Temperature behaviors of an arch bridge through integration of field monitoring and unified numerical simulation

Abstract

Varying temperature may cause a non-uniform temperature distribution of a bridge and lead to excessive movement and stresses of the structure. Traditional thermal analyses of bridges adopt a divide-and-conquer approach, which conducts a simplified 2D or local 3D heat-transfer analysis and then a global 3D structural analysis by inputting the calculated temperature into another bridge model. This process requires considerable manual intervention and is inefficient and may lead to inaccurate results. This study develops a unified approach of heat-transfer and structural analyses for the first time to calculate the temperature distribution and the associated responses of an entire structure by integrating the field monitoring data. The arch footbridge at the Hong Kong Polytechnic University is used as a testbed, and a detailed finite element model (FEM) of the bridge is established. The measured air temperature and solar radiation are used as the thermal boundary conditions. The hemisphere technique is adopted to calculate the view factor between different surfaces of the bridge, which are then used to obtain the solar radiation on all external surfaces in different instants on different dates. The 3D global hear-transfer analysis is conducted to obtain the temperature distribution of the entire bridge. The calculated temperature data of the bridge are then automatically input into the same FEM of the bridge to calculate the temperature-induced bridge responses via the structural analysis. The heat-transfer analysis and structural analysis share the same FEM while using different element types. Therefore, the manual intervention is avoided. The calculated and monitored temperature data and responses show a good agreement. The developed new unified approach enables an automatic and efficient analysis of thermal behaviors of bridges. This approach can be extended to other types of bridges.

Keywords

heat-transfer analysis temperature distribution temperature behaviors arch bridge structural health monitoring

Introduction

Bridges are subjected to daily and seasonal thermal actions induced by solar radiation and varying ambient air temperature. The temperature variation in bridge components considerably affects the deformation of the bridge. Furthermore, thermal stresses are typically generated due to the redundancy of the structure, the restriction of movement by bearings and expansion joints, and the nonuniform temperature distribution throughout the bridge (Xu et al., 2010; Xia et al., 2012, 2017; Zhou and Yi, 2013; Westgate et al., 2015; Zhao et al., 2019; Han et al., 2022; Zhang et al., 2021a, 2021b; Zhou et al., 2022).

Two approaches are commonly used to study the temperature behaviors of bridges. One is the correlation analysis between the temperature data and the responses of the structure measured at limited sensor points (Yang et al., 2018; Zhu et al., 2021; Wang et al., 2022). Given that the bridge responses are associated with the temperature distribution of the entire structure, the correlation analysis based on limited measurement points cannot capture the accurate relation between the structural responses and temperature and the most adverse temperature effect, particularly for large-scale bridges.

The other approach is the numerical simulation for obtaining the temperature distribution of the bridge; subsequently, the structural analysis is conducted for obtaining the temperature-induced bridge responses. The numerical simulation technique consists of finite difference methods and finite element (FE) methods. The finite difference method to solve the 1D heat flow equation was developed based on the assumption that the temperature varies along the depth of the cross section only (Moorty and Roeder, 1992). The 2D method has been developed to solve the heat-transfer equations for predicting the temperature distribution and thermal response (Elbadry and Ghali, 1983). Emanuel and Hulsey (1978) calculated the temperature distribution of a composite-girder highway bridge by performing FE analysis, in which the steady-state sinusoidal boundary condition representing actual weather patterns was adopted. These conventional heat-transfer analysis adopts a divide-and-conquer strategy. That is, the main components, such as the deck and towers, are modeled separately as 2D FE models by disregarding the temperature variation along the longitudinal direction (Zhou et al., 2016; Xia et al., 2019; Tan et al., 2021) or local 3D FE models (FEMs) (Xia et al., 2013, 2020; Zhu and Meng, 2017). These local models are analyzed individually to obtain the temperature distribution of the cross sections, which are then assembled manually and input into the global 3D FEM of the bridge for the structural analysis to calculate temperature-induced responses (Xia et al., 2013; Zhu and Meng, 2017). This process is inefficient because it requires considerable manual intervention. Moreover, the cross section of bridge components may be nonuniform, particularly those close to the intersection of the tower and beam. Simplified 2D/3D models fail to simulate the real temperature distribution of the entire bridge.

