Thermal behaviors of bridges — A literature review

Abstract

This work provides a comprehensive review of previous studies concerning the static thermal behaviors of various types of bridges, including beam bridges, arch bridges, cable-stayed, and suspension bridges. Given that thermal behaviors are closely associated with the temperature distribution of bridges, the basis of the heat transfer analysis is briefly introduced first. The studies of the temperature distribution and the temperature actions of each type of bridge are then reviewed from the perspective of theoretical analysis, numerical simulation, experimental tests, and field monitoring. Finally, some existing problems are discussed, and future research topics are recommended.

Keywords

Bridge temperature action temperature distribution heat transfer analysis temperature gradient

Introduction

With the rapid development of material science, computer technology, and construction technology, an increasing number of bridges have been built to respond to the booming of the social economy. These bridges form the key elements of transportation systems. The newly built bridges typically have long spans and complex configurations, resulting in increasingly complex loading conditions and structural behaviors. The thermal load is a significant source of load that varies with location and time. Temperature can affect the bridge behaviors on the material level [material properties (Adams and Bacon, 1973)] and structural level [e.g., structural deformation (Potgieter and Gamble, 1983) and stress (Dilger et al., 1982; Liu and Zuk, 1963)]. The other types of loads have different effects on the bridge behaviors. For example, wind loads can be divided into two components, namely, static and dynamic forces. The former can induce quasi-static behavior. The latter may cause abnormal and even catastrophic vibrations that have led to the collapse of the Tacoma Narrow Bridge (Amman et al., 1941). Earthquakes may result in significant inertial forces and severely damage bridges (Çelebi, 2006). Meanwhile, traffic loads may influence the static and dynamic behaviors of bridges (Prat, 2001).

Temperature actions are typically coupled with other loads and even mask the effects of other loads or structural damage. For example, temperature and wind velocity have a significant influence on certain modal frequencies of the Tianjin Yonghe Bridge (Li et al., 2010). Meanwhile, the daily variation of the first-order frequency of the Alamosa Canyon Bridge [up to 6% (Cornwell et al., 1999)] due to temperature variation was more significant than that caused by the simulated failure of a connection [maximum 1.11% for the first five modal frequencies (Doebling et al., 1996)]. Accordingly, the influence of temperature should be separated from other load effects to conduct an accurate assessment of the effect of individual load sources or damage levels (Su et al., 2017). Consequently, temperature monitoring should be incorporated into the structural health monitoring (SHM) systems. The structural temperatures and environmental factors, such as ambient temperature, wind speed and direction, and solar radiations, are monitored along with other loadings and structural responses. For example, a comprehensive SHM system devised for the Tsing Ma Bridge was installed in 1997 and has been operating since then, as shown in Figure 1 (Xu and Xia, 2011). A total of 276 sensors were installed, including six anemometers and 115 thermometers.

Figure 1.

Layout of sensors on the Tsing Ma Bridge (Xu and Xia, 2011).

Since the middle of the 20^th century, researchers have investigated the temperature behaviors of bridges (Zhou and Yi, 2013; Zuk, 1965). The research focuses are the temperature distribution in the bridges and the temperature actions (i.e., the influence of temperatures on the structural responses). The temperature distribution is usually investigated through numerical simulations (i.e., transient heat transfer analysis) or lab/field monitoring because it is practically difficult to obtain theoretical solutions. The temperature distribution of the bridges can be complex due to the complicated bridge configuration and environmental conditions.

The temperature of a bridge can influence its dynamic (or modal) properties and static behaviors. The former effect has been widely studied because the dynamic properties are widely used in vibration-based damage detection. Given that the variation in the modal properties caused by the varying temperatures may mask those caused by the structural damage (Farrar et al., 1994; Peeters et al., 2000, 2001), the effects of temperature on damage identification should be eliminated or minimized (Bao et al., 2012; Fan et al., 2020; Huang et al., 2018b). This topic has been reviewed by Xia et al. (2012), Su et al. (2020), and Luo et al. (2022). The emphasis of this work will be laid on the static temperature behaviors of bridges, including the temperature-induced deformations and stress/strain in the bridges.

There are four predominant types of bridges in service nowadays, namely, beam bridges, arch bridges, cable-stayed bridges, and cable-suspension bridges. These bridges differ from each other in terms of temperature behaviors. Beam bridges have simple structural composition and relatively small spans. They usually have a uniform temperature distribution along the longitudinal direction, a vertical temperature gradient, and a possible transverse temperature gradient on the cross-section. The temperature distribution feature generally causes the bridge displacement in the longitudinal direction and curvature in the vertical or transverse direction. The arch surfaces of an arch bridge along the circumferential and axial directions receive different radiations due to different orientations. Consequently, the temperature gradient exists in both circumferential and axial directions. In addition, the movement of both ends of the arch is generally constrained, causing the arch top to move upward or downward. The temperature behaviors of cable-suspension bridges are complicated because the main cables play an important role in the structural system. The main towers also have different temperature distributions and actions from those of the main cables and bridge deck. For cable-stayed bridges, the temperature actions of the bridge deck are subject to the constraint from multiple stay cables, which differs from the main cables’ thermal action of cable-suspension bridges. Moreover, the multiple stay cables increase the structural redundancies and the complexity of the thermal behaviors.

The paper is organized as follows. Previous studies on the temperature distributions and temperature actions are reviewed in the following two sections, respectively. In each section, studies are grouped according to the bridge types and further according to the study approaches (namely, the theoretical analysis, numerical simulation, lab tests, and field monitoring). The conclusions and future recommendations are drawn in the last section.

Temperature distribution

The temperature distribution is generally investigated through numerical studies or by deploying thermometers in lab specimens or real bridges. However, the temperature distributions cannot be fully captured due to the limited number of thermometers,. In this regard, the transient heat transfer analysis can be employed and verified by the measurement data. In this section, the theoretical basis of the heat transfer analysis will be given briefly, followed by the relevant studies on the temperature distributions of different types of bridges.

Theoretical basis for numerical simulation

Governing equations

The flow of heat in the isotropic solid is governed by the Fourier partial differential equations (Whitaker, 2013) as follows:

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 x, y, and z are the Cartesian coordinates in three directions, t is the time, T is the temperature of the solid, ρ is the material density, c is the specific heat of the solid (J/kg/°C), and k is the thermal conductivity (W/m/°C). The isotropy assumptions are valid for conventional bridge materials, such as concrete and metallic materials.

Thermal boundary conditions

The third type of thermal boundary conditions applies to bridges as follows (Elbadry and Ghali, 1983):

k (\frac{\partial T}{\partial x} n_{x} + \frac{\partial T}{\partial y} n_{y} + \frac{\partial T}{\partial z} n_{z}) + q = 0,

(2)

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

(3)

where (n_x, n_y, n_z) forms the unit outward normal vector of the boundary surface, and q is the heat exchange rate between the surface and the environment per unit area (W/m²). On a bridge surface, q may be divided into three components (Branco and Mendes, 1993), namely, thermal radiation intensity q_r, convection q_c, and solar radiation q_s, as shown in Figure 2. Each of these quantities varies with the boundary surface and time.

Figure 2.

Heat exchange of a bridge’s cross-section (Xu and Xia, 2011).

Radiation heat transfer

The nonlinear irradiation heat transfer between the bridge and the surrounding atmosphere due to longwave radiation (i.e., thermal irradiation) can be modeled by the Stefan–Boltzman radiation law (Dilger et al., 1982) as follows

q_{r} = C_{s} ε [{(T_{a} + K_{0})}^{4} - {(T_{a} + K_{0})}^{4}],

(4)

where C_s = 5.677 × 10⁻⁸ W/(m²K⁴) is the Stefan–Boltzman constant, ε ϵ (0,1) is the relative emissivity coefficient of the bridge surface, T_a is the ambient temperature, and K₀ = 273.15 converts the temperatures in degree Celsius to degree Kelvin. Equation (4) can be rewritten in a quasi-linear form for bridges (Elbadry and Ghali, 1983):

q_{r} = h_{r} (T_{a} - T),

(5)

where h_r is the radiation coefficient approximated as follows

h_{r} = ε [4.8 + 0.075 (T_{a} - 5)] .

(6)

Convection heat transfer

The convection heat transfer q_c depends on the difference between the structural temperature T and ambient temperature T_a. This factor can be computed following Newton’s convection law (Jiji, 2009) as follows

q_{c} = h_{c} (T_{a} - T),

(7)

where h_c is the convection coefficient associated with multiple environmental parameters.

Given the similar formation of equations (5) and (7), the heat transfer by convection and irradiation can be combined with an overall heat transfer coefficient as follows

h = h_{c} + h_{r} .

(8)

The overall heat transfer coefficient h significantly influences the heat transfer between the bridge and the environment. However, the convection coefficient h_c is associated with multiple environmental parameters, such as the wind speed v, flow configuration, surface roughness, thermal properties of the air, and temperature difference. Empirical formulae have been adopted for practical use (Table 1). The formulae can be classified into three types: a) wind-dependent, b) temperature and wind-dependent, and c) temperature and wind-independent (optimized constant values for different surfaces). The first type of formulae can be understood as two components, namely, the natural convection independent of the wind speed and the forced convection linearly/quasi-linearly dependent on the wind speed. The second type also incorporates the ambient temperature in calculating the coefficient. The last type represents constant heat transfer coefficients that are independent of ambient temperature and wind speed.

