Temperature Field and Gradient Effects for Concrete-Filled Steel Tubular Truss Arch Bridges Under Construction

Abstract

Long-span concrete-filled steel tubular truss arch bridges are extremely sensitive to thermal effects during cantilever construction, with non-uniform temperature distributions arising from mutual shading between members. The current standard JTG/T D65-06—2015 employs a simple gradient model that struggles to capture the temperature gradient characteristics of complex spatial trusses, failing to meet the demands of high-precision construction. Based on a truss-type steel arch bridge in Yunnan, a thermal conduction analysis framework is proposed to calculate the temperature field of the arch rib truss and its effects, and is validated by long-term monitoring data. The results indicate that the maximum temperature difference between the upper and lower chord tubes reaches 14.53 °C, significantly changing the secondary stress distribution. There is a significant negative correlation mechanism between arch rib elevation and solar radiation temperature, necessitating consideration of solar radiation temperature effects during arch rib assembly and closure. This study establishes an analytical method for the thermal effects of long-span steel truss arch ribs, laying the foundation for arch rib profile control and stress analysis.

1. Introduction

The initial internal force state of bridge structures and long-term operational safety are closely related to the control of the construction process [1]. During the construction of concrete-filled steel tubular (CFST) truss arch bridges, the cable-stayed suspension method is frequently employed. Bridge temperature fields are influenced by meteorological parameters such as solar radiation, environmental temperature, and wind velocity. The structural nonlinear temperature field is caused by the normal azimuth and inclination angle of its surface, which causes significant variations in the thermal load effect [2]. Furthermore, the structural temperature at closure directly determines the baseline state of the structure [3]. Closing the arch at an improper temperature will generate significant additional internal forces and excessive deformation. Therefore, the study of temperature effects on the arch during construction is particularly important.

Substantial advancements have been made in analytical methods based on numerical models and analytical theory to address the challenge of efficiently solving bridge temperature fields in complex environments. In terms of numerical models, prior research has formulated one-dimensional solar radiation temperature field models for steel-concrete composite girder bridges [4,5]. Furthermore, refined three-dimensional numerical models capable of capturing non-uniform thermal characteristics have been proposed for truss suspension bridges [6]. Simultaneously, analytical theories have been extensively operationalized for regular cross-sectional members due to their computational simplicity. For main cable structures, researchers have derived analytical solutions for steady-state and transient conditions under solar radiation [7,8], and proposed correction formulas for cross-sectional average temperature deviations [9]. For steel bridge structures, temperature distribution functions determined via steady-state thermodynamic equilibrium equations effectively reveal heat transfer mechanisms and critical influencing factors [10]. However, these simplified computational methods exhibit pronounced methodological constraints when applied to large-span CFST truss arch bridges. Specifically, these methods fail to adequately capture complex mutual shading effects or describe spatial non-uniformity in the temperature fields.

To balance engineering practicality and computational efficiency, simplified temperature gradient characterization methods based on finite element analysis have been widely used in the research of bridge temperature fields [11,12,13]. In the field of box girder bridges, research has evolved from establishing vertical gradient models [14] for steel–concrete composite beams to revealing the transverse thermal characteristics of concrete sections [15], then to examining the impact of longitudinal dimensions on temperature gradients [16,17]. Regarding arch bridges, scholars have proposed vertical temperature gradient models for box ribs [18] and corresponding correction methods for longitudinal inclination based on field monitoring and numerical simulations [19]. For concrete–steel composite structures with more complex thermal properties, refined investigations have progressed from basic solar radiation models [20,21] to more sophisticated models that incorporate circumferential variations in large-diameter members [22] and the effects of interfacial debonding [23]. Studies by Zhang et al. [24] and Liu et al. [25] have further elucidated the non-uniform thermal distributions in complex composite sections under solar exposure. Additionally, numerical frameworks have been refined through mechanical–hydration heat coupling [26,27,28,29] and fire resistance assessments under extreme temperatures [30,31,32]. However, these studies primarily focus on establishing static gradient models for the completed bridge state, which exhibits significant limitations in achieving precise control during the cantilever construction process. Specifically, the time-varying thermal–mechanical coupling mechanism during the construction phase is overlooked, making it difficult to meet the real-time control requirements for the main arch.

