# Research on the Internal Thermal Boundary Conditions of Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions

^{*}

## Abstract

**:**

## Featured Application

**The temperature distributions on concrete closed girder cross-sections can be accurately simulated by using the finite element models to establish air elements as the internal thermal boundary condition. Using the finite element models to establish air elements as the internal thermal boundary condition has a wide application range because it does not require the field measurement of the temperature inside the cavity, which is cost- and time-prohibitive. Moreover, this method can predict the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions.**

## Abstract

## 1. Introduction

_{mean}= 0.25 × (T

_{max,i}+ T

_{min,i}+ T

_{max,i−1}+ T

_{min,i−1})

_{mean}is the mean temperature over the previous two days, T

_{max,i}, T

_{min,i}, T

_{max,i−1}, and T

_{min,i−1}are the maximum and minimum temperatures of the ambient air temperature curves for the ith and (i

_{−}1)th days.

## 2. Finite Element Simulation

#### 2.1. Box Girder

^{3}, 1.5 W/(m·°C), and 900 J/(kg·°C), respectively, were used [14,15,17,47]. Based on the measured ambient air temperature, wind speed, and solar radiation obtained from the movable automatic meteorological station at the bridge site, the parameters associated with solar radiation, convection, irradiation, and the temperature of the surrounding fluid medium were used as the external thermal boundary conditions for the box girder in the FEM. The influence of solar radiation on different parts of the box girder was considered by using the following rules and assumptions: (a) the external surface of the top flange is influenced by both the beam and diffuse solar radiation; (b) the external surfaces of the two exterior webs in the shadow due to the top flange are influenced by the diffuse solar radiation and ground reflection; (c) the external surfaces of the two exterior webs not in the shadow are influenced by the beam solar radiation, diffuse solar radiation, and ground reflection; and (d) the undersides of the top and bottom flanges are influenced by the ground reflection alone. The environmental and climatic conditions existing on previous days must be considered to accurately determine the initial temperature distribution [13,14,17,25,39]. A 120-h period with the same environmental conditions imposed cyclically was considered in the analysis.

^{2}·°C) was set in the FEMs by using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method. For the Measured Temperature Method, the measured hourly temperature curves inside each cell of the box girder (13 August 2017) obtained from the temperature sensors C-1 to C-3 were chosen, as shown by the hollow points in Figure 3. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (13 August 2017) was used, as shown by the solid line in Figure 3. For the Mean Temperature Method, the mean temperature (30.7 °C) over the previous two days (12 and 13 August 2017) was calculated, as shown by the dashed line in Figure 3. The mesh size of the FEMs using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 37,017 and 35,659, respectively, as illustrated in Figure 2a. For the Air Element Method, the heat conduction between the air and girder cross-section can be simulated by establishing the air elements inside the cavities without the input of the temperature curves and the convection heat transfer coefficient inside the cavities. The air elements were also two-dimensional plane strain elements with the thermal parameters of air under 20 °C and one standard atmospheric pressure (density, heat conductivity, and specific heat of 1.205 kg/m

^{3}, 0.0259 W/(m·°C), and 1005 J/(kg·°C), respectively) [48]. The mesh size of the air element was set as 20 mm, which is the same as those of the girder cross-section. The total numbers of nodes and elements for the FEM using the Air Element Method were 52,458 and 51,669, respectively, as illustrated in Figure 2b.

#### 2.2. Small Box Girder

#### 2.3. Adjacent Box Girder

## 3. Influence of Different Simulation Methods for Internal Thermal Boundary Conditions

#### 3.1. Box Girder

#### 3.1.1. Influence on the Hourly Temperature Curves

#### 3.1.2. Influence on the Temperature Contour Plots

#### 3.1.3. Influence on the Average Effective Temperatures and Vertical Temperature Gradients

_{AVG}) of the box girder cross-sections can be estimated using the following rules. The temperature of each element in the FEM is multiplied by its respective element area, and the results for all elements in the cross-section are added together. Then, T

_{AVG}can be obtained by dividing the sum by the total area of the cross-section [13,22]. The highest T

_{AVG}values of the box girder cross-sections for 13 August 2017, obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions, are listed in Table 1. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest T

_{AVG}was small, with the maximum difference of 0.7 °C.