A detailed temperature distribution of the bridge components is required to quantitatively assess the temperature effects on the bridge. A numerical heat-transfer analysis is required to calculate detailed temperature information of the bridge that is not available through the monitoring system due to the limited number of sensors. Moreover, bridges are simultaneously subjected to multiple loads, such as winds and traffic. Separating these loading actions is necessary to assess each loading effect accurately. Furthermore, varying temperatures may cause more significant changes in structural vibration properties than damage (Sohn et al., 1999; Peeters and Roeck, 2001; Nayeri et al., 2008). Effects of temperature on damage detection or damage detection under different temperature conditions have been developed (Yan et al., 2005; Deraemaeker et al., 2008; Hios and Fassois, 2014; Hou et al., 2020; Wang et al., 2021). Therefore, an accurate investigation of the temperature behavior is crucial to the safety condition assessment of bridges. However, damage detection is beyond the scope of the present study and will not be discussed here. The present study focuses on the temperature distribution and temperature-induced static responses of bridges.

This study aims to investigate the temperature behaviors of a practical arch bridge (Xia et al., 2021) in real time by integrating field monitoring and unified numerical analysis. The unified analysis refers to that the heat-transfer analysis and structural analysis share the same FEM while using different element types. The temperature data of the bridge calculated from the heat-transfer analysis are automatically input into the same FEM for the structural analysis to calculate the temperature-induced bridge responses. The manual intervention is thus avoided. A 3D global FE modeling and the unified analysis of the bridge are conducted using the commercial FE analysis software ANSYS (ANSYS 16). The calculated and monitored temperature data and responses under the hottest and coldest days are compared and explained.

Footbridge and its health monitoring system

Footbridge descriptions

The footbridge linking Blocks Z and X in the Hong Kong Polytechnic University main campus was operated on October 15, 2019. The bridge has a total length of 84.24 m, consisting of a 64.26 m long main span that straddles the highway underneath and a 19.98 m long side span (Xia et al., 2021). As shown in Figures 1 and 2, the bridge deck is supported by two butterfly-shaped steel tube arches. One end of the arches meets the deck beams, which are supported on a tower at Block X. The other end of the deck beams rests on the podium floor of Block Z. The arches, beams, and columns are made of steel, whereas the bridge deck is composed of reinforced concrete.

Figure 1.

Bird’s view of the footbridge.

Figure 2.

Elevation of the footbridge.

Health monitoring system of the bridge

A monitoring system was installed on the bridge, along with the construction progress of the structure. The system started data collection on September 28, 2019, and worked normally from October 13, 2019, after testing and debugging were completed. The sensor subsystem consists of 88 permanently installed sensors of 13 types (Xia et al., 2021). The monitoring items are generally classified into three categories: environmental parameters (air temperature, air pressure, humidity, rain, and solar radiation), loads (winds, pedestrians, and structural temperature), and bridge responses (stress/strain, displacement, and acceleration). The detailed layout of sensors is shown in Figure 3. In particular, one pyranometer was installed on the roof of Block X to measure the real-time solar radiation intensity. Seven independent data acquisition units (DAUs) were installed on the roof of Block X and Sections A to F of the bridge to collect data from the surrounding sensors. Among them, 20 vibrating wire strain gauges were installed on the arch bridge and distributed on steel girders, arches and the concrete slab. The strain gauges measure the strain of the arch bridge members, which are used to calculate the internal force and then evaluate the safety and fatigue condition of the bridge.

Figure 3.

Layout of sensors and DAUs on the arch bridge.

Temperature distribution of the arch bridge

Air temperature variation in 2020

The measured temperature data in 2020 are first examined. The daily maximum and minimum temperature of all air temperature sensors in the year are shown in Figures 4 and 5. The highest temperature of the year appears in July and August, while the lowest is observed in January. The maximum temperature difference is 35°C. The two hottest and coldest days of the year are then selected to analyze the temperature and temperature-induced responses of the arch bridge.

