Analysis of temperature response in concrete box-girder bridges based on steel reinforcement thermal conductivity model

Abstract

In the thermal analysis of concrete box girder bridges, the high thermal conductivity of steel bars is often oversimplified, leading to deviations in temperature field prediction results. At high temperatures, the thermal conductivity of steel significantly alters the temperature gradient distribution within the section, thereby affecting thermal stress and the resistance to cracking. The box girder is widely used in long-span bridges due to its excellent torsional stiffness and economic benefits. However, the complex temperature field may cause cracking, warping, and durability problems. The significant thermal conductivity difference between steel and concrete will produce a localized temperature gradient in the area with dense reinforcement (which will induce thermal stress concentration and endanger the stability of the structure). Existing studies often overlook the thermal conductivity of steel bars, assuming a purely thermal conductivity process, which deviates from the actual situation. This study combines theoretical analysis, refined finite element modeling, parametric research (reinforcement diameter and stirrup spacing), and related stress analysis to propose the thermal siphon effect. The study shows that increasing the reinforcement diameter from 16 mm to 26 mm can significantly reduce the homogenization time of the structural temperature. When the stirrup spacing along the heat conduction direction is reduced from 170 mm to 30 mm, the reinforcement is dense, and the diameter is large. The structure can reach thermal equilibrium faster, and the temperature gradient is smaller. The temperature gradient causes the surface concrete to be under tension and the internal concrete to be under pressure, which may lead to surface cracking. In areas with dense reinforcement, accelerated heat transfer results in a significant increase in temperature, which can induce local stress concentration.

Keywords

thermal siphon effect temperature gradient and thermal stress sensitivity of reinforcement parameters heat transfer in composite structures structural response of box girders finite element modeling verification thermo mechanical coupling boundary conditions

Introduction

As primary load-bearing components, bridge structural engineering (Xiang, 2001) concrete box girders exhibit fire resistance that directly governs the overall structural safety. However, under fire or elevated temperature conditions, the distinct thermophysical properties of concrete and steel reinforcement induce non-uniform temperature distributions across the beam cross-section, triggering thermal stresses (Asadi et al., 2018; Sargam et al., 2020; Zhao et al., 2013), strength degradation, and even structural failure (Wang and Fang, 2009). Qi Xia et al. (2020) validated the significant influence of temperature gradients on the boundary stiffness of expansion joints in long-span suspension bridges based on long-term monitoring data from the Jiangyin Suspension Bridge. The high thermal conductivity of steel reinforcement establishes it as a “thermal bridge,” accelerating internal temperature rise in concrete. In contrast, the low thermal conductivity of concrete may delay heat dissipation, forming localized high-temperature zones (Li et al., 2023). Consequently, in-depth investigation into the temperature response of concrete box-girder bridges based on steel reinforcement thermal conductivity models is critical for accurately predicting mechanical performance degradation under high temperatures and ensuring structural safety. Hui Wang et al. (2022) investigated thermal stresses in concrete rectangular beams, attributing their formation to constraints imposed by microstructures, cross-sections, and supports. They proposed that temperature gradients induce complex thermal stress distributions in concrete beams through multi-scale constraint mechanisms, with micro-scale stress fluctuations being the primary factor affecting durability risks. Tanvir Hossain (2020) et al. evaluated cracking potential by comparing the restraining moments induced by temperature gradients with the cracking moments at beam ends. Their results showed that although the calculated restraining moments were lower than the cracking moments, the superposition of other long-term effects in real-world engineering could still lead to cracks at beam ends or continuous diaphragm locations. Enrique Mirambell (1990) analyzed the temperature and stress distributions in concrete box-girder bridges, investigating the influence of cross-sectional geometry on thermal response. Their findings confirmed that temperature distributions in concrete bridges are nonlinear. Linren Zhou et al. (2016) studied the temperature gradient behavior in the box girders of the long-span Humber Bridge in the UK through long-term monitoring and numerical simulations. They concluded that streamlined cross-sections, asphalt pavement layers, and bridge orientation have a significant impact on transverse temperature differences (TTD). Significant TTD may cause in-plane bending of the box girder, which impacts its structural performance.

Additionally, Linren Zhou et al. (2024) reviewed convective and radiative heat transfer between bridges and the external environment, emphasizing that structural characteristics, climatic conditions, and geographical factors influence temperature gradients. These gradients induce thermal stresses and deformations, which ultimately affect the load-bearing capacity and durability. Yushi Shan et al. (2023) established the global three-dimensional temperature distribution model of long-span cable-stayed bridge through the fusion of heat transfer analysis and field monitoring data, revealing the significant influence of environmental factors (such as solar radiation and wind speed) on the structural temperature gradient, providing a method reference for the thermodynamic analysis of concrete box girder in this study.

