# Prediction Study on the Alignment of a Steel-Concrete Composite Beam Track Cable-Stayed Bridge

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## Abstract

**:**

## 1. Introduction

## 2. Experiment Principles

#### 2.1. Design of Experiments

_{8}(8

^{8}) uniform design table based on the aspects of design complexity, environmental randomness and practical stability.

#### 2.2. Experiment Methods

#### 2.2.1. Response Surface Method

- 1.
- The independent variable matrix A
_{α × β}is a standardized treatment to obtain V_{0}and its column vectors V_{01},…,V_{0β}; the dependent variable matrix B_{α × δ}is a standardized treatment to obtain M_{0}and its column vectors M_{01},…,M_{0δ}, where V_{0}is expressed as V_{0}= (V_{01},…,V_{0β})_{α × β}and M_{0}is expressed as M_{0}= (M_{01},…,M_{0δ})_{α × β}. - 2.
- The transpose matrix of V
_{0}and M_{0}are denoted by ${V}_{0}^{T}$ and ${M}_{0}^{T}$. The characteristic vectors φ_{1}, φ_{2}corresponding to the maximum eigenvalues of ${V}_{0}^{T}$ M_{0}${M}_{0}^{T}$ V_{0}and ${V}_{1}^{T}$ M_{0}${M}_{0}^{T}$ V_{1}, respectively, are calculated. Then t_{1}and t_{2}are calculated, where t_{1}= V_{0}φ_{1}, V_{1}= V_{0}− t_{1}G_{1}′, ${G}_{1}\mathrm{\u2019}=\frac{{V}_{0}^{T}{t}_{1}}{{\Vert {t}_{1}\Vert}^{2}}$, t_{2}= V_{1}φ_{2}, V_{2}= V_{1}− t_{2}G_{2}′, and ${G}_{2}\mathrm{\u2019}=\frac{{V}_{1}^{T}{t}_{2}}{{\Vert {t}_{2}\Vert}^{2}}$. This step is repeated until step m when t_{m}, and V_{m−1}are obtained. - 3.
- To establish a suitable model, m finite components t
_{1}, …, t_{n}are selected. After executing to step m, the regression equation can be obtained, which is given as:$${M}_{0k}={\alpha}_{k1}{V}_{01}+\cdots +{\alpha}_{k\beta}{V}_{0\beta}+{M}_{k}$$_{k1}, …, α_{kβ}are the regression coefficients.

#### 2.2.2. Accuracy Test

^{2}test to judge the accuracy of the model. which is given as:

_{j}is the jth sample point response value, ${\widehat{f}\left(x\right)}_{j}$ is the calculated value of the response surface corresponding to the jth sample point, and $\stackrel{-}{f}\left(x\right)$ is the average value, which is obtained from $\stackrel{-}{f}\left(x\right)=\frac{1}{m}\sum _{j=1}^{m}{f\left(x\right)}_{j}$. The calculated R

^{2}is between 0 and 1, and the closer the value is to 1, the more accurate it is.

#### 2.2.3. Monte Carlo Sampling Analysis

_{i}(i = 1, 2, …, n) are the independent random parameters in the influence model function f(x), and each corresponding independent parameter x

_{i}(i = 1, 2, …, n) is randomly sampled to obtain a random sample of x

_{1}, x

_{2}, …, x

_{n}. Then, x

_{i}is substituted into a specific program for repeated random sampling and analysis. The response values of the M group structure are obtained, which are denoted as f(x)

_{1}, f(x)

_{2}, …, f(x)

_{m}. Furthermore, the average value μ and standard deviation σ of f(x) are calculated, which can be expressed as:

## 3. Application Analysis

#### 3.1. Project Profile

#### 3.2. Finite Element Analysis

#### 3.3. Model Establishment

#### 3.3.1. Selection of Parameters to Establish the Model

_{1}is the strength of the main beam, X

_{2}is the mean annual humidity of the main beam environment, X

_{3}is the concrete age at the beginning of main beam shrinkage, X

_{4}is the strength of the main tower, X

_{5}is the mean annual humidity of the main tower environment, X

_{6}is the concrete age at the beginning of the main tower shrinkage, X

_{7}is the volume weight of the main beam, and X

_{8}is the volume weight of the main tower.