Figure 4.

Daily maximum temperature of all temperature sensors in 2020.

Figure 5.

Daily minimum temperature in 2020.

Daily temperature variation

August 25 and February 17 represent the hottest and coldest day in 2020, respectively. The air temperature, solar radiation intensity, and average wind speed in the 2 days are shown in Figure 6. The air temperature varied between 28.5°C and 33.1°C on August 25, and 9.7°C and 17°C on February 17. The latter had a larger temperature variation. These environmental data will be used as the thermal boundary conditions for the transient heat-transfer analysis of the arch bridge.

Figure 6.

Meteorological data on 17 February and 25 August 2020. (a) Air temperature (b) Solar radiation intensity (c) Average wind speed.

Heat-transfer analysis of the arch bridge

Solar radiation

Solar radiation considerably affects large-scale structures in the open environment, especially long-span bridges or supertall buildings. The sun emits solar radiation, which is a form of radiant energy. Empirical formulas have been developed to estimate radiant energy. Sunshine duration and solar altitude are the most commonly used parameters for calculating the global solar radiation, which can be measured and calibrated (Al-Mostafa et al., 2014).

Environmental thermal condition

Thermal exchange q between the structure surface and the environment consists of convection $q_{c}$ , thermal irradiation $q_{r}$ , and solar radiation $q_{s}$ (Elbadry and Ghali, 1983).

q = q_{c} + q_{r} + q_{s}

(1)

The convectional heat transfer $q_{c}$ is proportional to the velocity of air particles and is dependent on the temperature difference between the air and the structure surface. Newton’s convection law approximates this relationship as

q_{c} = h_{c} (T_{a} - T_{s})

(2)

where

T_{a}

is the temperature of the air;

T_{s}

is the temperature of the structure’s surface; and

h_{c}

denotes the convection heat transfer coefficient, which is dependent on the wind speed, the roughness of the exposed surface, and the geometric configuration of the exposed surface (Branco and Mendes, 1993).

h_{c}

can be determined using empirical formulae in the heat flux problem of structures (Defraeye et al., 2011).

h_{c} = {\begin{array}{c} 3.7 υ + 6.0 (υ < 5.0 m / s) \\ 7.15 υ^{0.78} (υ \geq 5.0 m / s) \end{array}

(3)

where

υ

is the wind speed.

The thermal irradiation is the heat transfer between the surrounding atmosphere and the structural surface caused by long wave radiation. It can be characterized in a quasi-linear fashion as follows:

q_{r} = h_{r} (T_{a} - T_{s})

(4)

where

h_{r}

is the irradiation heat transfer coefficient, which is affected by the structural material, ambient temperature, and surface temperature.

h_{r}

can be calculated as (Mirambell and Aguado, 1990).

h_{r} = ε [4.8 + 0.075 (T_{a} - 5)]

(5)

where

ε

(0 <

ε

<1) is the structure surface’s emissivity coefficient. After the radiation coefficient

h_{r}

and the convection coefficient

h_{c}

are calculated, the heat fluxes due to convection and thermal irradiation can be combined to provide the overall heat transfer coefficient as

h = h_{c} + h_{r}

(6)

Heat-transfer equations and boundary conditions

The Fourier partial differential equation governs heat conduction in solids. A 3D heat flow equation can be used to define temperature T at each place in a structure at any time t (Whitaker 2013; Fu et al., 1990):

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

(7)

where

x

y

, and

z

denote Cartesian coordinates;

k

ρ

, and

c

denote the material’s isotropic thermal conductivity coefficient, density, and specific heat, respectively. When temperature change in one or two directions is assumed to be minimal, equation (7) can be simplified to a 2D or even 1D temperature field (John et al., 2006). Solving the Fourier partial differential equation under initial and boundary conditions yields the structural temperature distribution. The following three types of thermal boundary conditions are frequently encountered.

1) Specified temperatures on the structure boundary

{T |}_{Γ} = T_{Γ}

(8)

where Γ is the boundary surface.

