Temperature behavior of cable-stayed bridges. Part I — Global 3D temperature distribution by integrating heat-transfer analysis and field monitoring data

Abstract

Varying temperatures significantly affect long-span cable-stayed bridges. However, quantitative studies on their temperature behaviors are limited. Existing studies focus on 2D or 3D models of bridge segments only, exclude cables from heat-transfer analysis, and utilize inaccurate environmental conditions. For the first time, this study comprehensively and accurately investigates the global 3D temperature distribution of long-span cable-stayed bridges by integrating the heat-transfer analysis and field monitoring data. A navigation channel bridge of the Hong Kong‒Zhuhai‒Macao Bridge is used as the testbed. A global 3D refined finite element model of the entire bridge is established. The external thermal boundary conditions of the outer surfaces of the structure are carefully determined based on the real-time ambient temperature, wind, and solar radiation, which are tailored for each surface to reflect the influence of the geometric configuration. The internal thermal boundary conditions of the inner surfaces of the box girder and tower are dependent on the measured ambient temperature, considering the vertical temperature difference of the girder and the uniform temperature inside the tower. Then, the numerical heat-transfer analysis and field monitoring data are integrated to calculate the detailed temperature distribution of the entire bridge in different seasons. Results show that ambient temperature, wind, and solar radiation significantly affect the temperature distribution. For the girder, the vertical temperature difference is significant throughout the year, and the transverse temperature difference is nonnegligible in winter and summer, while the longitudinal temperature difference is trivial. The internal temperature of the tower remains stable owing to the insulation of the concrete. The temperatures of the cables vary from each other, which may cause stress redistribution within the structure. The calculated temperatures are in good agreement with their measured counterparts. The temperature results will be used to calculate the thermal-induced responses in the companion paper in a unified manner.

Keywords

Temperature distribution temperature behavior cable-stayed bridge heat-transfer analysis structural health monitoring

Introduction

Exposed to ambient temperature and solar radiation, bridges are subjected to daily and seasonal thermal effects (Dai et al., 2022; Xia et al., 2022; Li et al., 2023). Variations in structural temperature are likely to induce deformations and stresses, especially for long-span cable-stayed bridges. Their large-scale and high-indeterminacy features have led to an even more complex temperature behavior. Furthermore, studies have shown that structural behaviors may be significantly influenced by temperature more than vehicle loads or structural damage (Salawu, 1997; Xu and Xia, 2011; Xu et al., 2022). Therefore, the thermal effect must be separately explored to accurately assess other loading effects and evaluate the structural performance. With the booming construction of long-span cable-stayed bridges in recent years, in-depth investigations of their thermal behavior are increasingly demanded (Ding and Li, 2008; Cao et al., 2011; Guo et al., 2015; Xu, 2023). The thermal behavior of bridges includes the temperature distribution and temperature-induced responses; this paper (Part I) focuses on the former, whereas the companion paper (Part II) discusses the latter.

Studies on the temperature distribution of bridges date back to the 1950s (Naruoka et al., 1957). Zuk (1965) extensively examined field data of composite bridges and determined that air temperature, solar radiation, wind, humidity, and material type are the main influencing environmental factors of temperature distribution. Im and Chang (2004), who studied a composite box girder bridge in Seoul for more than 6 years, found that the transverse temperature difference can reach the same order as the vertical temperature difference and thus should not be ignored in the measurements. Early explorations of the temperature distribution of bridges mainly relied on field measurements (Capps, 1968; Churchward and Sokal, 1981; Ho et al., 1984; McClure et al., 1984; Shushkewich, 1998; Roberts-Wollman et al., 2002; Lucas et al., 2003; Li et al., 2004; Kulprapha and Warnitchai, 2012; Lee, 2012), which are limited by finite sensors and are thus not capable of obtaining the whole temperature field.

Since the 1970s, numerical analysis has been developed to calculate the temperature distribution of bridges based on heat-transfer analysis theory. 1D, 2D, and 3D finite element models (FEMs) can be adopted based on different assumptions. 1D FEMs assume that temperature only varies along a single direction of the cross-section, e.g., the vertical direction of the girder (Emerson, 1973; Cundy et al., 1979). 2D FEMs ignore the longitudinal temperature variation of bridge components. For the girder, both transverse and vertical temperature gradients are considered (Emanuel and Hulsey, 1978; Thurston et al., 1980; Tong et al., 2001, 2002). Xia et al. (2013) were the first to use 3D FEM to investigate the temperature distribution of the Tsing Ma Suspension Bridge by separately conducting heat-transfer analyses on refined models of the girder section and tower section.

Several suspension bridges have been studied via heat-transfer analysis (Westgate et al., 2015; Zhou et al., 2016). For cable-stayed bridges, which account for approximately 60% of long-span bridges with a main span of over 400 m, heat-transfer analyses are relatively limited. Li et al. (2019) used solid elements to simulate the temperature distribution of only a single tower of the Qingzhou Bridge and calculated its 3D temperature distribution. Liu et al. (2014) used plane elements to calculate the temperature distribution of the U-shaped girder section of a cable-stayed bridge. Zhu and Meng (2017) separately conducted heat-transfer analyses on a finely modeled girder and tower of a cable-stayed bridge. Tomé et al. (2018) constructed 2D thermal FEMs of one cross-section of the girder, three of the pylons, and three of the piers of the Central Sub-Viaduct of the Corgo Bridge prior to performing heat-transfer analysis. Wang et al. (2019) conducted a heat-transfer analysis of a girder segment of the Puxi Bridge and summarized its vertical temperature distribution patterns.

