# Temperature Response of Double-Layer Steel Truss Bridge Girders

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Analysis Model

#### 2.1. Environmental Temperature Model

_{max}and T

_{min}(°C), respectively, denote the average maximum and minimum temperatures of extreme high-temperature weather during the high-temperature season.

#### 2.2. Solar Radiation

#### 2.2.1. Direct Solar Radiation

_{0}(W/m

^{2}) is the solar constant, I

_{0}= 1367 [1 + 0.33 cos(360 N⁄365)], N is the daily serial number, k

_{a}(Pa) is the atmospheric relative pressure, m (m) is the optical path length, m = 1⁄sinβ, and α (°) is the solar radiation angle.

#### 2.2.2. Diffuse Solar Radiation

^{2}) is:

_{m}and P, respectively, represent the atmospheric transparency coefficient and the composite atmospheric transparency coefficient, and their expressions are ${P}_{m}={0.9}^{{t}_{u}{k}_{a}m}$, $P={0.9}^{{t}_{u}{k}_{a}}$, and $\beta $ (°) is solar altitude angle.

#### 2.2.3. Double-Deck Shading Model

_{s}(°) is the plane azimuth and solar azimuth angle, respectively, and δ (°) is the inclination angle of the plane.

#### 2.2.4. Total Solar Radiant

^{2}) of the sunshine area is different from that of the shaded area, so the total expression is:

#### 2.3. Simulation of External Thermal Boundary of Double-Layer Steel Truss

#### 2.3.1. Virtual Thermal Boundary

_{a}(°C) is the atmospheric temperature, α

_{s}is the radiation absorption coefficient of the material, and h is the comprehensive heat transfer coefficient.

#### 2.3.2. Convective Heat Transfer

^{2}) is the heat flux density of convective heat transfer, h

_{c}(W/m

^{2}/K) is the convective heat transfer coefficient, T

_{a}(°C) is the atmospheric temperature, and T

_{c}(°C) is the surface temperature of the structure.

#### 2.3.3. Radiation Heat Transfer

^{2}/K) is:

_{0}is the Stefan–Boltzman constant, C

_{0}= 5.67 × 10

^{−8}W/m

^{2}/K, and ε is the long-wave radiation emissivity of the component surface.

#### 2.4. Simulation of Internal Thermal Boundary of Double-Layer Steel Truss Box Members

_{Ai}(°C) is the comprehensive atmospheric temperature inside the box type component, and T

_{A}(°C) is the daily average temperature of the atmosphere, with a value of 34.5 °C in this study.

#### 2.5. Establishment and Verification of Finite Element Model

#### 2.5.1. Selection of Research Subjects

#### 2.5.2. Modeling

_{i}(m

^{4}) is the in-plane bending moment of inertia of each member; A

_{c}(m

^{2}) is the cross-section area of the diagonal web; E (N/m

^{2}) is the elastic modulus; and G (N/m

^{2}) is the Shear modulus.

#### 2.5.3. Temperature Analysis Verification

## 3. Time-Varying Temperature Field

#### 3.1. Time-Varying Temperature Field Distribution Law of The Whole Structure

#### 3.2. Temperature Distribution Model of Chord Section

#### 3.3. Distribution Law of Time-Varying Temperature Field on Double Deck

_{1}and L

_{2}is smaller than that within region L

_{2}, indicating a more rapid temperature change in region L

_{2}.

#### 3.4. Calculation Formula of Temperature Gradient of Component Section

#### 3.4.1. Proposed Formula

#### 3.4.2. Formula Validation

^{2}= 0.9889), thus confirming the formula’s feasibility.

## 4. Time-Varying Law of Temperature Response

#### 4.1. Structural Displacement

#### 4.1.1. Vertical Displacement

#### 4.1.2. Transverse Displacement

#### 4.1.3. Longitudinal Displacement

_{1}, N

_{2}, and N

_{3}are consistent. Under solar radiation, each point is heated and expanded, reaching a maximum of 54.14 mm at 14:00 and then entering the descending stage; for the lower bridge deck, the longitudinal displacements of N

_{4}, N

_{5}, and N

_{6}are consistent during 6:00~8:00. After 8:00, affected by the angle of solar radiation and shading, the degree of thermal expansion of the three positions was different. The girder end exhibits a complex structural configuration and experiences intricate stress states.