2) Specified heat flux on the structure boundary

{k \frac{\partial T}{\partial n} |}_{Γ} = q (t)

(9)

where n denotes normal to the surface, and

q (t)

denotes the boundary heat input or loss per unit area.

3) Heat flux is proportional to the temperature difference

{k \frac{\partial T}{\partial n} |}_{Γ} = h (T_{a} - T_{s})

(10)

For a bridge exposed to solar radiation and ambient thermal environment, the boundary condition for the thermal analysis is a combination of the second and third types.

{k \frac{\partial T}{\partial n} |}_{Γ} = q (t) + h (T_{a} - T_{s})

(11)

The heat flux on a boundary surface is described as

{k \frac{\partial T}{\partial n} |}_{Γ} = α I + h (T_{a} - T_{s})

(12)

Equation (12) can be rewritten as

{k \frac{\partial T}{\partial n} |}_{Γ} = h (T^{*} - T_{s})

(13)

T^{*} = T_{a} + \frac{α I}{h}

(14)

where temperature

T^{*}

is referred to as the equivalent air temperature given that it considers air temperature and solar radiation.

A thermal field model based on the abovementioned equations is required to estimate the time-dependent temperature across the entire structure.

View factor

The radiation view factor describes how one object can “see” another object, which can be used to compute the radiation heat load. Consider the heat exchange between elementary areas ${d A}_{i}$ and ${d A}_{j}$ of two surfaces $A_{i}$ and $A_{j}$ , respectively, is considered and shown in Figure 7(a). The view factor $F_{i j}$ between two surfaces $A_{i}$ and $A_{j}$ is the fraction of the energy emitted diffusely by surface $A_{i}$ that reaches surface $A_{j}$ . The view factor is a result of the orientation of surfaces in space. It depends mostly on the geometry of the body and the incident and reflected radiations on the body. The view factor is completely geometric and independent of surface qualities and temperature.

Figure 7.

View factor calculation. (a) Heat exchange between elementary areas (b) Hemi-cube method.

A structure consists of a large number of elements and the irradiation and shade faces of the elements change continuously with the sun’s orientation. Thus, applying the solar radiation intensity to the FEM is the most difficult and challenging task of the thermal analysis. Common methods are as follows: (1) direct integration method, (2) unit sphere method, (3) view factor by using the ray casting method, (4) cross string method, (5) view factor by using the Monte Carlo method, and (6) view factor by using the algebraic rule and matrix formulation. This study adopts the hemi-cube method (Gupta et al., 2017), as shown in Figure 7(b).

The view factor is calculated using the hemi-cube method as

F_{i j} = \sum_{n = 1}^{N} ∆ F_{n} = \frac{\cos θ_{i} \cos θ_{j}}{π r_{i j}^{2}} ∆ A_{j}

(15)

where

N

is the number of pixels and

∆ F

is the delta-view factor for each pixel.

The speed and accuracy of the hemicube method for the view factor computation can be affected by changing the size and number of discrete areas on the faces of the hemicube. All view factors are required to calculate the radiation exchange in the enclosure of surfaces. The view factors on surfaces can be written in the following matrix form:

[\begin{array}{c} \begin{array}{c} F_{11} & F_{12} \end{array} & \dots & F_{1 N} \\ \begin{array}{c} F_{21} & F_{22} \end{array} & \dots & ⋮ \\ \begin{array}{c} \begin{array}{c} ⋮ \\ F_{N 1} \end{array} & \begin{array}{c} ⋮ \\ \dots \end{array} \end{array} & \begin{array}{c} ⋱ \end{array} & \begin{array}{c} F_{N N} \end{array} \end{array}]

(16)

FEM of the footbridge

A 3D FEM was developed using the commercial FE analysis software ANSYS 16.0. The arches, main beams, deck and columns are modeled using solid elements for refined analyses, such as buffeting analysis, and global 3D heat-transfer analysis, etc. The full 3D FEM consists of 8785 elements and 22,585 nodes, as shown in Figure 8.

Figure 8.