Table 1.

Formulae for heat transfer coefficient.

Type	Formulae	Component	Ref.
Wind-dependent	$h_{c} = 4.59 v + 13.22$ ^*	Concrete box girder	Song et al. (2012)
	$h_{c} = {\begin{array}{c} 4.0 v + 5.6 & (v \leq 5.0 m / s) \\ 7.15 v^{0.78} & (v > 5.0 m / s) \end{array}$	Concrete bridge	Froli and Hariga (1993)
	$h_{c} = {\begin{array}{c} 4 v + 6 & (v \leq 5 m / s) \\ 7.4 v^{0.78} & (v > 5 m / s) \end{array}$	Concrete arch	Larsson and Karoumi (2011)
	$h_{c} = {\begin{array}{c} 3.95 v + 5.6 & (v \leq 5.0 m / s) \\ 7.6 v^{0.78} & (v > 5.0 m / s) \end{array}$	Concrete cable-stayed bridge	Sousa Tomé et al. (2018); Jonasson (1994)
	$h_{c} = 3.9 v + 5.58$	Concrete bridge tower	Li et al. (2019)
	$h_{c} = 3.88 v + 13.5$	Concrete girders	Potgieter and Gamble (1983); Potgieter and Gamble (1989); Moorty and Roeder (1992); Froli et al. (1996)
	$h_{c} = {\begin{array}{c} 3.83 v + 4.67 & (top surface) \\ 3.83 v + 2.17 & (bottom surface) \\ 3.83 v + 3.67 & (vrtical surface) \\ 3.5 & (inner surface) \end{array}$	Concrete box girder bridges	Mirambell and Aguado (1990); Abid (2015)
	$h_{c} \approx 3.7 v$	Concrete bridge	Branco and Mendes (1993)
	$h = {\begin{array}{c} 3.6 v + 11 & (top surface) \\ 9.5 & (bottom surface) \end{array}$	Concrete bridge	Ho and Liu (1989)*
Temperature- and wind-dependent	$h_{c} = 2.6 (1.54 v + \sqrt{\| T_{a} - T \|})$	Cable-stayed bridge	Liu et al. (2014)
	$h_{c} = 2.6 (1.54 v + 2 \sqrt{\| T_{a} - T \|})$	Steel box girder	Huang and Zhu (2021)
	$\begin{array}{l} h_{c} = 2.5 (1.54 v + \sqrt[4]{\| T_{a} - T \|}) \\ v = {\begin{array}{c} 2.5 & (h e i g h t \leq 10 m) \\ 2 . 5^{(h e i g h t / 10) c u} & (h e i g h t > 10 m) \end{array} \end{array}$ cu = surface roughness coefficient v = wind speed	Concrete bridge and cable-stayed bridge	Zhu and Meng (2017)
	$h_{c} = {\begin{array}{c} 2.6 (1.54 v + \sqrt[4]{\| T_{a} - T \|}) & (v \leq 5 m / s) \\ 6.31 v^{0.656} + 3.25 \times e^{- 1.91 v} & (v > 5 m / s) \end{array}$	Suspension bridge	Zhou et al. (2015)
	$\begin{array}{l} h_{c} = C \times 0.2782 \times {(T_{a} + 17.8)}^{- 0.181} \\ \times {\| T - T_{a} \|}^{0.266} \times \sqrt{1 + 2.8566 v} \\ C = {\begin{array}{c} 10.15 & (\begin{array}{l} hot bottom surface \\ or cool top surface \end{array}) \\ 15.89 & (vertical surface) \end{array} \end{array}$	Vertical concrete surfaces	Riding et al. (2007)
Temperature and wind independent*	$h_{c} = {\begin{array}{c} 18.5 & (suspension cable) \\ 26 & (deck) \\ 25 & (tower) \end{array}$	Suspension bridge with truss girder and extra stay cables	Westgate et al. (2015)
	$h = {\begin{array}{c} 6.8 & (steel) \\ 17.5 & (asphalt) \end{array}$	Steel girder	Tong et al. (2002)
	$h = {\begin{array}{c} 18.5 & (top surface) \\ 6.8 & (bottom surface) \\ 1.2 & (open air enclosure) \\ 0.25 & (closed air enclosure) \end{array}$	Steel girder	Tong et al. (2001)
	$h = {\begin{array}{c} 18.14 \sim 19.59 & (top surface) \\ 12.17 \sim 12.89 & (bottom surface) \\ 15.96 \sim 17.04 & (web) \\ 9.5 & (girder internal) \end{array}$	Concrete box-girder arch	Wang et al. (2016)

^*Including the convective and radiative parts.

The dependencies of the heat transfer coefficients on the wind speed significantly vary. The differences among these formulae may arise from the complicated wind field (e.g., turbulences) near the bridge and the wind sensor position, the material properties, the surface roughness conditions, and other bridge configurations.

Another important factor is that the wind speed on different surfaces may vary. A previous study empirically modified the measured wind speeds according to the incidence angle φ as follows (Zhou et al., 2015)

v = {\begin{array}{c} 0.8 v_{0} & (φ \leq 45 °) \\ 0.7 v_{0} & (45 ° < φ \leq 90 °) \\ 0.6 v_{0} & (φ > 90 °) \end{array},

(9)

where v₀ is the original wind measurement.

Alternatively, to obtain the wind speed on different surfaces and avoid manual modification, Huang and Zhu (2021) conducted computational fluid dynamics and estimated the wind speed at the thermal boundary layer before the heat transfer analysis.

Given the considerable divergence of these empirical formulae, more detailed and in-depth discussions of the heat transfer coefficient calculation are recommended. Alternatively, inverse parameter identification can also be conducted. Meanwhile, the wind field and wind speed variation in bridge surfaces for different bridge orientations and anemometer configurations also merit to consider.

For bridges with solid cross-sections, only the thermal boundary conditions on the outer surfaces need to be determined. By contrast, the thermal boundary conditions inside the girder/arch boxes also need to be carefully considered. Although the solar radiation and wind speed are zero inside the box, the air temperatures near different internal surfaces are different, especially between the top and bottom surfaces. However, the temperature inside the enclosure is either monitored with a few thermometers or not measured at all, which means that the temperature distribution across the enclosure is unavailable.

In this case, the inner boundary conditions either utilize the single measurement directly or are determined from the measurement manually. To address this issue, Larsson and Karoumi (2011) modeled the ambient temperatures inside the box arch using an iterative approach by combining equations (4) and (7). Another solution is to use the air elements inside the enclosure. For example, Zhou et al. (2015) used the conduction of air elements to simulate the convection heat transfer and the radiation matrix method to account for the radiative heat transfer. The results presented by the iterative method and the air element method agreed well with their measurement counterparts. However, the air element method is unsuitable for 3D models because it will introduce a large number of air elements.

Solar radiation and equivalent temperature

The solar radiation thermal heat transfer q_s can be computed as follows

q_{s} = α I_{t},

(10)

where α is the absorptivity coefficient, and I_t is the total radiation received by the bridge surface, including the direct, diffuse, and reflected solar radiations (Baba and Kanayama, 1987).

Then, equation (2) can be rewritten as follows

k (\frac{\partial T}{\partial x} n_{x} + \frac{\partial T}{\partial y} n_{y} + \frac{\partial T}{\partial z} n_{z}) = h (T_{e q} - T),

(11)

T_{e q} = T_{a} + α I_{t} / h,

(12)

where T_eq is referred to as the equivalent temperature.

The total radiation received by a surface can be categorized into three components, namely, direction radiation, diffuse radiation, and reflected radiation. The first two components or their summations for horizontal surfaces are available from the local meteorological bureau, while the third can be derived from them. These radiation data can also be measured with a pyranometer set, as shown in Figure 3. The total, diffuse, and reflected radiations received by the horizontal surface are measured by the pyranometer set. Then, these components of the tilted (or vertical) surfaces can be derived according to the geometric relationship and the atmospheric conditions following the methods presented by Yik et al. (2013). Consequently, the complete boundary conditions required in equation (11) are obtained, and the heat transfer analysis of the bridge can be conducted.

Figure 3.

Configuration of a set of pyranometers (Xu and Xia, 2011).

When the measurement of solar radiation is unavailable, the theoretical radiations can be assumed according to the location and configurations of the bridge following the clear-sky model presented by the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE, 2021) or similar variants (Chow et al., 2005; Iqbal, 2012; Sousa Tomé et al., 2018).

The solar radiation significantly influences the ambient temperature of different bridge surfaces. However, sometimes only the total radiation measurement of a horizontal surface is available. In this case, the total radiation received by different tilted surfaces needs to be modified from this value. The clearness/cloud condition, typically derived from the temperature difference between the shined and shaded surfaces, should be considered in the modification. For example, Westgate et al. (2015) considered the cloud condition from the temperature difference between the suspension cable and the truss. The temperature difference history with a duration of 7 weeks was processed to a normalized value and used to modify the solar radiation. Similarly, Xia et al. (2020a) used the temperature difference between the two anchors to modify the solar radiation.

According to the aforementioned issues, the thermal boundary conditions of a real bridge are related to several environmental factors in a non-straightforward way. Therefore, the thermal boundary conditions need to be carefully determined in the numerical simulation studies.

