Effect of solar temperature field on a sea-crossing cable-stayed bridge tower

Abstract

Based on the transient temperature field theory of heat conduction, the solar temperature field calculation model of a sea-cross cable-stayed bridge tower was established and calculated through transient heat transfer analysis, with the largest sea-cross construction project of Kong–Zhuhai–Macao Bridge as an example. The main parameters, including the solar elevation angle, azimuth angle, radiation absorption coefficient, and length of overhang, were incorporated. The proposed model was verified by comparing the measured data with the predicted values of both the temperature field and temperature deformation. The analysis results and monitoring data achieved a good agreement, which reveals that integrating numerical analysis with field monitoring data can be utilized to thoroughly understand the effects of a solar temperature field on a sea-crossing cable-stayed bridge tower. The research results guide the construction of the bridge.

Keywords

bridge engineering cable-stayed bridges sea-crossing sunshine radiation temperature field

Introduction

Thin-walled hollow concrete towers have been widely used in cable-stayed bridges, especially in long-span sea-crossing bridges. As the span increases, the height of the bridge tower would be over 300 m. These over-high concrete bridge towers would be exposed to the marine environment and suffer the daily, seasonally, and annually varying environmental thermal effects caused by solar radiation and the surrounding air temperature. During the construction and in-service stages, bridge towers would be affected by temperature fluctuations. Meanwhile, the marine environment has its unique characteristics, such as greater sunlight and larger temperature differences between day and night. Furthermore, the structural behavior of bridges would be more significantly affected by the environmental thermal effect, which is even more when compared to external operational loads. Therefore, it is crucial to consider the temperature effects on bridge design and construction.

Previous studies on the temperature effect on bridges have mainly focused on girder or slabs under service conditions. According to the knowledge of the authors, there are few studies about the solar temperature effects on bridge towers, and relevant regulations and bases of designing need to be complemented. Since the analysis of the solar temperature field is a complex nonlinear space transient problem, its boundary is more complex than the general analysis.

Although many researchers have reported on the temperature-induced variability of structural properties of bridges, less attention has been paid to cable-stayed bridge towers, especially under the conditions of the marine environment in the construction stage. Sohn et al. (1999) examined a linear adaptive model to discriminate the changes of modal parameters due to temperature changes from those caused by structural damage or other environmental effects. With this simple model, a confidence interval of frequencies for a new temperature profile could be established to discriminate natural variations due to temperature. Cai et al. (2012) investigated the effects of temperature variations on the in-plane stability of steel arch bridges. Mosavi et al. (2012) investigated the effect of temperature variations on the modal characteristics of a two-span steel–concrete composite bridge in North Carolina and addressed the extent and reason of the daily changes observed in its dynamic properties. Xia et al. (2013) investigated the temperature distribution and the associated responses of a long-span suspension bridge, that is, the 2132-m-long Tsing Ma Bridge, through the combination of numerical analysis and field monitoring. Zhou et al. (2016) investigated the temperature distribution of the Humber Bridge in the United Kingdom using numerical simulation and field measurements. Liu et al. (2014) investigated the solar temperature field of a cable-stayed bridge with a U-shaped section in a high-speed railway and obtained the nonlinear distribution characteristics and maximum cross-section temperature difference of the U-shaped section. Zhu and Meng (2017) proposed an effective simulation technology for finely predicting the temperature effect of bridges. Based on the ray tracing method, a three-dimensional (3D) sunlight-sheltering algorithm was developed to more precisely predict the temperature field. Wang et al. (2013) evaluated the temperature influence on a concrete-filled steel-tube arch bridge and made a comparison between the predictions and measurements of the bridge. Miao and Shi (2013) analyzed the temperature field variation law and distribution characteristics of an orthotropic flat steel box girder under sunny conditions through a field temperature test on the steel box girder of the operational Runyang Yangtze River Bridge (the suspension bridge part). Tian et al. (2016) presented a numerical approach to determine the temperature effects on train–bridge coupled dynamics. Gu et al. (2014) analyzed the temperature distributions of cold wave processes on a pre-stressed concrete box-girder bridge. Rodriguez et al. (2014) investigated the temperature effects on a box-girder integral-abutment bridge through field-measured data and evaluated the effects of temperature gradients on a typical integral-abutment bridge.