Currently, finite element-based temperature gradient models provide acceptable accuracy for standard engineering applications in CFST truss arch bridges. However, existing thermal models predominantly focus on the temperature differentials between upper and lower chords, while ignoring the vertical gradients within individual tubes. This simplification significantly compromises the precision of structural thermal effect calculations. Furthermore, conventional approaches are largely grounded in static design specifications, which fail to capture the transient and time-dependent nature of temperature gradients during the construction phase. Moreover, the high computational cost of high-fidelity finite element analysis precludes its application for real-time monitoring, as it cannot rapidly adapt to the evolving structural and environmental variables during construction. Consequently, establishing an equivalent temperature gradient model is of paramount importance. Such a model would facilitate the rapid evaluation of thermal effects, providing a robust framework for real-time alignment control in large-span bridges.

A calculation framework for the temperature field and its effects for truss steel arch bridges has been proposed and validated on an arch bridge in Yunnan. First, the calculation steps and theoretical analysis of the arch rib temperature gradient model are carried out in Section 2. Then, the numerical analysis of the temperature field and gradient model is carried out in Section 3. Finally, the analysis of temperature stress and deformation effects under gradient temperature loading is carried out in Section 4 and Section 5.

2. Calculation Method for Temperature Gradient of Arch Ribs

2.1. Physical Entity of CFST Truss Arch Bridge

A CFST truss arch bridge in Yunnan is selected as the subject for thermal field re-search. The arch ribs have a calculated span of 138 m with the rise-to-span ratio of 1/6. The arch axis adopts a catenary curve with an arch axis coefficient of 1.6. The arch, painted silver, consists of three arch ribs 3.32 m-wide, spaced 3.10 m apart (center-to-center) and formed of four A800 × 24 mm chord tube frames. The single-line cross-brace is installed between the arch ribs to ensure lateral stability. The chord tubes are connected by horizontal tie tubes (A426 × 16 mm) and diagonal tie tubes (A426 × 12 mm). The elevation of the arch bridge and the actual bridge structure are shown in Figure 1 and Figure 2, respectively. In the figures, “T” and “B” represent the sensor locations on the upper and lower chord tubes, respectively, and are numbered sequentially in the direction of water flow.

The bridge structure is situated in a mountainous gorge, with the bridge running in a north–south direction. The bridge is positioned at the geographical coordinates of 102.4° east longitude and 25.2° north latitude. Based on the terrain characteristics, the arch ribs are assembled by a diagonal tension-clamping method. Temperature and strain sensors are installed at control sections (arch base, L/4, and L/2) along the span of the main arch (Figure 3). Continuous monitoring is conducted using a wireless data acquisition system to investigate the distribution patterns of non-uniform temperature fields in the steel pipe arch ribs under solar radiation.

2.2. Temperature Field and Its Effects Analysis Framework

This paper proposes an integrated research framework combining monitoring data, heat transfer theory, and structural mechanics analysis to precisely reveal the temperature distribution and structural response behavior of truss arch bridges under solar radiation (Figure 4). The process is described as follows:

(1)

Data Foundation and Physical Entity Construction: The concrete-filled steel tube truss arch bridge equipped with monitoring system is treated as a physical entity, and structural temperature, ambient temperature, and temperature effects are collected.

(2)

High-Precision Computation of 3D Temperature Fields: The solar radiation heat flux density is calculated using structural geometry parameters and structural inclination angle, establishing a 3D temperature field for the arch rib chord tube based on the finite difference method. Compare the calculated results with the measured temperature data to verify the validity of the temperature distribution pattern.

(3)

Equivalent Gradient Temperature Models: The continuous string tube is divided axially into multiple representative cross-sections, with the temperature distribution at each cross-section condensed into a time-varying gradient temperature model.

(4)

Simulation and Analysis of Structural Temperature Effects: The equivalent gradient temperature model is applied as a time-varying load to the finite element model for thermal coupling analysis, calculating the structure’s stresses and deformations. Compare simulation results with monitoring data to reveal the time-varying behavior of solar radiation temperature effects and validate the overall accuracy of the computational workflow.