3D view of the FEM

In this study, the FEM will be used for the heat-transfer analysis and then structural analysis. The workflow is shown in Figure 9. In the heat-transfer analysis, the FEM consists of 3D thermal elements with appropriate thermal boundary conditions. The temperature distribution of the bridge will be calculated and verified using the measured temperature data. In the structural analysis, the configuration and mesh of the FEM remain unchanged, and the 3D thermal elements change to 3D mechanics elements. The calculated temperature in the previous step will be assigned to the FEM automatically, and the temperature-induced responses (such as displacement and stresses) will be obtained. The two analyses will be presented in Sections 5 and 6.

Figure 9.

Workflow of the unified FE simulation.

3D global heat-transfer analysis of the footbridge

Hottest day in 2020

As aforementioned, August 25, which was the hottest day in 2020, is studied. The most important task in the transient heat-transfer analysis is to determine the trajectory of the sun. First, according to the design of the footbridge, its azimuth is about 21° west-north, as shown in Figure 10. The geographical coordinates of the footbridge are about 114.17° east longitude and 22.3° north latitude. From the Hong Kong Observatory, sunrise and sunset on August 25, 2020 were at 6:03 a.m. and 6:46 p.m., respectively. The maximum solar altitude angle was 77.7° at about 12:26 p.m.

Figure 10.

Orientation of the footbridge.

The virtual solar orbit in different hours based on the abovementioned information is shown in Figure 11. In each hour, the view factors of all surface faces of the FEM are calculated according to equation (16). The heat exchange is then formulated using equation (15). The solar radiation and ambient temperature are measured from the monitoring system.

Figure 11.

Solar orbit on August 25, 2020.

The heat-transfer analysis is conducted on the FEM by using the boundary condition on the day. Solid70 elements are used. This type of element is 3D thermal solid element with eight nodes, each having one temperature degree of freedom. The temperature distribution of the whole bridge in each hour is obtained and illustrated in Figure 12. The main arches have higher temperature than other components because they receive the sunlight directly. As the solar orbit changes, the east side has higher temperature than the west side in the morning and lower in the afternoon.

Figure 12.

Temperature distributions of the footbridge on August 25, 2020.

Coldest day in 2020

From the Hong Kong Observatory, sunrise and sunset on February 17, 2020 occurred at 6:53 a.m. and 6:21 p.m., respectively. The maximum solar altitude angle is 55.1° at about 12:37 p.m. Figure 13 shows the solar orbit in different hours.

Figure 13.

Solar orbit on February 17, 2020.

The temperature distribution of the whole bridge on the day is then calculated and shown in Figure 14. The main arches again have higher temperature than other components because they receive the sunlight directly. The simulated structural temperature values will be compared with the measured ones in Section 7.1.

Figure 14.

Temperature distributions of the footbridge on February 17, 2020.

Temperature-induced responses of the bridge

Introduction of the structural analysis

As aforementioned, the structural analysis shares the same element mesh and configuration as the heat-transfer analysis in Section 5. The FEM for the heat-transfer analysis can be used for the structural analysis directly by revising the thermal elements as the mechanical elements, and the results of the heat-transfer analysis are transferred to the FEM as the input of the structural analysis. This unified heat-transfer and structural analyses have several advantages over the conventional divide-and-conquer approach: 1) re-building the FEM of the entire structure is avoided, 2) manual intervention and inputting temperature data are unnecessary, and 3) the 3D global heat-transfer analysis can obtain more accurate temperature distribution than the 2D or local 3D heat-transfer analysis.

The procedure of transferring the heat-transfer analysis to the structure analysis in ANSYS is described as follows:

1. Save the result file after the temperature field calculation.

2. Clear the physical environment variables and change the temperature unit to the structural unit.

3. Apply the structural boundary conditions. Bearing-X1 (BR-X1) can move in the X direction only. BR-X2, BR-Z1 and BR-Z2 can move in X and Y directions. All bearings can rotate freely. Four feet are fixed to the ground. The bearing location show in Figure 10.

4. The calculated temperature results are input to the FEM as the node temperature load. The time step and the time interval should be consistent with the temperature field calculation.

5. The element types are revised from Solid70 (3D thermal solid element) to Solid45 (3D structural solid element with eight nodes, each having three lateral displacement degrees of freedom).