Temperature distribution types

The temperature distribution on any bridge components can be divided into the uniform part and the nonuniform (linear and nonlinear) part (also referred to as temperature gradients), as shown in Figure 4. The uniform temperature variation induces uniform deformations, such as the longitudinal elongation/contraction. By contrast, the temperature gradients typically induce bending deformation in the bridge.

Figure 4.

Temperature components of a temperature profile in a generic section (CEN, 2003).

The temperature distribution in various types of bridges will be reviewed in the subsequent section, and these studies will be classified according to their study approaches. Given the difficulty in obtaining analytical solutions, numerical, experimental, and field monitoring studies are generally adopted.

Beam bridges

During the ancient era, people used to build bridges by placing a block of wood or stone on both sides of a valley or river for walking and transportation. With the development of science and technology, different types of bridges with larger spans emerged, forming the key elements in the transportation system. The typical types include beam bridges, arch bridges, cable-stayed bridges, and cable suspension bridges.

The beam bridges have simple structural compositions: abutments (piers) and girders (decks). The girders in the form of solid plates, boxes, or trusses are simply supported or continuously laid over the piers/abutments. Beam bridges, as the simplest and oldest bridge type, are widely used in small- and medium-span bridges because of their advantages of simple configuration, convenient construction, and low maintenance cost. A simply supported beam bridge generally crosses no more than 50 m long, whereas a continuous type can stride a span of up to 400 m.

Numerical studies

Dilger et al. (1982) conducted a series of numerical parametric studies to find the extreme temperature gradient in composite box girder bridges. The girder cross-section consisted of steel boxes and concrete decks. The considered parameters included the orientation and geometric configuration of the box cross-section (overhang length, slope of the webs, and box cell quantity), material properties (radiation absorptivity of steel and concrete surface under different surface conditions), and environmental conditions (daily temperature variations, wind speed, turbidity factor, and seasons). They reported extreme temperature gradients and the corresponding stress levels in the structures under the worst combination of these factors. Elbadry and Ghali (1983) conducted a similar numerical parametric study on reinforced concrete box bridges with similar parameters. They concluded that the temperature distribution and the corresponding thermal stresses and curvatures were similar for the solid, cellular slabs, and box girders.

Riding et al. (2007) systematically presented a method for determining the surface temperature of concrete. Complete radiation sources were considered, including solar, atmospheric, surface-emitted, and formwork radiation. In addition, wind speed, ambient temperature, and surface roughness were included in the convection model. Specially, a roughness multiplier of 1.52 was used for concrete, and a roughness multiplier of 1 was used for steel. The results obtained with the proposed model agreed well with the measurements obtained with 12 concrete members. Similar roughness multipliers were also adopted by Zhu and Meng (2017).

Experimental studies

Zuk (1965) comprehensively analyzed the temperature behaviors of composite bridges (concrete decks supported by steel or prestressed concrete beams). The vertical temperature distribution was determined by incorporating the steady-state solutions with the top and bottom surface temperatures obtained by the empirical formulae. Similar bridges insulated with a 1-inch-thick sprayed foam urethane coating on the bottom surfaces were also analyzed. They reported that the temperature gradients were significantly reduced by the insulation.

Berwanger (1983) conducted a numerical and experimental investigation of the temperature distribution and the temperature-induced moment, stress, and strain in the composite slab-beam bridges. A 0.354:1 scaled model of a three-span continuous bridge was cooled by placing ice on the top of the slap. The strain, temperature, and displacement were measured during the 77 min cooling period. The process was also simulated using a 2D FE model. High accuracy was obtained with a 0.99 chi-squared probability of correlation between the numerical and the measured temperatures. They also reported that the measured deflections were smaller than the computed values, which they attributed to concrete swelling due to water seepage.

Priestley (1978) presented an early review of the design codes regarding thermal loading in prestressed and reinforced bridges. They reported that the vertical temperature distribution pattern of a fifth-order temperature could accurately predict the critical conditions for all sections. Then, they developed a general analytical method for predicting temperature-induced stress in concrete bridges. The method was validated by the measurement collected in a one-quarter scale box girder model. The method enabled the designers to accurately estimate the thermal effects for given locations and geometrics.

Field monitoring studies

Hoffman et al. (1980) presented the field monitoring results of an experimental segmental box girder bridge. Despite the bridge being curved in the south–north direction, no significant longitudinal and transverse temperature differences were observed. The observations indicated that the critical vertical temperature distribution could be approximated by a fifth-order polynomial, which was consistent with the New Zealand specifications and the conclusion by Priestley (1978). The temperature distribution on the slab above the box cell followed a linear distribution. They remarked that the temperature distribution and curvature pattern provided by the prestressed concrete institute and post-tensioning institute did not match.

Statistical analysis

The above-mentioned studies generally focus on an individual bridge (bridge model). The statistical characteristics of bridges have also been studied.

Au et al. (2002) comprehensively investigated the effective temperature and temperature gradients of composite bridges in Hong Kong, a tropical city. The extreme value analysis was carried out based on the statistics of extremes and numerical integration. The factors affecting the design effective temperature were also reviewed, and suitable values for Hong Kong were proposed. The design temperature profiles, namely, the fifth-order polynomials, for various types of composite bridge decks with bituminous surfacing and concrete slabs of different thicknesses were prosed. The method could also be applied to create zoning maps for temperature loadings for large countries with significant differences, such as the United States.

Potgieter and Gamble (1989) used the data of the 26 solar meteorology stations to compute the probable temperature distributions of bridges located in different parts of the United States. The combination of the weather conditions of clear days with high solar radiation and low wind speed would result in extreme temperature distributions in bridges. They reported that the nonlinear temperature distributions in the bridges along the coast were much less critical in design.

Tong et al. (2002) conducted the extreme value analysis based on the statistics of extremes and the numerical analysis of typical plate-girder and box-girder sections. They developed the fifth-order design temperature profiles for open and closed steel sections, with adjustments for various thicknesses of bituminous surfacing. The method could facilitate the development of site-specific temperature profiles for code documents and be applied to create zoning maps for temperature loading for large countries, such as the United States, as an extension of the work performed by Potgieter and Gamble (1989).

Typically, the temperature along the transverse direction is assumed as uniform. However, Song et al. (2012) reported that the transverse temperature difference was non-negligible. They studied the temperature distribution and action of a high-performance concrete box girder with deep box cross-sections and compared them with the Chinese and international design codes. The existing codes overestimated the temperature effects and were inappropriate for the considered girder. A method was proposed to determine the temperature actions for the girder based on the stationary binomial probability model of a random process. The standard values for the maximum vertical and transverse temperature gradients corresponding to the cumulative probability of 98% and their equivalent linear counterparts were summarized.

These studies analyzed the extreme values of the long-term structural and environmental temperature data of single or multiple zones, which could be adopted in the code documents of design.

Generally, the temperature distributions in the beam bridges were studied using numerical approaches, lab tests, and field monitoring techniques. The uniform components and the gradients are mainly concerned. Similar studies on temperature distributions in beam bridges were also conducted by other researchers (Abid, 2015; Branco and Mendes, 1993; Dai et al., 2022; Fu et al., 1990; Froli et al., 1996; Ho and Liu, 1989; Kennedy and Soliman, 1987; Kromanis et al., 2015; Mirambell and Aguado, 1990; Potgieter and Gamble, 1983; Roeder, 2003; Tong et al., 2001; Wollman et al., 2002; Zhang et al., 2021a). These studies are summarized in Table 2.

Table 2.

Temperature distributions in beam bridges.

Bridge	Focus	Study approach	Ref.
Composite box girders	Extreme temperature gradients in the composite box girder bridges of different configurations	Numerical parametric study	Dilger et al. (1982)
Concrete box girders	Extreme temperature gradients in the concrete bridges of different configurations	Numerical	Elbadry and Ghali (1983)
Concrete members	Surface temperature of concrete considering full boundary conditions	Numerical	Riding et al. (2007)
Composite bridges	Temperature behaviors under different insulation conditions	Experimental and numerical	Zuk (1965)
Composite slab-beam bridge	Temperature distribution and temperature-induced moment, stress, and strain	Experimental and numerical	Berwanger (1983)
Concrete box and π beam bridge	Vertical distribution of the thermally induced stress in bridges	Theoretical and experimental	Priestley (1978)
Curved prestressed concrete box girder	Temperature distribution and deflection in a curved box girder bridge	Field monitoring	Hoffman et al. (1980)
Steel I beam and box girder bridge	Thermal behavior of steel bridges in the tropical region	Numerical and statistical	Au et al. (2002)
Various bridges	Temperature distributions in the bridges in different locations	Statistical and numerical	Potgieter and Gamble (1989)
Steel I beam and box girder bridges with concrete decks	Thermal behavior of steel bridges in the tropical region	Numerical and statistical	Tong et al. (2002)
Concrete box girder with deep box	Vertical and transverse temperature gradients and their statistical characteristics	Numerical, field monitoring, and statistical	Song et al. (2012)
Kishwaukee river bridge	Structural responses to the nonlinear temperature distributions	Numerical and field monitoring	Potgieter and Gamble (1983)
Concrete bridges	Temperature distribution and inspection of the code specifications	Numerical parametric study	Branco and Mendes (1993)
Casilina bridge	Temperature distribution	Numerical and field monitoring	Froli et al. (1996)
Concrete box girder bridges	Temperature distribution	Numerical and field monitoring	Mirambell and Aguado (1990)
Concrete box girder	Temperature distributions and variations in concrete bridges under environmental thermal loads	Numerical and experiment	Abid (2015)
Various highway bridges	Statistical model of extreme thermal loadings in highway bridges	Numerical and statistical	Ho and Liu (1989)
Steel box and π girder bridges	Temperature distribution	Numerical and experimental	Tong et al. (2001)
Various bridges	Statistics of temperatures in various bridges	Statistical	Roeder (2003)
High-speed railway bridge	Extreme value model for uniform temperatures in bridge piers	Field monitoring and statistical	Dai et al. (2022)
Composite concrete slab on steel I beam bridge	Simplification of temperature distribution along the depth	Mathematical simplification	Kennedy and Soliman (1987)
Composite bridges	Numerical parametric study of temperature distribution and induced stresses	Numerical and parametric	Fu et al. (1990)
Concrete box girder bridge	Temperature gradients and correlations between ambient and structural temperatures	Field monitoring, correlation analysis	Wollman et al. (2002)
Cleddau bridge	Quasi-static temperature effects on the bearing movements	Numerical and field monitoring	Kromanis et al. (2015)
Xiaoqing river no. 3 bridge	Effect of encased concrete on the temperature gradient	Field monitoring and statistical	Zhang et al. (2021a)