In this study, the largest sea-cross construction project of the Kong–Zhuhai–Macao Bridge was taken as an example. According to the field-measured temperature data and thermal analysis prediction results by meteorological condition, the temperature effect on cable-stayed bridge towers was calculated and analyzed using a commercial software ABAQUS. The main parameters, including the solar elevation angle, azimuth angle, radiation absorption coefficient, and length of the overhang, were incorporated.

The bridge description

The Hong Kong–Zhuhai–Macao Bridge project is the biggest crossing-sea project in the world, which is located at the entrance of the Pearl River in China (Figure 1) and is worth more than ¥ 80 billion. There are three navigation channel bridges, including two cable-stayed bridges with double towers and a cable-stayed bridge with three towers. The main navigable bridge, named Qingzhou Navigable Bridge, is the biggest of these three navigable bridges (Figure 2) and has a span arrangement of 110+236+458+236+100 m. The main span reaches 458 m, and the height of the tower is 167 m. The top crossbeam is designed as a “Chinese Knot” steel structure which is first used in the cable-stayed bridge (Figure 3). The detailed installation of this special steel structure needs to obtain the exact spatial position of the tower under different complex conditions. It is necessary to analyze the deformation of the bridge tower under sunshine temperature field in the marine environment.

Figure 1.

Location of the Hong Kong–Zhuhai–Macao Bridge.

Figure 2.

Span arrangement of the Qingzhou Navigable Bridge (dimensions in meters).

Figure 3.

Arrangement of the bridge tower.

Intensity of solar radiation and the thermal parameters

Solar radiation

According to the heat conduction theory (Song et al., 2006), the intensity of solar radiation $I$ depends on the solar declination, real solar time, solar latitude angle, azimuth, incidence angle, atmospheric quality, coefficient of atmospheric transparency, material properties, and other parameters. In engineering practice, $I$ includes the direct solar radiation $I_{1}$ and diffuse solar radiation $I_{2}$ by considering the main factors, which can be calculated as follows (Elbadry and Ghali, 1983)

I = I_{1} + I_{2}

(1)

where $I_{1}$ is the direct radiation, which can be calculated as

I_{1} = I_{d} \cos θ

(2)

I_{d} = I_{0} P_{1}^{m}

(3)

m = \frac{1}{\sin (h_{s})}

(4)

where $I_{d}$ is the ground sunshine radiation intensity of normal direction $(w / m^{2})$ , θ is the solar incident angle, $m$ is the light path, $P_{1} (= 0.632)$ is the constant coefficients of air transparency, and $I_{0}$ is the solar constant.

Diffuse solar radiation is the portion of solar radiation energy that reaches the Earth’s surface after being scattered by the atmosphere. Diffuse radiation on structure surfaces depends on the solar altitude angle and atmospheric transparency coefficient. However, the atmospheric transparency coefficient is not contingent on the azimuth of the surface. The following empirical formula can be used to determine the value of diffuse sky radiation, which is determined as

I_{2} = I_{s} (\frac{1 + \cos α}{2})

(5)

I_{s} = I_{0} (0.2710 - 0.2913 P_{1}^{m}) \sin (h_{s})

(6)

where $I_{s}$ is the scattered radiation intensity of the horizontal plane, $α$ is the angle between the inclined and horizontal planes, and $h_{s}$ is the altitude angle of the Sun. The geometric relationship is presented in Figure 4.

Figure 4.

Relationship of the sun radiation angle.

Heat transfer analysis

The Fourier partial differential equation can be used for the flow of heat. The heat of the pillar will simultaneously increase due to sun radiation and air convection, and this exchange process is shown as

- k \frac{\partial T}{\partial n} | r = h (T_{c} - T)

(7)

h = h_{c} + h_{r}

(8)

where $T_{c}$ is the comprehensive temperature, $h$ is the comprehensive heat convection coefficient $(w / m^{2} ° C)$ , $h_{c} (= 5.58 + 3.9 v$ , with v being the wind speed) is the convective heat transfer coefficient, and $h_{r}$ is the radiant heat transfer coefficient, which can be calculated as

h_{r} = c_{s} ε [{(T^{*} + T_{c})}^{4} - {(T^{*} + T)}^{4}]

(9)

where $c_{s} (= 5.677 \times 10^{- 8})$ is the constant $(W / (m^{2} K^{4}))$ , $ε$ is the radiance ratio, $T^{*} (= 273.15)$ is a constant, and $T$ is the air temperature.

