Analysis of thermal behavior on concrete box-girder arch bridges under convection and solar radiation

Abstract

The two-dimensional finite element method can be used to calculate the thermal field distribution of prestressed concrete bridge girders. However, this method is not appropriate for the concrete box-girder arch bridges, because they have different azimuth and dip angles at the top and bottom flanges along arch axis. Thus, an experimental and analytical study was conducted on a concrete box-girder arch bridge located in Guizhou, China, to investigate the thermal behavior under convection and the solar radiations. To determine the vertical temperature gradients, a two-dimensional plane finite element model was used to calculate the thermal field based on meteorological parameter methods. In addition, the three-dimensional beam finite element model was proposed to study the thermal stress and displacement of arch bridges using the vertical temperature gradients. Finally, the thermal behavior of the concrete box-girder arch bridges determined by the two-dimensional plane and three-dimensional beam finite element model was verified by three-dimensional solid finite element model.

Keywords

box-girder concrete arch bridges convection finite element model solar radiation stress thermal behavior

Introduction

Concrete box-girder arch bridges (CBGAB) are exposed to environmental conditions such as solar radiation, air temperature, and wind. As they are hyperstatic structures, the temperature-related problems such as temperature self-equilibrated longitudinal and transverse stresses, thermal inferior stress, and excessive displacements can be seriously affected by nonlinear temperature gradients along the depth of the box-girder (Hendry, 1976; Zhou et al., 2015). The thermal load can equal to the dead load in some cases (Inaudi et al., 2002). With the rapid development of high speed and heavy load in Chinese railways, it is rather urgent to build long span concrete box-girder railway arch bridges in mountainous area of southwestern China. However, there are few studies that pay attention to the thermal load on the CBGAB, although the thermal load can affect the bridge safety during construction and service stage.

The thermal effects on the bridge cross-section of single box with single cell have been studied by many researchers (Hedegaard et al., 2012; Mondal and DeWolf, 2007; Shushkewich, 1998), but few papers have been published for studying the thermal behavior on a bridge cross-section of single box with three cells (Hedegaard et al., 2012; Priestley, 1976). Also, the thermal behavior in simply supported beams, prestressed concrete girder bridges, and concrete filled steel tube arch bridges has been studied by many researchers, such as Priestley (1976), Elbadry and Ghali (1983), and Chen and Liu (2009). However, the thermal effects on CBGAB were different from those on prestressed concrete bridges, because the azimuth and the dip angles at top and bottom flanges of the box-girder were varied along the axis of arch, so the absorbed solar radiation by them could be different. In addition, many researchers have assumed that the temperatures are constant along the length of the bridge. In this case, the thermal effect can be determined by two-dimensional (2D) finite element model (FEM) (Elbadry and Ghali, 1983; Hedegaard et al., 2012; Oskar and Thelandersson, 2011; Priestley, 1976). However, this assumption is not suitable for the CBGAB.

For arch bridges, Inaudi et al. (2002) monitored the temperatures, deformations, and curvatures of Siggenthal concrete arch bridge. But the authors did not show analysis of the measured data; Zhang (2012) studied the thermal effects on CBGAB by the 2D FEM based on meteorological parameter method; Lai (2013) showed that the temperature gradients in American Association of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Design (LRFD) Specifications (AASHTO, 2012) might not be applicable to open spandrel reinforced concrete arch bridges. However, previous studies listed above failed to propose the appropriate method to solve the thermal stress and displacement in CBGAB. All in all, the temperature distribution of the CBGAB was not fully understood, and the present method was also not suitable to solve this problem; it needs to find a new way to study the temperature distribution along the axis of the arch.

At present, there are three vertical temperature gradient models: the AASHTO’s (2012) vertical temperature gradient, the New Zealand’s Bridge Manual (Bridge Manual SP/M/022, 2013), and the China Concrete Railway Bridge Design Code (TB 10002.3-99, 2005). But no specifications specified the temperature gradients along the depth or width of the cross-section and thermal stress calculation method in CBGAB.

As the vertical temperature gradients are known, the thermal stresses and displacements of concrete bridges can be analyzed easily (Lee, 2012; Shushkewich, 1998; Song et al., 2012). The thermal stress in concrete bridge was treated by analytic method in previous studies. Based on the plane assumption, Shushkewich (1998), Barr et al. (2005), and Mondal and DeWolf (2007) derived the equations to calculate the stresses induced by temperatures of concrete box-girder, and Lee and Kalkan (2012) proposed one-dimensional beam theory to get the thermal deformations of prestressed box-girder concrete beams. But all the methods are quite complex or not suitable to get the temperature self-equilibrated longitudinal and transverse stresses, thermal inferior stress, and excessive displacements of CBGAB. The thermal behavior study on CBGAB is still scarce, and further research is needed to find a method which has a fast speed and high accuracy to calculate the thermal stress and displacement in CBGAB by FEM.