When the temperature load is applied to the FEM, the structural analysis is conducted to obtain the structural responses (stresses and displacement). The results are shown and discussed in the following subsections. The heat transfer analysis and structural analysis over a 24-h period (one point per each hour) take 86 s and 76 s, respectively, using a laptop with a CPU of Intel® Core 17–1260P and 16 GB memory.

Stress of the bridge on the hottest day in 2020

August 25 was the hottest day in 2020. The calculated structural stress of the whole bridge from 8 a.m. to 7 p.m. is shown in Figure 15. The main arches exhibit higher structural stresses than other components because they have a higher temperature. As the solar orbit changes, the stress of the main arch changes accordingly. The two legs also have a relatively high stress, which is probably due to that the deformation of the feet is constrained.

Figure 15.

Structural stress distributions of the footbridge on August 25, 2020.

The maximum tensile stress of the whole footbridge is 31.7 MPa at 1 p.m. on the upper flange of the eastern beam. At the same time, the maximum compressive stress of the whole footbridge is 90.2 MPa at the bottom of the east column.

Stress of the bridge on the coldest day in 2020

February 17 was the coldest day in 2020. The calculated structural stress distribution of the bridge from 8 a.m. to 7 p.m. is shown in Figure 16. Again, the main arches exhibit a higher structural stress than other components. As the solar orbit changes, the structural temperature and the stress distribution of the main arches change accordingly. The two legs also have a relatively high stress, as explained above.

Figure 16.

Structural stress distributions of the footbridge on February 17, 2020.

The maximum tensile stress of the whole footbridge is 44.6 MPa at 1 p.m. on the upper flange of the eastern beam. At the same time, the maximum compressive stress of the whole footbridge is 82.8 MPa at the bottom of the east column. The maximum tensile stress on the coldest day is higher than that on the hottest day. Conversely, the maximum compressive stress on the hottest day is higher than that on the coldest day. Nevertheless, the maximum tensile stress on the coldest and hottest days occurs in the same locations. The same is observed on the maximum compressive stress. Furthermore, the simulated structural response values will be compared with the measured ones in Sections 7.2 and 7.3.

Comparison of numerical simulation and field measurement

Temperature comparison

The simulated temperature data are compared with the measured results. The temperature data at the eastern beam, western beam, and the concrete slab of Section C are compared as an example. The location of the strain gauges is shown in Figure 17. In Section C, SG–C2, SG–C4, and SG–C6 are under the top beam flange of GEOKON 4000 type, and SG–C7, SG–C8, and SG–C9 are inside the concrete deck of GEOKON 4200 type.

Figure 17.

Layout of the vibrating wire strain gauges.

Figure 18(a) shows the temperature of the eastern beam of the footbridge on August 25, 2020. The measured lowest temperature is 30.1°C and the highest temperature is 35.6°C, and the simulated counterparts are 30.9°C and 35.1°C. Their discrepancy is less than 1.0°C, and the variation trends are very close. Similarly, the temperature of the western beam is shown in Figure 18(b). The measured lowest temperature is 29.8°C and the highest is 36.2°C, and the simulated counterparts are 30.2°C and 35.5°C. Their variation trends are also consistent. The measured temperature variation of the beams on the day is about 6°C.

Figure 18.

Comparison of the measured and simulated temperature of the eastern and western beams on August 25, 2020.

The temperature at the middle height of the concrete slab is shown in Figure 19. The simulation results agree well with the measured ones, with a difference of less than 0.5°C. The measured temperature variation on the day is less than 3°C, which is smaller than that of the steel beams. The reason is that the concrete slab is paved by timber panels, and thus, the temperature variation is smaller.

Figure 19.

Comparison of the measured and simulated temperature of the concrete slab on August 25, 2020.