Despite significant progress in numerical analysis, existing studies are still constrained to 2D or local 3D levels, separately calculating the temperature distribution of bridge components. However, these simplified approaches cannot describe the temperature distribution of decks and towers with varying cross-sections and deck-tower intersections. In addition, the temperature variation of stay cables has not been examined, because they are generally excluded from the heat-transfer analysis and no thermometers are installed on the cables. Moreover, most heat-transfer analyses assume a constant wind speed and simulate the ambient temperature and solar radiation in a simple manner, causing thermal boundary conditions to deviate from real conditions on site. For example, the top plate of a box girder generally has a much higher temperature than the bottom plate, whereas the internal air temperature of the box is assumed to be uniform. The inaccurate approximations of these environmental factors, which are highly related to the heat dissipation and accumulation process, may cause significant errors in the calculated temperature distributions.

The aim of this study is to investigate the global temperature distribution of long-span cable-stayed bridges. The Qingzhou Bridge, a navigation channel bridge of the Hong Kong‒Zhuhai‒Macao Bridge (HZMB), is taken as a case study. Instead of a local FEM, a 3D global FEM of the entire bridge is established. Then, heat-transfer analysis is conducted to calculate the detailed temperature distribution of the entire bridge by using real-time ambient temperature, wind, and solar radiation data as thermal boundary conditions. This paper is organized as follows. Heat-transfer analysis theory is briefly described in The Heat-transfer analysis theory, followed by the introduction of the Qingzhou Bridge and its structural health monitoring (SHM) system. The development and calibration of a global 3D FEM of the bridge, including a refined cable model, is presented in The Global 3D FEM for heat-transfer analysis. The thermal boundary conditions are carefully determined based on the measured environmental factors. Subsequently, in The Temperature distribution of Qingzhou Bridge, the heat-transfer analysis is conducted for representative periods of four seasons. Then, the calculated temperature results of the whole bridge are analyzed and compared with the measured counterparts. Finally, conclusions are drawn in The Final. The calculated temperature will be transferred to the thermal load for the subsequent structural analysis in the companion paper.

Heat-transfer analysis theory

In general, for a point with Cartesian coordinates of (x, y, z) at time instant t in a structure, its heat flow is governed by Fourier differential equations (Xu and Xia, 2011).

ρ c \frac{\partial T}{\partial t} k (\frac{\partial^{2} T}{{\partial x}^{2}} + \frac{\partial^{2} T}{{\partial y}^{2}} + \frac{\partial^{2} T}{{\partial z}^{2}})

(1)

where ρ is the density, c is the specific heat capacity of the material, and k is the isotropic thermal conductivity coefficient. Assuming that heat transfer is negligible in some directions, the above equation can be simplified for 2D or 1D analysis. Initial conditions and boundary conditions need to be specified to solve this partial differential equation.

The boundary conditions of the heat-transfer analysis can be categorized into three types of conditions (Xu and Xia, 2011): (1) the structural temperature at the boundary is known; (2) the heat flow at the boundary is known; and (3) the heat flow at the boundary is proportional to the temperature difference between the ambient temperature and the structural temperature. For the heat-transfer analysis of bridges, the heat flow q on the bridge surface consists of three components: q_c due to convection, q_r due to thermal irradiation, and q_s due to solar radiation. q_c and q_r are proportional to the difference between the ambient temperature and structural temperature, while q_s can be calculated based on solar radiation data. Therefore, at the boundaries, the second and third types of boundary conditions can be combined as follows:

k \frac{\partial T_{s}}{\partial n} = h (T_{a} - T_{s}) + q_{s}

(2)

h = h_{c} + h_{r}

(3)

where n is normal to the structural surface, T_a is the ambient temperature, T_s is the structural surface temperature, and h is the overall heat-transfer coefficient by combining the convection heat-transfer coefficient h_c and the irradiation heat-transfer coefficient h_r. Both h_c and h_r can be calculated according to the following empirical formulae (Froli and Hariga, 1993)

h_{c} = {\begin{array}{c} 4 v + 5.6 \\ {7.15 v}^{0.78} \end{array} \begin{array}{c} (v < 5 m / s) \\ (v \geq 5 m / s) \end{array}

(4)

h_{r} = ε [4.8 + 0.075 (T_{a} - 5)]

(5)

where h_c is related to wind speed, and h_r is the function of the emissivity coefficient ε of the surface material and ambient temperature. Finally, q_s can be determined by

q_{s} = α I

(6)

where α is the absorptivity coefficient of the surface material, and I is the total radiation received by the surface, including direct, diffuse, and reflected solar radiation.

Equation (2) can be written in a simpler form as follows:

k \frac{{\partial T}_{s}}{\partial n} = h (T_{e q} - T_{s})

(7)

T_{e q} = T_{a} + \frac{α I}{h}

(8)

where T_eq incorporates ambient temperature and solar radiation and is termed the equivalent ambient temperature.