Arch bridges

Arch bridges enlarge the span of bridges by utilizing the material’s compressive capacity and the arch’s geometry stiffness. The longest arch bridge in the world at present is the Chaotianmen Bridge, which was completed in 2009 in China with a main span of 552 m. Arch bridges are built in the valleys or have strong abutments due to the relatively large horizontal forces at the foot supports. These bridges can also be self-anchored. The temperature may considerably influence the deformation and stresses of arch bridges.

Numerical studies

The temperature distribution of arch bridges has been less studied than other types of bridges. Most of the studies are numerical.

Larsson and Karoumi (2011) investigated the temperature distribution of the hollow concrete arch of the New Svinesund Bridge, in which the ambient temperature inside the enclosure was unavailable. They calculated the ambient temperatures inside the box arch using an iterative manner by combining equations (4) and (7). Moreover, they also calculated the temperature distribution with the temperature measurement inside the cavity. The difference in the concrete temperature obtained with the two methods was insignificant. They reported that the vertical temperature gradients were more significant than the horizontal gradients in the arch.

Wang et al. (2016) numerically analyzed the temperature distribution and thermal stress in the arch of a concrete box arch bridge. The temperature distribution along the arch axis was non-uniform because of the different azimuth and inclination angles at the top and bottom of each cross-section. They showed that the thermal field was influenced by material and meteorological parameters, such as total heat transfer coefficients and comprehensive atmospheric temperatures. The vertical temperature gradient in the middle web was expressed as a negative index function, while that in the end web as a cubical parabola function. Horizontal temperature gradients also existed near the foot of the arch where the width was large.

Field monitoring studies

Xia et al. (2022) studied the temperature behavior of an arch footbridge through the integration of 1-year field monitoring data and unified numerical simulation. The sensor layout of the bridge is illustrated in Figure 5 (Xia, 2019). They reported that the difference between the measured highest and lowest temperatures at the bridge on the hottest and coldest days was about 6 °C and 5 °C, respectively. The root mean square error of the temperature computed from the global 3D unified numerical simulation and the measured temperature in each section was less than 1.2 °C. In addition, the maximum stress discrepancy between the FE analysis and the SHM measurement was less than 1.5 MPa, demonstrating the unified model’s accuracy in capturing arch bridge temperature behavior.

Figure 5.

Layout of sensors on the arch bridge (Xia, 2019).

Statistical analysis

Zhou et al. (2017b) developed a generalized Pareto distribution-based extreme value model for the thermal gradients in the prestressed concrete girder of Jiubao Bridge using the Bayesian estimation. They reported that the conventional normal and logistic distributions were incapable of capturing the statistical characteristics of the temperature gradients. In addition, the generalized extreme value distribution and the Gumbel distribution could not describe the daily maximum temperature gradients. Accordingly, they proposed to use the generalized Pareto distribution to fit the excesses of the thermal gradients over a chosen threshold. The distribution parameters were obtained using the Bayesian estimation and showed superiority over results obtained using the maximum likelihood estimation. The extreme values of the vertical temperature gradients were less than 15.1 °C within a 100-year return period because both sides of the girder could absorb the solar radiation due to the north–south direction of the bridge.

Zhu et al. (2021) statistically studied the effective temperature and temperature gradients in the truss arch and deck of Dashengguan Yangtze River Arch Bridge. They concluded that the probability distribution of the effective temperature could be described by the normal distribution. Meanwhile, the distribution of temperature gradients could be described by a weighted sum of two lognormal distributions. Furthermore, extreme values of the effective temperature and temperature gradients did not concurrently occur. The temperature of the shaded components could be assumed uniform, while the effective temperature and temperature gradients of those components exposed to solar radiation significantly varied, highlighting the influence of the solar radiation.

In summary, the temperature distributions in arch bridges have been numerically investigated and validated by the measurement counterparts. The statistical characteristics of the measurements were derived. These studies are summarized in Table 3.

Table 3.

Temperature distribution in arch bridges.

Bridge	Focus	Study approach	Ref.
New Svinesund bridge	Temperature distribution in the hollow concrete arch	Numerical and field monitoring	Larsson and Karoumi (2011)
Concrete box arch bridge	Temperature distribution and the thermal stress	Numerical	Wang et al. (2016)
Steel arch bridge with hangers	Temperature distribution and temperature-induced stress and deflection	Numerical and field monitoring	Xia et al. (2022)
Jiubao bridge	Extreme value model for the thermal gradients using the Bayesian estimation	Statistical and field monitoring	Zhou et al. (2017b)
Dashengguan Yangtze river arch bridge	Probability distribution of effective temperature and temperature gradients	Statistical and field monitoring	Zhu et al. (2021)

Cable-stayed bridges

Cables are widely used in modern structures. They allow for lightweight solutions with high material efficiency and contribute to aesthetically appealing designs. A cable-stayed bridge is a structural system with a bridge girder supported by stay cables from towers. The stay cables can be viewed as alternatives to the piers, which reduce the girder span and thus the moments. Multiple towers may be arranged, and continuous spans can be realized, such as the Millau Viaduct in France (Cachot et al., 2015), which has seven towers and eight continuous spans (204 + 342 × 6 + 204 m span arrangement). The bridge girders may also have multiple choices, such as the slab type, box type, and double deck made of trusses. Multiple cables significantly increase the redundancy of the bridge, increasing the mechanical complexity of the structure. Consequently, the temperature behaviors may be more complicated. Comprehensive SHM systems (Cao et al., 2011; Wong, 2004, 2007) have been installed to evaluate the performance of these bridges.

Given the difficulties in fabricating physical models of the cable-stayed bridges, numerical simulations and field monitoring have been conducted to study the temperature characteristics of cable-stayed bridges.

Numerical studies

Li et al. (2019) numerically studied the temperature behavior of the concrete tower of the Qingzhou Navigation Bridge, which is one of the three cable-stayed box girder bridges of the Hong Kong−Zhuhai−Macau Bridge. The FE model of one tower was built in ABAQUS, and a detailed heat transfer analysis considering all theoretical/assumed environmental conditions was conducted. The radiative and conductive heat transfer were not combined following the method previously introduced; instead, the structural temperatures in equations (4) and (7) were iteratively solved. A good agreement between the measured and the computed temperatures was obtained. A large temperature gradient along the thickness direction was reported due to the large cross-section of the tower and low thermal conductivity.

Field monitoring studies

Wang et al. (2019) studied the temperature field of the prestressed concrete girder in the Puxi Cable-stayed Bridge and the thermal effects on the cable forces by using the field monitoring data when the bridge was not open to the public. They reported a three-piece vertical temperature distribution pattern instead of a two-piece one as specified by the Chinese code. During sunny days, the vertical temperature gradient between the top surface and webs might reach a maximum value of 15.4 °C, and the vertical temperature gradient between the bottom surface and webs might reach 10.3 °C.

Statistical analysis

Mei et al. (2021) presented the temperature monitoring results of the Qingshan Yangtze River Bridge during the construction period. They proposed to use the cubic spline function to describe the temperature time history of the bridge based on the daily extreme temperature values and the temperature change rate. They reported that the maximum daily temperature difference of the top plate of the mid-span (steel box girder part without pavement) was 42.8 °C during the construction period. In addition, the top plate temperature might rise at a rate of 9.6 °C/h and drop at 19.73 °C/h, indicating a rapid heat transfer. The side spans of the bridge consisted of steel–concrete composite beams, and the temperature variation decreased after the concrete was cast. In particular, the PE sheath’s color significantly influenced the stay cables’ temperature characteristics. Specifically, the maximum daily variation of the center temperature in the cable with 127 wires and the white sheath was 19.10 °C, while that in the cable with 187 wires (thus a larger cross-section) and the blue sheath was 23.93 °C.