The solutions of equations (7) to (9) need an iterative process. The iteration method is shown in Figure 5.

Figure 5.

Iterative solution process of T_c.

The thermal parameters

The material used for the Qingzhou Navigable Bridge tower is C40 concrete, and the thermal parameters, thermal conductivity, heat capacity, and other parameters, were acquired from the corresponding codes listed in Table 1.

Table 1.

Thermal properties of materials.

Parameter	Value	Unit
k—thermal conductivity	1.74	W/(m °C)
c—heat capacity	920	J/(kg °C)
ρ—density	2500	kg/m³
E—modulus of elasticity	3.45 × 10⁴	MPa
ν—Poisson ratio	0.2	–

Analysis of the temperature field

Direction of the cable tower and radiation angle

The Qingzhou Navigable Bridge is located in the Lingdingyang Bay, which is located at 113.83°, E22.27°N. The angle characteristic of this tower is shown in Figure 6.

Figure 6.

Angle characteristic of the tower.

According to the angle characteristic, the radiation of solar heat from 06:00 to 18:00 can be calculated (Tables 2 and 3). Based on that, the sunshine radiation intensity is presented is Tables 4 and 5.

Table 2.

Azimuth angle of the lower parts of the tower.

Time	Timeangle $(ω)$	Altitude angleof the Sun (h_s)	Angle of the wall of the tower $(θ)$
			1 (North)	2 (East)	3 (South)	4 (West)
			sin θ₁	sin θ₂	sin θ₃	sin θ₄
6:00	−90.00	−0.150	−0.379	0.912	0.362	−0.923
7:00	−75.00	0.070	−0.449	0.889	0.459	−0.883
8:00	−60.00	0.274	−0.514	0.805	0.550	−0.784
9:00	−45.00	0.450	−0.571	0.665	0.628	−0.632
10:00	−30.00	0.585	−0.614	0.480	0.688	−0.437
11:00	−15.00	0.670	−0.641	0.262	0.726	−0.212
12:00	0.00	0.699	−0.650	0.026	0.739	0.026
13:00	15.00	0.670	−0.641	−0.212	0.726	0.262
14:00	30.00	0.585	−0.614	−0.437	0.688	0.480
15:00	45.00	0.450	−0.571	−0.632	0.628	0.665
16:00	60.00	0.274	−0.514	−0.784	0.550	0.805
17:00	75.00	0.070	−0.449	−0.883	0.459	0.889
18:00	90.00	−0.150	−0.379	−0.923	0.362	0.912

Table 3.

Azimuth angle of the middle and upon parts of the tower.

Time	Time angle $(ω)$	Altitude angle of the Sun (h_s)	Angle of the wall of the tower $(θ)$
			1 (North)	2 (East)	3 (South)	4 (West)
			sin θ₁	sin θ₂	sin θ₃	sin θ₄
6:00	−90.00	−0.150	−0.379	0.918	0.352	−0.918
7:00	−75.00	0.070	−0.449	0.887	0.461	−0.887
8:00	−60.00	0.274	−0.514	0.795	0.563	−0.795
9:00	−45.00	0.450	−0.571	0.649	0.651	−0.649
10:00	−30.00	0.585	−0.614	0.459	0.718	−0.459
11:00	−15.00	0.670	−0.641	0.238	0.760	−0.238
12:00	0.00	0.699	−0.650	0.000	0.774	0.000
13:00	15.00	0.670	−0.641	−0.238	0.760	0.238
14:00	30.00	0.585	−0.614	−0.459	0.718	0.459
15:00	45.00	0.450	−0.571	−0.649	0.651	0.649
16:00	60.00	0.274	−0.514	−0.795	0.563	0.795
17:00	75.00	0.070	−0.449	−0.887	0.461	0.887
18:00	90.00	−0.150	−0.379	−0.918	0.352	0.918

Table 4.

Sunshine radiation intensity of the lower parts of the tower (W/m²).