#### 3.2. Small Box Girder

#### 3.2.1. Influence on the Hourly Temperature Curves

#### 3.2.2. Influence on the Temperature Contour Plots

#### 3.2.3. Influence on the Average Effective Temperatures and Vertical Temperature Gradients

_{AVG}values of the small box girder cross-sections for 17 July, 2015 obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions, are listed in Table 2. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest T

_{AVG}was small, with the maximum difference of 1.1 °C.

#### 3.3. Adjacent Box Girder

#### 3.3.1. Influence on the Hourly Temperature Curves

#### 3.3.2. Influence on the Temperature Contour Plots

#### 3.3.3. Influence on the Average Effective Temperatures and Vertical Temperature Gradients

_{AVG}values of the adjacent box girder cross-sections for 24 July 2017 obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions are listed in Table 3. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest T

_{AVG}was small, with a maximum difference of 0.8 °C.

## 4. Temperature Distributions on Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions

#### 4.1. Meteorological Data

_{max}and T

_{min}) using the following Equation [15]:

_{i}= asinω

_{i}+ β

_{i}is the ambient air temperature at the ith hour, a and β are the model coefficients calculated by a = (T

_{max}− T

_{min})/2 and β = (T

_{max}+ T

_{min})/2, and ω

_{i}is the hour angle at the ith hour.

_{max}and T

_{min}for the 100-year return period can be predicted, as listed in Table 4. The monthly average daily temperature range can be obtained from the China Meteorological Administration [51]. The idealized hourly temperature curve on the hottest day was based on T

_{max}, and the corresponding lowest temperature of the idealized curve was selected by subtracting the monthly average daily temperature range in July. The idealized hourly temperature curve on the coldest day was based on T

_{min}, and the corresponding highest temperature of the idealized curve was selected by adding the monthly average daily temperature range in January [13,53]. The times at which T

_{max}and T

_{min}occurred in the idealized hourly temperature curves under historically extreme temperature conditions were assumed to be 14:00 for the idealized hourly temperature curve on the hottest day and 06:00 for the idealized hourly temperature curve on the coldest day. According to the bridge locations introduced in Chapter 2, the hourly ambient air temperature curves (continued for five days) under historically extreme temperature conditions in Shenzhen, Handan, and Fuzhou are illustrated in Figure 22.

#### 4.2. Average Effective Temperatures of Cross-Sections

_{E}). The one-year measured average effective temperatures of the cross-sections can be obtained from the field test (hereafter referred to as T

_{M}). The maximum and minimum effective temperatures of highway concrete bridges were recommended by the Chinese code JTG D60-2015 [50] (hereafter referred to as T

_{C}). China is partitioned into four climatic regions that are categorized as severe cold, cold, warm, and “other” regions [50]. Handan is in the cold region, with the highest T

_{C}of 34 °C and the lowest T

_{C}of −10 °C; Shenzhen is in the warm region, with the highest T

_{C}of 34 °C and the lowest T

_{C}of −3 °C; and Fuzhou is in the “other” region, with the highest T

_{C}of 34 °C and the lowest T

_{C}of 0 °C. Comparisons of T

_{M}, T

_{C}, and T

_{E}are illustrated in Figure 24.