Existing studies on the effects of bridge temperature generally overlook the influence of the thermal conductivity of steel reinforcement on the temperature field distribution. Neglecting this factor treats the bridge structure as a homogeneous material, simplifying the temperature field analysis model. While this simplification remains applicable for most bridge types, particularly open-section structures such as T-beams or I-beams, the thermal conductivity of steel reinforcement has a minor impact on the overall temperature distribution. Neglecting this factor can still yield reasonably accurate analysis results. However, for closed-frame structures such as concrete box girders with complex thermodynamic boundary conditions, ignoring the thermal conductivity of steel reinforcement can lead to significant deviations between simulated and actual temperature distributions. The high thermal conductivity of the reinforcement may cause uneven temperature distribution within the girder, particularly in densely reinforced zones (e.g., support or mid-span regions), resulting in a “thermal bridge effect.” Only a limited number of studies have considered the thermal conductivity difference between steel and concrete in structural temperature distribution. Building on Linren Zhou et al. (2020) findings of steel-cable-dominated thermal conduction in suspension bridges, the present study demonstrates the existence of a thermal siphoning effect in the steel reinforcement of box girders. Yamaguchi (1993) proposed that since steel’s thermal conductivity is over 30 times higher than concrete’s, The presence of steel reinforcement can reduce the internal temperature gradient in rectangular beams, lowering temperatures near the heated surface while raising temperatures in regions farther from it, as validated by finite element model. B.E. (2023) measured the thermal conductivity of concrete cylinders with 4% and 8% reinforcement ratios following industry standards, demonstrating steel’s significant influence on the composite structure’s thermal properties. Zhang, H.S. et al. (2023). Investigated a steel-box-concrete-filled steel tube (CFST) bridge pylon under controlled laboratory loading. Their experiments revealed that as the steel box temperature increased, the CFST core exhibited a slower thermal response, resulting in relative displacement and interfacial gaps between the steel and concrete due to their differing thermal conductivities.

The influence of temperature gradient on box girder stresses is a crucial research topic in bridge engineering. Its underlying mechanism primarily arises from thermal expansion and contraction of materials due to temperature variations, which generate internal stresses under structural constraints. Studies have shown that when temperature gradients exhibit non-uniform distribution across the cross-section or along the longitudinal direction of a box girder, they induce bending stresses and axial stresses. Stress concentrations may occur near supports and mid-span regions with high constraint levels, potentially leading to cracking or structural deformation. Researchers worldwide have conducted in-depth investigations through theoretical analysis, numerical simulations, and experimental studies. As early as 1959, E. S. Barrekette (1960) proposed a three-dimensional thermoelastic stress calculation method based on one-dimensional stress-strain constitutive relations and the governing equations of thermoelasticity. This approach enables the computation of stress distribution in any cross-section under arbitrary temperature conditions, thereby validating the thermo-stress-strength theory. Qiang Jing et al. (2024) proposed a unified global analysis method to separate the effects of typhoons and temperature, demonstrating the dominant role of temperature gradients in the displacements and stresses of box girders. Zeng Qingxiang et al. (2010). Analyzed the temperature effects in prestressed concrete box girder bridges and found that thermal stresses distribute unevenly across the cross-section. The maximum tensile stress occurs at the lower edge of the top slab, while the maximum compressive stress appears at the upper edge of the top slab. This nonlinear temperature distribution induces self-equilibrating stress distributions, which are correlated with the temperature distribution across the cross-section.

To investigate the influence of steel reinforcement on the temperature distribution of concrete box-girder structures, this study begins by examining the heat transfer mechanism. Based on the principle of energy conservation during structural-environmental thermal interactions, the heat propagation process in concrete box girders is analyzed. It is revealed that the steel reinforcement mesh, due to its significantly higher thermal conductivity compared to concrete, achieves thermal equilibrium faster than the concrete matrix, thereby serving as the dominant pathway for heat transmission. A novel concept, termed the “siphonic effect,” is proposed for reinforced concrete structures, defined as the phenomenon where pronounced differences in thermal conductivity between steel and concrete induce preferential rapid heat transfer through the reinforcement, consequently altering the global temperature distribution pattern of the structure. Utilizing monitoring data, including concrete temperature, solar radiation, ambient temperature, and wind speed (Kairbek, 1981), a refined temperature analysis methodology is developed that incorporates the effects of steel reinforcement on concrete box-girder temperature fields. Finite element models are established to validate both the siphonic effect during heat propagation and the distribution of heat flux density within concrete box girders. The study investigated the influence of parameters describing reinforcement density, such as rebar diameter and stirrup spacing, on the temperature distribution in concrete, as well as the stress induced by temperature gradients.