Cao et al. (2010) reported the SHM results of the Zhanjiang Bay Cable-Stayed Bridge. The concrete temperature significantly lagged behind the ambient temperature (5–6 h), and no noticeable temperature lag can be observed between the stay cables and the tower. The daily temperatures of the cables were between those of the tower and ambient air. Similarly, Yang et al. (2018a) reported that the structural temperature lagged behind the ambient temperature by 3.2 h, while the girder deflection lagged the structural temperature by 2.2 h in the Huanggang Yangtze River Cable-stayed Bridge.

In addition to the temporal statistics, extreme temperatures have also been studied. Xu et al. (2021) predicted the site-specific extreme wind and structural temperature loads of the Nanjing Dashengguan Yangtze River Cable-Stayed Bridge using the SHM data. They considered the uncertainties, such as environmental variability, measurement noise, and parameter estimation, in the Bayesian context. The result indicated that only the design value of the uniform temperature of the bridge was sufficient according to the 1-year temperature measurement. By contrast, the horizontal and vertical temperature gradients and the tower gradient were insufficient, with the exceeding probabilities by 99.96% or almost 100%.

The temperature distributions in cable-stayed bridges have been investigated through numerical and field monitoring studies. Obvious temperature delays among different structural components have been reported. These studies on the temperature distributions of the cable-stayed bridges are summarized in Table 4.

Table 4.

Temperature distribution in cable-stayed bridges.

Bridge	Focus	Study approach	Ref.
Qingzhou bridge	Temperature behavior of the concrete tower	Numerical	Li et al. (2019)
Puxi cable-stayed bridge	Temperature field in the prestressed concrete girder	Numerical and field monitoring	Wang et al. (2019)
Qingshan Yangtze river bridge	Temperature change pattern description using cubic spline functions	Mathematical	Mei et al. (2021)
Zhanjiang Bay cable-stayed bridge	Temperature temporal and spatial behaviors and structural deformations	Field monitoring and numerical	Cao et al. (2010)
Huanggang Yangtze river bridge	Time lag of the structural temperature and girder deflection	Field monitoring	Yang et al. (2018a)
Dashengguan Yangtze river cable-stayed bridge	Site-specific extreme wind and structural temperature loads	Statistical and field monitoring	Xu et al. (2021)

Cable-suspension bridges

The cable suspension bridge is another efficient bridge type. A suspension bridge consists of main cables, towers, stiffening girders/trusses, hangers (or suspenders), and anchorages. The main cables carry most loads of the bridge deck through the hangers/suspenders and transfer their tensions to the foundations and the towers in the middle. Suspension bridges are the most cost-effective type as the main span exceeds 1000 m. The 1915 Canakkale Bridge, which opened on 18 March 2022 with a main span of 2023 m, is the longest of all types at present.

Numerical studies

Zhou et al. (2015) analyzed the temperature-induced static response of the Humber bridge through the integration of numerical simulation and field monitoring. The heat convection coefficients of the external surfaces were determined according to the wind incidence angles. In addition, they simulated the convection by the conduction of air elements and modeled the radiation through the radiation matrix method in ANSYS. A good agreement between the numerical analysis and the measurement was obtained. The SHM data showed that the vertical and transverse temperature gradients were larger than 14 °C and 18 °C, respectively. They also investigated the effect of the footpath’s vertical position on the temperature behavior. When the footpath was at the mid-height of the section, the maximum transverse temperature gradient would be produced. However, the asphalt cover could reduce this temperature gradient.

Xia et al. (2020a) analyzed the temperature distribution in the Jiangyin Suspension Bridge using a time-varying solar radiation model modified on the basis of the temperature difference between the south and north anchors. The temperatures obtained from the heat transfer analysis agreed well with the measurement and showed superiority over the traditional radiation model in terms of simulation accuracy. They reported that a large temperature gradient existed between the interior and the exterior of the bridge tower. The thermal stress of the bridge tower should be considered in practical engineering.

Field monitoring studies

The temperature variation in the longitudinal direction is generally ignored despite being reported in several studies. For example, Xia et al. (2017b) reported from the 1-year SHM data that the temperature distribution along the longitudinal direction of the Jiangyin Yangtze River Bridge was nonuniform, possibly due to different airflow conditions along the main span. Wang et al. (2018) studied the temperature behavior of the Aizhai Suspension Bridge using FE simulation and SHM data. The bridge is featured by two main towers of different heights (70 and 120 m) because it crosses a deep valley. In addition, the main girder is directly connected to two tunnels, and the mountains at the ends of the bridge resulted in different radiation shading conditions within a day. The field measurements showed that the vertical, transverse, and longitudinal temperature gradients were more significant than the Chinese code specifications. The transverse and longitudinal temperature gradients were mainly concentrated on the steel structural parts. According to the numerical simulation, the temperature-induced tensile stress in the concrete deck was also more significant than the current specifications.

Statistical analysis

Ding et al. (2012) explored the thermal characteristics of the thermal fields of the steel box girder of the Runyang Suspension Bridge using long-term SHM data. The extreme temperatures obtained from different sensing points in a cross-section were presented. They formulated the probability functions of the vertical temperature gradient and the horizontal temperature gradients in the top and bottom plates as a weighted sum of a Weibull distribution and a Gaussian distribution based on these extreme values. They predicted the extreme temperature gradients with a 100-year return period and found that they agreed with the expected values specified by the British standard. Given that these extreme values did not concurrently occur, they further analyzed the correlation of these extreme values and developed the critical temperature difference models of the flat steel box girder.

Zhang et al. (2021c) proposed a novel copula-based probabilistic model to establish the temperature gradient model for the steel box girder of the Nanxi Yangtze River Suspension Bridge. The correlations between adjacent maximum and minimum values of daily temperature differences were established using a t-copula function. The bridge temperature gradient model was extrapolated for the service life estimation by using the Monte Carlo method. They reported that the vertical temperature gradient in the bridge was 96% of the design specification.

Similar studies have also been conducted by other researchers (Deng et al., 2018; Mao et al., 2018). The temperature distributions are generally studied through numerical and field monitoring due to the structural complexities. The statistics of the temperature measurements were also investigated. In addition, large horizontal and longitudinal temperature gradients have been reported. These studies are summarized in Table 5.

Table 5.

Temperature distribution in suspension bridges.

Bridge	Focus	Study approach	Ref.
Humber bridge	Temperature distribution	Numerical and field monitoring	Zhou et al. (2015)
Jiangyin Yangtze river suspension bridge	Temperature distribution with a modified solar radiation model	Numerical and field monitoring	Xia et al. (2020a)
Jiangyin Yangtze river suspension bridge	Temperature distribution in the longitudinal direction	Statistical and field monitoring	Xia et al. (2017b)
Aizhai bridge	Temperature distribution in the bridge	Statistical and field monitoring	Wang et al. (2018)
Runyang suspension bridge	Probability functions of the temperature gradients and extreme values	Statistical and field monitoring	Ding et al. (2012)
Nanxi Yangtze river suspension bridge	Probabilistic model for temperature gradients	Statistical and field monitoring	Zhang et al. (2021c)
Nanxi suspension bridge	Extreme value analysis and temperature distribution	Statistical and field monitoring	Deng et al. (2018)
Sutong bridge	Temperature distribution and variation pattern	Field monitoring and statistical	Mao et al. (2018)

The studies covered the temperature distribution patterns in various types of bridges, facilitating the development of design codes and the studies of the temperature actions of bridges in the next section.

Temperature actions

The temperature causes structural deformation and strain/stress in the bridges, which may even mask the influence of other loads. For example, the temperature-induced vertical deflection at the mid-span of the Tsing Ma bridge reached 2313 mm in 1 year, and the longitudinal of the girder reached 903 mm according to the measurement (Xu et al., 2010). The peak-to-peak strain difference was around 400 με in the Astoria–Megler Bridge, which was 10 times higher than the traffic-induced maximum strains according to the 1-year monitoring data (Catbas et al., 2008). The studies regarding temperature-induced deformation and strain/stress for each type of bridge will be reviewed in this section. The temperature actions are studied with theoretical, numerical, and field monitoring techniques.

Beam bridges

Theoretical studies

Zhou et al. (2021a) proposed unified analytical formulae to calculate the vertical temperature gradient-induced deflection of beams with any number of spans, as shown in Figure 6. The influences of the structural geometry, material property, and temperature change on the beam deflection were investigated through detailed parametric analyses. The figure depicts that a beam with odd-numbered spans has distinct thermal deformation characteristics from that with even-numbered spans. The side spans on both ends of an equal-span continuous beam underwent the largest deformation, while the middlemost spans the least. The mid-span deflection of each span quickly converged to the limit value with the increase in the total span number. The limits for the outermost and middlemost spans were $D_{0} \cdot (\sqrt{3} - 1) / 2 (\approx 0.366 D_{0})$ and zero, respectively, where D₀ is the thermal deflection at the mid-span of a simply supported beam with the same span length, as shown in Figure 6(a). Their study enhanced the understanding of the thermal behavior of beam bridges.

Figure 6.

Thermal deflection of the continuous beams with 1–6 spans (positive temperature gradients, equal spans, not to scale) (Zhou et al., 2021a).

