# Research on Torsional Characteristic and Stiffness Reinforcement of Main Girder of Half-Through Truss Bridge

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## Abstract

**:**

## 1. Introduction

_{1}

_{,}and the actual buckling load is designated as F

_{2}.

_{1}is 1, the actual calculation result F

_{2}is only 0.53. We use F

_{2}/F

_{1}to represent the stability performance of the structure. The closer this ratio (F

_{2}/F

_{1}) is to 1, the better the stability of the bridge.

## 2. Equivalent Calculation of Torsional Moment of Inertia

#### 2.1. Equivalent Model of Section of Main Girder

_{z}, and the thickness of the bottom wall is designated as t

_{x}. The two side walls and the bottom wall together form an open thin-walled member.

_{z}is the thickness of the equivalent side wall; t

_{x}is the thickness of the equivalent bottom wall; H is the height of the bridge; and b is the width of the bridge.

#### 2.2. Equivalent Wall Thickness

_{z}.

_{z}of the N-shaped main truss [16] can be calculated according to Equation (2):

_{x}is the cross-sectional area of the diagonal web member; E is the elastic modulus of the material; and G is the shear modulus of the material.

_{x}of the X-shaped bottom frame [16] can be calculated according to Equation (3):

_{p}is the cross-sectional area of the diagonal web member of the X-shaped frame; E is the elastic modulus of the material; and G is the shear modulus of the material.

## 3. Example and Finite Element Analysis

#### 3.1. Example

#### 3.2. Finite Element Analysis

#### 3.3. Analysis of Calculation Results

## 4. Calculation Correction Considering Restraint Torsional Effect

#### 4.1. Torsional Characteristics of Half-Through Truss Girder

#### 4.2. Correction of Theoretical Value of Torsional Stiffness

_{x}, and take the cantilever beam as the boundary condition. Calculate the torsional angle under the action of torque T, and then bring the torsional angle into Equation (4) to calculate J

_{x}.

_{s}the vertical bending moment of the inertia of the single-sided main truss; x is the length of the cantilever beam; E is the elastic modulus of the aluminum alloy; b is the width of the bridge; T is the torque acting on the bridge; and the relation between T and F is shown in Equation (6).

_{s}can be composed of the sum of the vertical bending moment of the inertia of the upper and lower chords and the parallel displacement axis value of the Y axis (horizontal center line of vertical web member) in Figure 13, I

_{s}can be calculated as shown in Equation (8):

_{s}the vertical bending moment of the inertia of the single-sided main truss.

_{x}is

_{s}the vertical bending moment of the inertia of the single-sided main truss; and x is the length of the cantilever beam.

_{xz}of the main beam can be obtained as follows:

_{z}is the thickness of the equivalent side wall; t

_{x}is the thickness of the equivalent bottom wall; H is the height of the bridge; b is the width of the bridge; I

_{s}the vertical bending moment of inertia of the single-sided main truss; and x is the length of the cantilever beam.

## 5. Reinforcement Method of Torsional Stiffness and Effect Assessment

#### 5.1. Discussion of Reinforcement Method of Torsional Stiffness

- (1)
- Change the dimension parameters of bridge structure

- (2)
- Installation of reinforcing members

#### 5.2. Test of the Reinforcement Method

#### 5.3. Effect Assessment

- (1)
- Resisting erosion.

- (2)
- Resisting earthquake and shock.

## 6. Conclusions

- (1)
- Based on the principle of making truss bridges equivalent to thin-walled members, the calculation formula of the free torsional moment of inertia of the half-through truss bridge is deduced. The correction formula of the torsional moment of inertia of the main truss against warping deformation is deduced, and the analytical results are compared with the finite element results. The results show that the modified formula can accurately characterize the torsional principle of the half-through truss bridge, but because the transverse stiffness of the bottom chords is not considered, the error of the analytical solution is more than 10% when the width-span ratio is large. However, it is still applicable to the half-through truss bridge with a general width-span ratio and does not affect explaining the torsional characteristics of the half-through truss bridge.
- (2)
- The main truss strengthens the torsional stiffness of the whole bridge by resisting warping, which increases rapidly with the increase of the width-span ratio. The corrected value of the torsional moment of inertia is far higher than the uncorrected result, which is the dominant factor of the torque resistance of the half-through truss bridge. The most important parameter affecting the torsional stiffness of the half-through truss bridge is the width-span ratio. The torsional stiffness of the bridge will increase with an increase in bridge width and decrease rapidly with a decrease in bridge span.
- (3)
- Strengthening the torsional stiffness of the half-through truss pedestrian bridge by blindly strengthening the size of bridge components or increasing the volume of the bridge is not suitable for existing bridges. By analogy with the open thin-walled member, it is found that the torsional stiffness of the whole bridge can be improved by adding X-shaped frames between the transverse girders of the half-through truss pedestrian bridge. The effect of adding X-shaped reinforcing frames is remarkable, which can greatly improve the torsional stiffness. The strengthened torsional stiffness not only can improve the deformation performance of the bridge, but it can also improve the stability of the bridge. This method does not change the appearance and material of the structure and does not require large-scale disassembly. The strengthened aluminum alloy bridges not only can maintain the advantage of corrosion resistance, but can also be improved in resisting extreme loads, such as earthquakes and shock.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The simplified bridge cross section and the equivalent section of the open thin-walled girder: (

**a**) bridge cross section; (

**b**) equivalent open thin-walled beam section.

**Figure 5.**Shear deformation of two types of trusses: (

**a**) N-shaped main truss; (

**b**) X-shaped bottom horizontal frame.

**Figure 10.**Calculation results of torsional moment of inertia with changing x (length of cantilever beam).

**Figure 11.**Warping normal stress and equivalent moment on the section of the open thin-walled member and the spatial warpage deformation.

**Figure 12.**Torsional deformation of the half-through truss bridge: (

**a**) torsional deformation; (

**b**) transverse decomposition; (

**c**) vertical decomposition.

**Figure 17.**Schematic diagram of the thin-walled member with X-shaped battens and the half-through truss bridge installed with torsion-resistant frame.

Component | Cross Section of Component (mm) |
---|---|

Upper and bottom chord | Double 230 × 90 × 12 |

Web member of the main girder | □ 120 × 160 × 10 |

Upper member of the transverse girder and Web member of the transverse girder | □ 70 × 70 × 5 |

Bottom member of the transverse girder | Double □ 70 × 70 × 5 |

The X-shaped bottom horizontal frame | Double □ 120 × 160 × 10 |

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**MDPI and ACS Style**

Yue, Z.; Wen, Q.; Ding, Y.
Research on Torsional Characteristic and Stiffness Reinforcement of Main Girder of Half-Through Truss Bridge. *Sustainability* **2022**, *14*, 6628.
https://doi.org/10.3390/su14116628

**AMA Style**

Yue Z, Wen Q, Ding Y.
Research on Torsional Characteristic and Stiffness Reinforcement of Main Girder of Half-Through Truss Bridge. *Sustainability*. 2022; 14(11):6628.
https://doi.org/10.3390/su14116628

**Chicago/Turabian Style**

Yue, Zixiang, Qingjie Wen, and Youliang Ding.
2022. "Research on Torsional Characteristic and Stiffness Reinforcement of Main Girder of Half-Through Truss Bridge" *Sustainability* 14, no. 11: 6628.
https://doi.org/10.3390/su14116628