Basic theory of heat transfer by conduction

Heat transfer is one of the three fundamental modes of thermal energy transfer, primarily including conduction, convection, and radiation. Conduction refers to the heat transfer process between directly contacting objects or within a single object resulting from a temperature gradient. Its essence lies in energy transfer caused by the thermal motion of microscopic particles (such as molecules, atoms, or free electrons). Convection describes the heat exchange between a fluid (gas or liquid) and a solid surface, which can be categorized into natural convection (driven by density differences) and forced convection (propelled by external forces). Radiation is the only heat transfer method that does not require a medium, occurring through electromagnetic waves that transfer heat between objects. Its efficiency depends closely on surface properties, temperature, and relative positions of the objects. In practical engineering applications, these three heat transfer mechanisms often coexist, but the dominant mechanism varies with specific operating conditions.

The thermal conduction mechanism in reinforced concrete structures is crucial for understanding their temperature-related effects. In natural environments, heat transfer in concrete box girder structures is primarily influenced by external environmental factors such as solar radiation, air temperature variations, and wind speed. The heat transfer process within the structure follows Fourier’s law of heat conduction (Narasimhan, 1999; Liu, 1990; Yang and Tao, 2010), meaning heat flows from high-temperature zones to low-temperature zones at a rate proportional to both the temperature gradient and the material’s thermal conductivity coefficient.

Governing differential equation of heat conduction

\frac{\partial}{\partial x} (k_{x x} \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (k_{y y} \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (k_{z z} \frac{\partial T}{\partial z}) + q = ρ c \frac{d T}{d t}

(1)

In which:

\frac{d T}{d t} = \frac{\partial T}{\partial t} + V_{x} \frac{\partial T}{\partial x} + V_{y} \frac{\partial T}{\partial y} + V_{z} \frac{\partial T}{\partial z}

(2)

In the equation,

V_{x}

V_{y}

, and

V_{z}

represent the medium’s thermal conduction rates.

Heat conduction

Heat conduction can be defined as the exchange of internal energy between two objects in complete contact or between different parts of a single object due to a temperature gradient. Heat conduction follows Fourier’s Law:

q^{*} = - K_{n n} \frac{d T}{d n}

(3)

In this equation,

q^{*}

represents the heat flux density

(W / m^{2})

;

K_{n n}

denotes the thermal conductivity

[W / (m \cdot ℃)]

; and

d T / d n

is the temperature gradient along the n-direction. The negative sign indicates that heat flows in the direction of decreasing temperature.

Reinforced concrete structures are a typical type of composite structure where the thermal conductivity exhibits a stepwise change at the interface between steel reinforcement and concrete during heat transfer. In practical engineering structures, factors such as construction quality, shrinkage, and creep may lead to voids or poor contact (e.g., debonding). To simplify the analysis, it can be assumed that the steel and concrete are in perfect contact, satisfying the following continuity conditions for temperature and heat flux density:

t_{I} = t_{I I}, {(λ \frac{\partial t}{\partial n})}_{I} = {(λ \frac{\partial t}{\partial n})}_{I I}

(4)

In the equation, t denotes temperature, and

λ \partial t / \partial n

represent the heat flux density.

ρ c \frac{\partial t}{\partial τ} = \frac{\partial}{\partial x} (λ \frac{\partial t}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial t}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial t}{\partial z}) + F

(5)

In the equation,

F

represents heat generated by heat source per unit volume per unit time.

When only relatively mild heat sources, such as solar radiation and wind speed, are considered in natural environments, the thermal conductivity, specific heat capacity, density, and other physical parameters of the steel reinforcement and concrete materials can be assumed to be constant. Under these conditions, equation (5) can be rewritten as:

\frac{\partial t}{\partial τ} = a (\frac{\partial^{2} t}{\partial x^{2}} + \frac{\partial^{2} t}{\partial y^{2}} + \frac{\partial^{2} t}{\partial z^{2}}) + \frac{F}{ρ c}

(6)

In the equation,

a = λ / (ρ c)

is defined as the thermal diffusivity (also referred to as thermal diffusion coefficient), where

λ 、 ρ 、 c

represents the material’s thermal conductivity, density, and specific heat capacity, respectively. According to ISO 10456:2007 (2007), typical values of these physical parameters for steel reinforcement and concrete are listed in the following table (Table 1):

Table 1.