2.3. Theory of Temperature Gradient Model

Arch bridges absorb solar radiation, exchange heat with their surroundings, and experience internal heat conduction. The 3D temperature field is established to consider axial heat conduction caused by the variation of the arch rib surface normal angle along the axial direction. In the transient 3D temperature distribution method, the temperature field T can be expressed as a function of coordinates (r, θ, z) of a point in the section and the specific time t as

$$T=f\left(r,\theta ,z\right)$$

(1)

According to the first law of thermodynamics, the structural internal energy change equals the heat inflow at its boundaries within any time step Δt. The transient heat conduction equation and external boundary balance equation of chord tube under column coordinates are Equations (2) and (3), respectively:

$${\displaystyle \frac{\partial T}{\partial t}}=\lambda \left({\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^{2}T}{\partial {\theta }^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}\right),\lambda ={\displaystyle \frac{k}{\rho c}}$$

(2)

$$-k{\displaystyle \frac{\partial T}{\partial r}}=q_{sur}, q_{sur}=q_s+q_c+q_r$$

(3)

where λ is the thermal diffusivity, which is related to its thermal conductivity k; ρ is density; c is the specific heat; q_sur is the heat flux on the structural surface, composed of the solar radiation q_s, atmospheric convective heat transfer q_c and radiative heat transfer heat flux densities q_r.

$$q_s=a_s\left(G_s+G_r+G_g\right), q_c=h_c\left(T_a-T\right), q_r=\epsilon C_0\left[{\left(T_a\right)}^4-{\left(T\right)}^4\right]$$

(4)

where a_s is the absorptivity of the object surface; G_s, G_r, and G_g are the total solar radiation intensity, and sky-scattered and ground reflected radiation intensity, respectively; h_c is the film coefficient; ε is the emissivity of the object surface; C₀ is the Stefan–Boltzmann constant; T_a is the surrounding ambient temperature. The daily temperature variation process is considered as a sinusoidal function [33].

$$T_a(t)={\displaystyle \frac{1}{2}}\left(T_{a,\max }+T_{a,\min }\right)+{\displaystyle \frac{1}{2}}\left(T_{a,\max }-T_{a,\min }\right)\sin {\displaystyle \frac{\left(t-t_0\right)\pi }{12}}$$

(5)

where T_a,max and T_a,min are the daily maximum and minimum temperatures, respectively; t₀ is the time when extreme temperature occurrence.

The arch axis is a catenary curve, which is composed of numerous straight segments. Therefore, the axial thermal conduction problem in a catenary chord tube can be equivalently modeled as a straightened one. The transient temperature distribution of the chord tubes is calculated by the finite difference method. The chord tube is uniformly divided into n, m, and p nodes along the axial, radial, and circumferential directions, respectively. The spatial node temperature derivatives are discretized to gain the node temperature update equations at internal and boundary locations. The discrete form of the first derivative for the temperature field T_r,θ,z at time t and radial radius r can be expressed as

$${\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{T_{r,\theta ,z}^{t+1}-T_{r,\theta ,z}^t}{\Delta t}}, {\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}}={\displaystyle \frac{1}{r}}\cdot {\displaystyle \frac{T_{r+1,\theta ,z}-T_{r-1,\theta ,z}}{2\Delta r}}$$

(6)

The second derivatives of the temperature field T_r,θ,z with respect to the axial, radial, and circumferential steps Δz, Δr, and Δθ are discretized as

$${\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}={\displaystyle \frac{T_{r,\theta ,z+1}-2T_{r,\theta ,z}+T_{r,\theta ,z-1}}{{\left(\Delta z\right)}^2}},{\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}={\displaystyle \frac{T_{r+1,\theta ,z}-2T_{r,\theta ,z}+T_{r-1,\theta ,z}}{{\left(\Delta r\right)}^2}},{\displaystyle \frac{{\partial }^{2}T}{\partial {\theta }^2}}={\displaystyle \frac{T_{r,\theta +1,z}-2T_{r,\theta ,z}+T_{r,\theta -1,z}}{{\left(\Delta \theta \right)}^2}}$$

