# Performance Analysis of Short-Span Simply Supported Bridges for Heavy-Haul Railways with A Novel Prefabricated Strengthening Structure

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology

## 3. Theoretical Model

_{1}represents the center distance between the bridge bearing and its adjacent NPSS, the factors l

_{t}and F stand for the bogie’s wheelbase and the axle load, respectively. l

_{2}is the difference between l

_{z}and l

_{t}(see Figure 2), and l

_{z}denotes the center distance between the back bogie of the first vehicle and the front bogie of the second vehicle. F denotes the wheel weight. Furthermore, the factors k

_{1}and k

_{2}in order are the vertical stiffness values of the bridge bearing and the NPSS (i.e., bearing stiffness and support stiffness for short, respectively). Please see Appendix A for formula derivation.

_{y}) is evaluated by:

## 4. Finite Element Model and Verification

#### 4.1. Finite Element Model

#### 4.2. Experiments

^{4}MPa. Figure 5 shows the detailing of steel bars.

#### 4.3. Comparison between Model and Experimental Data

_{Ci}, F

_{Ti}stand for the i-th calculation and test data with the same deflection. n denotes the number of comparison data. Figure 10 shows the relation between the error and yield strength of the steel bars.

## 5. Discussion

#### 5.1. Comparison Studies with and without NPSS

_{1}is set equal to 0.8 m, and support stiffness k

_{2}is set as 0.5 times of the bearing stiffness (k

_{1}) which is 2.46 × 10

^{9}N/m. The calculation parameters of the track structure are also achievable in Refs. [42,43].

_{2}/k

_{1}should be less than 0.15 by virtue of Equation (3). The excessive support force of the NPSS not only intensifies the difficulty in the structural design and the possibility of pier concrete collapse, but also leads to partial function loss of the bridge bearing. Therefore, it is necessary to optimize the support stiffness to ensure that the NPSS does not undergo too much load while improving the bridge stiffness and bearing capacity.

#### 5.2. Influence of the Support Stiffness

_{1}and k

_{1}in order are kept fixed at 0.8 m and 2.46 × 10

^{9}N/m, and five levels are taken for the ratio k

_{2}/k

_{1}(i.e., 0.1, 0.2, 0.3, 0.4, and 0.5). The results by the calculation model are illustrated in Figure 14. Figure 14a displays the support force amplitudes of both the bridge bearing and the NPSS as a function of the ratio k

_{2}/k

_{1}. The support pressure force of the bridge bearing is viewed as positive, otherwise negative. Figure 14b is the relational curve between the midspan bending moment and deflection amplitude and k

_{2}/k

_{1}.

_{2}/k

_{1}in the range of 0.1–0.5, the support force amplitude of the NPSS increases by 302.9 kN, indicating a growth of 132.0%. For the case of k

_{2}/k

_{1}= 0.1, the support force amplitude of the NPSS and the positive support force amplitude of the bridge bearing in order are 229.5 kN and 212.1 kN. These values are approximately equal, representing that the NPSS and the bridge bearing collaboratively undergo the train load. Additionally, the negative support force amplitude of the bridge bearing is only 30.1 kN, revealing that the whole structure is in good stress performance.

_{2}/k

_{1}in the interval of 0.1–0.3, the midspan deflection and bending moment amplitudes decrease by 1.2 mm and 154.2 kN·m, respectively (i.e., a drop of 26.0% and 14.9%). However, as k

_{2}/k

_{1}increases from 0.3 to 0.5, the midspan deflection and bending moment amplitudes only exhibit a reduction of 0.5 mm and 73.5 kN·m (i.e., fall by 15.2% and 8.4%). In the case of k

_{2}/k

_{1}= 0.1, the midspan deflection amplitude is obtained as 4.7 mm, which does not exceed the usual value of 4.9 mm under the action of design load [44].

#### 5.3. Influence of the Installation Location

_{1}is almost small, from 0.4 m to 0.8 m. Under the calculation conditions, the ratio k

_{2}/k

_{1}is kept fixed at 0.1. The obtained results have been demonstrated in Figure 16.