Parameters related to steel bars and concrete.

Physical parameters	Steel reinforcement	Concrete
Thermal conductivity $λ (w / m \cdot k)$	450	1.15∼2.5
Density $ρ (k g / m^{3})$	7800	1800∼2400
Specific heat capacity $c (j / kg \cdot k$ )	450	1000

The thermal diffusivity of steel is approximately 24.7 times that of concrete, indicating that steel responds to temperature changes far more rapidly than concrete. In transient heat transfer scenarios (e.g., fire exposure or solar radiation), steel reinforcement rapidly conducts heat, causing localized temperature spikes in the surrounding concrete and creating significant spatiotemporal temperature gradients.

Steel’s high thermal diffusivity makes it the dominant pathway for heat transfer. When thermal energy flows from high-temperature zones (such as fire-exposed surfaces) to cooler regions, the majority of heat propagates through the steel reinforcement network, with only a minor fraction diffusing through the concrete matrix. This “thermal siphoning effect” leads to steep temperature gradients in reinforcement-dense areas (e.g., web-flange junctions), accelerating thermal stress concentrations.

Thermal convection

Thermal convection is one of the fundamental modes of heat transfer, referring to the heat exchange process between a solid surface and the surrounding fluid (gas or liquid) due to temperature differences. Based on the driving mechanism of fluid motion, thermal convection can be classified into two main types: natural convection and forced convection. Convection is generally applied as a surface boundary condition. Newton’s law of cooling describes it:

q^{*} = α (T_{S} - T_{B})

(7)

In the equation, α is the convective heat transfer coefficient, T_S is the temperature of the solid surface, and T_B is the temperature of the surrounding fluid.

Thermal radiation

Thermal radiation is the process by which objects transfer energy through electromagnetic waves (primarily in the infrared spectrum). Unlike heat conduction and thermal convection, thermal radiation plays a dominant role in high-temperature and vacuum environments, making it a critical heat transfer mechanism in fields such as spacecraft thermal control and high-temperature equipment.

In engineering applications, thermal radiation typically involves interactions between two or more objects, where each body simultaneously emits and absorbs heat. The net heat transfer between them can be calculated using the Stefan-Boltzmann equation:

Q = ε σ A_{1} F_{12} (T_{1}^{4} - T_{2}^{4})

(8)

In the equation,

Q

represents the heat flux rate,

ε

is the absorptivity (blackness), and

σ

denotes the Stefan-Boltzmann constant, approximately equal to 5.67 × 10^-8

W / (m^{2} \cdot K^{4})

Boundary conditions of beam segment

Convective heat transfer boundary conditions

Based on Newton’s cooling law, the surface heat flux is determined by the convective heat transfer coefficient $α$ , the ambient temperature $T_{\infty}$ , and the surface temperature $T_{s}$

q_{conv} = α (T_{\infty} - T_{s})

(9)

Internal heat conduction boundary conditions

The perfect contact assumption is adopted for the steel-concrete interface to ensure continuity of temperature and heat flow. The internal pore thermal resistance of concrete is treated by the equivalent thermal conductivity method, and the measured porosity is converted into the thermal conductivity reduction factor of 0.85-0.92.

T_{steel} = T_{concrete}, λ_{steel} {\frac{\partial T}{\partial n} |}_{steel} = λ_{concrete} {\frac{\partial T}{\partial n} |}_{concrete}

(10)

T_{steel}

: Temperature of steel at the interface (unit:

k

T_{concrete}

: Temperature of concrete at the interface (unit:

k

λ_{steel}

: Thermal conductivity of steel (unit:

w / (m \cdot k)

λ_{concrete}

: Thermal conductivity of concrete (unit:

w / (m \cdot k)

{\partial t / \partial n |}_{steel}

: Normal temperature gradient at the steel side (unit:

k / m

{\partial t / \partial n |}_{concrete}

: Normal temperature gradient at the concrete side (unit:

k / m

Finite element model and analysis

Boundary condition treatment

Heat conduction boundary conditions

(1) A three-dimensional transient heat conduction equation is established based on fourier law, taking into account the difference in thermal conductivity between concrete and reinforcement (the thermal conductivity of reinforcement is 44.5 $w / (m \cdot k)$ , and that of concrete is 1.8 $w / (m \cdot k)$ . The interface continuity condition (equation (4)) forces the reinforcement concrete interface to meet the requirements of temperature continuity and heat flux conservation. It quantifies the dominant effect of the reinforcement network on the heat conduction path through the equivalent thermal diffusion coefficient ( $\partial = λ / ρ \cdot c$ ) (the thermal diffusion coefficient of reinforcement is 24.7 times that of concrete).