#### 4.2. Structural Stress

#### 4.2.1. Stress of the Support Sections

#### 4.2.2. Stress of Chords

#### 4.3. Steel Truss Girder Rotation Angles

#### 4.3.1. Transverse Rotation Angle of Girder End

#### 4.3.2. Vertical Rotation Angle of Girder End

## 5. Conclusions

- A model analyzing the impact of solar radiation on bridge structures was developed. This model, integrating time-varying thermal boundary conditions and support scenarios, led to an effective temperature analysis framework for the double-layer steel truss continuous girder. Validation efforts revealed that the temperature model’s predictions deviate from experimental data by a mere 2.22%, demonstrating the model’s reliability and effectiveness.
- The study identified distinct vertical, horizontal, and longitudinal temperature gradients within the structure. The vertical gradient, most pronounced on the truss sides, showed a maximum temperature difference of 19.27 °C. The horizontal gradient, concentrated on the lower deck, varied with solar radiation angles, reaching a peak difference of 29.73 °C. The longitudinal gradient, less evident and located at the chord junctions, exhibited a temperature variation within 1.87 °C under solar influence.
- The proposed temperature distribution model of the chord section under shielding encompasses five vertical temperature gradient distribution models and four horizontal temperature gradient distribution models. These models are primarily influenced by the environmental temperature, solar radiation, and panel heat exchange. A noteworthy finding is the grid-like temperature field distribution in the double deck under shading, with a distinct temperature boundary on the lower deck influenced by the solar altitude angle. Additionally, the study introduces a methodology for determining temperature gradients at any member section time point.
- Shading was observed to significantly influence the displacements of the upper and lower decks, leading to notable disparities. The most considerable vertical displacement difference occurred at noon (22.58 mm), while the lateral and longitudinal displacements showed the maximum differences of 6.50 mm and 7.49 mm, respectively, at different times of day. Uneven transverse temperature distribution was found to alter the maximum stress location in the lateral fulcrum section over time. The study also highlighted that the girder end’s rotational behavior, both transversely and vertically, is subject to the intensity and angle of solar radiation, with a lag in response to radiation intensity changes.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**Longitudinal temperature distribution of steel truss. (

**a**) Vertical distribution temperature of west side. (

**b**) Vertical distribution temperature of east side. (

**c**) Transverse temperature distribution (morning). (

**d**) Transverse temperature distribution (afternoon). (

**e**) Longitudinal temperature distribution on the east side of steel truss.

**Figure 15.**Temperature distribution of east chord section. (

**a**) Temperature distribution of T

_{1}. (

**b**) Temperature distribution of T

_{6}. (

**c**) Temperature distribution of T

_{2}. (

**d**) Temperature distribution of T

_{5}.

**Figure 17.**Temperature distribution of lower deck. (

**a**) 8:00. (

**b**) 10:00. (

**c**) 12:00. (

**d**) 14:00. (

**e**) 16:00. (

**f**) 18:00.

**Figure 21.**Vertical displacement curve of double-layer bridge deck at different time points. (

**a**) 10:00. (

**b**) 12:00. (

**c**) 14:00. (

**d**) 16:00.

**Figure 24.**Longitudinal displacement. (

**a**) The northern end of the continuous girder. (

**b**) The southern end of the continuous girder.

**Figure 25.**Stress distribution cloud of continuous girder of ultra-wide cross-section double-layer steel truss at 12:00.

Position | h_{c} (W/m^{2}/K) |
---|---|

Girder surface | 15 |

Bridge side surface | 15 |

Bridge bottom surface | 10 |

Q370qE | Numeric Value |
---|---|

Mass density ρ | 7850 kg/m^{3} |

Thermal expansion coefficient α | 1.2 × 10^{−5} °C^{−1} |

Poisson’s ratio υ | 0.31 |

Specific heat capacity c | 434 J/kg·°C |

Isotropic thermal conductivity | 60.5 W/(m·°C) |

Elastic modulus E | 2.06 × 105 MPa |

Component Cross-Section | Formula |
---|---|

Top chord section | $\left\{\begin{array}{l}a=-0.0042{t}^{4}+0.21{t}^{3}-3.63{t}^{2}+24.15t-53.85\\ b=0.034{t}^{4}-1.79{t}^{3}-33.59{t}^{2}-259.42t-704.34\\ c=-0.047{t}^{4}+2.52{t}^{3}-47.71{t}^{2}+375.08t-1037.31\\ d=-0.0092{t}^{4}-0.51{t}^{3}-10.88{t}^{2}-103.81t-309.65\end{array}\right.$ |

Bottom chord section | $\left\{\begin{array}{l}a=0.15{t}^{4}-6.08{t}^{3}+90.08{t}^{2}-590.31t+1443.31\\ b=-0.51{t}^{4}+20.51{t}^{3}-303.73{t}^{2}-1990.4t-4866.54\\ c=0.56{t}^{4}-22.51{t}^{3}+333.11{t}^{2}-2182.93t-5337.25\\ d=-0.21{t}^{4}+8.26{t}^{3}-122.32{t}^{2}+801.48t-1959.67\end{array}\right.$ |