This article investigated the temperature distribution and thermal stress using field monitoring data in CBGAB located in GuiZhou, China. Based on the meteorological parameter methods, a 2D plane FEM was used to study the thermal field, boundary conditions, and vertical temperature gradients of CBGAB. In order to find a simple method to study the thermal stresses and displacements of the arch bridge, the article proposed a three-dimensional (3D) beam FEM. All the results were verified by the 3D solid FEM and the experimental data at last. The calculating method of 2D thermal field and 3D thermal stress was extracted using the commercial finite element software ANSYS.

Thermal behavior analysis methods

Heat flow governing equations and boundary conditions

Temperature distribution within a bridge can be governed by the well-known Fourier’s heat flow equation (Elbadry and Ghali, 1983; John and William, 2013; Priestley, 1976)

ρ c \frac{\partial T}{\partial t} = k_{x} \frac{\partial^{2} T}{\partial x^{2}} + k_{y} \frac{\partial^{2} T}{\partial y^{2}} + k_{z} \frac{\partial^{2} T}{\partial z^{2}}

(1)

where ρ is the density (kg/m³); c represents specific heat (J/kg K); T is the bridge temperature (°C); t is time; and k_x, k_y, and k_z represent thermal conductivities corresponding to the x, y, and z Cartesian axes, respectively (W/m K).

Equation (2) is recommended by Elbadry and Ghali (1983) and Peng (2007) to estimate the boundary conditions of equation (1) $(h_{c} + h_{r})$ is the total heat transfer coefficient, $[T_{a} + (q_{s} - q_{r a}) / (h_{c} + h_{r})]$ is the comprehensive atmospheric temperature.

(h_{c} + h_{r}) (T_{a} + \frac{q_{s} - q_{ra}}{h_{c} + h_{r}} - T) + k \frac{\partial T}{\partial n} = 0

(2)

where h_c is convection coefficient between the concrete surfaces and the ambient air (W/m² K); h_r is radiant heat transfer coefficient (W/m² K); T_a is ambient air temperature around the box-girder which can be expressed as sinusoidal function (°C) (Elbadry and Ghali, 1983); q_s is total heat flux gained by solar radiation at the external boundaries (W/m²), which is expressed by equation (3); q_ra is a coefficient (W/m²) that is given by equation (4); α_s is shortwave solar radiation absorption coefficient of the bridge surfaces, 0.50–0.77 (Oskar and Thelandersson, 2011); n is the normal direction of the surface; and k is thermal conductivity (W/m K)

q_{s} = α_{s} (I_{D ϕ} + I_{D β} + I_{f})

(3)

q_{ra} = \frac{1 + \cos β}{2} (1 - ε_{a}) ε C {(273 + T_{a})}^{4}

(4)

where I_Dφ, I_Dβ, respectively, represent reflected radiation flux of the direct and diffuse solar radiation in tilt surface (W/m²); I_f is the reflected radiation from the surroundings (Kim et al., 2009; Peng, 2007) (W/m²); β is the surface inclination angle, between the incline and horizontal plane, 0°≤ β ≤ 180°, when β > 90°, it means that the surface does not absorb solar radiation; ε_a is long-wave solar radiation absorption coefficient, which is often taken as 0.82; C is Stefan–Boltzmann constant, equals 5.67 × 10⁻⁸ (W/m² K⁴) (John and William, 2013); and ε is object emissivity.

Thermal field analysis theory

The 2D plane FEM and 3D solid FEM are the two methods used to study the thermal field of CBGAB. The 2D plane FEM has been presented in Peng (2007) and Elbadry and Ghali (1983) for pre-stress concrete box-girder bridges. By improvement, the 2D plane FEM can be used for CBGAB. In this method, the arch axis needs to be divided into several parts. Each part has different boundary conditions. The total heat transfer coefficients and comprehensive atmospheric temperature are applied to different parts of the 2D plane FEM, so the thermal field can be obtained.

Based on the boundary conditions obtained from the 2D plane FEM, the flow chart of the 3D solid FEM for thermal field analysis is summarized in Figure 1. The advantages of the 3D solid FEM are that it can consider the temperature transferring along the axis direction, and it is more convenient to calculate the thermal stress by transferring the SOLID70 elements into SOLID185. The disadvantages of the 3D solid FEM are that it requires more memory and computing time. It is more complicated to build a FEM, compared with 2D plane FEM.

Figure 1.

The flow chart of the 3D solid FEM thermal field analysis.

Temperature stress analysis theory

There are two methods used to calculate the thermal stress of CBGAB: the first is the 3D solid FEM and the second is 3D beam FEM. For 3D solid FEM, while the thermal distribution of arch bridge was obtained depending on the 3D solid thermal FEM, the thermal element SOLID70 was translated into structural element SOLID185. The temperatures of thermal element can be applied to the structural element automatically. Therefore, the thermal stress of the arch can be determined accurately. But this method needs more computer memory and time to finish the analysis. For 3D beam FEM, based on the beam elements BEAM189 and the temperature gradients obtained by 2D plane FEM, an analytical method that could calculate the thermal stress and displacement of the CBGAB is derived in this study.

