3 | 30 | 2.0 | 2.0 | ③ | 1.99 | 1.57 | 5.2013 |

4 | 40 | 1.2 | 1.8 | ③ | 2.66 | 1.85 | 5.0639 |

5 | 40 | 1.6 | 2.0 | ① | 2.14 | 1.78 | 8.3185 |

6 | 40 | 2.0 | 1.6 | ② | 2.64 | 1.47 | 8.2158 |

7 | 50 | 1.2 | 2.0 | ② | 2.35 | 1.90 | 8.3485 |

8 | 50 | 1.6 | 1.6 | ③ | 2.70 | 1.61 | 5.0858 |

9 | 50 | 2.0 | 1.8 | ① | 2.24 | 1.51 | 8.4667 |

Test Index | Item | Factor | |||
---|---|---|---|---|---|

R (m) | D (m) | H (m) | U | ||

Maximum stress | k_{1} | 2.43 | 2.65 | 2.76 | 2.44 |

k_{2} | 2.48 | 2.40 | 2.42 | 2.45 | |

k_{3} | 2.43 | 2.29 | 2.16 | 2.45 | |

R | 0.05 | 0.36 | 0.60 | 0.01 | |

Stress variation coefficient | k_{1} | 1.66 | 1.84 | 1.61 | 1.68 |

k_{2} | 1.70 | 1.68 | 1.67 | 1.67 | |

k_{3} | 1.67 | 1.52 | 1.75 | 1.68 | |

R | 0.04 | 0.32 | 0.14 | 0.01 | |

Torsion vibration fundamental frequency | k_{1} | 6.9362 | 7.0488 | 7.0119 | 8.1731 |

k_{2} | 7.1994 | 7.0925 | 7.1346 | 8.1459 | |

k_{3} | 7.3003 | 7.2946 | 7.2894 | 5.1170 | |

R | 0.3641 | 0.2458 | 0.2775 | 3.0561 |

Factor | R (m) | D (m) | H (m) | U | |
---|---|---|---|---|---|

Weight | ${T}_{{\sigma}_{max}}$ | 0.0490 | 0.3529 | 0.5882 | 0.0098 |

${T}_{\lambda}$ | 0.0784 | 0.6275 | 0.2745 | 0.0196 | |

${T}_{{f}_{d}}$ | 0.0923 | 0.0623 | 0.0704 | 0.7750 | |

Q | 0.0780 | 0.2763 | 0.2509 | 0.3948 |

Factor | R (m) | D (m) | H (m) | U | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Level | 30 | 40 | 50 | 1.2 | 1.6 | 2.0 | 1.6 | 1.8 | 2.0 | ① | ② | ③ | |

Weight | ${M}_{{\sigma}_{max}}$ | 0.0165 | 0.0161 | 0.0165 | 0.1082 | 0.1195 | 0.1252 | 0.1721 | 0.1963 | 0.2199 | 0.0033 | 0.0033 | 0.0033 |

${M}_{\lambda}$ | 0.0264 | 0.0258 | 0.0262 | 0.1898 | 0.2079 | 0.2298 | 0.0952 | 0.0918 | 0.0876 | 0.0065 | 0.0066 | 0.0065 | |

${M}_{{f}_{d}}$ | 0.0299 | 0.0310 | 0.0314 | 0.0205 | 0.0206 | 0.0212 | 0.0230 | 0.0234 | 0.0239 | 0.2955 | 0.2945 | 0.1850 | |

P | 0.0257 | 0.0260 | 0.0264 | 0.0848 | 0.0922 | 0.0994 | 0.0783 | 0.0837 | 0.0888 | 0.1502 | 0.1497 | 0.0949 |

Analysis Method | Factor | Index | |||||
---|---|---|---|---|---|---|---|

R (m) | D (m) | H (m) | U | ${\mathit{\sigma}}_{\mathbf{\text{max}}}$ (MPa) | $\mathit{\lambda}$ | ${\mathit{f}}_{\mathit{d}}$ (Hz) | |

Overall balance method | 50 | 2.0 | 1.8 | ① | 2.24 | 1.51 | 8.4667 |

Comprehensive weight analysis method | 50 | 2.0 | 2.0 | ① | 1.94 | 1.54 | 8.5642 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, H.; He, X.; Wang, X.; Jiao, Y.; Song, G.
An Optimization Algorithm for the Design of an Irregularly-Shaped Bridge Based on the Orthogonal Test and Analytic Hierarchy Process. *Algorithms* **2016**, *9*, 74.
https://doi.org/10.3390/a9040074

**AMA Style**

Liu H, He X, Wang X, Jiao Y, Song G.
An Optimization Algorithm for the Design of an Irregularly-Shaped Bridge Based on the Orthogonal Test and Analytic Hierarchy Process. *Algorithms*. 2016; 9(4):74.
https://doi.org/10.3390/a9040074

**Chicago/Turabian Style**

Liu, Hanbing, Xin He, Xianqiang Wang, Yubo Jiao, and Gang Song.
2016. "An Optimization Algorithm for the Design of an Irregularly-Shaped Bridge Based on the Orthogonal Test and Analytic Hierarchy Process" *Algorithms* 9, no. 4: 74.
https://doi.org/10.3390/a9040074