_{M}, T

_{C}, and T

_{E}for the box girder cross-section are illustrated in Figure 24a. It can be observed that the highest T

_{M}was close to the highest T

_{C}(a difference of 0.9 °C), and these values were lower than the highest T

_{E}(differences of 4.9 °C for T

_{M}and 5.8 °C for T

_{C}). The lowest T

_{M}was much higher than the lowest T

_{E}(a difference of 12.2 °C) and the lowest T

_{C}(a difference of 17.8 °C). The variation of T

_{C}was close to that of T

_{E}(a difference of only 0.2 °C), and they were larger than that of T

_{M}(differences of 16.9 °C for T

_{C}and 17.1 °C for T

_{E}). It can be predicted that the longitudinal thermal movement of the concrete box girder under historically extreme temperature conditions would be underestimated by considering the variation of T

_{M}. The variations of T

_{C}(37.0 °C) and T

_{E}(37.2 °C) were almost the same. However, assuming that the datum temperatures (T

_{0}) were the same, the thermal expansion of the concrete box girder calculated by the difference between the highest T

_{C}and T

_{0}would be smaller than that calculated by the difference between the highest T

_{E}and T

_{0}, while the thermal contraction of the concrete box girder calculated by the difference between T

_{0}and the lowest T

_{C}would be larger than that calculated by the difference between T

_{0}and the lowest T

_{E}.

_{M}, T

_{C}, and T

_{E}for the small box girder cross-sections are illustrated in Figure 24b. It can be observed that the highest T

_{E}(44.7 °C) was significantly higher than the highest T

_{M}(a difference of 8.8 °C) and the highest T

_{C}(a difference of 10.7 °C); meanwhile, the lowest T

_{E}(−18.2 °C) was significantly lower than the lowest T

_{M}(a difference of 20.2 °C) and the lowest T

_{C}(a difference of 8.2 °C). The variation of T

_{E}(62.9 °C) was significantly larger than that of T

_{M}(a difference of 29.0 °C) and T

_{C}(a difference of 18.9 °C), which was possibly because Handan is in Northern China, where the variation of ambient air temperature is large. It can be predicted that the longitudinal thermal movement of the concrete small box girders under historically extreme temperature conditions would be underestimated by considering variations of T

_{M}or T

_{C}.

_{M}, T

_{C}, and T

_{E}for the adjacent box girder cross-sections are illustrated in Figure 24c. It can be observed that the highest T

_{E}(39.8 °C) was higher than the highest T

_{M}(a difference of 4.9 °C) and the highest T

_{C}(a difference of 5.8 °C); meanwhile, the lowest T

_{E}(0.1 °C) was lower than the lowest T

_{M}(a difference of 8.4 °C) and similar to the lowest T

_{C}(a difference of only 0.1 °C). The variation of T

_{E}(39.8 °C) was larger than that of T

_{M}(a difference of 10.8 °C) and T

_{C}(a difference of 5.8 °C). It can be predicted that the longitudinal thermal movement of the concrete adjacent box girders under historically extreme temperature conditions would be underestimated by considering variations of T

_{M}or T

_{C}.