Although the thermal stress in CBGAB can be calculated by BEAM189 element, only the linear temperature gradients along the depth of section can be applied to the elements. However, the temperature gradients in CBGAB are usually nonlinear. The nonlinear temperature gradients must be translated into linear temperature gradients, so that the thermal stress can be calculated by the BEAM189 element.

In order to develop the analytical model, the basic assumptions are as follows. (1) Transverse temperature gradients and negative temperature gradients in the bottom flange of the box-girder are not considered. (2) The fibers represented by the nonlinear and linear curve functions have the same thermal elongation length at the same height of the cross-section, as shown in equation (5). (3) The curve function of the nonlinear temperature gradients is assumed to a negative exponential function, as shown in equation (6). (4) Linear temperature gradients curve function can be expressed as in equation (7). (5) The differences of vertical temperature gradient curves in middle and edge web are not considered.

Based on the assumption, the calculating procedures of the 3D beam FEM are summarized as follows: (1) maximum temperature differentials T₀ are determined from the 2D plane FEM, and elongation coefficients of the ith cross-section height of linear temperature gradient curve α_i are obtained by equation (8); (2) the BEAM189 element section type is performed by plane element PLANE32 in each section, and the section type is divided into n parts along the height of the cross-section, as shown in Figure 2. The elongation coefficient α_i is applied to the ith part. Then the different section types along the arch axis are established; (3) based on the section type, the 3D beam FEM is established by the BEAM189 element; and (4) when maximum temperature differentials T₀ and minimum temperature differentials T₀·EXP(−b·h) are applied to the BEAM189 element, the thermal stresses and displacements can be obtained

α T_{1} (y) = α_{i} T_{2} (y)

(5)

T_{1} (y) = T_{0} e^{- by}

(6)

T_{2} (y) = T_{0} e^{- bh} + \frac{T_{0} - T_{0} e^{- bh}}{h} (h - y)

(7)

α_{i} = \frac{α T_{1} (y)}{T_{2} (y)}

(8)

where T₁(y) is the nonlinear temperature gradients curve function (°C), T₀ is the maximum temperature differentials (°C), b is constant, T₂(y) is linear temperature gradients curve function (°C), α is elongation coefficient of nonlinear temperature gradients curve (/°C), α_i is the elongation coefficient in the ith height of linear temperature gradients curve in (/°C), h is the height of arch box-girder section (m), and y is the distance from the top surface (m). From equation (8), the authors know that the elongation coefficients of linear temperature gradients have no relationship with the maximum temperature differentials, they just relate to the coefficient b.

Figure 2.

The profiles of the divided parts in the cross-sections and vertical temperature gradients.

Finite-element simulation

The 2D plane, 3D beam, and 3D solid FEM were developed to investigate the thermal behavior of CBGAB. The models were established by ANSYS, which allowed for thermal and structural analyses. In order to analyze 2D plane thermal field, several 2D plane FEMs of different cross-sections were developed with PLANE55 elements, as shown in Figure 3(a). To verify the correctness of the 2D plane FEM, a 3D solid FEM was established, as shown in Figure 3(b).

Figure 3.

Finite element model of the CBGAB: (a) cross-section and (b) 3D solid FEM.

In this study, the transient temperature field was simulated by ANSYS. (1) Thermal conduction element PLANE55 is adopted to model the single box with three-cell section. This element has four nodes with a single temperature degree of freedom. Figure 3(a) shows the finite elements in the crown section of the arch. (2) The plane element PLANE32 is used to define the section type used in BEAM189 beam element. Every section is divided into several parts along the height of section, and each part adopts different elongation coefficients. (3) Beam element BEAM189 is used to simulate the arch bridge. It has three nodes and 6 structure degrees of freedom at each node. The element is adopted to calculate longitudinal self-restraint temperature stress and thermal displacements. (4) The solid element SOLID70 which has eight nodes and a single temperature degree of freedom at each node is adopted to model the bridge structure. The total heat transfer coefficients and the comprehensive atmospheric temperatures are specified on the solid element SOLID70 just as the thermal conduction element PLANE55. The element is replaced by an equivalent structural element SOLID185 when the model is used for structural analysis. The SOLID185 element has eight nodes and 3 degrees of freedom at each node. (5) The CBGAB simulated in this study is a hingeless arch bridge, so the displacement boundary conditions in the arch spring is fully fixed, and the dead load, live load, or the creep and shrinkage of concrete are not considered in this study except the thermal load. (6) The thickness of the box-girder was only 14 cm, the temperature in box-girder transferred quite quickly, so the initial temperature of the bridge was assumed to be uniformly distributed as 22°C at 0:00 on 21 August 2014. The errors caused by this assumption were eliminated by several times running when the thermal element suffered from the same boundary conditions. (7) The environment conditions at the bridge sites were predicted by the same method discussed by Elbadry and Ghali (1983) or were directly referred from the measured data. Wind speed was applied as a constant which is the daily average speed. The meteorological parameters shown in Table 2 are transformed to the comprehensive air temperature and comprehensive atmospheric heat transfer coefficient (Peng, 2007) as shown in Table 1. So it is easy to apply to ANSYS for thermal field analysis. The steel tube has little effect on the thermal behavior, so it was not considered in the 3D beam and 3D solid FEM.