Liu and Zuk (1963) conducted a series of theoretical thermoelastic analyses of four types of prestressed flexural members by assuming a vertical temperature gradient and constant horizontal temperature. The parabolic and straight tendons were considered. A moderate temperature variation of 13.9 °C along the depth of concrete could influence the initial prestress by 3%–5%. The deflections were generally less than 0.04% of the bridge span.

Numerical studies

Moorty and Roeder (1992) studied the temperature distribution and temperature-induced deformation in skew and curved bridges with steel box concrete decks through numerical analysis. The results suggested that temperature ranges and thermal movements considered for concrete bridge designs were sometimes smaller than predicted in practice. Skew and curved bridges frequently displayed movement patterns that differed from what was expected. They provided recommendations for thermal movements and placement of bearings and expansion joints.

Field monitoring studies

Fu and DeWolf (2004) studied the effect of temperature gradients on the overall behavior of a three-span continuous curved post-tensioned concrete bridge. Specially, the tilt data were collected to monitor the transverse rotations. They reported that the transverse rotations due to the temperature gradient of the bridge at the interior columns were one of the causes of the torsion cracks in the box girders. In addition, the temperature changes in the bridge induced torsional deformations of the columns. Kromanis et al. (2015) reported the plan twist at the roller bearings of the Cleddau Bridge.

Catbas et al. (2008) established the reliability model of the main truss components and the entire structural system of a long-span cantilever truss bridge. The structural responses to the temperature change were difficult to model using conventional analysis methods due to the complexity. They reported that the peak-to-peak strain difference was around 400 με, which was 10 times higher than the traffic-induced maximum strains according to the 1-year monitoring data. The temperature was then incorporated into the reliability analysis and found to have a significant influence.

Hedegaard et al. (2013) investigated the effects of thermal gradients on longitudinal stresses and bridge curvature of the I-35W St Anthony Falls Bridge. They found that stresses and deformations calculated using the bilinear design gradient model specified by the Load and Resistance Factor Design (AASHTO, 2010) of the American Association of State Highway Officials were considerably lower than those derived from the measured results. They concluded that the design thermal gradient was not necessarily conservative for structures in all regions.

The above-mentioned studies revealed the mechanism of bridge responses induced by uniform temperature and temperature gradients through correlation analysis and theoretical analysis. These studies are summarized in Table 6.

Table 6.

Temperature action of beam bridges.

Bridge	Focus	Study approach	Ref.
Beam bridges with different configurations	Deflections of beam bridges due to the vertical temperature difference	Theoretical	Zhou et al. (2021a)
Prestressed concrete beam	Prestress and deflection change due to temperature gradient	Theoretical	Liu and Zuk (1963)
Steel box bridges with concrete decks	Temperature distribution and temperature-induced deformation	Numerical and field monitoring	Moorty and Roeder (1992)
Curved post-tensioned concrete bridge	Temperature distribution and induced displacement	Numerical and field monitoring	Fu and DeWolf (2004)
Cleddau bridge	Quasi-static temperature effects on the bearing movements	Numerical and field monitoring	Kromanis et al. (2015)
Long-span truss bridge	Temperature-induced strain; reliability assessment	Statistical and numerical	Catbas et al. (2008)
Three-span post-tensioned concrete dual-box bridge	Temperature distribution and induced stress	Numerical and field monitoring	Hedegaard et al. (2013)
Lieshihe bridge	Temperature-induced strain and correlation between strain and temperature	Statistical and field monitoring	Yang et al. (2019)

Arch bridges

The temperature actions of the arch bridges are generally analyzed with numerical and field monitoring methods.

Numerical studies

Yuksel (2009) investigated the temperature behavior of the non-prismatic members subjected to temperature changes using an FE parametric study. The non-prismatic form is usually used in arch bridges. He used the 2D plane stress elements and reported that the frame elements could lead to significant errors. The design equations and coefficients were proposed based on a parametric study. The fixed-end actions of the non-prismatic members (having parabolic and straight haunches) due to temperature changes were determined using the proposed approach without a detailed FE model.

Field monitoring studies

Zhao et al. (2019) investigated the temperature effects on the deflection of the deck of the Dashengguan Yangtze River Arch Bridge. The temperature-induced and train-induced deflection was separated using the wavelet transform. The principal component analysis (PCA) was used to verify the temperature sensitivity of the girder deflection. They reported that the train-induced deflection would increase with the temperature increase, but by no more than 3%. They also found that the downward and upward of the train-induced girder deflections followed the t location-scale distribution. Then, they developed a real-time warning method using the girder deflection to indicate the deterioration of track irregularity or damaged bridge components with a triggering time of shorter than 10 s.

Zhou et al. (2017a) reported preliminary monitoring results of the comprehensive SHM system of the Jiubao Bridge, including the structural temperatures, main girder deformations, and the nonlinear relationship between the structural temperature and the main girder deformation. Later, they (Zhou et al., 2018) formulated the relation as a weighted sum of the fitted functions of the predominant thermal measurements according to the clustering analysis results. Three effective temperature variables (effective temperatures of the main girder, suspender, and main arch rib), two temperature differences (between the suspender and girder and between arch and girder), and one temperature gradient (vertical temperature gradient of the girder) were clustered. The midspan deflection was respectively fitted as the second and linear polynomials of these variables. The effective temperature of the main arch rib and the vertical temperature gradient in the girder was determined as the predominant variables according to the mean impact values. Consequently, the vertical deflection was correlated to these two variables. A prediction error less than 2 mm was obtained from the model.

Duan et al. (2011) conducted the strain-temperature correlation analysis of a tied arch bridge to distinguish the abnormal bridge response changes caused by the damage from those induced by environmental fluctuations. The 1-year measurement data of strain and temperature from the SHM system were used to establish a linear correlation model. Evident hysteresis was obtained in the regression model, but almost all the temperature-normalized strain response pairs fell within the area with a confidence interval of 95%. After the linear and nonlinear parts of the temperature effect were removed from the strain measurement, the processed strain was used for novel detection and overload alarming.

Yarnold et al. (2012) developed a temperature-based structural identification method using the FE model updating technique using the 168-m long Tacony-Palmyra Bridge as the example, with the local temperature measurements as inputs and the local strain and displacements as output. The correlation model was used to establish the long-term performance criteria for the bearings, expansion joints, decks or substructure cracking, member/connection overstresses, and linearity of the structure. Later, this method was compared with the conventional vibration-based structural identification method. The sensitivity advantage of the current method on three damage scenarios was confirmed by Yarnold and Moon (2015) and Yarnold et al. (2015).

Ding et al. (2017) extracted the strain influence line and the dynamic load factor from the SHM data of the Dashengguan Yangtze River Arch Bridge with the adaptive finite impulse response filter. They found that the effect of temperature on the influence line could be described by the fitted third-order polynomials, while the dynamic load factor of the majority of members was independent of temperatures.

The temperature actions of arch bridges were investigated through numerical and field monitoring studies. In addition to traditional correlation analysis, mathematical techniques such as clustering analysis and PCA were also adopted. These studies are summarized in Table 7.

Table 7.

Temperature action of arch bridges.

Bridge	Focus	Study approach	Ref.
Non-prismatic solid structure	Temperature distribution and design suggestions	Numerical	Yuksel (2009)
Dashengguan Yangtze river arch bridge	Temperature effects on the deflection	Field monitoring and PCA	Zhao et al. (2019)
Jiubao bridge	SHM system and preliminary monitoring results	Field monitoring	Zhou et al. (2017a)
Jiubao bridge	Correlation of temperature and structural deformation	Statistical	Zhou et al. (2018)
Tied arch bridge	Correlation of temperature and induced strain and structural health monitoring	Statistical	Duan et al. (2011)
Tacony-Palmyra bridge	Correlation of temperature and strain/deformation	Numerical and field monitoring	Yarnold et al.(2012, 2015); Yarnold and Moon (2015)
Dashengguan Yangtze river arch bridge	Correlation of temperature and strain influence line and dynamic load factors	Statistical and field monitoring	Ding et al. (2017)

Cable-stayed bridge

Theoretical studies

Zhou and Sun (2019a; 2019b) theoretically investigated the mechanisms underlying the temperature effect on the deformation of cable-stayed bridges. Five temperature models, namely, the variations in the average girder temperature, girder vertical temperature gradient, cable temperature, average tower temperature, and tower temperature gradient, were studied, as illustrated in Figure 7. For example, the rise in the average girder temperature would induce the elongation and hunch-up of the girder and the symmetric horizontal deformations of the two towers (Figure 7(a)). By contrast, the horizontal temperature gradient in the towers would lead to unsymmetric bending of the towers (Figure 7(e)). These analytical results were validated by the field monitoring data. Later, a set of generalized formulae were proposed to quantify the bridge displacements with respect to these temperature variables (Zhou et al., 2020b). These studies presented a theoretical explanation of the mechanism of thermal actions on cable-stayed bridges. The formulae have the advantages of clear concept, simple calculation, and general applicability.

Figure 7.

Analytical thermal models of a twin tower cable-stayed bridge (Zhou and Sun, 2019b).