Qingzhou Bridge and its monitoring system

Qingzhou Bridge

HZMB is known as the world’s longest sea-crossing bridge; it connects three regions with a total length of 55 km. The Qingzhou Bridge, one of the three main navigation channel bridges of HZMB, located at 22.28° N and 113.73° E, is an east‒west cable-stayed bridge with a total length of 1150 m and a main span of 458 m, as shown in Figure 1. The steel box girder, covered by 70 mm-thick asphalt, carries a dual three-lane highway and is continuously supported by two H-shaped towers, two auxiliary piers, and two transitional piers. As high as 163 m, the H-shaped tower is composed of two concrete box legs, a concrete lower transom, and a steel upper transom that resembles the shape of a Chinese knot.

Figure 1.

Configuration of the Qingzhou bridge.

SHM system of Qingzhou Bridge

An SHM system was designed for HZMB and has been operating since 2018 when the bridge was first opened to the public. The SHM system consists of 262 sensors, including hygro-thermometers, thermometers, anemometers, GPS rovers, accelerometers, liquid leveling systems, displacement transducers, cable tensiometers, strain gauges, corrosion sensors, and reaction dynamometers, to monitor the environmental factors, external loads, and structural responses of the bridge. In the heat-transfer analysis, thermometers for measuring ambient and structural temperatures and anemometers for recording wind data are taken into account. The 3D layout of the sensors is shown in Figure 2.

Figure 2.

3D layout of the sensors of the Qingzhou Bridge.

The locations of the sections with thermometers are shown in Figure 3. The internal ambient temperatures of the girder and tower are measured by hygro-thermometers, which are located inside the box girder (G04, G09, and G14) and inside the tower legs (T02 and T06). Each section has two hygro-thermometers, one on the south side and the other on the north side. The structural temperatures are recorded by fiber optic thermometers, which are located on the girder sections of the mid-span (G09), on the auxiliary piers (G03 and G15), on the towers (G05 and G13), and on the lower columns of the towers (T03 and T07). Sections G05 and G13 are supported on the towers and thus have no tuyere due to the spatial limitation of the tower column. The sensor layouts of the thermometers used to measure the structural temperature on girder section G09 (with tuyere) and tower sections (T03 and T07) are shown in Figures 4 and 5, respectively.

Figure 3.

Location of sections with thermometers.

Figure 4.

Layout of thermometers on girder section G09 (with tuyere).

Figure 5.

Layout of thermometers on tower sections (T03 and T07).

Two propeller anemometers measure the wind velocity and wind direction on top of the towers, and two ultrasonic anemometers are installed at the mid-span of the girder to measure the wind velocity in three orthogonal directions.

A weather station installed at the mid-span of the girder records the ambient temperature of the external air, wind conditions, air pressure, humidity, and rainfall by integrating certain types of sensors. Among them, the ambient temperature of the external air is taken into account. As the SHM system of the bridge does not have a pyranometer, the solar radiation data collected at the Hong Kong Observatory located approximately 45 km away from the bridge are adopted for this study.

Global 3D FEM for heat-transfer analysis

General information

Previous studies on the temperature distribution of long-span bridges were limited to simplified 2D or 3D bridge segments. In the present study, the thermal distribution of the whole bridge is comprehensively and accurately simulated by establishing the global 3D FEM of the Qingzhou Bridge using the general FE software package ANSYS (ANSYS 16.1). The refined model consists of the following parts: 493,941 nodes and 520,422 elements, including 170,336 solid elements for the steel box girder, the asphalt concrete layer, tower legs, lower tower transoms, and piers; 349,962 shell elements for the diaphragms, U-ribs, and upper tower transoms; and 124 link elements for the cables and bearings. The FEM of the bridge and close-range illustrations of some components are shown in Figure 6.

Figure 6.

Global 3D FEM of the Qingzhou bridge.

The webs, diaphragms, and U-ribs of the girder simulated with shell elements contain 305,455 nodes and 344,650 elements, which constitute most of the nodes and elements of the model. The upper transom of the tower, which is in the shape of a Chinese knot and composed of steel, is also carefully modeled with shell elements. The bearings are modeled with link elements which connect the central nodes of the stiffened surfaces of the girder and the pier or the tower. For the connection between the tower and stay cables, the nodes of the concrete pylon, the steel anchor box, and the cables are joined together. Close views of these parts are also shown in Figure 6.

Previous studies of the temperature behaviors of cable-stayed bridges generally excluded cables from the heat-transfer analysis. In this study, a refined FEM of four cables is established with solid elements for simulating the steel wire inner core and the polyethylene (PE) sheath to calculate the temperature distribution of the cables. The shortest and longest side cables (S1 and S14) and middle cables (M1 and M14) anchored on the south leg of the tower on the Zhuhai side are studied for simplicity (Figure 7). The temperature of other cables is interpolated from the temperature of these four cables.

Figure 7.

Refined FEM of stay cables. (a) Elevation, (b) Close-range view.

SOLID70, SHELL57, and LINK33 are assigned to the solid, shell, and link elements, respectively, for the heat-transfer analysis. The FEM contains a total of 493,941 nodes, each of which has one temperature degree of freedom.

The material parameters used in the heat-transfer analysis are listed in Table 1.

Table 1.

Material parameters for heat-transfer analysis.

Parameter	Notation	Asphalt	Steel	Concrete	PE
Density	ρ (kg/m³)	2,365	7,850	2,600	965
Heat capacity	c (J/kg/°C)	1075	460	925	2300
Thermal conductivity	k (W/m/°C)	1.8	55	2.71	0.44
Emissivity coefficient	ε	0.92	0.8	0.88	0.90
Absorptivity coefficient	α	0.90	0.685	0.65	0.65

Convergence analysis

The timestep of the analysis and mesh size of the FEM are likely to affect the computational efficiency and accuracy (Beckmann et al., 2013). Thus, convergence analysis is conducted on a 3D segmental model, as shown in Figure 8.