Table 1.

Optimized meteorological parameters.

Total heat transfer coefficients (W/m² K)	Comprehensive atmospheric temperatures (°C)
On the top surfaces	On the top surfaces
On the bottom surfaces	On the bottom surfaces
On the exterior vertical surfaces	On the west exterior vertical surfaces
Inside the box-girder	On the east exterior vertical surfaces
	Inside the east box-girder
	Inside the middle box-girder
	Inside the west box-girder

The meteorological parameters and material properties used in this study are shown in Tables 2 and 3. All the material parameters were determined from the reviewed literatures or experimental results.

Table 2.

Related meteorological parameters.

Parameters	Values
Date	21 August 2014
Time range	0–24 h
Longitude and latitude	25.93°N, 105.26°E
Location altitude (m)	695
Initial temperature (°C)	22
The atmospheric transparency coefficient, P	0.65
Short wave absorption coefficient	0.65
Ground reflection coefficient	0.20
Atmospheric radiation coefficient	0.82
Concrete emissivity	0.88
Wind speed (m/s)	1.0

Table 3.

Material properties used in FEM.

Material properties	Values
Concrete density (kg/m³)	2500
Steel density (kg/m³)	7850
Concrete thermal conductivity (W/m K)	2.25
Steel thermal conductivity (W/m K)	80.00
Concrete specific heat (J/kg K)	1020
Steel specific heat (J/kg K)	465
Concrete elastic modulus (MPa)	3.72 × 10⁴
Concrete Poisson ratio	0.21

FEM: finite element model.

Experimental plan

One CBGAB located in GuiZhou, China, was taken as an example to study the thermal effect, as shown in Figure 4. The bridge location is 25.93°N, 105.26°E. The arch bridge axis direction is from north to east 55°. The bridge span is 59.333 m; arch rise is 13.413 m; and arch axis equation is catenary and arch axis coefficient m = 1.612. The detail dimensions of the cross-section are shown in Figure 5(a). Its height is 1.2 m, and its width is 2.400–3.733 m from the arch spring to position of 9.165 m horizontal distance from the arch spring. The arch bridge was made of C60 fine aggregate high-strength concrete. The C60 fine aggregate concrete has the same mechanical properties (such as strength and elastic modulus) as the normal C60 concrete, but the maximum diameter of coarse aggregate is 9.5 mm compared with the normal C60 concrete with 5–20 mm maximum diameter of coarse aggregate. The most important purpose of the arch model bridge shown in Figure 4 was to prove the safety of bridge arch construction process. But when the model bridge was finished, in order to know the thermal field and thermal stress distribution in the arch bridge, the temperature behavior of the model bridge was monitored on 21 August 2014. The results of the measurements could be used to verify the correctness of the established FEMs.

Figure 4.

The experimental concrete box-girder arch bridge.

Figure 5.

Locations of the vibrating wire strain gauges (mm): (a) arch bridge sections (mm) and (b) arch bridge axis.

Several vibrating wire (VW) strain gauges were installed in the arch cross-section, and they could be used to measure the temperatures and the corresponding strains at those points. Figure 5 shows the locations of VW strain gauges in the bridge section along the arch axis. H series means that the thermocouples are embedded in concrete inside and G series means that the thermocouples are placed on the surface of the steel tube. There were also thermocouples installed in and out of the box-girder to measure the atmosphere temperature of different parts of the box-girder. The temperatures and strains are measured every 2 h.

The strains measured by the VW are the sum of the transient elastic strain, creep and shrinkage strain, thermal expansion strain, strain caused by thermal stresses and the sensor itself, and strain induced by temperature increments. At the measuring time, there is no other load except the convection and solar radiation. The increased elastic strains can be assumed to zero. Besides, the concrete used in the CBGAB was finished about 131 days and it was C60 high-strength concrete, so there is little shrinkage and creep strains at that measured day, hence the shrinkage and creep strains can also be ignored. The strains caused by thermal expansion were calculated by αΔT. Also, the temperature at 0:00 was the reference temperature. At that time, there is no thermal stress. The modified formula used to calculate strains affected by temperatures themselves are provided by the sensor supplier. The thermal stresses shown in this article are the stress increments based on the measured results at 0:00.