(7)

By substituting Equations (6) and (7) into Equation (2), the updated formula of internal node temperature T_in is Equation (8):

$$T_{\mathrm{in}}^{t+1}\left(r,\theta ,z\right)=T_{\mathrm{in}}^t\left(r,\theta ,z\right)+F{o}_r\cdot {\left(\Delta r\right)}^2\cdot {\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}+F{o}_{rn}\cdot 2\Delta r\cdot {\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}}+F{o}_{\theta }\cdot {\left(\Delta \theta \right)}^2\cdot {\displaystyle \frac{{\partial }^{2}T}{\partial {\theta }^2}}+F{o}_z\cdot {\left(\Delta z\right)}^2\cdot {\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}$$

(8)

where Fo_r, Fo_rn, Fo_θ, and Fo_z are the radial, radially corrected, circumferential, and axial Fourier numbers, respectively.

$$F{o}_r={\displaystyle \frac{\lambda \Delta t}{{\left(\Delta r\right)}^2}}, F{o}_{rn}={\displaystyle \frac{\lambda \Delta t}{2r\cdot \Delta r}}, F{o}_{\theta }={\displaystyle \frac{\lambda \Delta t}{{\left(\Delta \theta \right)}^2}}, F{o}_z={\displaystyle \frac{\lambda \Delta t}{{\left(\Delta z\right)}^2}}$$

(9)

The outer surface of the arch ribs is affected by the combined effects of solar radiation heating, convective heat transfer cooling, and thermal radiation cooling. Based on the equilibrium equations for the Neumann boundary condition and Robin boundary condition, the temperature at the outer surface nodes T_B,out is updated as

$$T_{B,\mathrm{out}}^{t+1}\left(r,\theta ,z\right)=T_{B,\mathrm{out}}^{t+1}\left(r-2,\theta ,z\right)-{\displaystyle \frac{2\Delta r}{k}}\cdot \left(q_s+q_{\mathrm{c}}+q_r\right)$$

(10)

Since there is no air circulation on the internal surface of the arch ribs, the inner boundary, considered an adiabatic boundary (T_B,in), is

$$T_{B,\mathrm{in}}\left(r,\theta ,z\right)-T_{B,\mathrm{in}}\left(r+1,\theta ,z\right)=0$$

(11)

The temperature distribution of the chord tube is obtained through the cyclic iteration of the updated internal temperature equation (Equation (8)) and boundary conditions (Equations (10) and (11)) for each time step Δt. Based on the plane section assumption, the nonlinear temperature field within the cross-section is modeled as a time-varying temperature gradient model (T₀(t) and k_z(t)), which enhances the computational efficiency of the structural temperature effects in the model.

$$T_0\left(t\right)={\displaystyle \frac{{\displaystyle \underset{S}{\iint }{T}_{r,\theta ,z}\left(t\right)dA}}{A_s}}, k_z\left(t\right)={\displaystyle \frac{{\displaystyle \underset{S}{\iint }z\cdot T_{r,\theta ,z}\left(t\right)dA}}{I_y}}$$

(12)

where T₀ is the equivalent overall temperature; k_z is the vertical temperature gradient along the cross-section; A_s is the cross-sectional area; I_y is the inertia moment of a cross-section at the y-axis.

3. Analysis of Temperature Field and Gradient Model

3.1. Temperature Field Analysis

The chord tube is uniformly divided into 10, 32, and 500 nodes in the radial r, circumferential θ, and axial z directions, respectively. The heat flux density of solar radiation at different locations along the arch segment is influenced by the varying inclination and circumferential angles of the arches. Based on an extensive literature investigation [5,23,34], the reasonable values of thermal properties are determined and listed in

Table 1. Thermal Property.