Figure 8.

A 3D segmental model.

The period of July 5–7, 2019 (3 days) is selected for the convergence analysis. First, the heat-transfer analysis is conducted for one step per hour, totaling 72 load steps. The mesh division is one layer in the thickness direction of the steel girder and the asphalt, and the default mesh size is 2.5 m in the longitudinal direction. The total computational time is the recorded wall clock time on a computer with an i9-10900K CPU, a basic frequency of 3.70 GHz, and RAM of 32 GB. The root mean square error (RMSE) of the temperature result is calculated by comparing with the result of using a timestep of 1 minute and mesh size denser by six times in both the thickness and longitudinal directions, which is assumed to be accurate. The calculation accuracy is measured based on the average RMSE of all effective sensor data with areas weighted on the mid-span section of the girder during the 3 days.

Four different timesteps (60, 30, 20, and 10 min) are selected. The total computational times and the RMSEs of the temperature results of the mid-span girder section are calculated and compared in Figure 9. As expected, a longer timestep consumes less computational time but also entails a larger error. When the timestep is 30 min, the average RMSE is nearly reduced by half, with the total calculation time increasing by only approximately 5 s compared with the timestep of 60 min. Thus, the timestep of 30 min is used hereafter.

Figure 9.

Timestep convergence analysis. (a) Computational time for different timesteps, (b) Calculation accuracy for different timesteps.

Four different mesh sizes are investigated, corresponding to the deck with one to four layers of meshes in the thickness direction with longitudinal sizes of 2.5, 1.25, 0.833, and 0.625 m, with 5,666, 26,726, 74,615, and 159,820 elements in total, respectively. Figure 10 shows the total computational times and the RMSEs of the four mesh schemes. In general, more elements take longer computational time and lead to more accurate results. However, as the mesh size is refined two times, although the RMSE is reduced by over one-third, the time cost increases as much as fivefold. Therefore, the original mesh size is adopted in the subsequent analysis.

Figure 10.

Mesh convergence analysis. (a) Calculation time for different numbers of elements, (b) Calculation accuracy for different numbers of elements.

Environmental conditions

The measured environmental data, including ambient temperature, wind, and solar radiation data, provide boundary conditions for the heat-transfer analysis. The first weeks of January, April, July, and October in 2019 are selected for studying the temperature distribution in different seasons. The specific periods are January 1–6, April 1–7, July 1–7, and October 1–7, 2019. Summer is selected as the representative period because it is the hottest time of the year with the highest solar radiation level, and it entails extreme meteorological conditions. The hourly measured wind and ambient temperature data obtained from the SHM system and the solar radiation data obtained from the Hong Kong Observatory on July 1–7, 2019 are shown in Figure 11.

Figure 11.

Environmental conditions on July 1–7, 2019. (a) Wind, (b) Solar radiation, (c) Ambient temperature.

On July 1–7, 2019, the wind speed varied greatly and was mainly from the south and east. The solar radiation conditions of consecutive days were not identical. The internal ambient temperature of the girder was higher than that of the external air and had a similar variation pattern as the solar radiation, whereas the internal temperature of the tower changed only slightly owing to the good insulation of the concrete tower wall.

As July 3, 2019 was a rainy day and the rain effect was not considered in the heat-transfer analysis, July 1–4 was then taken as the preanalysis and the results on July 5–7 were examined. A preanalysis is adopted because the initial thermal condition of the model is unknown. After several days of heat-transfer analysis with the measured environmental conditions, the effect of the incorrect initial condition on the temperature distribution is determined to be negligible (Zhou et al., 2016).

Thermal boundary conditions

As the girder and tower are box-sectioned, both external and internal thermal boundary conditions need to be determined.

External thermal boundary conditions

For the external thermal boundaries, all outer surfaces of the main components of the whole bridge are categorized into four subsets according to their geometric configurations and materials, as listed in Table 2. For instance, the outer surfaces of the box girder include the upper surfaces of the asphalt and the side and bottom surfaces of the steel girder. The configuration is simplified into 7×2 surfaces, which are doubled in consideration of the longitudinal slope. The seven surfaces are shown in Figure 12. The eighth and ninth surfaces constitute the outer surfaces of the girder segments without tuyere.

Table 2.

Outer surfaces for external thermal boundary conditions.

Component	Material	Number of surfaces
Box girder	Asphalt concrete, steel	14
Pylon and lower transom of tower	Concrete	113
Upper transom of tower	Steel	154
Cable	PE	96

Figure 12.

Outer surfaces of the main girder.

The ambient temperature of all outer surfaces is taken as identical to that of the external air, whereas the wind speed and solar radiation imposed on each surface are taken differently.