2D plane FEM

Based on the 2D plane thermal FEM and measured temperatures, the thermal field in CBGAB was studied. This part illustrates the calculated temperatures in the steel tube, concrete box-girder, out surface of the box-girder, and the boundary conditions such as the daily comprehensive atmospheric temperatures and the total heat transfer coefficients. In addition, the vertical temperature gradients were also discussed. The temperature boundary conditions and vertical temperature gradients obtained in this part could be used to calculate temperature distribution of the 3D solid FEM and the thermal stress of the 3D beam FEM.

Thermal field in the concrete box-girder

Figures 6 to 8 show the comparisons of temperatures calculated by 2D plane FEM and measured results for selected thermocouples in Sections 3, 5, and 8. Not all the measured points were shown here in order to save space. In these figures, FEM and Exp refer to the predicated and measured results at different measuring points, respectively. Those figures show that the temperatures predicated using the 2D plane FEM agree well with those experimental data. The 2D plane FEM can be used to calculate the thermal field of CBGAB.

Figure 6.

Comparison of temperatures calculated by 2D plane FEM and measured results in Section 3: (a) steel tube measuring point, (b) the top and bottom flange, (c) top of concrete web, and (d) bottom of concrete web.

Figure 7.

Comparison of temperatures calculated by 2D plane FEM and measured results in Section 5: (a) steel tube measuring point, (b) the top and bottom flange, (c) top of concrete web, and (d) bottom of concrete web.

Figure 8.

Comparison of temperatures calculated by 2D plane FEM and measured results in Section 8: (a) steel tube measuring point, (b) the top and bottom flange, (c) top of concrete web, and (d) bottom of concrete web.

In most cases, temperatures on the top surface of cross-section were higher than those on the bottom surface due to the top surface receiving more solar radiation. The nearer to the out top surface the measured points, the higher the maximum temperatures. The nearer to the internal surface the measured points, the more time lags the maximum temperatures.

The temperatures were not only different in different positions of the same section, but also different in same position of different sections, as shown in Figures 6 to 8. There were differences among the temperatures in the top flange of different sections. The maximum temperatures at point H2 (top flange) in Sections 3 and 5 were 40.1°C, but they were 36.9°C at point H2 in Section 8 (arch spring in north side), which was lower than in Sections 3 and 5. But, the temperature differences were small at the bottom of the web such as H7–H14, as shown in Figures 6(c), 6(d), 7(c), 7(d), 8(c), and 8(d) in various sections. The inclination and azimuth angles of web surface were the same in different sections along the arch axis, so they had the same ability to absorb the solar radiation at the same time. But the top or bottom surfaces in different cross-sections along the arch axis had different inclination and azimuth angles, so Section 8 absorbed less solar radiation than Sections 3 and 5 on top surface. Although the total heat transfer coefficients of different cross-sections were different, they had little influence on the thermal distribution when compared with the ability of absorbing the solar radiation.

From the above discussion, the authors knew that it was not suitable to assume the temperature as constant along the axis of arch bridge. The maximum temperature of top flange in Section 5 was 4°C higher than in Section 8. All the results showed that one must calculate the thermal field in CBGAB by the improved 2D plane FEMs with different thermal boundary conditions.

Figure 9 shows the comparison of measured and calculated temperatures outside of the top surface, bottom surface, east surface, and west surface of the box-girder. The comparison of this figures revealed that the temperatures calculated by 2D plane FEM exhibited coincident variation tendency with the measured temperatures. The top surfaces of Sections 3 and 5 faced south could absorb more solar radiation. So the temperature on those top surfaces was 43.8°C which was larger than Section 8 (38.2°C), as shown in Figure 8(a) and (b). But the temperatures outside of the web in different sections were nearly the same. The top surface of box-girder had the longest time to absorb solar radiation, and the maximum temperatures occurred at about 14:00. The receiving solar radiation time on east web was earlier than on west web, so the time of maximum temperature occurred at 13:00 on east web and 17:00 on west web. Also, the maximum temperature on east side of the web was larger than on west side. There was no doubt that the lateral temperature gradients would be emerged in top or bottom flange of box-girder.

Figure 9.

Variations in the temperature on the surfaces of the box-girder in Sections (a) 3, (b) 5, and (c) 8.

Daily comprehensive atmospheric temperatures

Figure 10 illustrates the optimized comprehensive atmospheric temperatures in the outside of top surface, bottom surface, web, and box internals determined by the 2D plane FEM and equation (2). They were the basis to calculate the thermal distribution of the 3D solid FEM.

Figure 10.

Variations in the daily comprehensive atmospheric temperature in Sections (a) 3, (b) 5, and (c) 8.