Compare two typical working conditions

(1) Constant heat flux (5800 $w / m^{2}$ ): simulate the continuous sunshine conditions to verify the redistribution of heat flux density caused by the “thermal siphon effect” of the reinforcement network

(2) Constant temperature (343.15 $k$ ): Simulate an extremely high-temperature environment and quantify the impact of reinforcement diameter (16-26 mm) and stirrup spacing (30-170 mm) on temperature uniformity.

Constraint condition

The mechanical boundary adopts the free deformation assumption; however, the self-equilibrium stress field caused by the temperature gradient is revealed through the thermal stress sequence coupling analysis. Stress concentration is observed in areas with dense reinforcement, such as the web flange junction.

Parametric boundary

Material properties: take values according to ISO 10456 standard and consider the anisotropy of concrete thermal conductivity.

Geometric constraint: The thickness of the protective layer (30 mm) and reinforcement ratio shall be set according to the GB 50010-2010 specification to ensure that the model conforms to engineering practice.

Model establishment

To investigate the temperature distribution and thermal siphon effect in concrete box girders under different heat sources, a finite element model was developed using COMSOL Multiphysics 6.2 for transient heat conduction analysis (Jia et al., 2017; Jiang, 2006; Wilson and Nickell, 1966) The model incorporates the thermophysical properties of concrete (thermal conductivity, specific heat capacity, and density) and accounts for the thermal influence of steel reinforcement. Appropriate thermal boundary conditions were applied to simulate realistic environmental heat exchange processes.

A single-cell box girder segment model with the cross-section shown in the figure was developed. The box girder features a symmetrical hollow section with overall dimensions of 600 mm (width) × 400 mm (height) and a flange height of 100 mm. The concrete cover thickness was set to 30 mm, with longitudinal reinforcement bars of 16 mm diameter and stirrups of 8 mm diameter. The 3D box girder has a depth of 1060 mm (Figure 1).

Figure 1.

(a) Box Girder Cross-Section, Reinforcement Layout, and Measurement Point Arrangement (Unit: mm) (b) 3D finite element model.

Validate the siphon effect

Constant heat flux

The thermal conduction process in a reinforced concrete box girder structure was simulated using a constant heat flux as a stable heat source. By varying the position of the heat source, the siphon effect of heat conduction was investigated under different heat source locations. The heat flux was set at 5800 $w / m^{2}$ , with the thermal conductivities of concrete and steel reinforcement being 1.8 $w / (m \cdot k)$ and 44.5 $w / (m \cdot k)$ , respectively.

The standard deviation of the temperature distribution at each measurement point is used as a temperature distribution uniformity index to quantify the dispersion of temperature data across different points simultaneously, reflecting the uniformity of the temperature distribution in the concrete structure. Its unit is h. The formula is as follows:

S T D = \sqrt{\frac{\sum_{i = 1}^{n} {(t_{i} - \bar{t})}^{2}}{n}}

(11)

Where them,

t_{i}

is the temperature at each measurement point,

\bar{t}

is the average temperature of the measurement points, and

n

is the number of measurement points.

As shown in Figure 2, when the top slab is heated for 12 hours, the temperatures at the same measuring points of the reinforced concrete box girder structure are significantly higher when the thermal conductivity of the steel reinforcement is considered, except for measuring point 1. The reason lies in the fact that heat is concentrated and propagated backward along the steel reinforcement mesh, accelerating the temperature rise of the concrete.

Figure 2.

Temperature variations at measurement points with and without consideration of steel reinforcement thermal conductivity under constant heat flux.

For measuring point 1, when the influence of the steel reinforcement is neglected, heat is primarily transferred downward uniformly through the concrete rather than being concentrated along the steel reinforcement network. This results in the concrete absorbing more heat, leading to a higher measured temperature compared to the scenario where the thermal conductivity of the steel reinforcement is considered. This phenomenon confirms that the “thermal siphon effect” of steel reinforcement has a significant impact on the temperature distribution in concrete. Specifically, the high thermal conductivity of the steel reinforcement preferentially directs heat propagation along its path, thereby reducing the temperature rise of the concrete itself.