Time	Location of the wall of the tower
	1 (North)		2 (East)		3 (South)		4 (West)
	I ₁	I ₂	I ₁	I ₂	I ₁	I ₂	I ₁	I ₂
6:00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
7:00	0.00	13.91	1.64	13.25	0.85	13.21	0.00	13.25
8:00	0.00	43.88	205.27	41.80	140.33	41.68	0.00	41.80
9:00	0.00	55.23	326.23	52.61	307.94	52.45	0.00	52.61
10:00	0.00	59.72	297.95	56.89	426.65	56.73	0.00	56.89
11:00	0.00	61.50	179.71	58.59	496.95	58.42	0.00	58.59
12:00	0.00	61.98	18.37	59.04	520.28	58.87	18.37	59.04
13:00	0.00	61.50	0.00	58.59	496.95	58.42	179.71	58.59
14:00	0.00	59.72	0.00	56.89	426.65	56.73	297.95	56.89
15:00	0.00	55.23	0.00	52.61	307.94	52.45	326.23	52.61
16:00	0.00	43.88	0.00	41.80	140.33	41.68	205.27	41.80
17:00	0.00	13.91	0.00	13.25	0.85	13.21	1.64	13.25
18:00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

Table 5.

Sunshine radiation intensity of the middle and upon parts of the tower (W/m²).

Time	Location of the wall of the tower
	1 (North)		2 (East)		3 (South)		4 (West)
	I ₁	I ₂	I ₁	I ₂	I ₁	I ₂	I ₁	I ₂
6:00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
7:00	0.00	13.91	1.64	12.78	0.85	13.91	0.00	12.78
8:00	0.00	43.88	202.80	40.30	143.69	43.88	0.00	40.30
9:00	0.00	55.23	318.21	50.72	319.00	55.23	0.00	50.72
10:00	0.00	59.72	284.62	54.85	445.09	59.72	0.00	54.85
11:00	0.00	61.50	162.70	56.48	520.42	61.50	0.00	56.48
12:00	0.00	61.98	0.00	56.92	545.51	61.98	0.00	56.92
13:00	0.00	61.50	0.00	56.48	520.42	61.50	162.70	56.48
14:00	0.00	59.72	0.00	54.85	445.09	59.72	284.62	54.85
15:00	0.00	55.23	0.00	50.72	319.00	55.23	318.21	50.72
16:00	0.00	43.88	0.00	40.30	143.69	43.88	202.80	40.30
17:00	0.00	13.91	0.00	12.78	0.85	13.91	1.64	12.78
18:00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

System temperature analysis

The temperature at each moment needs to be obtained for the temperature field analysis. For this study, the meteorological formula can be used to predict the variation of temperature in the daytime. When the highest and lowest temperatures are obtained, the temperature change can be approximately expressed by a sine function, as follows

\begin{matrix} T = \frac{T_{max} + T_{min}}{2} + \frac{T_{max} - T_{min}}{2} \sin \frac{(h + 30) π}{24}, (0 \leq h < 6) \\ T = \frac{T_{max} + T_{min}}{2} + \frac{T_{max} - T_{min}}{2} \sin \frac{(h - 10) π}{16}, (6 \leq h < 14) \\ T = \frac{T_{max} + T_{min}}{2} + \frac{T_{max} - T_{min}}{2} \sin \frac{(3 h - 22) π}{40}, (14 \leq h \leq 24) \end{matrix}}

(10)

where $T_{max}$ and $T_{min}$ are the daily maximum and minimum temperatures, respectively, and $h$ is the hour.

A comparison between the measured and calculated temperatures is shown in Figure 7, and the maximum error was less than 2.1%. Based on the sunshine radiation intensity, atmospheric temperature, and comprehensive heat transfer coefficient, the temperature of each tower wall is presented in Figures 8 and 9.

Figure 7.

Comparison between the measured and calculated temperatures.

Figure 8.

Comprehensive temperature of the lower parts of the tower wall: (a) north side, (b) east side, (c) south side, and (d) west side.

Figure 9.

Comprehensive temperature of the middle and upon parts of the tower wall: (a) north side, (b) east side, (c) south side, and (d) west side.

Transient finite element numerical analysis of the temperature field

The tower of the Qingzhou navigable cable-stayed bridge was analyzed using the ABAQUS software. ABAQUS is not a professional heat conduction analysis software, but it can efficiently complete the thermal analysis function of civil structures by defining the thermal conduction properties of materials and the boundary of the temperature field. During the transient analysis, the DC3D8 element was adopted. Thermal analysis is achieved by discrete geometry into diffused heat conduction units and the equivalent integrated temperature was considered as a boundary in the finite element analysis (FEA) model, as shown in Figure 10.

Figure 10.

Transient analysis model of the temperature field: (a) numerical analysis model and (b) mesh generation.