Property	Steel
Density ρ/(kg/m3)	7850.0
Thermal conductivity k/(W/m·°C)	55.0
Specific heat c/(J/kg·°C)	475.0
Absorptivity as	0.5
Emissivity ε	0.8

. Figure 5 shows the temperature field at the L/2 cross-section, where the cross-sectional circumferential temperature varies with solar incidence angle. The cross-sectional temperature rises gradually from sunrise, reaching its maximum value (44.80 °C) at 14:00 before decreasing, until it becomes uniformly distributed.

The arch ribs in the transverse direction have the same inclination, and it can be considered that the temperature distribution is identical. However, the inclination of the chord tubes along the axial directions varies, resulting in differing projections of solar radiation along their surface normal directions, which leads to non-uniform temperature distribution (Figure 6). The maximum temperature (45.08 °C) of the cross-section occurs at the L/2 cross-section, which is 3.79 °C and 2.57 °C higher than the arch foot and L/4 section, respectively. The measured temperature variation trends at each section (arch foot, L/4, and L/2) are generally consistent with the numerical results, and the root mean square error between the two is less than 1.78 °C. The maximum temperatures of the arch ribs on the left and right banks occur with a 1h difference. During the heating phase, the solar orientation aligns closely with the normal direction of the right bank arch rib surface, causing its heating rate and heat flux density to be greater than the left bank.

3.2. Temperature Gradient Model

According to the “Specifications for the Design of Highway Concrete-Filled Steel Tubular Arch Bridge” [35], the temperature difference between the upper and lower chords of truss arches with silver surfaces is calculated to be 5 °C. Specifications are determined by probabilistic statistical methods to establish the most unfavorable temperature values during the design reference period. The truss section assumes that solar radiation primarily impacts the top of the upper chord, while the web and lower chord receive insufficient solar radiation. The temperature gradient is simplified to the temperature difference between the average temperatures of the upper and lower chords [3] (Figure 7a). However, the solar elevation during the day causes real-time variations in cross-sectional temperature, making it difficult to directly apply the temperature gradient from the specification to control the thermal effects on the arch. Furthermore, the specification gradient model fails to account for the temperature gradient differences caused by solar radiation, which will prevent accurate calculation of the linear and stress changes in the structure. Therefore, based on the temperature field characteristics of the truss-type arch ribs under solar radiation, a gradient temperature model (Figure 7b) suitable for this structure is proposed, where “U” and “L” represent the upper and lower chord tubes, respectively. The temperature gradient model will be validated through the temperature gradient effect in the subsequent sections.

Figure 8 shows the equivalent temperatures (T_0,U and T_0,L) and the temperature gradient model parameters (k_z,U and k_z,L) of the upper and lower chord tubes of the left bank arch rib. While the equivalent temperatures T₀ across different sections are relatively similar, the gradient temperatures k_z exhibit significant differences. The maximum gradient temperature of the upper chord tube is 53.05% higher than the lower one. The temperature gradient model shows the same trend as the cross-sectional temperature (Figure 6).

4. Effects Under Solar Gradient Temperature

4.1. Thermal Effects Analysis

The vertical deformation of the arch due to temperature effects is primarily caused by four factors: an overall rise in the temperature of the stay cables and the arch rib; a temperature difference between the upper and lower chord tubes; and a chord tube temperature gradient (Figure 9). All heating factors induce elongation deformation in the arch ribs (Δx_c, Δx_a, and Δx_dg). Specifically, cable heating, temperature difference, and gradient in the chord tube cause downward deflection (Δz_c and Δz_dg), while heating the arch rib elevates the arch ribs (Δz_a). When Δx_a + Δx_dg < Δx_c, the arch rib undergoes downward deflection; conversely, when Δx_a + Δx_dg > Δx_c, the significant elongation of the arch rib is constrained by the stay cables, causing notable additional stresses in both the upper and lower chord tubes. Thermal secondary stresses can reach magnitudes comparable to loading stresses, masking the true stress state of the structure and compromising the accuracy of structural condition assessments.

