Combined effect representative value determination method of sectional temperature differences for bridge box girder and ballastless track slab

Abstract

Significant asynchrony exists between the extreme sectional temperature differences (STDs) of the high-speed railway (HSR) box girder and the ballastless track slab. Structural flexural deformation analysis based solely on univariate extreme temperature gradient may lead to distortion and redundancy. To address this issue, a method based on the maximum entropy principle is proposed for the unbiased estimation of the combined representative STDs for the box girder and track slab. Firstly, a model transforming spatial temperature measurements from multiple points into a one-dimensional equivalent linear STD for the structure is established. Secondly, an unbiased fitting method is proposed for the estimation of extreme STD based on the maximum entropy principle. Using measured data samples, the optimal marginal distributions for the univariate values are fitted. Thirdly, based on long-term measured temperature of the box girder and track slab system, a method for determining the combination coefficients of extreme STDs over multiyear return periods is proposed. The study reveals that due to the differences in structure size and shielding effect, the magnitude and seasonal pattern of the STD of box girder are significantly different from those of track slab. The joint exceedance probability model of the STDs of box girder and track slab, fitted using the maximum entropy principle, can quantitatively characterize the time lag effect of their temperature fields over multi-year return periods. Compared to the traditional extreme value analysis models, the proposed method based on maximum entropy principle can provide a more unbiased probability distribution function without making any distribution assumptions in advance. The proposed method can be used to determine the combined STD effects of HSR box girder and track slab, improving the accuracy of temperature effect analysis and providing support for more economical and safer service of HSR.

Keywords

high-speed railway box girder ballastless track slab temperature difference combined effect

Introduction

China HSRs predominantly adopt the bridges instead of roads to adapt to various terrain conditions and conserve arable land. The 32 m standard box girder has become the preferred standard type for HSR bridges due to its advantages of fast construction speed, reliable safety, and favorable dynamic characteristics (He et al., 2017). In addition, ballastless track slabs are used on the girders to further ensure the smoothness of HSR rails, they can offer several advantages such as better smoothness, higher stability, and reduced maintenance.

The ballastless track slab, which has a certain thickness, is directly exposed to the natural environment. Due to the effects of atmospheric temperature and solar radiation, an uneven temperature difference forms within the concrete ballastless track slab. This uneven temperature difference induces stress in the ballastless track slab (Subramaniam et al., 2010; Zhou et al., 2023, 2024), directly affecting the durability of the concrete track slab (Subramaniam et al., 2010; Zhou et al., 2023). The temperature difference also cause warping deformation of the slab (Cai et al., 2019; Chen et al., 2019; Huang et al., 2021; Zhou et al., 2023). This warping deformation leads to geometric changes in the track system on the bridge, further impacting the smoothness of the HSR line (Chen et al., 2024; Zhang et al., 2022). The box girder, as the primary structural form of railway bridges, has a cross-sectional height significantly greater than that of the track slab, resulting in a more pronounced vertical nonlinear STD. Under the effect of the STD, the girder is prone to vertical flexural deformation (Dai et al., 2017; Li et al., 2023; Zhou et al., 2024), which not only affects the safety of upper track structure but also poses a threat to the HSR line smoothness. Therefore, when designing ballastless track slabs on bridges, it is necessary not only to consider the additional rail forces caused by the elongation and contraction of the girder under uniform temperature changes (Wang et al., 2010, 2013), but also to analyze the coupling effect of the girder flexural deformation and track slab warping deformation on the HSR line smoothness (Zhang et al., 2023). Studying the combined STDs of the bridge girder and the track slab is a prerequisite for accurately analyzing the coupling effects of girder flexural deformation and track slab warping deformation. This is of great significance for the design of the ballastless track structure and the subsequent maintenance efforts.

Currently, the AASHTO specification uses polyline function to represent the temperature gradient of bridges (Rodriguez et al., 2014). The parameter values of its temperature gradient mode need to be determined based on the structural materials and climate zones, taking into account the influence of the concrete pavement thickness. China’s Code for Design of Continuous Welded Rail on Railway Bridges (House, 2013) stipulates that when considering the temperature load on bridges, the girder with ballastless track slab should be assigned a uniform temperature change of ±30°C annually. Additionally, the Code for Design of High-Speed Railways prescribes the positive and negative temperature gradient values for ballastless track slabs as 90°C/m and −45°C/m, respectively (House, 2014). However, these recommended values differ significantly from the actual temperature fields observed (Ren et al., 2024; Sheng et al., 2022). Moreover, structural system deformation analysis based solely on the univariate extreme STD of the box girder and track slab may result in distortion and redundancy. Existing studies (Zhang et al., 2022; Zhou et al., 2023) have pointed out that, when analyzing the temperature effects of railway bridge-track slab system, the combined temperature fields of both bridge and track slab should be considered, rather than separately analyzing the temperature field of individual structural section.