Wind speed is the dominant factor of the convection heat-transfer coefficient and is thus expected to significantly affect the heat-transfer results. Previous studies generally used constant wind speed in their simulations, but real structural surfaces are affected by varying and different wind conditions due to diverse wind incident angles. Furthermore, the actual wind speed near the structural surface is lower than the measurement of the anemometer, which is generally located several meters away from the structure. Here, empirical equations are introduced to calculate the effective wind speed on different structural surfaces with respect to the wind incident angle (Zhou et al., 2016).

v = {\begin{array}{c} {0.8 v}_{0} \\ {0.7 v}_{0} \\ {0.6 v}_{0} \end{array} \begin{array}{c} φ \leq 45 ° \\ 45 ° < φ \leq 90 ° \\ φ > 90 ° \end{array}

(9)

where v is the effective wind speed, v₀ represents the incident wind speed, and φ refers to the wind incident angle calculated with respect to the wind direction and the normal of the outer surface.

The solar radiation data obtained from the Hong Kong Observatory are the sum of the direct and diffuse solar radiation on a horizontal surface. They need to be decomposed and transformed to the direct, diffuse, and reflected solar radiation on tilted surfaces of the structure. The decomposition is conducted according to the diffuse ratio as follows (Yik et al., 1995):

\frac{I_{i h}}{I_{t h}} = {\begin{cases} {1 - 0.435 k}_{T} \\ {1.41 - 1.695 k}_{T} \\ 0.259 \end{cases} \begin{array}{l} 0 \leq k_{T} < 0.325 \\ 0.325 \leq k_{T} \leq 0.679 \\ 0.679 < k_{T} \end{array}

(10)

I_{d h} = I_{t h} - I_{i h}

(11)

where I_ih, I_th, and I_dh refer to the diffuse, total, and direct solar radiation on a horizontal surface, respectively. k_T is the clearance index that can be calculated as follows:

k_{T} = \frac{I_{t h}}{I_{e x}}

(12)

I_{e x} = I_{S C} \sin ψ

(13)

I_{S C} = 1367 \times (1 + 0.033 \cos \frac{2 π N}{365})

(14)

where I_ex is the extraterrestrial radiation,

I_{S C}

is the solar constant, ψ denotes the solar altitude angle, and N is the number of the day of the year, counted starting January 1.

The direct, diffuse, and reflected solar radiation on tilted surfaces of the structure can be obtained as follows (Yang and Lu, 2007):

I_{d t} = I_{d h} R_{b}

(15)

I_{i t} = 0.5 I_{i h} (1 + \cos β) (1 - A_{I}) (1 + f \sin^{3} (\frac{β}{2})) + I_{i h} A_{I} R_{b}

(16)

I_{r t} = r_{e} (I_{d h} + I_{i h}) \frac{1 - \cos β}{2}

(17)

R_{b} \frac{\cos θ}{{\cos θ}_{z}}

(18)

A_{I} = \frac{I_{d h}}{I_{e x n}} = \frac{\frac{I_{d h}}{{\cos θ}_{z}}}{\frac{I_{e x}}{{\cos θ}_{z}}} = \frac{I_{d h}}{I_{e x}}

(19)

f = \sqrt{\frac{I_{d h}}{I_{t h}}}

(20)

where I_dt, I_it, and I_rt refer to the direct, diffuse, and reflected solar radiation on a tilted surface with angle β, respectively; I_dn and I_exn are the direct and extraterrestrial radiation on a surface always perpendicular to the solar rays, respectively; R_b is the geometric factor related to the solar incident angle θ and the solar zenith angle θ_z; A_I and f are intermediate parameters; and r_e represents the reflected coefficient of the tilted surface.

The external thermal boundaries on all outer surfaces can be determined and expressed as the overall heat-transfer coefficients h and the equivalent ambient temperatures T_eq. The external thermal boundaries of the asphalt deck surface and the south-side surface, the north-side surface, and the bottom surface of the steel girder, corresponding to surfaces 2, 5, 8, and 9 in Figure 12, are illustrated in Figure 13.

Figure 13.

Thermal boundary conditions of the four outer surfaces of the girder. (a) Overall heat-transfer coefficient, (b) Equivalent ambient temperature.

As shown in Figure 13(a), the overall heat-transfer coefficients of the four surfaces differ from each other due to different wind incident angles. The overall coefficient of the south-side surface is generally the highest, whereas that of the north-side surface is the lowest because the wind mainly comes from the south and the east, as depicted in Figure 11. Meanwhile, as shown in Figure 13(b), the deck surface has the highest equivalent ambient temperature, whereas the bottom surface has the lowest. This finding can be explained by the deck surface being subject to the most direct solar radiation whereas the bottom surface receiving only a slight reflected solar radiation. The equivalent ambient temperature of the north-side surface is higher than that of the south-side surface and exhibits a sudden peak at sunset. This difference can be explained by the Qingzhou Bridge being located slightly lower than the Tropic of Cancer, and thus, the northern surface receives more solar radiation than the southern surface during the period, particularly at sunset when the solar incident angle reaches the maximum.

Obviously, ambient temperature, wind, and solar radiation are highly related to the heat dissipation and accumulation process and thus play significant roles in the heat-transfer analysis of long-span cable-stayed bridges. Therefore, in situ environmental condition measurements are necessary to accurately calculate the temperature distribution of bridges.

Internal thermal boundary conditions

The internal thermal boundary conditions of the box girder and the tower also need to be determined. As the wind speed and solar radiation inside the box section are zero, the equivalent ambient temperature equals the ambient temperature inside the box section.

The temperature of the inner surface of the tower is approximately evenly distributed owing to the good insulation effect of the thick concrete tower wall. Therefore, the monitored structural temperature is applied to the inner surfaces of the tower as the equivalent ambient temperature.