Numerical studies

Sousa Tomé et al. (2018) conducted a 2D FE heat transfer analysis and inputted the extracted temperatures to a simplified 3D model as thermal loads by assuming a constant temperature distribution in the longitudinal direction. They showed that the radiative heat transfer was essential to guarantee the analysis accuracy. The uniform variations in temperature components were responsible for about 90% of the structural responses (bearing displacement, vertical displacements, pylon/tower rotations, and stay cable forces). Furthermore, the temperature gradient contributed little to the longitudinal bearing displacement. The optimal deployment of thermometers was discussed to more accurately estimate the bridge responses.

Zhang et al. (2021b) numerically studied the temperature effects on a cable-stayed bridge with a separated side-box steel-concrete composite girder. The bridge was separately simulated by two models with different element compositions. The beam model consisted of only beam elements; thus, only longitudinal temperature gradients were considered. By contrast, the hybrid model consisted of various element types according to the bridge configurations. The maximum thermal stress of the bridge deck in each section of the main girder produced by the hybrid model was generally 20% larger than that by the beam model. The direct solar radiation greatly affected the temperature-induced stress of the steel box girder, and the maximum stress on the steel box girder was 31 MPa in summer. In addition, the maximum temperature stress time was not entirely synchronized with the maximum temperature gradient time. Therefore, adopting only the maximum temperature gradient for structural thermal analysis would underestimate the temperature stress value.

Field monitoring studies

Xu et al. (2019) studied the thermal effects on the girder deflection of the Dashengguan Cable-stayed Bridge by using a multivariate linear-based model. The weight coefficients, defined as the normalized responses to each type of temperature action (uniform temperature, horizontal and vertical temperature gradients, and the temperature gradient of the tower), were obtained from the numerical models and updated with the measured temperature and displacement data collected during the bridge closure time. The maximum prediction error was 5.6 mm and satisfied the requirements for practical applications. They also reported that the actual girder temperature profile agreed well with that recommended by the British standard, while the transverse temperature difference of the deck was non-negligible.

Wang et al. (2022) investigated the temperature effect on the tower deformation using the SHM data of a cable-stayed bridge. A strong linear relationship between the air temperature and the tower deformation was observed. The two towers departed from each other as the temperature rose with correlation coefficients of 0.89 and −0.92. A similar pattern was also observed in the effective lengths of the outermost cables (represented by the tower-girder distance), with correlation coefficients of 0.93 and 0.94.

Wang et al. (2019) reported that the thermal effects on cable forces of the Puxi Bridge during extreme weather events were more adverse than those during average weather. The field monitoring data showed that tensions of the cables at a few locations (namely, near the side pier, in the center of the side span, and near the towers) were the most seriously affected by temperature.

These studies generally established the correlation models between the temperature variables (uniform temperature, temperature gradients, etc.) and the bridge deformations (girder deflection, displacement of the expansion joints, horizontal displacement of the towers, etc.). These patterns were then applied to possible damage detection of the bridge components. Similar studies have also been conducted by Yang et al. (2018b; 2018a), Zhu and Meng (2017), Wang and Ye (2019), Le and Nishio (2015, 2019), Cao et al. (2010), Huang et al. (2018a), Li et al. (2019), and Wang and Ye (2019). These studies on the temperature actions of cable-stayed bridges are summarized in Table 8.

Table 8.

Temperature action of cable-stayed bridges.

Bridge	Focus	Study approach	Ref.
Shanghai Yangtze river bridge	Temperature-induced deformations of the bridges	Theoretical and field monitoring	Zhou and Sun (2019a; 2019b); Zhou et al. (2020b)
Corgo bridge	Temperature distribution and its influences on bridge responses	Numerical	Sousa Tomé et al. (2018)
A cable-stayed bridge	Temperature distribution and induced stresses	Numerical	Zhang et al. (2021b)
Dashengguan Yangtze river cable-stayed bridge	Thermal effects on the girder deflection using a multivariate linear-based model	Field monitoring and mathematical	Xu et al. (2019)
A cable-stayed bridge	Temperature effects on the tower deformation	Field monitoring and statistical	Wang et al. (2022)
Puxi bridge	Temperature effects on the cable forces	Numerical and field monitoring	Wang et al. (2019)
Huanggang Yangtze river bridge	Correlation of structural temperature and girder deflection	Statistical and field monitoring	Yang et al. (2018a)
Anqing Yangtze river bridge	Correlation of structural temperature and the tower deformation	Statistical and field monitoring	Yang et al. (2018b)
Qingzhou bridge	Temperature behavior of the tower	Numerical	Li et al. (2019)
Liuzhuang bridge	Temperature distribution and induced deformation; computation method	Numerical	Zhu and Meng (2017)
Zhanjiang Bay cable-stayed bridge	Temperature temporal and spatial behaviors and structural deformations	Field monitoring and numerical	Cao et al. (2010)
Zhijiang bridge	Cable damage detection through abnormal variation of the girder deflection-temperature correlation envelope curve	Numerical and statistical	Wang and Ye (2019)
Can tho bridge	Condition evaluation using Mahalanobis distance	Numerical and field monitoring	Le and Nishio (2015, 2019)
Zhaobaoshan bridge	Damage detection of expansion joints	Statistical	Huang et al. (2018a)

Cable suspension bridges

Theoretical studies

Zhou et al. (2021b, 2020c, 2022) systematically analyzed the temperature-induced deformations of general suspension bridges. They investigated the temperature-induced mid-span deflection and tower-top horizontal displacement in the ground-anchored bridges and developed a set of formulae to calculate these deformations by the superposition of the thermal effects of different components. The result indicated that the cable temperature played the dominant role in the mid-span deflection and tower-top horizontal displacement. They then derived a set of general, succinct analytical formulae of the thermal deformation of three-span suspension bridges by introducing the concept of equivalent span length (Zhou et al., 2022). The SHM data of the Tsing Ma Bridge and Akashi Kaikyo Bridge were adopted to verify the formulae. They demonstrated that the neglect of the sag effect of the side-span cables would overestimate the thermal deformation and then characterized it by a modification coefficient. The formulae were then expanded to the multi-span suspension bridges considering the tower lateral stiffness and the elastic deformation of cables caused by the tension changes. These theoretical studies enabled engineers to extensively understand the behavior of suspension bridges and predict the thermal deformation in a simple, generalized manner.

Xia et al. (2020b) analyzed the condition of the expansion joints in the Jiangyin Suspension Bridge, considering the thermal effect through a Gaussian process metamodel-based model updating. The relationship between the longitudinal boundary stiffness (modeled by elastic springs) and the structural temperature was formulated, and the corresponding stiffness values were updated. The updated FE model predicted longitudinal deformation with high accuracy compared with the measurement, demonstrating the effectiveness of the model in reflecting the condition of the expansion joints.

Xia et al. (2018) developed a simple and efficient analytical method of calculating/separating thermal stress from the SHM data of the Jiangyin Suspension Bridge. The thermal stresses under the uniform temperature and the linear/nonlinear temperature gradients considering the partial axial/rotational constraints were calculated by modeling the main girder as a simply-supported beam. The results agreed well with the numerical calculations. The temperature-induced stresses were more significant than those induced by vehicles at high temperatures.

Numerical studies

Westgate et al. (2015) investigated the temperature-induced behavior (thermal expansion and contraction cycles) of the Tamar Bridge with FE simulation and field monitoring data. The bridge was originally a pure suspension bridge. Cantilever decks were added, and nine pairs of stay cables were installed during a strengthening exercise after 50 years of service, resulting in a complex structural composition. The cloud condition was derived from the temperature differences between the truss and the suspension cable. The result demonstrated that the behavior of the Saltash tower was almost unaffected by the diurnal thermal behavior of the girder because it is located near the main span expansion gap. However, a seasonal thermal behavior was observed from the early morning samples of the tower. The sway of the Plymouth tower was linked to the thermal expansion of the deck, which pulled it toward the bridge midspan as the bridge warmed up.

Jesus et al. (2019) presented a probabilistic structural identification of the Tamar Bridge using a detailed FE model with a modular Bayesian framework. The initial strain of the main cables and stay cables and the stiffness of the linear springs of the thermal expansion gaps were identified from the bridge’s displacement and natural frequency response by jointly considering the effects of temperature and traffic as a driving excitation. Multi-response Gaussian process was used to simulate the model response surface and the model discrepancy. The Metropolis–Hastings algorithm was used as an expansion for multiple parameter identification. A close agreement between the parameters and test data was obtained.

Xia et al. (2013) numerically analyzed the thermal effects of the Tsing Ma Bridge. A detailed global FE model and heat transfer analysis are time-consuming because of the large size of the bridge. Accordingly, the components of the girder, tower and cross-frame segments were modeled separately, as shown in Figure 8(a). Heat transfer analyses were conducted to calculate their temperatures, which were close to the measurements. These temperature results were input into a 3D FE model of the entire bridge (Figure 8(b)) to analyze the temperature-induced responses. The displacement results agreed well with their measurement counterparts. The temperature-induced stresses were separated from the total strain and found to be generally small. The numerical approach can be used to eliminate the temperature effects in bridge condition evaluation or as damage indicators.

Figure 8.

FE models of the Tsing Ma Bridge.

This divide-and-conquer approach is efficient in terms of computation. However, this approach requires manual intervention. Moreover, it cannot accurately simulate the temperature in the deck-tower connections. Nonetheless, the temperature in the deck section is non-uniform. With the fast development of computer techniques, the glocal FE modeling of the bridge and heat-transfer analysis are possible.