#### 4.3. Vertical Temperature Gradients

## 5. Summary

- (1)
- The Measured Temperature Method can reflect the actual temperatures inside the cavities, but measurements on site are cost- and time-prohibitive. Therefore, the application range of the Measured Temperature Method is limited. When there is no measurement on site to obtain the temperature inside the cavity, the Ambient Temperature Method, Mean Temperature Method, or Air Element Method can be used as alternative methods.
- (2)
- The influences of different simulation methods for the internal thermal boundary conditions on the numerical hourly temperature curves of the parts of cross-sections far from the cavities are negligible. Compared with the measured hourly temperature curves, the numerical hourly temperature curves of the parts of the cross-sections near the cavities calculated by the Measured Temperature Method provide the closest agreement. When there is a lack of measured temperature inside the cavity, the numerical hourly temperature curves calculated by the Air Element Method provide a closer agreement with the measured curves than the curves calculated by the Ambient Temperature Method and Mean Temperature Method. When the Ambient Temperature Method is used, the trends of the numerical curves of the bottom flanges and the webs near the bottom flanges are similar to the measured hourly ambient air temperature curve, because the temperature change in the hourly ambient air temperature curve was much larger than the temperature changes in the measured curves inside the cavities.
- (3)
- The comparisons of the temperature contour plots obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions indicated that the temperature distributions on the parts near the cavities calculated by the Measured Temperature Method and Air Element Method were close to the measured values.
- (4)
- The influences of different simulation methods for the internal thermal boundary conditions on the highest hourly average effective temperature of concrete closed girder cross-sections and the trends of the vertical temperature gradients for the box girder and adjacent box girder cross-sections were small. The maximum vertical temperature gradients calculated by the Air Element Method on the top and bottom flanges were larger than those calculated by the Ambient Temperature Method and Mean Temperature Method for the small box girder cross-sections.
- (5)
- The Air Element Method can be used as a simulation method for the internal thermal boundary conditions in the FEM to predict the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions. The longitudinal thermal movement of concrete closed girders calculated by the one-year measured average effective temperature of the cross-sections or by the Chinese-code-specified effective temperatures for the highway bridge structures would be smaller than those under historically extreme temperature conditions, which are thus unconservative for engineering applications. It is suggested that the average effective temperature of concrete closed girder cross-sections under historically extreme temperature conditions should be calculated for each city using the FEM with the Air Element Method, as described in this paper. The comparisons of vertical temperature gradients under historically extreme temperature conditions indicate that the Chinese-code-specified vertical temperature gradients are conservative for the bridge deck surfaces and unconservative for the bottom flanges.
- (6)
- Further research has been carried out to analyze the average effective temperatures and vertical temperature gradients of concrete closed girder cross-sections under historically extreme temperature conditions for each city in China using the FEM with the Air Element Method described in this paper.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Finite element model (FEM) of the box girder cross-sections: (

**a**) FEM without air elements; (

**b**) FEM with air elements.

**Figure 3.**Comparison of different simulation methods for the temperature curves inside the cavities.

**Figure 5.**FEM of the small box girder cross-sections: (

**a**) FEM without air elements; (

**b**) FEM with air elements.

**Figure 6.**Comparison of different simulation methods for the temperature curves inside the cavities.

**Figure 8.**FEM of the adjacent box girder cross-sections: (

**a**) FEM without air elements; (

**b**) FEM with air elements.

**Figure 9.**Comparison of different simulation methods for the temperature curves inside the cavities.

**Figure 10.**Comparisons of the hourly temperature curves of the box girder cross-sections obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions: (

**a**) T-5 in the top flange; (

**b**) T-4 in the top flange; (

**c**) B-3 in the bottom flange; and (

**d**) W-7 and W-9 in the web.

**Figure 11.**The root-mean-square error (RMSE) and mean absolute percentage error (MAPE) of the numerical hourly temperature curves of the box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions: (

**a**) RMSE; (

**b**) MAPE.

**Figure 12.**Temperature contour plots of the box girder cross-sections at 15:00 on 13 August 2017: (

**a**) Measured Temperature Method; (

**b**) Ambient Temperature Method; (

**c**) Mean Temperature Method; (

**d**) Air Element Method.

**Figure 13.**Comparisons of the maximum vertical temperature gradients of the box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions.

**Figure 14.**Comparisons of the hourly temperature curves of the small box girder cross-sections obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions: (

**a**) 7-T in the top flange; (

**b**) 7-W in the web; (

**c**) 7-B in the bottom flange.

**Figure 15.**The RMSE and MAPE of the numerical hourly temperature curves of the small box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions: (

**a**) RMSE; (

**b**) MAPE.

**Figure 16.**Temperature contour plots of the small box girder cross-sections at 15:00 on 31 July 2014: (

**a**) Ambient Temperature Method; (

**b**) Mean Temperature Method; (

**c**) Air Element Method.

**Figure 17.**Comparisons of the maximum vertical temperature gradients of the small box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions.