For the temperature field of concrete box girder, Hu et al. (2023) using an experimentally validated numerical model to simulate the temperature field of concrete box girders in various cities, showing that the temperature gradient exceeded the recommended value in specification. Therefore, for the temperature field of structure in special environment, separate measurements need to be taken and reliability analysis needs to be conducted to obtain representative extreme values with multi-year return periods. Cai et al. (2022) employed a three-dimensional finite element (FE) model to predict the thermal effects of concrete box girders in different regions, the extreme value analysis (EVA) were further conducted to determine the temperature gradients based on the simulated temperature field. Research on the temperature field of track slabs has gradually increased because of the demand of HSRs development and ballastless track slab maintenance, Zhao et al. (2023) conducted a detailed study on the effect of the track slab temperature gradient on its interlayer contact area, revealing the mechanisms of stiffness softening and stiffness hardening in interlayer contact caused by the temperature gradient. Zhong et al. (2018) analyzed the influence of STD on the deformation and interface separation of CRTS II ballastless track system based on FE numerical model. Chen et al. (2019) derived analytical expression for the warping deformation of track slab under temperature gradient based on differential equilibrium equations, the effect of temperature distribution on the warping deformation of track slab was discussed. Yang et al. (2017) established a FE model that simultaneously considers geographic location and environmental conditions, the simulation result showed that the STD in the track slab under sustained high temperatures is relatively large, which can lead to bending deformation in longitudinally connected track slabs. Shi et al. (2022) proposed a method for determining the STD of track slabs based on FE model and machine learning methods, which can automatically accounts for environmental parameters such as air temperature, latitude, and average wind speed. The above studies demonstrated that the value of temperature gradient is crucial for evaluating the warping deformation of track structures, it is necessary to conduct extreme reliability analysis. Lou et al. (2018) analyzed the temperature distribution of a box girder-track slab structural system based on 1 year measured data using high-order moment method, they find the shielding effect of the track slab has a significant impact on the surface temperature of the box girder, proving the necessity of studying the correlation between the temperature fields of box girder and track slab. Zhou et al. (2023) developed a three-dimensional FE model to simulate the temperature effect of the box girder-track slab system. Gou et al. (2024) analyzed the effect of STD of the ballastless track-girder structural system on the safety of train operations on the arch bridge. They pointed out that the temperature gradients of steel box girder and concrete slab are significantly different, demonstrating the importance of studying the combined temperature effects of box girder and track slab.

A review of existing research reveals that the vertical STDs of box girder and track slab cause significant warping deformation in the track structural system. However, there remain several urgent issues to be addressed regarding the STD mode of the HSR girder-track slab system: (1) Many studies have conducted load effect analysis based on short-term monitored temperature or recommended values from codes. However, short-term temperature loads are not representative, which leads to distortion in the analysis of thermal effects. Therefore, it is necessary to study how to provide unbiased estimate of representative temperature load value for the service life return period. (2) Most vertical STDs are fitted based on one-dimensional vertical measurement points located in the middle of the structure, but the cross-sections of box girder and track slab are planar. Ignoring lateral temperature measurement data will introduce bias in the calculation of the overall vertical STD of the cross-section. Thus, it is important to analyze how to account for the influence of multiple spatial measured data on the overall STD. (3) The box girder and track slab have different spatial positions and dimensions, and there is a significant asynchrony between the extreme STDs of the two structures. Deformation analysis based on univariate extreme STDs in the box girder and track slab leads to distortion and redundancy. Therefore, it is necessary to further develop a method for determining representative values of the combined effect when the STDs of the box girder and track slab occur simultaneously.

In response to these issues, this paper first establishes an equivalent linear STD model based on multiple spatial temperature measurement points, converting the measured temperature data into a one-dimensional equivalent linear STD for the structure. Then, a joint probability distribution model of linear STDs for the box girder and track slab is constructed using a two-dimensional maximum entropy model. Finally, a method for determining the combination coefficients of extreme STDs for a certain return period is proposed. The proposed method enables a more realistic determination of the combined STDs of the HSR box girder and track slab, improving the accuracy of structural thermal effect analysis.