For the inner surface of the girder, the vertical temperature gradient is generally significant. However, almost all SHM systems (including the one presented in this study) have a single thermometer (even none in others) inside the box to measure the air temperature. Moreover, previous numerical studies did not consider the air temperature difference between the top plate and bottom plate. In this study, the inner surface of the girder is divided into four components, namely, the bottom surface of the top plate, the surface of the U-ribs on the top plate, the surface of the U-ribs on the bottom plate, and the top surface of the bottom plate, abbreviated as TP (top plate), TU (top U-rib), BU (bottom U-rib), and BP (bottom plate), respectively. Different thermal boundaries are assigned to these surfaces, as discussed below.

As the internal periodic temperature variation of the girder is similar to a sine function, five-piecewise sine functions are constructed to simulate the daily air temperature variations. Figure 14 shows a sketch of a constructed equivalent ambient temperature in 1 day for a single inner surface. The magnitude difference and the phase difference between the equivalent ambient temperature of the four surfaces and the measured internal ambient temperature are jointly considered. Four amplification coefficients and four phase indices for indicating two extreme points and two inflection points are determined iteratively according to the monitored structural surface temperature. For the above four surfaces, a total of 16 amplification coefficients and 16 phase indices needs to be determined.

Figure 14.

Constructed equivalent ambient temperature in 1 day.

A temperature difference also exists in the transverse direction, but it is not obvious in the longitudinal direction inside the girder box. Therefore, the internal thermal boundaries on the south- and north-side surfaces of the box girder are based on the longitudinally averaged ambient temperature on the south and north sides, respectively.

The overall heat-transfer coefficients and the equivalent ambient temperatures of the four inner surfaces of the girder (i.e., TP, TU, BU, and BP) on the south and north sides are shown in Figure 15.

Figure 15.

Thermal boundary conditions of the four components of the inner surfaces of the girder. (a) Overall heat-transfer coefficient, (b) Equivalent ambient temperature.

The vertical temperature gradient is reflected by a descending equivalent ambient temperature of TP, TU, BU, and BP. As the overall heat-transfer coefficient is positively related to the ambient temperature according to equation (5), the patterns of the corresponding curves are also similar. Both the overall heat-transfer coefficient and the equivalent ambient temperature on the north side exceed those on the south side because the heat accumulation on the former, which is caused by solar radiation, is slightly greater than that on the south side during the calculation period. This aspect is also explained in the last section.

On the basis of the monitored ambient temperature, wind, and solar radiation data, the thermal boundary conditions, which are characterized by the overall heat-transfer coefficients and the equivalent ambient temperatures of all outer and inner surfaces, are calculated and input into the FEM in ANSYS. Then, transient heat-transfer analysis is conducted, and the temperature distribution of the entire bridge is obtained. The findings are discussed in the subsequent section.

Temperature distribution of Qingzhou Bridge

On the basis of the meteorological conditions, January 4–6, April 5–7, July 5–7, 2019, and October 2–3, 2019 are selected as the representative periods of the four seasons. Here, the findings for July 5–7, 2019 are selected for a thorough discussion.

Temperature distribution of the main girder

Section G09 at the middle of the main span, regarded as a typical section of the girder, is examined. The locations of the sections with thermometers and the sensor layout are shown in Figures 3 and 4, respectively.

Thermometers are used to measure the temperatures of TP (top plate), TU (top U-rib), BU (bottom U-rib), and BP (bottom plate). The initial data processing and data integrity checking show that data from point 22 exhibit abnormality and are thus discarded. Then, the characteristics of the temperature distribution are explored by selecting the following seven points as the representative measurement points of the section: points 8 and 18 for TP on the south and north sides, respectively; points 7 and 17 for TU on the south and north sides, respectively; points 3 and 13 for BP on the south and north sides, respectively; and point 21 for BU.

The temperatures of the representative points on July 5–7, 2019 are compared with the corresponding measurements, as presented in Figure 16. The curves with a legend ending with the letter M denote the measured temperatures, while the curves with a legend ending with C are the calculated ones from the FEM.

Figure 16.

Measured and calculated temperatures of G09 on July 5–7, 2019. (a) TP, (b) TU, (c) BP, (d) BU.

The following results can be derived from Figure 16:

(1) The simulated temperatures are in good agreement with the monitored counterparts. The average RMSE is approximately 1.1°C for the whole girder section, with all effective sensor data weighted with the areas around the sensors. Thus, the effectiveness of the model and the associated heat-transfer analysis is verified.

(2) The temperature of TP in summer can reach as high as 53°C with a variation of approximately 23°C in a day. Meanwhile, the maximum temperature of BP is approximately 32°C with a variation of 3°C in a day. TP has a much higher temperature and more diurnal variation than BP due to the solar radiation contribution.

(3) On the upper plate, the temperature of TU is lower than that of TP with a lag effect. The vertical temperature difference between TP and TU can reach approximately 9°C. On the lower plate, the temperature of BU is higher than that of BP with a lag effect, and the maximum vertical temperature difference between them is approximately 4°C. A comparison of TP and BP indicates that the vertical temperature difference can reach as high as 22°C.

(4) The temperature on the north side is higher than that on the south side for two reasons: (i) the subsolar point is on the north side of the bridge during the calculation period and (ii) the wind is mainly blowing from the south, leading to a larger heat dissipation in the south. The measured transverse temperature difference is 4°C for the representative points on the TP.