As can be seen from Figure 3, After heating the top slab for 12 hours, the standard deviation of temperatures at each measuring point was used to reflect the degree of temperature dispersion within the structure.

Figure 3.

The standard deviation of temperature at each measurement points with and without thermal conductivity is considered.

In Case 1 (With the thermal conductivity of steel reinforcement), the standard deviation increased gradually, indicating that the steel reinforcement accelerated heat diffusion due to its high thermal conductivity, leading to a more uniform temperature distribution.

In contrast, Case 2 (without the thermal conductivity of steel reinforcement) exhibited a rapid increase in standard deviation, indicating that the low thermal conductivity of concrete led to delayed heat transfer, resulting in persistent and worsening temperature non-uniformity. This further verifies the thermal siphon effect of steel reinforcement in concrete structures. Its conductive network significantly suppresses temperature gradients by efficiently redistributing heat.

When the top slab of the box girder is heated, heat is initially transferred to the concrete and steel reinforcement near the heated surface. Due to the high thermal conductivity of the steel reinforcement, heat rapidly spreads along both its longitudinal and transverse directions, preventing excessive heat accumulation near the heated surface. The heat conduction through the steel reinforcement promotes a more uniform temperature distribution, avoiding significant temperature differentials within the girder.

As shown Table 2, when a constant heat flux is applied as the heat source, regardless of the heating location in the box girder structure, the heat flux density of the reinforcement mesh is significantly higher than that of the concrete. This indicates that heat is primarily conducted through the reinforcement mesh in a backward direction, verifying the siphon-like heat transfer mechanism. The simulation results demonstrate that under constant heat flux, the temperature field of the box girder exhibits a distinct gradient distribution. High-temperature regions are mainly concentrated near the heat source, while low-temperature regions are located farther away. Due to the significantly higher thermal conductivity of steel compared to concrete, heat is rapidly transferred through the reinforcement, resulting in faster temperature changes in the steel regions and relatively slower changes in the concrete regions.

Table 2.

Comparison of constant heat flux source locations.

Heat source location	With the thermal conductivity of steel reinforcement	Without the thermal conductivity of steel reinforcement
Left side
Top slab
Inner wall
Base slab

Furthermore, due to differences in geometric shape and heat source location, the temperature distribution varies significantly among the top slab, web, and bottom slab of the box girder. As a stable heat source, the constant heat flux leads to a heat flux density distribution characterized by rapid heat transfer through the reinforcement, making the steel bars the primary pathway for heat flow. In contrast, the heat flux density in the concrete remains relatively low. Heat flows from high-temperature regions (near the heat source) to low-temperature regions (away from the heat source), aligning with the temperature gradient. In areas with dense reinforcement, the heat flux density is higher; in concrete-dominated regions, it is lower.

Constant temperature

This section investigates the temperature and heat flux distribution in concrete structures under a constant-temperature heat source, further validating the aforementioned conclusions. The study continues to focus on heating the top slab, with an initial temperature of 293.15 K and a constant heating temperature set at 343.15 K. The heating duration is 12 hours.

The simulation results are consistent with those obtained using a constant heat flux as the thermal source. The findings demonstrate that under constant temperature conditions, the box girder structure exhibits a distinct temperature gradient distribution.

As shown in Figure 4, when considering the thermal siphon effect of the reinforcement, the temperature of the reinforced concrete box girder after 12 hours of heating is still significantly higher than when ignoring the influence of the reinforcement on the temperature distribution of the concrete. The high thermal conductivity of the reinforcement significantly accelerates heat transfer, resulting in a faster temperature rise within the structure. Without accounting for the thermal conductivity of the reinforcement, heat transfer primarily relies on the concrete, which has a lower thermal diffusivity, resulting in a slower temperature increase.

Figure 4.

Temperature variations at measurement points with and without consideration of steel reinforcement thermal conductivity under constant temperature conditions.

It can be observed from Figure 5, that, in the measurement of temperature dispersion, the dispersion exhibited a significant downward trend after approximately 2.3 hours of thermal loading. Increasing the diameter of steel bars from 16 mm to 26 mm can significantly accelerate the process of structural temperature homogenization, indicating that the high thermal conductivity of the reinforcement accelerated temperature homogenization within the structure, thereby reducing the temperature gradient.

Figure 5.

The standard deviation of temperature with and without consideration of thermal conductivity under constant heat flux.