Cross beam section | $\left\{\begin{array}{l}a=-0.0056{t}^{4}+0.28{t}^{3}-4.84{t}^{2}+32.05t-71.87\\ b=0.019{t}^{4}-0.96{t}^{3}+16.47{t}^{2}-108.91t+244.19\\ c=-0.021{t}^{4}+1.06{t}^{3}-18.28{t}^{2}+120.88t-271.08\\ d=-0.0079{t}^{4}-0.39{t}^{3}+6.81{t}^{2}+45.01t-100.91\end{array}\right.$ |

Box-shaped web member | $\left\{\begin{array}{l}a=0.037{t}^{4}-1.48{t}^{3}+21.14{t}^{2}-128.77t-294.41\\ b=-0.034{t}^{4}+1.36{t}^{3}-19.51{t}^{2}+119.56t-275.05\\ c=0.014{t}^{4}-0.57{t}^{3}+8.24{t}^{2}-51.11t+118.86\\ d=-0.0002{t}^{4}+0.0082{t}^{3}-0.12{t}^{2}+0.69t-1.57\end{array}\right.$ |

I-shaped web member | $\left\{\begin{array}{l}a=-0.049{t}^{4}-1.94{t}^{3}+27.61{t}^{2}-168.08t-384.15\\ b=-0.042{t}^{4}+1.66{t}^{3}-23.61{t}^{2}+143.75t-328.56\\ c=-0.015{t}^{4}-0.61{t}^{3}+8.54{t}^{2}-52.01t+118.90\\ d=-0.0003{t}^{4}-0.01{t}^{3}-0.16{t}^{2}+1.03t-2.35\end{array}\right.$ |

Location | 7:00 | 10:00 | 12:00 | 14:00 | 16:00 | 19:00 | |
---|---|---|---|---|---|---|---|

East-side upper chord | ① pier | 14.95 | 38.25 | 44.18 | 43.12 | 28.45 | 4.02 |

② pier | 11.64 | 32.61 | 42.49 | 41.93 | 28.41 | 1.10 | |

③ pier | 11.60 | 32.21 | 42.05 | 41.98 | 29.49 | 1.14 | |

④ pier | 14.99 | 39.61 | 44.28 | 43.42 | 28.79 | 3.86 | |

West-side upper chord | ① pier | 6.57 | 31.75 | 44.71 | 45.75 | 35.65 | 8.86 |

② pier | 2.80 | 30.69 | 43.64 | 44.71 | 32.30 | 8.52 | |

③ pier | 2.86 | 30.68 | 43.94 | 44.12 | 32.55 | 8.37 | |

④ pier | 6.37 | 31.59 | 43.86 | 45.99 | 35.53 | 8.77 | |

East-side lower chord | ① pier | 10.41 | 55.38 | 84.88 | 80.13 | 51.65 | 4.86 |

② pier | 14.69 | 29.52 | 35.91 | 40.89 | 38.90 | 24.62 | |

③ pier | 17.85 | 32.85 | 43.96 | 45.38 | 40.78 | 33.94 | |

④ pier | 10.31 | 55.91 | 83.84 | 81.34 | 52.53 | 4.07 | |

West-side lower chord | ① pier | 5.98 | 53.74 | 81.46 | 85.58 | 66.84 | 7.96 |

② pier | 9.95 | 24.66 | 28.15 | 35.51 | 26.98 | 15.76 | |

③ pier | 11.96 | 30.61 | 36.07 | 40.44 | 34.71 | 28.80 | |

④ pier | 4.85 | 52.46 | 80.22 | 86.08 | 67.47 | 7.91 |

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## Share and Cite

**MDPI and ACS Style**

Wang, S.; Zhang, G.; Li, J.; Wang, Y.; Chen, B.
Temperature Response of Double-Layer Steel Truss Bridge Girders. *Buildings* **2023**, *13*, 2889.
https://doi.org/10.3390/buildings13112889

**AMA Style**

Wang S, Zhang G, Li J, Wang Y, Chen B.
Temperature Response of Double-Layer Steel Truss Bridge Girders. *Buildings*. 2023; 13(11):2889.
https://doi.org/10.3390/buildings13112889

**Chicago/Turabian Style**

Wang, Shichao, Gang Zhang, Jie Li, Yubo Wang, and Bohao Chen.
2023. "Temperature Response of Double-Layer Steel Truss Bridge Girders" *Buildings* 13, no. 11: 2889.
https://doi.org/10.3390/buildings13112889