Regarding the other three seasons, the temperature results for the representative days in January, April, and October are shown in Figure 17, Figure 18, and Figure 19, respectively. The dates with unfavorable weather and the dates used for the preanalysis are not shown in the figures.

Figure 17.

Measured and calculated temperatures of G09 on January 4–6, 2019. (a) TP, (b) TU, (c) BP, (d) BU.

Figure 18.

Measured and calculated temperatures of G09 on April 5–7, 2019. (a) TP, (b) TU, (c) BP, (d) BU.

Figure 19.

Measured and calculated temperatures of G09 on October 2–3, 2019. (a) TP, (b) TU, (c) BP, (d) BU.

Figures 17, 18 and Figure 19 show the high absolute temperatures and daily temperature variations for most of the year, with significant vertical temperature differences. Table 3 summarizes the characteristic values of the measured temperature of the main girder, including the maximum temperatures T_max of TP and BP, the maximum daily temperature variations ΔT_max of TP and BP, and the maximum temperature differences between TP and BP (i.e., T_{diff_max}) for the different seasons in 2019.

Table 3.

Measured temperature of the main girder in different seasons in 2019 (unit: °C).

Period	TP		BP		T _{diff_max}
Period	T _max	ΔT_max	T _max	ΔT_max	T _{diff_max}
January 4–6	27	9	19	3	10
April 5–7	47	22	27	5	20
July 5–7	53	23	32	3	22
October 2–3	53	24	33	4	21

For TP, the maximum temperature is approximately 50°C, with a daily variation larger than 20°C except in winter when the corresponding values are 27°C and 9°C. For BP, the maximum temperature is approximately 30°C in the last three seasons, which is 10°C higher than that in winter, with a daily variation of approximately 4°C for the whole year. The vertical temperature differences between TP and BP for most of the year are higher than 20°C except in winter (lower by 10°C). Apart from the diurnal variation, TP also experiences a higher seasonal temperature variation than BP. However, the maximum temperatures and daily variations do not necessarily occur in summer because the wind speed on July 5–7, 2019 was relatively high, indicating rapid heat dissipation.

The simulated and measured temperatures are consistent on all days. The discrepancies over some periods are due to the solar radiation data obtained from the Hong Kong Observatory, which may differ from the in situ conditions. The RMSEs of the temperature results of G09 in the four seasons are summarized in Table 4. The average RMSEs of the effective temperature do not exceed 1.4°C. TP has the largest difference because the top plate receives more heat flow from solar radiation. BU has the smallest difference, as its temperature is mainly controlled by the internal air temperature.

Table 4.

RMSEs of the simulated temperature of G09 in 2019 (unit: °C).

Period	TP	TU	BU	BP	Average
January 4–6	1.5	0.8	0.2	0.6	1.0
April 5–7	2.0	1.0	0.4	1.2	1.4
July 5–7	1.8	1.4	0.2	0.5	1.1
October 2–3	2.1	1.1	0.6	1.0	1.4

As mentioned above, the vertical temperature difference is significant, and the transverse temperature difference is nonnegligible. Figure 20 shows the maximum temperature of each sensor on the selected days as a means of illustrating the transverse temperature profiles of TP, TU, and BP in different seasons. The transverse temperature profile of BU is unavailable because it has only one sensor point on each section.

Figure 20.

Transverse temperature profiles of TP, TU, and BP. (a) Winter, (b) Spring, (c) Summer, (d) Autumn.

The vertical temperature difference is significant for most of the year, and the difference between TP and TU constitutes over one-third of that between TP and BP, indicating a rapid and nonlinear temperature decrease with distance from the top plate. The transverse temperature profiles of TP and TU are similar throughout the year because of the high heat conductivity of steel. In winter, the temperatures of TP and TU on the south side are higher than those on the north side, with a maximum difference of approximately 3°C. The opposite is true in summer, with a maximum difference of approximately 7°C, while the difference is not obvious in spring and autumn. This phenomenon can be attributed to the Qingzhou Bridge being located slightly lower than the Tropic of Cancer. The relative position change in the subsolar point to the bridge site in different seasons causes varied transverse heat accumulation. The transverse temperature difference of BP is trivial because the bottom part has a small difference in solar radiation and wind on the two sides. In addition, the transverse temperature profiles in the three girder sections of G03, G09, and G15 almost coincide with each other, indicating a negligible temperature difference in the longitudinal direction.

Temperature distribution of the main tower

Tower section T07 is selected as the representative section to investigate the temperature distribution of the tower. The locations of the sections with thermometers and the sensor layout are shown in Figures 3 and 5, respectively.

The temperatures of points 1 and 6 for the four seasons are compared with the corresponding measurements, as shown in Figure 21. The monitored and calculated internal temperatures of the tower both remain almost unchanged owing to the good insulation effect of the concrete. A good agreement exists between the simulation and monitoring. The average RMSEs of all measuring points in January, April, July, and October are 0.8°C, 0.9°C, 0.9°C, and 1.7°C, respectively. The large RMSE in October is due to the significant difference of point 6. A further investigation shows that the significant difference in point 6 continued after Oct. 2019. This may be due to the sensor drift after that period. As thermometers are only installed inside the lower column of the concrete tower wall, the calculated temperature results of the upper part and the outer surfaces of the tower cannot be verified and thus are not shown here.