The measured and theoretical validations

Comparison of the theoretical and actual temperature fields

In order to validate the accuracy of the temperature field analysis, eight temperature sensors were embedded at 10 m from the pile caps in each tower branch (Figure 11).

Figure 11.

Location of the monitoring section.

The distribution of transient temperature of the cable-stayed bridge tower at different time points is shown in Figure 12. The calculation results show that as time went by, the largest temperature differential between the inner and outer walls occurred at 12:00 noon. On account of the thickness of the tower wall, there was a marked temperature gradient along the thickness direction.

Figure 12.

Temperature distribution of the monitoring section: (a) 8:00, (b) 10:00, (c) 12:00, (d) 14:00, and (e) 16:00.

Comparison of the theoretical and actual temperature deformations

The comparison of the theoretical and actual temperature fields has proven the accuracy of the transient temperature analysis. After that, the deformation of the tower under the transient temperature field was analyzed, with emphasis on the longitudinal and lateral deformation law of the tower. A comparison of the computational and measured lateral deformations is presented in Figure 13, which shows that the rules of the calculated lateral deformation corresponded with that of the practical deformation. The value of the lateral deformation gradually increased after sunrise and reached a maximum of 3.63 cm at 11:00 (Figure 14).

Figure 13.

Comparison between the calculation and measurements: (a) Point No. 4 temperature sensor, (b) Point No. 2 temperature sensor, (c) Point No. 5 temperature sensor, and (d) Point No. 8 temperature sensor.

Figure 14.

Comparison of the computational and measured lateral deformations.

The comparison of the computational and measured longitudinal deformations is presented in Figure 15, which shows that the rules of the calculated longitudinal deformation corresponded with that of the actual deformation. The longitudinal deformation was toward two opposite directions in the morning and afternoon. The deformation reaches 2.00 cm to the west at 10:00 (Figure 16) and 1.97 cm to the east at 15:00 (Figure 17).

Figure 15.

Comparison of the computational and measured longitudinal deformations.

Figure 16.

Longitudinal deformation of the tower at 10:00 (unit: meters).

Figure 17.

Longitudinal deformation of the tower at 15:00 (unit: meters).

Results and discussion

This study investigated the temperature behavior of the largest sea-cross construction project, the Kong–Zhuhai–Macao Bridge, in which a high-resolution FE model was established. The thermal boundary conditions were calculated with consideration of the solar elevation angle, azimuth angle, radiation absorption coefficient, and other parameters. The temperature distribution and the associated responses were compared with the field monitoring counterparts. The following results could be drawn:

The comparison between the transient temperature field simulation and measured data revealed that the difference between the actual temperature and the theoretical value at the measurement points was almost within 2°C, which demonstrates the accuracy of the transient temperature field simulation analysis. The proposed model can be utilized for the thorough understanding of the effect of the solar temperature field on the towers of sea-crossing cable-stayed bridges.

The four side walls of the tower exhibited different temperature field changes during the day. Due to the thermal conductivity, a large temperature gradient will be formed in the direction of column thickness. The temperature on the different surfaces of the bridge tower appears at different times during the day.

The temperature stresses in the test section were mainly compression, and a tensile stress region appeared at the inner side of the tower column. The maximum tensile stress appeared inside the southwest corner.

The theoretical and measured values of temperature deformation of the tower column along the transverse and longitudinal bridges were almost the same. After sunrise, the lateral deformation of the tower gradually increases, and the deformation direction was northward. The direction of longitudinal deformation was westward.

The deformation directions of two tower columns of the H-frame were the same, and the deformation rules were similar. The detailed understanding of the structural deformation of the bridge tower under the temperature field can provide a foundation for the installation of the steel top crossbeam of the bridge.

Footnotes

Acknowledgements

The help of engineers and technicians in the Key Laboratory of Bridge Detection Reinforcement Technology Ministry of Communications, Chang’an University is highly appreciated. These supports are gratefully acknowledged. The valuable comments of the anonymous reviewers on this paper are also acknowledged.

Author’s Note

Yuan Li and Shuanhai He are also affiliated to Key Laboratory of Transport Industry of Bridge Detection Reinforcement Technology, Chang’an University, Xi’an, China.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the financial support provided by the Fundamental Research Funds for the Central Universities of China (Grant No. 300102218214) and Guangdong Provincial Transportation and Transportation Office Technology Funding Project (Grant No. 2016-02-016).

ORCID iD

Yuan Li

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