In these figures, top, bottom, east out, and west out refer to the comprehensive atmospheric temperatures outside of the top flange, bottom flange, east web, and west web, respectively. The comprehensive atmospheric temperatures outside of top surface of different cross-sections were almost the same at night, but they increased more sharply in Section 3 than in Section 8 at sunrise, because Section 3 absorbed more solar radiations than Section 8. All the cross-sections had nearly the same comprehensive atmospheric temperature outside of the bottom and web at the same time. They have the same rule as the calculated temperatures on the surface of box-girder which is shown in Figure 9. East in, West in, and Middle in refer to the comprehensive atmospheric temperatures of the east, the west, and the middle box-girder internals, respectively. The comprehensive atmospheric temperatures in the box internals were little influenced by atmosphere temperatures and the solar radiation in the open air. They were almost the same in different sections. The time of maximum temperature occurred in the east, west, and middle girder internals was 15:00, 17:00, and 20:00, respectively. All in all, the comprehensive atmospheric temperatures had much influence on the temperature distribution in CBGAB. The accuracy of the predicated comprehensive atmospheric temperature is the basis of simulating the temperature field distribution of concrete box-girder.

Total heat transfer coefficients

Table 4 demonstrates the total heat transfer coefficients calculated by equation (2) and the 2D plane FEM. It was found that the total heat transfer coefficients outside of the top surface, bottom surface, and web in different sections were various. Section 5 which was the arch crown section had the largest heat transfer coefficient compared with other sections. But the differences of heat transfer coefficients in different sections were little because of their little altitude range. Besides that, the total heat transfer coefficients in girder internals were not affected by wind speed, so they were constant along the axis of arch bridge. The total heat transfer coefficients had little influence on the temperature distribution of CBGAB in this study. But, if the altitude range between the arch spring and the arch crown is larger than 50 m, the wind speed between them will be much different, so the thermal field may be influenced by the total heat transfer coefficients.

Table 4.

Total heat transfer coefficients in different parts of cross-section (W/m² °C).

Section no.	Top surface	Bottom surface	Web	Girder internals
3	18.95	12.57	16.56	9.50
5	19.59	12.89	17.04	9.50
8	18.14	12.17	15.96	9.50

Temperature gradients

The vertical temperature gradients could be determined by the thermal field. Figure 7 reveals that the maximum temperature of top surface in Section 5 occurred at about 15:00. The maximum vertical thermal differential emerged at the same time. Table 5 shows the maximum vertical thermal differentials and maximum, minimum, and average temperatures in all the measured sections. The maximum vertical thermal differentials on the middle web were about 4°C larger than on edge web in all sections. The maximum vertical thermal differentials in Sections 1–5 whose azimuth angles faced south were larger than in Sections 6–9 whose azimuth angles faced north. While the middle web had large maximum vertical temperature gradient, it had small maximum average temperature. This was because the temperature of the middle web changed little compared with edge web.

Table 5.

Vertical temperature gradients at 15:00 (°C).

Location		Differentials	Average	Min	Max
1	Middle web	12.96	31.05	29.26	42.22
1	Edge web	10.22	36.45	33.55	43.77
2	Middle web	13.61	29.44	28.02	41.62
2	Edge web	9.85	35.76	33.56	43.41
3	Middle web	13.31	29.83	28.34	41.65
3	Edge web	9.41	36.03	33.93	43.34
4	Middle web	13.50	30.12	28.54	42.03
4	Edge web	9.62	36.18	33.93	43.55
5	Middle web	12.90	30.30	28.60	41.50
5	Edge web	8.34	36.30	34.65	42.99
6	Middle web	11.41	29.11	27.62	39.03
6	Edge web	7.41	35.59	33.40	40.81
7	Middle web	10.18	28.75	27.35	37.54
7	Edge web	6.35	35.37	33.12	39.47
8	Middle web	8.38	28.48	27.25	35.63
8	Edge web	4.93	35.12	32.83	37.75
9	Middle web	7.03	28.09	26.96	33.99
9	Edge web	3.95	34.25	32.12	36.07

The numbers in the left column means the section number as shown in Figure 5(b). Section 1 means the south arch spring of arch bridge, and Section 9 means the north arch spring of arch bridge.

Figure 11 shows the maximum vertical temperature gradient curves of middle web and edge web in Section 5. All of the other sections had the similar vertical temperature gradient curves, they are not listed here. From Figure 11, it is known that there are many differences in the maximum vertical temperature gradient curves between middle web and edge web. The vertical temperature gradient curve of middle web can be expressed as a negative index function, which is shown as

T (y) = 28.61 + 13.61 e^{- 7.8 y}

(9)

where y is the distance from the top surface (m), and the temperature gradient curves of edge web can be written as cubical parabola function, which is shown as

T (y) = 41.58 - 29.25 y + 46.03 y^{2} - 23.04 y^{3}

(10)

Figure 11.