Figure 21.

Measured and calculated temperatures of the tower. (a) Winter, (b) Spring, (c) Summer, (d) Autumn.

Temperature distribution of cables

In this study, the heat-transfer analysis is first conducted on cables to obtain their real-time temperature variation using the refined model. As thermometers are not installed on the cable, only the calculated results are presented in this paper. The temperature of the cable is uniformly distributed along the whole steel inner core owing to the high heat conductivity of steel. Then, the calculated temperatures of the center points of the four simulated cables are extracted, as shown in Figure 22. The S1C and M1C curves refer to the calculated temperatures of the cables S1 and M1, which are the two shortest cables on the side span and the middle span, respectively, while S14 C and M14 C are those of the two longest cables (S14 and M14). The locations of the cables are shown in Figure 7(a).

Figure 22.

Calculated temperatures of the center points of four cables. (a) Winter, (b) Spring, (c) Summer, (d) Autumn.

As presented in Figure 22, the temperature of the cable can reach approximately 42°C in autumn, and the maximum variation in a day is 21°C in spring. By contrast, the temperature of the cable in summer is relatively low, which is mainly due to the high wind speed and thus the fast heat dissipation during the calculation period. Furthermore, the temperature curves of the cables have spikes and are less smooth than those inside the girder. These features differ from those of the tower and girder because the cables’ diameter is only approximately 20 cm and thus are more sensitive to external conditions. Moreover, the two longest cables generally have a single peak temperature at noon, whereas the two shortest cables tend to have one more peak at sunset when they are subject to a large incident angle and receive significant solar radiation. Different cables with varying patterns may cause stress redistribution within the structure.

Conclusions and discussions

An efficient and accurate temperature distribution analysis for long-span cable-stayed bridges is developed by integrating numerical simulation and field monitoring data. The Qingzhou Bridge of HZMB is taken as a case study. A global 3D FEM of the bridge is established and calibrated with different timesteps and meshes. Then, the external and internal thermal boundary conditions are carefully determined based on the ambient temperature, wind, and solar radiation data. Finally, the temperature distribution of the whole bridge is calculated and compared with the monitored counterparts. The following conclusions can be drawn from the findings:

(1) The simulated temperatures are in good agreement with the monitored counterparts for the four seasons. The average differences are within 1.4°C for the girder and 1.7°C for the tower, which verifies the effectiveness of the proposed approach to simulating the temperature distribution throughout the year. The discrepancies in the numerical results may be attributed to the solar radiation data obtained from the Hong Kong Observatory that differ from the in-situ conditions.

(2) For the box girder, the temperature of the top plate can reach as high as 53°C with a variation of approximately 24°C in a day, while the highest temperature of the bottom plate is approximately 33°C with a variation reaching 5°C in a day. The vertical temperature difference of the girder is significant for most of the year, with the maximum between the top and bottom plate of approximately 22°C in summer and 10°C in winter. The difference between the top plate and the top U-rib constitutes over one-third of the total vertical difference, indicating a rapid and nonlinear temperature decrease with distance from the top plate. The various heat sources of different components and the insulation effect of the box girder contribute to the temperature difference and the lag effect.

(3) The transverse temperature difference of the girder is nonnegligible for the top plate of the girder, while it is trivial for the bottom plate. The temperature of the north side is lower than that of the south side with a maximum difference of approximately 3°C in winter, and it is the opposite in summer with a maximum difference of approximately 7°C, which can be attributed to the Qingzhou Bridge being located slightly lower than the Tropic of Cancer. By contrast, the longitudinal temperature difference is trivial. The transverse temperature difference is geographically dependent. The pattern is expected to differ for bridges in other geographical locations.

(4) The internal temperature of the tower remains almost unchanged owing to the good insulation effect of the concrete.

(5) The temperature variations of cables are not identical to each other because of different inclination angles. Moreover, the cables have a relatively small diameter and thus are more sensitive to external conditions.

In the thermal simulation, determining the boundary conditions is the most critical step, as it quantifies the effect of the environment on the temperature field of the structure. Among all environmental factors, wind, ambient temperature, and solar radiation are the most influential, with wind influencing heat dissipation and ambient temperature and solar radiation influencing heat accumulation. In situ real-time measurement of those environmental factors is required to accurately simulate the temperature field. The simulation accuracy can be improved by taking into account the following measurement parameters for the SHM system:

(1) Accurate in situ solar radiation data are critical, and thus, pyranometers should be included in the SHM system of bridges.

(2) The internal air temperature of the box girder is nonuniform. More thermometers should be installed at different locations along the vertical and transverse directions.

(3) More thermometers need to be installed on the outer surface and the upper part of the tower to acquire detailed information about its temperature distribution.

(4) Stay cables generally do not have thermometers. Typical cables, for example, the longest and shortest cables, are better equipped with thermometers inside the cable.

The obtained temperature in this study will be input into the same global FEM for calculating the temperature-induced responses. The unified approach will be highlighted in the companion paper.

Footnotes

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: This research was supported by the National Key R&D Program (Project No. 2019YFB1600700), the Key-Area Research and Development Program of Guangdong Province (Project No. 2019B111106001), and RGC-GRF (Project No. 15206821).

Correction (December 2023):

Article updated to correct the order of the funding information in the Funding Section since its original publication.

ORCID iDs

Yushi Shan

Lingfang Li

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