**Figure 18.**Comparisons of the hourly temperature curves of the adjacent box girder cross-sections obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions: (

**a**) 3-T in the top flange; (

**b**) 3-W in the web; (

**c**) 3-B in the bottom flange.

**Figure 19.**The RMSE and MAPE of the numerical hourly temperature curves of the adjacent box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions: (

**a**) RMSE; (

**b**) MAPE.

**Figure 20.**Temperature contour plots of the adjacent box girder cross-sections at 15:00 on 24 July 2017: (

**a**) Measured Temperature Method; (

**b**) Ambient Temperature Method; (

**c**) Mean Temperature Method; (

**d**) Air Element Method.

**Figure 21.**Comparisons of the maximum vertical temperature gradients of the adjacent box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions.

**Figure 22.**Idealized hourly ambient air temperature curves under historically extreme temperature conditions: (

**a**) Shenzhen; (

**b**) Handan; (

**c**) Fuzhou.

**Figure 23.**Numerical hourly beam solar radiation and diffuse solar radiation under the highest historical temperature conditions: (

**a**) Shenzhen; (

**b**) Handan; (

**c**) Fuzhou.

**Figure 24.**Comparisons of T

_{M}, T

_{C}, and T

_{E}: (

**a**) Box girder cross-section; (

**b**) small box girder cross-sections; (

**c**) adjacent box girder cross-sections.

**Figure 25.**Comparison of the vertical temperature gradients of the box girder cross-section: (

**a**) Positive vertical temperature gradient; (

**b**) negative vertical temperature gradient.

**Figure 26.**Comparison of the vertical temperature gradients of the small box girder cross-sections: (

**a**) Positive vertical temperature gradient; (

**b**) negative vertical temperature gradient.

**Figure 27.**Comparison of the vertical temperature gradients of the adjacent box girder cross-sections: (

**a**) Positive vertical temperature gradient; (

**b**) negative vertical temperature gradient.

**Table 1.**Comparisons of the highest T

_{AVG}of the box girder cross-sections obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions (unit: °C).

Methods | Measured Temperature Method | Ambient Temperature Method | Mean Temperature Method | Air Element Method |
---|---|---|---|---|

Measured value | 34.9 | |||

Numerical value | 35.0 | 35.0 | 34.6 | 35.6 |

**Table 2.**Comparisons of the highest T

_{AVG}of the small box girder cross-sections obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions (Unit: °C).

Methods | Ambient Temperature Method | Mean Temperature Method | Air Element Method |
---|---|---|---|

Measured value | 28.4 | ||

Numerical value | 29.5 | 28.6 | 28.6 |

**Table 3.**Comparisons of the highest T

_{AVG}of the adjacent box girder cross-sections obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions (Unit: °C).

Methods | Measured Temperature Method | Ambient Temperature Method | Mean Temperature Method | Air Element Method |
---|---|---|---|---|

Measured value | 37.5 | |||

Numerical value | 37.0 | 37.8 | 37.1 | 36.7 |

**Table 4.**Meteorological parameters of the hourly ambient air temperature curves under historically extreme temperature conditions (Unit: °C).

Cross-Section | Location | Ambient Air Temperature | Monthly Average Daily Temperature Range | ||
---|---|---|---|---|---|

Max | Min | July | January | ||

Box girder | Shenzhen | 37.9 | −0.3 | 6.1 | 7.4 |

Small box girder | Handan | 43.1 | −21.3 | 8.9 | 8.9 |

Adjacent box girder | Fuzhou | 40.6 | −1.9 | 8.7 | 7.0 |

© 2020 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**

Lin, J.; Xue, J.; Huang, F.; Chen, B.
Research on the Internal Thermal Boundary Conditions of Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions. *Appl. Sci.* **2020**, *10*, 1274.
https://doi.org/10.3390/app10041274

**AMA Style**

Lin J, Xue J, Huang F, Chen B.
Research on the Internal Thermal Boundary Conditions of Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions. *Applied Sciences*. 2020; 10(4):1274.
https://doi.org/10.3390/app10041274

**Chicago/Turabian Style**

Lin, Jianhui, Junqing Xue, Fuyun Huang, and Baochun Chen.
2020. "Research on the Internal Thermal Boundary Conditions of Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions" *Applied Sciences* 10, no. 4: 1274.
https://doi.org/10.3390/app10041274