_{1}, the supporting force amplitudes of the bridge bearing and the NPSS vary approximately linearly, where the positive supporting force amplitude of the bridge bearing lessens and the supporting force amplitude of the NPSS grows. For the case of l

_{1}= 0.73 m, the support forces of the bridge bearing and the NPSS are approximately equal. As the value of l

_{1}ranges from 0.4 m to 0.8 m, the support forces of the bridge bearing and the NPSS demonstrate the changes of 60.7 kN and 75.0 kN, respectively (i.e., the percentage alterations of 22.2% and 48.5%).

_{1}such that the reduction rates in order are 2.4 mm/m and 291.8 kN·m/m. As the value of l

_{1}increases from 0.4 m to 0.8 m, the midspan deflection and bending moment amplitudes lessen by 1.0 mm and 116.6 kN·m, respectively, signifying a reduction of 17.1% and 10.1%.

_{2}/k

_{1}= 0.1, l

_{1}must be greater than 0.7 m to meet the requirements of the bridge deflection.

#### 5.4. Parameters Optimization

_{2}and l

_{1}is beneficial in reducing both the deflection and bending moment of the beam, it results in a higher support force of the NPSS and a smaller positive support force of the bridge bearing, which is unfavorable to the NPSS. Therefore, the amplitudes of the positive support force of the bridge bearing and the support force of the NPSS should be equal as far as possible. Meanwhile, the midspan deflection and the negative support force of the bridge bearing should be as small as possible to determine the optimal values of k

_{2}and l

_{1}. By virtue of the refined calculation model and the numerical test, the RSM is implemented to determine the explicit functional relationship between the mechanical indexes of the bridge with NPSS and support stiffness, and the installation location of NPSS. Thereby,

_{2}/k

_{1}, w represents the midspan deflection amplitude, F

_{b}denotes the absolute value of the difference between the positive support force amplitude of the bridge bearing and that of the NPSS, and F

_{y}is the negative support force amplitude of the bridge bearing. The multiple correlation coefficient, modified multiple correlation coefficient, and R

^{2}(prediction) of the fitting results are all over 0.95. By taking the minimum of w, F

_{b}, and F

_{y}as the objective functions, a multi-objective optimization model could be constructed as:

_{1}.

_{1}for the optimal design is fixed at 0.8 m, which is the maximum allowable value. For the optimal boundary given in Figure 16a, the analysis hierarchy process-fuzzy comprehensive evaluation (AHP-FCE) method is employed to determine the reasonable support stiffness of the NPSS for engineering applications. Since w, F

_{b}, and F

_{y}are all negative indicators, the data standardization scheme shown in Equation (11) is adopted. Take midspan deflection amplitude w as an example:

_{i}and r

_{i}represent the values before and after data standardization, respectively.

_{b}, and F

_{y}determined based on the AHP-FEC are 0.15, 0.48, and 0.37, respectively. The weight coefficients remain unchanged with the variation of the selected optimal boundary points. At this time, the values of x and l

_{1}associated with the optimal scheme are 0.1018 and 0.8, which means that the support stiffness value is 2.5 × 10

^{8}N/m, and the center distance between the bridge bearing and its adjacent NPSS is 0.8 m. According to the optimal factors, the midspan deflection-load curve of the bridge based on the four-point bending test has been presented in Figure 18. The plot of the load-bearing capacity of the strengthened bridge demonstrates an increase of 19.5%. The slope of the curve in the elastic deformation stage of the bridge with the NPSS also rises from 104.7 kN/mm to 126.7 kN/mm, which reveals that the vertical stiffness of the bridge has been enhanced by 21.0%.

#### 5.5. Dynamic Performance Verification

^{2}, which indicates a reduction of 23.4% and 25.2%, respectively. Due to the improvement of the vertical stiffness of the bridge with NPSS, the car’s vertical vibration acceleration and wheel unloading rate intensify by 1.31 m/s

^{2}and 0.01, which still satisfies the demand for heavy-haul train running safety.

## 6. Conclusions

^{8}N/m and 0.8 m, which could enhance the load-bearing capacity and the vertical stiffness of the bridge by 19.5% and 21.0%, respectively, and reduce the midspan dynamic deflection amplitude and vertical vibration acceleration amplitude of the bridge by 23.4% and 25.2%, respectively.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{1}and F

_{2}represent the support force of the bridge bearing and the NPSS. The bending moments at different positions are expressed by:

_{1}–C

_{8}are undetermined constants. The curve of the first derivative of deflection and the deflection are continuous at l

_{1}, L/2 − l

_{t}− l

_{2}/2 and L/2 − l

_{2}/2. The relation among C

_{1}~C

_{8}can be determined.