#### 3.3.2. Selection of Samples to Establish the Model

_{1/2}, S

_{1}, S

_{2}, S

_{3}, S

_{5}, and S

_{10}, respectively, and the uniformity experiment was carried out according to the uniform design table of U

_{8}(8

^{8}). Eight sample points were obtained by testing each set of data, with the deformation of the main beam defined as negative when vertically downward and positive when vertically upward. In this way, the sample points of the uniform design (Table 2) and the structural response data were obtained (Table 3).

#### 3.4. Response Surface Analysis

#### 3.4.1. Response Surface Model Fitting

#### 3.4.2. Response Surface Model Validation

#### 3.5. Deformation Data Analysis

#### 3.5.1. Deformation Data Analysis since Closure of the Main Span

#### 3.5.2. Predictive Analysis of Deformation after Bridge Completion

## 4. Conclusions

- Combining different influencing factors with the alignment of the entire large-span track SCCB cable-stayed bridge, the RSM method was used to solve the problem of low accuracy of alignment prediction for large-span SCCB track cable-stayed bridges. Accurate alignment prediction was realized under the randomness of influencing factors.
- The MC method was combined with the RSM method for analysis. The partial least squares method was used to verify the analysis according to the influencing factors combined with the uniform design method. The long-term deformation for different years after bridge completion was predicted. In addition, the finite element calculated value interval of (−86.5963, −21.3299) mm is included in the 95% confidence interval of (−133.64, −33.68) mm.
- RSM can be used to analyze different influencing factors. In this paper, eight influencing factors, such as the strength of the main beam, were analyzed, to realize the alignment prediction of the bridge. In addition, other influencing factors could be analyzed and studied quickly and effectively through this method, reducing the test cost and time, and improving work efficiency.
- RSM was used to predict and analyze the alignment of SCCB cable-stayed bridges, and the predicted values at different years over 10 years were obtained. Alignment prediction is an important consideration in the field of bridge health monitoring. The prediction method based on RSM provides a theoretical basis and engineering application reference for reasonable alignment control and state evaluation of bridges.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Statistical Characteristics | Strength of the Main Beam X_{1}/(N/mm^{2}) | Mean Annual Humidity of the Main Beam Environment X_{2}/(%) | Concrete Age at the Beginning of Main Beam Shrinkage X_{3}/(day) | Strength of the Main Tower X_{4}/(N/mm^{2}) | Mean Annual Humidity of the Main Tower Environment X_{5}/(%) | Concrete Age at the Beginning of the Main Tower Shrinkage X_{6}/(day) | Volume Weight of the Main Beam X_{7}/(N/mm^{3}) | Volume Weight of the Main Tower X_{8}/(N/mm^{3}) |
---|---|---|---|---|---|---|---|---|

Type of distribution | Normal distribution | Normal distribution | Normal distribution | Normal distribution | Normal distribution | Normal distribution | Normal distribution | Normal distribution |

Average value | 60 | 79 | 3 | 50 | 79 | 3 | 26 | 26 |

Coefficient of variation | 0.15 | 0.20 | 0.22 | 0.14 | 0.12 | 0.26 | 0.11 | 0.11 |

Serial Number | X_{1}/(N/mm^{2}) | X_{2}/(%) | X_{3}/(day) | X_{4}/(N/mm^{2}) | X_{5}/(%) | X_{6}/(day) | X_{7}/(N/mm^{3}) | X_{8}/(N/mm^{3}) |
---|---|---|---|---|---|---|---|---|