A similar study was conducted by Farreras-Alcover et al. (2015). The strain/stresses induced by the temperatures were separated from the total strain measurement either from the correlation model or with the decomposition method. The damage detection based on the abnormal strain/stress patterns changes was thus possible.

Field monitoring studies

Koo et al. (2013) reported that the thermal expansions of the girder, main cables, and additional stay cables were the major drivers of the girder deformation, whereas wind effects were relatively insignificant from the SHM data.

Xu et al. (2010) reported the statistics of temperature responses of the Tsing Ma Bridge and their correlations with the bridge displacement in three directions. The effective deck temperature was linearly correlated to the longitudinal displacement of bridge towers, and the longitudinal and vertical displacements of the girder deck sections and the main cables. The temperature did not significantly influence the lateral deformations. Specifically, the vertical displacement of the mid-span girder and cables changed by more than 47 mm due to a one-degree Celsius change in the effective temperature, highlighting the dominant effect of the temperature on the bridge deformation.

Xu et al. (2020b) separated the temperature-induced and traffic-induced mid-span deflection of the Xihoumen Suspension Bridge using the multi-resolution wavelet-based method. The daily and seasonal thermal responses were identified in the subspaces and removed from the measured signals. The reconstructed signal showed a similar variation pattern with the daily vehicle gross mass, demonstrating the effectiveness of the proposed method. The separated deflection under the normal operating condition was then used for the damage detection of the bridge (Xu et al., 2020a). The threshold for anomaly detection was determined as the statistical value corresponding to a 95% guarantee rate of the generalized Pareto distribution with a 100-year return period. Specially, the threshold was gradually updated with the latest SHM data to account for the increasing traffic volumes and the degradation of the bridge.

Kim et al. (2005) presented the monitoring system and the early monitoring results of the Yeongjong Suspension Bridge. A total of 33 thermometers were installed to monitor the temperature of the cable, deck, and tower. They reported that the average hanger tension, the horizontal displacement of expansion joints, and the vertical and lateral displacement of the girder were essentially governed by the yearly and daily temperature variations according to the monitoring data. An ARX model was constructed to predict the displacement with high accuracy. The difference between displacement prediction and measurement was assumed to be in the normal distribution and used to establish the thresholds for structural change or damage warning.

Xia et al. (2017b) analyzed the correlations of temperature and structural responses, including the strain, displacement, and suspender frequency/force, of the 1385-m main span of Jiangyin Yangtze River Bridge using 1-year monitoring data. The suspender force varied by 5% due to the temperature variation. The temperature distribution along the longitudinal direction was nonuniform, possibly because the airflow condition along the main span might be non-uniform. Furthermore, Xia et al. (2017a) proposed a Euclidean distance-based damage identification method to assess the bridge condition after a ship collision. The temperature-induced strain was separated from the measured strain with ensemble empirical mode decomposition technology, and the structural transfer function was defined by taking the temperature variation as the input and temperature-induced strains as the output. The inherent structural characteristics were revealed through this transfer function.

Many similar studies on the temperature-induced deformations of the suspension bridges have been conducted by other researchers (Battista et al., 2015; Deng et al., 2010, 2022; Guo et al., 2015; Yu and Ou, 2017; Zhou et al., 2020a). These studies generally formed the correlation models of the temperature variants (e.g., effective temperature and temperature gradients) and the structural deflection and deformations using the conventional least squares method or relatively new methods, such as neural networks, control charts, and deep learning. These correlation models were used to characterize the bridge behaviors under normal operating conditions. The deviated structural behaviors would be categorized as abnormal and raise alerts for inspections. The studies on the temperature actions of the suspension bridges are summarized in Table 9.

Table 9.

Temperature action of suspension bridges.

Bridge	Focus	Study approach	Ref.
Various suspension bridges	Temperature-induced bridge deformations	Theoretical	Zhou et al.(2020c, 2021b, 2022)
Jiangyin suspension bridge	Relationship between the longitudinal boundary stiffness and the structural temperature	Mathematical	Xia et al. (2020b)
Jiangyin suspension bridge	Analytical method of calculating/separating thermal stress from the monitoring data	Theoretical	Xia et al. (2018)
Tamar bridge	Temperature distribution and temperature-induced behavior	Numerical	Westgate et al. (2015)
Tamar bridge	Probabilistic structural identification	Mathematical	Jesus et al. (2019)
Tsing Ma bridge	Temperature-induced stress	Numerical and field monitoring	Xia et al. (2013)
Great Belt bridge	Regression analysis of temperature and structural response	Statistical	Farreras-Alcover et al. (2015)
Tamar bridge	Thermal effects on structural deformation	Field monitoring	Koo et al. (2013)
Tsing Ma bridge	Temperature behavior	Statistical and field monitoring	Xu et al. (2010)
Xihoumen suspension bridge	Temperature-induced and traffic-induced mid-span deflection and damage detection	Mathematical and field monitoring	Xu et al. (2020a; 2020b)
Yeongjong suspension bridge	Correlation of structural responses and temperature and damage warning	Field monitoring and mathematical	Kim et al. (2005)
Jiangyin suspension bridge	Temperature effects on suspender forces	Statistical and field monitoring	Xia et al. (2017b)
Jiangyin suspension bridge	Damage detection using temperature and temperature-induced strain	Statistical and field monitoring	Xia et al. (2017a)
Humber bridge	Temperature-induced static response	Numerical and field monitoring	Zhou et al. (2020a)
Runyang suspension bridge	Seasonal correlation of frequency-temperature and expansion-temperature; damage detection	Mathematical	Deng et al. (2010)
Three long-span steel bridges	Expansion joint displacement and wear	Mathematical	Guo et al. (2015)
Tamar bridge	Correlation between the deck extension and temperature measurement	Correlation and field monitoring	Battista et al. (2015)
Aizhai suspension bridge	Linear relations between the structural deformations and the temperature	Mathematical and field monitoring	Yu and Ou (2017)
Nanxi suspension bridge	Deflection–vehicle load–temperature correlation model using deep learning	Mathematical and field monitoring	Deng et al. (2022)

Conclusions and future recommendation

In this work, previous studies on temperature distributions and the temperature actions of various types of bridges were reviewed, including beam bridges, arch bridges, cable-stayed bridges, and suspension bridges. This work extends the existing reviews (Luo et al., 2022; Xia et al., 2012; Zhou and Yi, 2013) by focusing on temperature distribution and static temperature actions. The investigation methods cover the site/laboratory test, theoretical analysis, numerical simulation, and field monitoring.

As for the temperature distribution, one type of study attempted to reveal the temperature distributions across the bridges, especially along the vertical directions. Different distribution patterns, such as the fifth-order polynomials or simplified piecewise linear patterns, were proposed and later adopted in the design specifications. Another type of study checked the temperature distributions on a long-term scale and their statistical characteristics to assess the safety and reliability of the bridges. Numerical, lab tests and field monitoring techniques were usually adopted in these studies.

The temperature actions were also intensively inspected, especially in long-span cable-stayed bridges and cable suspension bridges. Typically, the temperature-induced deformation and stresses of the girders, towers, cables, and arches were investigated via the three manners. Many studies established the relations between the responses and the temperature variables as linear regression patterns with the least square methods or statistical methods, such as ARX models and neural networks. The second batch of studies adopt the FE analysis, which can obtain the detailed quantitative temperature-induced responses. Finally, a few theoretical studies revealed the mechanisms underlying the temperature-induced deformations in the beam, cable-stayed, and suspension bridges.

In spite of the enormous studies and achievements, the following issues are worth further investigation.

(1) Although the simple formulae for the deformation of beam bridges, cable-stayed bridges, and cable suspension bridges have been available, the corresponding formulae for arch bridges are still unavailable. The development of such formulae is worthwhile for understanding the thermal mechanism of arch bridges.

(2) The numerical analyses of the temperature behaviors usually follow a divide-and-conquer strategy, and the uniform longitudinal distribution assumption is usually adopted. However, earlier studies reported nonuniform longitudinal distribution (Xia et al., 2017b), especially near the tower-girder joints and side spans (Wang et al., 2018). The proper “divide” rules are worth investigating. Global 3D analyses are usually used for small-scale bridges limited by computational capability. 3D analysis of large-scale bridges is worth exploring.

(3) As summarized in Table 1, the calculation of the heat transfer coefficients used in different studies varied significantly. The formulae were generally empirically summarized from practice, with possibly different wind flow configurations, surface roughness, thermal properties of the air, and temperature difference. Accurate measurement of these parameters or inverse parameter identification can be conducted.

(4) The current SHM systems usually incorporate thermometers for both the environmental and structural temperatures. However, the quantities and deployment plan might be optimized to better capture the temperature distribution and reveal the underlying mechanism. In addition, as required in the boundary conditions of the internal surfaces of box cross sections of the numerical simulations, the proper thermometer arrangement for the ambient temperature measurement inside the box is preferred. Finally, most SHM systems do not have pyranometers to monitor the solar radiations. This should be considered in the future.

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 work is supported by the RGC-GRF (Project No. 15206821) and PolyU Postdoctoral Matching Fund (Project No. 1-W172).

ORCID iDs

Lingfang Li

Yi Zhou

Xiaoqing Zhou

Yong Xia

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