The time-dependent stress patterns induced by solar radiation must be studied to accurately analyze structural stresses during construction. Figure 10 shows the stress monitoring data of the upper and lower chord tubes at the left arch foot over 4 continuous days. During the installation of the arch segment, the thermal deformation of the main arch is constrained by the stay cables. The upper and lower chords at the arch feet generate significant compressive and tensile stresses under the combined effects of self-weight and stay cable forces. According to the “Technical Specifications for Construction Monitoring and Control of Highway Bridges” [36], the error limit of steel structure stress calculations does not exceed ±10 MPa when the stress value is below 100 MPa. The stress limit in the standard is induced by construction loads. However, the stress variation caused by temperature exceeds 10 MPa (Figure 10), demonstrating the significant impact of solar radiation on stress. Therefore, the effect of thermal residual stresses should be eliminated when analyzing stress.

4.2. Shielding Effect Influence

The arch ribs and stay cables are simulated using Beam elements (B31) and Truss elements (T3D2), respectively, when calculating temperature effects. Rigid connections are employed between the arch ribs, the web tubes, and the stay cables. Fixed constraints are applied at the arch feet, while the stay cable endpoints utilize hinged constraints at the mountain anchors (Figure 11). The stay cables exposed in the natural environment are affected by variations in atmospheric temperature and sunlight. During heating and cooling cycles, the cable temperature typically ranges within ±5 °C of the environmental temperature, exceeding it during heating and falling below it during cooling [37].

The relative position between the bridge structure and the sun changes over time. The pre-correction solar radiation calculations failed to account for structural shading effects caused by variations in this relative position. The complex truss arch bridge components block sunlight, thereby reducing solar radiation on the lower chord tubes. The shading coefficient μ is introduced to reduce solar radiation, enabling accurate calculation of temperature effects [38]:

$$q_{sur,shade}=\left(1-\mu \right)q_s+q_{\mathrm{c}}+q_r$$

(13)

The shading coefficient μ represents the ratio of shaded area to total structural area. A higher shading coefficient indicates a larger proportion of shaded area, resulting in reduced direct solar radiation exposure. When no shading exists, μ = 0; when fully shaded, μ = 1.0; and, for partial shading, 0 < μ < 1.0. A reasonable shading coefficient is obtained by comparing the structural temperature load stress effects under different shading coefficients with measured values. The shading coefficient shows a linear relationship with overall temperature and temperature gradient (Figure 12). The curve slope at each time shows variations influenced by the solar azimuth angle.

As the primary load-bearing member, the axial thermal expansion of the lower chord directly affects the overall geometry of the arch ribs. Realistic simulation ensures the accuracy of thermal deformation predictions for the arch ribs. By comparing results for different shading coefficients (μ = 0.2, 0.4, 0.6), the simulation results at μ = 0.4 most closely approximated the measured values (Figure 13). In the three specified operating conditions, μ = 0.4 is the optimal parameter, demonstrating the smallest errors in Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Symmetric Mean Absolute Percentage Error (SMPE) between the simulation and measured values during daytime (

Table 2. Error evaluation indicators under different shading coefficients.

Error Indicators	μ = 0.2	μ = 0.4	μ = 0.6
MAE	1.052	0.499	0.647
RMSE	1.461	1.027	1.173
SMAPE	0.084	0.063	0.082

). When accurate ray-tracing models are unavailable, the shadowing coefficient μ = 0.4 is recommended as the preferred choice for calculating the temperature of the lower chord tubes in similar truss arch bridges.

4.3. Thermal Stress Effects

During the installation of left bank arch segment 6 and right bank arch segment 3, arch foot stresses are calculated using the equivalent temperature gradient (Figure 14a). The stress variations exhibit a clear correlation with the structural temperature. The simulation results demonstrate excellent consistency with measured trends, validating the effectiveness of the adopted temperature gradient model. Peak stresses at the arch feet on both sides exhibit a significant time lag of 1~2 h, indicating that the thermal effects on the structure are spatially non-synchronous. Significant differences exist in the peak stresses between the two banks, with the maximum difference reaching 9.16 MPa. Furthermore, the stay cables constrain the main arch and inhibit free thermal expansion, leading to greater tensile stress in the lower chord tubes on the right bank compared to those on the left; this is evidenced by the higher cross-sectional temperatures on the right bank (Figure 14b).