Sensitivity analysis of reinforcement parameters

Steel bar diameter

This section analyzes the influence of steel bar diameter on the thermal siphon effect. To achieve this, the same steel reinforcement layout is adopted, with only the diameter of the steel bars varied and described using the reinforcement ratio. To ensure that all reference diameters comply with the regulations on the precise spacing between steel bars, the concrete cover thickness is set to 30 mm, and the stirrup spacing is set to 200 mm by the “Code for Design of Concrete Structures” GB 50010-2010. The longitudinal reinforcement diameters are 16 mm, 18 mm, 20 mm, 22 mm, 24 mm, and 26 mm, while the stirrup diameter is 8 mm.

Establish a three-dimensional finite element model of a concrete box girder structure, using a constant temperature as a stable heat source to analyze the influence of steel bar diameter on the structural thermal equilibrium. Based on heat transfer theory, analyze the impact of different reinforcement ratios on the thermal siphon effect under varying locations of the heat source.

Condition 1: Heat source temperature gradient perpendicular to the heat conduction direction of longitudinal reinforcement bars

As illustrated in Figure 6(b), when the temperature gradient of the heat source is perpendicular to the heat conduction direction of the longitudinal reinforcement bars, the influence of rebar diameter on structural temperature uniformity exhibits the following characteristics: Although increasing the rebar diameter leads to a certain degree of reduction in the temperature uniformity index, this effect remains relatively limited. Numerical simulation results indicate that under typical working conditions when the rebar diameter increases from 16 mm to 26 mm, the decrease in the temperature uniformity index does not exceed 12%. This demonstrates that the impact of rebar diameter variation on structural thermal equilibrium is a secondary factor, though not entirely negligible.

Figure 6.

(a) Apply a constant temperature to the top plate. (b) Temperature standard deviation with and without thermal conductivity consideration under the top plate.

Condition 2: Heat source temperature aligned with the heat conduction direction of longitudinal reinforcement bars

As illustrated in Figure 7(b), when the direction of the heat source temperature gradient aligns with the heat conduction path of the steel reinforcement, increasing the rebar diameter significantly accelerates structural temperature homogenization. Due to steel’s high thermal diffusivity, larger diameter rebars substantially enhance axial heat transfer capacity. According to heat transfer principles, increasing rebar diameter causes a quadratic growth in conductive cross-sectional area, thereby markedly improving heat flux transmission efficiency under identical temperature gradients.

Figure 7.

(a) Apply a constant temperature to the left side. (b) Temperature standard deviation with and without thermal conductivity consideration under left side.

Increasing the diameter of reinforcement from 16 mm to 26 mm can significantly accelerate the process of structural temperature homogenization. Consequently, in engineering applications that require rapid thermal equilibrium, increasing the rebar diameter appropriately constitutes an effective technical measure. However, careful consideration must be given to its impact on the performance of concrete cover.

Stirrup spacing

According to the Code for Design of Concrete Structures (GB 50010-2010), the maximum stirrup spacing shall not exceed 200 mm when the cross-sectional height exceeds 300 mm. Therefore, the following reference spacings were selected: 30 mm, 50 mm, 70 mm, 90 mm, 110 mm, 130 mm, 150 mm, and 170 mm, with 16 mm-diameter longitudinal rebars adopted.

As shown in Figure 8, under conditions where the temperature gradient of the heat source propagates along the structural heat conduction direction, higher reinforcement density and larger steel bar diameters result in faster temperature equilibrium and more significant attenuation of the temperature gradient across the structure. This phenomenon primarily stems from the following mechanism: As a high thermal conductivity medium, the increased cross-sectional area and distribution density of steel bars significantly enhance the equivalent thermal conductivity of the structure, leading to a nonlinear improvement in heat flow transfer efficiency.

Figure 8.

Temperature uniformity index under varying stirrup spacing.

Thermal gradient-induced stress

This section examines the thermal stress resulting from the coupling of heat transfer and stress fields, with roof heating as the research subject. The standard deviation of temperature measurements at each monitoring point reflects the degree of dispersion of internal structural temperatures.

Under constant temperature conditions, after applying a 12-h thermal load, the temperature difference between the top and bottom slabs of the box girder creates a thermal gradient. The top slab has a higher temperature, while the bottom slab remains cooler. This non-uniform temperature distribution induces bending stress in the box girder. Temperature gradients can cause non-uniform deformation in box girder cross-sections, leading to expansion of the top slab and contraction of the bottom slab, thereby inducing bending deformation and generating additional bending moments.