Vertical temperature gradients in Section 5.

All the results shown here are similar to Priestley’s (1976) conclusions. For simplification, the authors did not consider the differences in vertical temperature gradient curves in middle web and edge web. Only one vertical temperature gradient curve was applied to BEAM189 elements in ANSYS.

3D beam FEM

Thermal stress

In order to verify the correctness of the 3D beam FEM that was used to calculate the thermal stress of arch bridge, the comparison of thermal stresses along the depth of cross-section calculated by 3D beam FEM and 3D solid FEM is shown in Figure 12, when the maximum temperature gradients occurred at 15:00. Those results are one part of the time-dependent stress history shown in Figures 15 and 16. The lateral temperature gradient of the box-girder was not considered in this research. The reference temperature was the temperature 22°C at 0:00 on 21 August 2014.

Figure 12.

Vertical temperature stress distribution in Sections (a) 3, (b) 5, and (c) 8.

At the top surface of the box-girder, the largest compressive stress emerged. The compressive stress became tensile stress in the middle depth of web, at the bottom surface of the box-girder, the tensile stress turned to be smaller. All the results shown here have the same conclusions with Shushkewich (1998) and Hedegaard et al. (2012). The maximum compressive and tensile stresses calculated by 3D solid FEM were −2.9 and 2.2 MPa, respectively. And the maximum compressive and tensile stresses calculated by 3D beam FEM were −4.3 and 2.2 MPa. The maximum thermal stress calculated by 3D beam FEM was larger than by 3D solid FEM. However, the thermal stress curves along the depth of the cross-section calculated by 3D beam FEM were similar to the results calculated by 3D solid FEM. The advantages of 3D beam FEM are that its element number is relatively small and it has a faster calculation speed. The 3D beam FEM method can be used to assess the structure safety.

The shapes of the thermal stress curves along the depth of the cross-section are similar to each other in different sections, as shown in Figure 12. But there were little differences between the maximum thermal stresses in various sections. Sections 3 and 5 had larger thermal stresses than Section 8. They had good relationship with the vertical temperature gradients. The larger the vertical temperature gradients, the greater the thermal stress.

Although there were 2.2 MPa thermal tensile stress which might lead to cracks on concrete box-girder surface, the stress induced by dead or live load in each section was compressive stress in most times. The whole compressive stress might decrease part of the thermal tensile stress. So, the thermal tensile stress has little effect on the safety of the arch bridge, but the thermal compressive stress may have bad effect on its safety. On the primary design stage of bridge structure, it is necessary to calculate the thermal stress by 3D beam FEM, although this method may overestimate the thermal stress. If the bridge designers want to know the thermal stress more accurately, a 3D solid FEM is needed.

Thermal displacement

Table 6 shows the comparison of vertical thermal displacements calculated by 3D beam FEM and 3D solid FEM. The vertical thermal displacements calculated by 3D beam FEM had the same tendency with 3D solid FEM. The 3D beam FEM can well predicate the vertical displacements. The direction of whole arch displacement was upward on the effect of the maximum temperature gradients. The vertical thermal displacement was about 8.2 mm in the arch crown calculated by 3D solid FEM, which was less than 1/5000 when compared with the span of the arch. It was so small that can be ignored in the bridge design stages.

Table 6.

Vertical displacements at 15:00 (mm).

Section no.	3D beam FEM	3D solid FEM
Section no.	3D beam FEM	West	East	Average
2	1.81	1.60	1.86	1.73
3	4.78	4.27	4.77	4.52
4	7.94	6.89	7.56	7.23
5	8.85	7.86	8.58	8.22
6	7.22	6.82	7.49	7.16
7	4.66	4.24	4.74	4.49
8	1.67	1.50	1.75	1.63

3D: three-dimensional; FEM: finite element model; West: west middle web; East: east middle web.

3D solid FEM

Compared with the 2D plane FEM and 3D beam FEM, the 3D solid FEM has more accuracy, so a 3D solid FEM was established to verify the correctness of the results. All the thermal boundary conditions and concrete thermal properties used in this model came from the 2D plane FEM. From section “Thermal field in the concrete box-girder,” the authors knew that the steel tube on cross-section had little effect on the thermal field in concrete box-girder. For simplification, the steel tube was not considered in the 3D solid FEM thermal field calculation.

Thermal field

Figures 13 and 14 show the variations of temperatures calculated by 2D plane FEM and 3D solid FEM in Sections 3 and 5 on 21 August. In these figures, H7 2D and H7 3D"means the temperatures in measuring point H7 calculated by 2D plane FEM and 3D solid FEM, respectively. The temperatures of different measured points predicated using the 2D plane FEM agree well with those calculated by 3D solid FEM. The concrete temperatures in the edge measuring points (H7, H10, H11, and H14) predicated by 3D solid FEM were larger than those predicated by 2D plane FEM. But in conversely, the temperatures in the middle measuring points (H8, H9, H12, and H13) obtained by 3D solid FEM were smaller than those obtained by 2D plane FEM. The differences in the temperatures between the two FEMs were less than 1.3°C (4.2%) because there was heat transfer along the axis of arch bridge in the 3D solid FEM. All in all, the 2D plane FEM could well predict the thermal field in CBGAB.