_{1}, F

_{2}and w(x) and the principle of force balance, Formulas (A8)–(A10) can be obtained.

_{1}, F

_{2}and M(L/2) can be obtained as:

_{1}should be greater than zero, namely:

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**Figure 5.**Detailing of steel bars. (

**a**) Midspan section; (

**b**) stressed steel bars of midspan section; (

**c**) center half section of beam stem (unit: mm).

**Figure 12.**The calculated results with and without NPSS. (

**a**) Midspan deflection; (

**b**) midspan bending moment; (

**c**) support force.

**Figure 14.**Effects of the support stiffness. (

**a**) Support force; (

**b**) deflection and bending moment amplitudes.

**Figure 16.**Effects of the installation location. (

**a**) Support force; (

**b**) deflection and bending moment amplitudes.

**Figure 17.**The plotted results of the multi-objective optimization. (

**a**) Pareto optimal solution; (

**b**) optimization variable value.

**Figure 20.**Dynamic responses. (

**a**) Midspan deflection; (

**b**) midspan’s vibration acceleration; (

**c**) car’s vibration acceleration; (

**d**) wheel unloading rate.

Component | Element Type | Mesh Sizes | Amount |
---|---|---|---|

rail | beam188 | 0.01–0.09 m | 2574 |

sleeper | shell63 | 0.05 m | 16,223 |

beam | solid185 | 0.05 m | 134,750 |

steel bars | mesh200/reinf264 | 0.05 m | 55,847 |

fastener | combin14/combin39 | - | 208/104 |

ballast bed | combin14/combin39 | - | 39,310/19,655 |

bearing | combin14/combin39 | - | 196 |

NPSS | combin39 | - | 196 |

Name | Model | Range | Sensitivity | Accuracy |
---|---|---|---|---|

pressure sensor | JMZX3420 | 0–2000 kN | 2.0 mV/V | 2 kN |

displacement sensor (bearings) | CDP-10 | 0–10 mm | 1000 με/mm | 0.001 mm |

displacement sensor (midspan) | SDP-50C | 0–50 mm | 100 με/mm | 0.01 mm |

Type | Parameter | Value | Type | Parameter | Value |
---|---|---|---|---|---|

steel bars | Young’s modulus (Pa) | 2.1 × 10^{11} | bridge section | area (m^{2}) | 1.18 |

Poisson’s ratio | 0.3 | vertical moment of inertia (m^{4}) | 0.087 | ||

yield strength (MPa) | 400/300 | single bearing | length × width × height (m) | 0.3 × 0.3 × 0.05 | |

tangent modulus (Pa) | 1.6 × 10^{9} | Young’s modulus (Pa) | 6.84 × 10^{8} | ||

diameter (mm) | 8–25 | vertical stiffness (N/m) | 1.23 × 10^{9} |

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## Share and Cite

**MDPI and ACS Style**

Xie, K.; Liu, B.; Dai, W.; Chen, S.; Wang, X. Performance Analysis of Short-Span Simply Supported Bridges for Heavy-Haul Railways with A Novel Prefabricated Strengthening Structure. *Buildings* **2023**, *13*, 876.
https://doi.org/10.3390/buildings13040876

**AMA Style**

Xie K, Liu B, Dai W, Chen S, Wang X. Performance Analysis of Short-Span Simply Supported Bridges for Heavy-Haul Railways with A Novel Prefabricated Strengthening Structure. *Buildings*. 2023; 13(4):876.
https://doi.org/10.3390/buildings13040876

**Chicago/Turabian Style**

Xie, Kaize, Bowen Liu, Weiwu Dai, Shuli Chen, and Xinmin Wang. 2023. "Performance Analysis of Short-Span Simply Supported Bridges for Heavy-Haul Railways with A Novel Prefabricated Strengthening Structure" *Buildings* 13, no. 4: 876.
https://doi.org/10.3390/buildings13040876