The measured and simulated temperature of the eastern and western beams on the coldest day are compared and shown in Figure 20. The measured lowest temperature at the east beam is 12°C and the highest is 16.7°C, and the simulated counterparts are 12.8°C and 16.8°C. Similarly, the measured lowest temperature at the west beam is 12.2°C and the maximum is 22.6°C, and the simulated counterparts are 12.5°C and 22.7°C. The variation trend of simulated temperature is consistent with that of measured temperature. The west beam has a higher temperature than the east beam because several buildings in the southeast of the bridge block the sunshine in the morning of winter. The west beam also has a more significant temperature variation (about 10°C) than the latter. In addition, the temperature variation on that day is larger than that of on August 25 because the air temperature variation on the former is larger than the latter (Figure 6(a)).

Figure 20.

Comparison of the measured and simulated temperature of the eastern and western beam on February 17, 2020. (a) Eastern beam (b) Western beam.

The temperature of the middle of the concrete slab of the footbridge is shown in Figure 21. The simulation results agree well with the measured results, with a difference of less than 0.5°C. The temperature variation of the slab is about 3.0°C, which is smaller than that of the beams because of the timber pavement on the concrete slab.

Figure 21.

Comparison of measured and simulated temperature of the concrete slab on February 17, 2020.

The measured and simulated temperature data at all cross sections of the bridge are compared and shown in Table 1. Only maximum and minimum values are given for simplicity. A significant discrepancy between the measured and simulated temperature is observed in Section A because GEKON 4150 sensors are installed in the section due to the small space there. The measurement accuracy of the sensors is inaccurate. These sensors will be replaced in the future. Except that in Section A, the maximum discrepancy is 1.4°C on the west side of Section F. The root mean square error (RMSE) of the whole day is also calculated for different sections. Most RMSE values are less than 1.0, with the largest being less than 1.2.

Table 1.

Measured and simulated temperature of each section (unit: °C).

Cross section (Direction)	August 25					February 17
	Maximum		Minimum		RMSE	Maximum		Minimum		RMSE
	Mea	Sim	Mea	Sim	RMSE	Mea	Sim	Mea	Sim	RMSE
A (E)	35.5	36.5	28.6	30.7	1.894	—	17.5	—	12.4	—
A (W)	38.9	36.7	30.8	28.3	2.433	16.3	18.3	12.4	10.7	1.616
B (E)	35.9	35	30.2	30.9	0.852	15.8	16.3	11.7	12.8	0.675
B (W)	36.4	37.2	29.9	30.8	1.043	19.2	20.3	11.9	12.4	1.155
C (E)	35.6	35.1	30.2	30.9	0.698	16.7	16.8	12	12.8	0.604
C (W)	36.2	35.5	29.8	30.2	0.524	22.6	22.6	12.2	12.4	0.299
D (E)	35.6	35.2	30.3	30.9	0.703	16.7	16.8	12.2	12.8	0.44
D (W)	39.1	39.2	32.1	31.3	0.857	21.9	21.4	12.2	12.6	0.658
E (E)	37.3	37.1	30.2	30.9	0.747	17	16.7	12.4	12.9	0.421
E (W)	34.1	34.5	29.9	30.8	0.987	22	22.3	12.4	12.4	0.705
F (E)	35.8	36.6	30.7	30.2	0.908	17.5	16.8	12.8	12.8	0.402
F (W)	35.4	34.1	30.8	30.7	0.369	18	17.4	13.9	12.5	0.634

Note: E is for the east side; W for the west; Mea and Sim denote measured and simulated data, respectively; RMSE refers to the root mean square error.

Stress comparison

Considering that temperature tends to cause static strain in a bridge, whereas pedestrians and winds cause dynamic strain, the hourly mean strain is calculated to remove the dynamic strain. The stresses at sensors C2 (western beam) and C4 (eastern beam) in Section C on the 2 days are shown in Figure 22. Here, the measured stress is calculated from the strain after the free expansion of the component is removed. In addition, the stress change is plotted by using the stress at 1 a.m. as the reference. The numerical and measurement results are close, and the variation trend is consistent. The stresses of the beams decrease when the temperature rises in the morning (in compression). The stresses of the beams increase when the temperature decreases in the afternoon (in tension). The stress variation of the west beam occurs 4–6 h later than that of the west beam. The reason is that the temperature variation of the west beam lags the east beam. Moreover, the stress variations of the east and west beams are 4 and 6 MPa, respectively, in the summer and 7 and 10 MPa, respectively, in the winter. This result is due to that the west beam has a higher temperature change than the east beam on both days, and winter has a higher temperature variation than summer, as shown in Table 1. Therefore, winter is more critical than summer in this case.