Unbiased estimation of STD representative value based on the maximum entropy model

The box girder and track slab have significant spatial dimensions, while the number of actual temperature measurement points is limited. To more accurately assess the overall temperature effects on the structure, it is necessary to quantify the contribution of different temperature measurement points in space. This involves converting the nonlinear spatial STD of the cross-section into a one-dimensional equivalent variable that can be used for extreme value statistical analysis. Subsequently, using a fitting method that does not require distribution assumptions to achieve unbiased estimation of STD representative values and co-occurrence joint distributions.

Equivalent linear STD model

Although the nonlinear vertical STD across the girder cross-section better reflects the actual temperature difference distribution, its practical application in engineering is rather cumbersome. It requires nonlinear functions for multi-point numerical fitting, and the corresponding representative value of the STD cannot be directly calculated. In this section, based on the equivalent principle of flexural deformation curvature, the nonlinear temperature difference can be converted into an equivalent linear STD.

The following assumptions are made: (1) The structural material is homogeneous, isotropic, and follows the laws of elastic deformation. (2) The structural temperature is uniformly distributed along the longitudinal direction. (3) The structural flexural deformation complies with the plane section assumption. Based on these assumptions, an attempt is made to transform the three-dimensional temperature field effect into a plane strain problem.

Taking an arbitrary cross-section for analysis, as shown in Figure 1, the temperature change within the cross-section follows a curve T (y) along the section height. The vertical coordinate of the neutral axis is y_c. When the longitudinal fibers are not constrained and can expand freely, the temperature self-stress $σ_{s} (y)$ corresponding to the constraint between the longitudinal fibers is given as follow:

σ_{s} (y) = E \cdot ε_{s} (y) = ε_{T} (y) - ε_{a} (y) = E \cdot [α \cdot T (y) - (ε_{0} + χ y)]

(1)

Figure 1.

Stress-strain analysis of arbitrary cross-section under temperature change T(y).

The strain corresponding to the constraint between longitudinal fibers is $ε_{s} (y)$ , $ε_{T} (y)$ is the free expansion or contraction strain at each point along the section height, $ε_{a} (y)$ represents the deformation at each point along the height under the plane section assumption, $ε_{0}$ represents the strain at the bottom of the cross-section y = 0, $χ$ represents the curvature of the section after bending deformation. There is no external load acting on the cross-section, and the temperature self-stress $σ_{s} (y)$ on the cross-section can be seen as a self-balancing stress. As shown in equation (2), the curvature of the cross-section after flexural deformation can be calculated based on the moment M = 0 at the centroid of the cross-section, and the expression of the curvature $χ$ is obtained in equation (3).

\begin{array}{l} M = E \int_{h} ε_{s} (y) \cdot b (y) (y - y_{c}) d y = E \int_{h} [α T (y) - (ε_{0} + χ y)] \cdot b (y) \cdot (y - y_{c}) d y \\ = E [α \int_{h} T (y) \cdot b (y) (y - y_{c}) d y - χ I] = 0 \end{array}

(2)

χ = \frac{α}{I} \int_{h} T (y) \cdot b (y) (y - y_{c}) d y

(3)

where

I = \int_{h} b (y) \cdot y \cdot (y - y_{c}) d y

represents the moment of inertia of the cross-section. When the equivalent linear temperature difference is T_E, then the free expansion strain

ε_{T} (y)

at any point of the cross-section is:

ε_{T} (y) = α \cdot T (y) = α \cdot \frac{T_{E}}{h} y

(4)

where y is the distance from the bottom of the cross-section. According to the deformation equivalence relationship

ε_{T} (y) = ε_{a} (y)

, there is:

χ = α \cdot \frac{T_{E}}{h}

(5)

The linear temperature gradient T_E can be obtained by combining equations (3) and (5):

T_{E} = \frac{h}{I} \int_{h} T (y) \cdot b (y) (y - y_{c}) d y = \frac{h}{I} \int_{h} T (y) \cdot (y - y_{c}) d A

(6)

When n temperature measurement points are arranged on the section, the uniform temperature of the section can be expressed as follow (Liu et al., 2019).