Figure 13.

Comparison of temperature calculated by 2D plane FEM and 3D solid FEM in Section 3.

Figure 14.

Comparison of temperature calculated by 2D plane FEM and 3D solid FEM in Section 5.

Thermal stress

The thermal stresses measured by VW strain gauges in Sections 3 and 5 are shown in Figures 15 and 16. In these figures, H2 FEM and H2 Exp means the thermal stresses in measuring point H2 calculated by 3D solid FEM and the experimental results, respectively. The measured thermal stresses shown here are the stress increments based on the measured results at 0:00. The measured stresses in major points of box-girder were periodically changed under the influence of the nonlinear temperature field in the concrete box-girder. Generally speaking, the maximum stress occurred when the maximum temperature gradients emerged. With the solar radiation and air temperature increasing, the thermal stresses in the measured points H2, H10, and H14 in the top and east web surfaces of the box-girder which had the longest time to absorb the direct radiations from the sun changed from tensile stress to compressive stresses. The maximum thermal compressive stresses in H2, H10, and H14 were −2.92, −2.45, and −2.41 MPa at 18:00, 16:00, and 12:00 in Section 3, respectively. Besides, the compressive stresses in Section 3 were much larger than in Section 5. On the other hand, the maximum thermal stresses in H7, H8, H9, H11, H12, and H13 decreased to −1.66 MPa during the night and increased to 2.41 MPa at around 16:00 during the day. The total range of the thermal stress in the measured points of all measured points was under 3.37 MPa.

Figure 15.

Comparison of thermal stress calculated by 3D solid FEM and measured by test in Section 3.

Figure 16.

Comparison of thermal stress calculated by 3D solid FEM and measured by test in Section 5.

The thermal stresses predicated by the 3D solid FEM were compared with the measured data, as shown in Figures 15 and 16. They can correlate well with measured stresses. There were little differences between the measured and predicated thermal stresses at each point. Thus, the 3D solid FEM could be used to calculate the thermal stress in CBGAB.

Conclusion

A CBGAB model was established to study its thermal field and thermal stress under the solar radiation and convention. The thermal field distribution, thermal stresses, and displacements were also investigated by the 2D plane, 3D beam, and 3D solid FEM based on meteorological parameter methods. Some conclusions are summarized as follows:

For the CBGAB, the temperature distribution along the arch axis was non-uniform compared with that on prestressed concrete box-girder bridges because of their different azimuth and inclination angles in top and bottom of each cross-section.

The 2D plane FEM was the basis of the 3D beam and 3D solid FEM. It can provide temperature gradients for 3D beam FEM and temperature boundary condition for 3D solid FEM. All the analytical studies showed that the thermal fields were influenced by meteorological parameters such as total heat transfer coefficients and comprehensive atmospheric temperatures. It was very important to select the right thermal boundary conditions for analyzing the thermal effect on CBGAB.

The temperature gradients between middle and edge web of the single box with three cells were different. The vertical temperature gradient in middle web was expressed as negative index function, but in edge web was expressed as cubical parabola function. There are also lateral temperature gradients if the width of box-girder is large.

The 3D beam FEM developed in this research can be used to calculate the thermal stress and deformation in a more simple way. The thermal stresses along the depth of the cross-section are nonlinear distribution. The compressive stress in the top surface of the box-girder can affect the structural safety.

A 3D solid FEM was also proposed to verify the correctness of 2D plane and 3D beam FEM. The time-dependent thermal stresses showed periodic changes. The maximum thermal stress occurred when the maximum temperature gradients occurred. The vertical displacements of the arch were asymmetric distribution along the arch. Compared with the 3D solid FEM, the 2D plane FEM and 3D beam FEM can be used to assess the thermal field and thermal stress on the initial stage of bridge structure design. But design engineers could use 3D solid FEM for check, particularly during the construction stage of long span CBGAB.

However, the bridge used in this study is a model bridge which has thin wall, small span, and arch rise; the thermal field and thermal stress between the model bridges and prototype bridges may be different, and it still needs further research. And there are several broken thermocouples or strain sensors in the arch bridges, but they have little impact on the general conclusions studied in this article.

Footnotes

Appendix

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 experiment test in this work is sponsored by The China Railway Eryuan Engineering Group Co. Ltd. It is financially supported by the National Natural Science Foundation of China (51208431), the Fundamental Research Funds for the Central Universities (SWJTU12CX064), and Science and Technology Project of the Ministry of Railways in China (2010G018-A-1-09).

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