As shown in Figure 9(b), the temperature uniformity index considering the thermal conductivity of steel bars is higher than that without considering the thermal conductivity of steel bars. In Case 1, the steel reinforcement acts as a heat conduction path, accelerating heat transfer and resulting in a more uniform temperature distribution. In Case 2, the temperature field is primarily determined by the thermal conductivity of the concrete, resulting in a larger temperature gradient and less uniform temperature distribution. During roof heating, a significant temperature difference is observed between scenarios with and without consideration of thermal conductivity, resulting in a notable distinction in thermal stress.

Figure 9.

(a) Two-dimensional stress surface distribution. (b) Under the two working conditions, the stress uniformity index changes with time.

Summary and discussion

This study systematically analyzed the temperature field distribution characteristics and the variation law of heat flux density of reinforced concrete box girder bridge segments under solar radiation using the finite element method. By establishing numerical models under two boundary conditions—constant heat flux and constant temperature —the phenomenon of the ‘siphon effect’ during heat conduction was verified in box girder structures for the first time. Furthermore, related studies on reinforcement parameters revealed that:

(1) The thermal conductivity of steel is 24.7 times that of concrete, forming an efficient heat transfer network. This results in a rapid rise in temperature near the reinforcement and a slow rise away from it, creating a significant temperature gradient. Under the constant temperature boundary condition, increasing the diameter of the steel bar from 16 mm to 26 mm can significantly reduce the homogenization time of the structural temperature, primarily due to the improvement in thermal conductivity resulting from the increase in steel cross-sectional area.

(2) The reinforcement density has a key influence on the temperature distribution. When the stirrup spacing along the heat conduction direction is reduced from 170 mm to 30 mm, the structure with dense reinforcement and a large diameter can reach heat balance faster, and the temperature gradient is smaller.

(3) Along the direction of heat conduction of the structure, the denser the reinforcement, the larger the diameter, the faster the overall temperature balance of the structure, and the smaller the temperature gradient. Therefore, in the design process, the number of steel bars can be appropriately increased, and the diameter of the steel bars can be reduced to improve the temperature uniformity of the concrete box girder section and achieve a balanced speed.

(4) The temperature difference between the surface and the inside of the box girder leads to tension-compression stress in the concrete, and stress concentration is more likely to occur in areas with dense reinforcement, which aggravates cracking and degrades the bond performance between the reinforcement and concrete.

Currently, with the trend toward long-span and high-performance development in bridge engineering, the temperature field effects of reinforced concrete box girder bridges have become a critical factor affecting structural safety and durability. However, this article remains limited to surface-level research, and the following shortcomings exist in the analysis of the temperature field in reinforced concrete box girder bridges, which need to be addressed in future studies:

(1) Solar radiation simulation employs constant heat flux and temperature as thermal sources, neglecting dynamic factors such as actual solar angle variations and cloud cover. Comparative analysis with measured data suggests the need to introduce meteorologically driven non-uniform thermal boundary conditions, incorporating spatiotemporal variations in convective heat transfer coefficients.

(2) The “thermal siphon effect” has only been validated through numerical simulations, lacking full-scale experimental or long-term monitoring data. Further attention should be given to measured temperature gradients in reinforcement-dense zones such as web-flange junctions.

Footnotes

Acknowledgements

The authors greatly appreciate the financial support from the National Natural Science Foundation of China (No.52178272, NO. 52478300), Youth Project of Science and Technology Research Program of Chongqing Education Commission of China (NO. KJ202300790032498), and the Chongqing Municipal Education Commission Science and Technology Key Project (KJZD-K202305501).

Jinglin Pan

Benniu Zhang

Jigang Yu

Hongmei Cui

Author Contributions

J.L.P. led the finite element modeling, parametric studies, and numerical validation of the thermosiphon effect, while also conducting result analysis and manuscript drafting. B.N.Z. (corresponding author) conceived the research, proposed the theoretical framework, and supervised the overall research direction, participating in manuscript revision and academic oversight. J.G.Y. focused on the theoretical interpretation of thermal stress analysis and temperature gradient distribution, assisting in data visualization. H.M.C. contributed to refining the manuscript’s structural framework and participated in revision guidance. All authors discussed the results, reviewed the manuscript, and approved the final version.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the financial support from the National Natural Science Foundation of China (No.52178272, NO. 52478300), Youth Project of Science and Technology Research Program of Chongqing Education Commission of China (NO. KJ202300790032498), and the Chongqing Municipal Education Commission Science and Technology Key Project (KJZD-K202305501)

Declaration of conflicting interests

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

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