T (y) = \frac{\sum T (x, y)}{n} = \frac{\sum_{i = 1}^{n} T_{i} A_{i}}{n A_{i}}

(7)

According to equations (6) and (7), for a finite number of temperature measurement points arranged in a plane space, the corresponding linear STD T_E is:

T_{E} = \frac{h}{I} \int_{h} T (y) \cdot (y - y_{c}) d A = \frac{h}{I} \sum_{i = 1}^{m} T_{i} A_{i} (y_{i} - y_{c})

(8)

By using equation (8), the spatial temperature distribution of the structural cross-section can be converted into an equivalent linear STD (T_E). According to the derivation process of this equation, it can be concluded that using only one-dimensional vertical measuring points to fit the vertical STD of the section is not accurate enough.

Extreme value unbiased estimation method based on maximum entropy model

In practical engineering applications, long-term statistical analysis of structural loads is required, and the representative value for a certain return period is extrapolated as the design standard value. For example, for a bridge with a design life of 100 years, the representative value of the once-in-a-century temperature load should be used as the design reference. This involves statistically describing the extreme values of temperature loads based on the occurrence probabilities of different temperature levels. Currently, scholars have used extreme value statistics theories, such as the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD) (Madi and Raqab, 2004), to analyze historical temperature data of structures. However, these probabilistic models introduce additional distribution assumptions, which often compromise the impartiality of the probability estimates. On the contrary, the maximum entropy principle infers unknown information based solely on known data without making any distribution assumptions. It selects the probability distribution model with the maximum entropy based on given constraints. Compared to the GEV and GPD models, the maximum entropy model provides a more unbiased and objective probability distribution function. Moreover, the maximum entropy model is not sensitive to the choice of threshold, offering greater stability and lower variability (Dai et al., 2022; Saad and Ruai, 2019).

The probability distribution function with maximum entropy can be expressed in equation (9) (Saad and Ruai, 2019) under known data constraints equation (10):

\max (H (x)) = - \int_{D} [f (x) \ln f (x)] d x

(9)

{\begin{cases} \int_{D} f (x) d x = 1 \\ \int_{D} [g_{i} (x) f (x)] d x = m_{i}; i = 1, 2, \cdot \cdot \cdot, m \end{cases}

(10)

In the above Eqs, f (x) represents the probability density function of random variable x, and H (x) is the information entropy. D represents the integration interval, g_i(x) is a function about x, m represents the order of the moment used, and m_i is the origin moment of the i-th random variable.

Lagrange multipliers can be introduced to transform the constrained optimization into unconstrained optimization using the following function L (Saad and Ruai, 2019).

L = - \int_{D} [f (x) \ln f (x)] d x + (λ_{0} + 1) \cdot [\int_{D} f (x) d x - 1] + \sum_{i = 1}^{m} λ_{i} [\int_{D} [g_{i} (x) f (x)] d x - m_{i}]

(11)

In equation (11), λ_i represents the Lagrange multipliers. To obtain the extreme value of the function, let the partial derivative of L with respect to f (x) be equal to 0, and the maximum entropy probability density function (PDF) and cumulative probability distribution function (CDF) can be obtained as follow:

{\begin{cases} f (x) = \exp [λ_{0} + \sum_{i = 1}^{m} λ_{i} g_{i} (x)], i = 1, 2, 3, \cdot \cdot \cdot, m \\ F (x) = \int_{D} \exp [λ_{0} + \sum_{i = 1}^{m} λ_{i} g_{i} (x)] d x, i = 1, 2, 3, \cdot \cdot \cdot, m \end{cases}

(12)

The Lagrange multipliers λ_i (i = 1, 2, …, m) can be iteratively computed using the improved Newton iteration method (Zhao and Dong, 2021), and the iteration stops once the required accuracy is achieved.

Based on the above theory and computational process, it is evident that using the maximum entropy distribution function for extreme value fitting does not require a specific distribution assumption. By directly applying the maximum entropy principle and incorporating the constraint conditions of the data sample, an optimal fit based on information entropy can be obtained. This method generates representative values that are consistent with the characteristic of the data sample, offering the advantages of objectivity, impartiality, and broad applicability.

Joint probability distribution model of equivalent STDs

To analyze the structural STDs, researchers typically first identify the moment when the extreme STD occurs and then fit a nonlinear temperature distribution function based on the vertical temperature data at that moment. This method considers the simultaneous occurrence of temperatures at different measurement points, but the structural STD distribution functions are independent of each other, failing to account for the time-varying correlations between different structural STDs. Due to the thermal conduction effect within the structure, the STDs of the box girder and track slab are not independent but exhibit time-lagged correlations. When studying the combined effect of box girder STD and track slab STD on track system deformation, it is crucial to establish a joint probability distribution using bivariate return period for the simultaneous occurrence of STDs.