1 | 59.40 | 78.2 | 2.12 | 49.44 | 78.530 | 1.96 | 25.56 | 25.56 |

2 | 59.55 | 78.4 | 2.34 | 49.58 | 78.648 | 2.22 | 25.67 | 25.67 |

3 | 59.70 | 78.6 | 2.56 | 49.72 | 78.766 | 2.48 | 25.78 | 25.78 |

4 | 59.85 | 78.8 | 2.78 | 49.86 | 78.884 | 2.74 | 25.89 | 25.89 |

5 | 60.00 | 79.0 | 3.00 | 50.00 | 79.002 | 3.00 | 26.00 | 26.00 |

6 | 60.15 | 79.2 | 3.22 | 50.14 | 79.120 | 3.26 | 26.11 | 26.11 |

7 | 60.30 | 79.4 | 3.44 | 50.28 | 79.238 | 3.52 | 26.22 | 26.22 |

8 | 60.45 | 79.6 | 3.66 | 50.42 | 79.356 | 3.78 | 26.33 | 26.33 |

Serial Number | S_{1/2} | S_{1} | S_{2} | S_{3} | S_{5} | S_{10} |
---|---|---|---|---|---|---|

1 | −21.810 | −37.231 | −50.194 | −61.070 | −76.402 | −98.043 |

2 | −22.271 | −34.782 | −50.784 | −61.656 | −77.336 | −99.256 |

3 | −22.732 | −35.310 | −51.350 | −62.803 | −78.614 | −100.498 |

4 | −23.231 | −35.639 | −52.549 | −64.103 | −80.249 | −101.795 |

5 | −25.981 | −39.213 | −55.915 | −67.192 | −83.015 | −104.833 |

6 | −25.245 | −38.547 | −55.291 | −66.585 | −82.416 | −104.202 |

7 | −17.659 | −27.846 | −36.506 | −41.716 | −48.309 | −55.856 |

8 | −14.690 | −24.974 | −33.767 | −39.047 | −45.754 | −52.834 |

Models | R^{2} | Models | R^{2} |
---|---|---|---|

S_{1/2} | 0.8993 | S_{5} | 0.9455 |

S_{1} | 0.8714 | S_{10} | 0.9425 |

S_{2} | 0.8788 | Average value | 0.9091 |

S_{3} | 0.9173 |

Working Conditions | Design Expected Values (mm) | Measured Values (mm) | Statistical Characteristic Values | ||
---|---|---|---|---|---|

Average Values (mm) | Standard Deviation | ||||

Time after bridge completion | 1/2 year | −21.33 | −20.02 | −21.70 | 3.54 |

1 year | −33.56 | −25.13 | −34.19 | 4.77 | |

2 years | −47.28 | / | −48.29 | 7.86 | |

3 years | −56.60 | / | −58.02 | 10.40 | |

5 years | −69.46 | / | −71.51 | 14.31 | |

10 years | −86.60 | / | −89.66 | 20.52 |

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## Share and Cite

**MDPI and ACS Style**

Li, X.; Luo, H.; Ding, P.; Chen, X.; Tan, S.
Prediction Study on the Alignment of a Steel-Concrete Composite Beam Track Cable-Stayed Bridge. *Buildings* **2023**, *13*, 882.
https://doi.org/10.3390/buildings13040882

**AMA Style**

Li X, Luo H, Ding P, Chen X, Tan S.
Prediction Study on the Alignment of a Steel-Concrete Composite Beam Track Cable-Stayed Bridge. *Buildings*. 2023; 13(4):882.
https://doi.org/10.3390/buildings13040882

**Chicago/Turabian Style**

Li, Xiaogang, Haoran Luo, Peng Ding, Xiaohu Chen, and Shulin Tan.
2023. "Prediction Study on the Alignment of a Steel-Concrete Composite Beam Track Cable-Stayed Bridge" *Buildings* 13, no. 4: 882.
https://doi.org/10.3390/buildings13040882