Figure 22.

Stress comparison at Section C on August 25 and February 17, 2020. (a) East beam on August 25 (b) Western beam on August 25 (c) Eastern beam on February 17 (d) Western beam February 17.

Displacement comparison

The cameras were debugged in 2020. Thus, the displacement data of the arches are unavailable. Only the displacement of the expansion joint is compared here. The measured expansion joint displacement and simulation results on the 2 days are shown in Figure 23. The displacements of the expansion joint are very small, which are about 1.2 and 1.5 mm on August 25 and 1 February 17, respectively. Figures 19 and 21 show that the temperature change of the concrete slab within 1 day (summer and winter) is very small, which is about 2°C. The resulting free thermal expansion of the deck can be estimated as 84×2×1.0×10⁻⁵ = 1.7×10⁻³ m, which is close to the measured value of 1.5 mm. This finding verifies the measurement and numerical results.

Figure 23.

Displacement of the expansion joint on August 25 and February 17, 2020. (a) Displacement on August 25 (b) Displacement on February 17.

Conclusions

Numerical analysis of heat conduction is an effective and important method for studying the temperature effect of structures. However, traditional thermal analyses of bridges adopt a divide-and-conquer approach. This approach conducts a simplified 2D or local 3D heat-transfer analysis and then a global 3D structural analysis by inputting the calculated temperature into the other bridge model with considerable manual intervention, which brings unavoidable manual calculation errors. This study develops a unified approach of heat-transfer and structural analyses for the first time to calculate the temperature distribution and the associated responses of an entire structure by integrating the field monitoring data. The arch bridge at PolyU is used as the testbed of the study. The calculated results of the temperature analysis and structural response agree well with the monitoring results. The main contributions and conclusions of this study are summarized as follows.

1. The temperature distribution of this arch bridge was studied on the basis of the 1-year field monitoring data during the service stage of the arch bridge. The differences between the measured highest and lowest temperatures at the beams on the hottest and coldest days are about 6°C and 5°C, respectively. On both days, the east and west beams have the similar temperature, whereas the maximum temperature occurs at different times due to the influence of the bridge orientation.

2. A fine 3D FEM was developed using the commercial FE analysis software ANSYS. The FEM facilitates detailed global 3D heat-transfer and structural analyses with the same meshes, such that the temperature data of the entire bridge can be directly used to calculate the temperature-induced responses. This study is the first to realize the 3D global temperature analysis in the field. The heat-transfer analysis can capture the temperature of the bridge accurately. In different seasons, the discrepancy of the measured and simulated temperatures is nearly less than 1.0°C, and their variation trends are similar. The RMSE of the temperature in each section is less than 1.2°C.

3. Subsequently, the structural analysis of the arch bridge was conducted without manual intervention. The calculated responses agree well with the measurement data. The maximum discrepancy of the stress does not exceed 1.5 MPa. The displacement of the expansion joint on the hottest day and the coldest day are the same, which is mainly due to the temperature variation of the concrete slab. It plays a major controlling role. Overall, the comparison results show the accuracy and practicability of the numerical method without human intervention proposed in this study.

The proposed unified approach of heat-transfer and structural analyses can calculate the detailed temperature distribution and the associated temperature-induced responses of the arch bridge accurately and efficiently. It can also be applied to other types of bridges.

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 study was supported by RGC-GRF (Project No. PolyU 152125/17E), PolyU Strategic Development Special Projects (Project Nos. 1-ZVFH and 1-ZVJN), the Jiangsu Provincial Double-Innovation Doctoral Program (Grant No. JSSCBS20210129), the Natural Science Foundation of Jiangsu Province (Grant No. BK20220852), and the Fundamental Research Funds for the Central Universities (Grant No.3205002208A1).

ORCID iDs

Qi Xia

Xiao-qing Zhou

You-lin Xu

Yong Xia

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