4.4. Thermal Deformation Effects

Figure 15a shows the measured elevation of the right bank arch rib referenced to the 8:00 AM data and temperature values. The elevation exhibits a significant negative correlation with the temperature. During the 11:00 to 16:00 PM period, the structure exhibited two instances of “reverse arching”, while the temperature of the arch ribs showed a pronounced downward trend throughout this timeframe. The “reverse arching” phenomenon is caused by weather conditions (cloudy weather) blocking the structure, leading to a rapid drop in the surface temperature of the arch ribs and consequently causing the elevation of control points to trend upward. Furthermore, the deflection trends of the arch rib segments are consistent, and the maximum values occur at the same time (14:00). The maximum deflections of arch segments 4 to 6 are 11.52 mm, 18.69 mm, and 25.85 mm, respectively, and the influence of solar radiation must be considered during tangential assembly of the arch ribs. The most elevation errors can be controlled within 3 mm, demonstrating that the temperature gradient model possesses reliable computational accuracy.

Some researchers consider the temperature gradient of the truss arch ribs as the temperature difference between the upper and lower chord tubes [39,40]. The temperature field across the cross-section is calculated as an average value, with a maximum average temperature difference of 2.6 °C between the upper and lower chord tubes (Figure 15a). The maximum deflections recorded at control points 4–6 of the arch segment are 0.48 mm, 0.78 mm, and 2.20 mm, respectively—significantly lower than the deflections at control points under gradient temperature conditions (11.52 mm, 18.69 mm, and 25.85 mm). However, considering only the average temperature weakens the impact of the vertical temperature gradient, thereby severely underestimating the thermal effects on the arch ribs. Therefore, the influence of temperature gradients across the cross-section must not be ignored during the installation of the arch ribs.

The elevation at the cantilever ends of both banks is consistent (Figure 15b), showing two prominent “reverse arching” phenomena. The main arches on both sides display temperature differences (5.36 °C), which results in greater elevation changes on the right bank compared to the left (6.27 mm). Sunlight significantly affects the elevation difference between the two banks. Sunlight significantly affects the elevation difference between the two banks, and the closure of the arch ribs must be scheduled before sunrise. The relative errors between simulated and measured elevation at the cantilever ends of the left and right banks are controlled within 11.55% and 10.29%, respectively, which validated the effectiveness of the temperature gradient model and provided a reliable basis for controlling the closure profile of the main arches on both banks.

5. Conclusions

A thermal conduction analysis framework is proposed to calculate the temperature field of the CFST truss arch and its effects. The proposed computational method is validated using monitored temperature effects data. The primary conclusions are drawn as follows:

(1)

The maximum temperature difference between the upper and lower chord tubes of the arch rib reached 14.53 °C, while the maximum longitudinal temperature difference along the rib arch was 3.79 °C. The calculated temperature field values for the chord tubes showed satisfactory agreement with the measured values. Furthermore, the proposed gradient temperature model accounts for the vertical temperature gradient within a single tube, accurately reflecting the temperature field characteristics of the chord tubes.

(2)

The stress at the arch foot exceeded the tolerance limit (±10 MPa) specified in specification JTG/T 3650-01-2022. During arch bridge monitoring, the impact of solar radiation temperature must be fully considered to prevent thermal secondary stress masking the true structural stress. When accurate ray-tracing models are unavailable, the shadowing coefficient μ = 0.4 is recommended as the preferred choice for calculating the temperature of the lower chord tubes in truss arch bridges of the same type.

(3)

The structural elevation exhibits a negative correlation with solar radiation temperature. The maximum deflection of the arch ribs (25.85 mm) occurs at 14:00. Additionally, the elevation of the arches on the left and right banks shows a significant difference (6.27 mm). The closure moment should be selected under conditions of minimal elevation difference.

Future research should deeply integrate artificial intelligence technologies to further enhance the intelligent analysis and control of solar radiation temperature effects on complex bridge structures. Combining long-term monitoring data with meteorological records, high-precision temperature field prediction models are established using machine learning or physical information neural networks to achieve dynamic temperature field forecasting that accounts for multidimensional environmental factors.