Commonly used two-dimensional probability models include the bivariate Gumbel distribution and the bivariate Lognormal distribution. These models generally use a family of bivariate distribution functions to establish the two-dimensional joint distribution, requiring that the marginal distribution of each variable be characterized by the same distribution function. In reality, different variables often exhibit non-consistent distribution characteristics. Therefore, it is advisable to use the two-dimensional maximum entropy model, which imposes less stringent requirements on marginal distributions and offers better fitting stability. This model can be applied to analyze the joint distribution of STDs of different structures, providing a reference for determining the simultaneous STDs in the box girder-track slab structural system.

Assuming the probability density functions of continuous random variables X and Y are f_x(x) and f_y(y), respectively, and the joint probability density function is f(x, y). When g_i(x, y) represents a function about x and y, the constraint conditions can be obtained (Singh et al., 2012):

{\begin{cases} \iint_{D} f (x, y) d x d y = 1 \\ E [g_{i} (x, y)] = \iint_{D} g_{i} (x, y) \cdot f (x, y) d x d y = b_{i}, i = 1, 2, . . ., m \end{cases}

(13)

Based on the above constraints, taking the maximum value of two-dimensional joint information entropy as the objective function and introducing Lagrange multipliers (Singh et al., 2012), equation (14) can be obtained:

H = - \iint_{D} f (x, y) \ln f (x, y) d x d y + (λ_{0} + 1) (\iint_{D} f (x, y) d x d y - 1) + \sum_{i = 1}^{m} λ_{i} [\iint_{D} g_{i} (x, y) f (x, y) d x d y - b_{i}]

(14)

Let the partial derivative of H with respect to f (x,y) be equal to 0, a nonlinear system of equations with (m+1) Lagrange multipliers can be obtained using the constraint conditions equation (13), as shown in equation (15):

{\begin{cases} \underset{D}{∭} f (x, y) d x d y = \iint_{D} \exp (λ_{0} + \sum_{i = 1}^{m} λ_{i} g_{i} (x, y)) = 1 \\ \iint_{D} g_{i} (x, y) f (x, y) d x d y = \iint_{D} g_{i} (x, y) \exp (λ_{0} + \sum_{i = 1}^{m} λ_{i} g_{i} (x, y)) d x d y = b_{i}, i = 1, 2, . . ., m \end{cases}

(15)

The final Lagrange multipliers can be iteratively solved using the improved Newton’s iteration method and the two-dimensional Gaussian numerical integration method (Rockinger and Jondeau, 2002), as described in equation (12). The overall calculation flowchart is shown in Figure 2.

Figure 2.

Modeling flowchart for combined STDs of box girder and track slab based on maximum entropy theory.

The HSR structural system STD analysis based on the proposed method

Taking the measured temperature field of a HSR structural system composed of box girder and track slab as an example, this section specifically illustrates the application process of the aforementioned theoretical methods.

Engineering application and measurement point description

A standard 32 m simply supported box girder with a CRTS I type ballastless track system on the Hefei-Fuzhou HSR passenger line was selected as the monitoring object. The structural temperature field was monitored by embedding temperature sensors inside the structure. The temperature measurement components used BGK-3700 resistive temperature sensors, with a measurement accuracy of ±0.2°C. The terminal of the temperature sensors was connected to a BGK-Mirco40 data acquisition device, enabling automatic data collection and storage. The monitoring instrument operates with an automatic sampling interval of 30 min and supports remote wireless data transmission, utilizing a solar-powered energy system, To eliminate the boundary effects, the section with a longitudinal distance of 15 m from the girder end was selected as the testing section. Considering the longitudinal orientation of the track system at 9.5° to the north-south axis, the measurement points were arranged on one side. A total of 37 temperature measurement points were arranged across the box girder and track slab, with specific locations and numbering shown in Figure 3. For the box girder, four groups of measurement points were set along the transverse direction of the top deck, the inside and outside of the web, and along the girder centerline. In the track system, eight groups of temperature measurement points were arranged transversely and vertically, as shown from points 22 to 37 in Figure 3.

Figure 3.

Measurement points arranged in the box girder and track slab.

Time-varying pattern of structural STDs and unbiased estimation of extreme representative values

The on-site temperature monitoring spanned the entire construction process of the structural system, covering all seasons throughout the year. The casting of the box girder was essentially completed on December 2, 2013, and temperature monitoring began from this date. The installation of the track slab was finished on June 7, 2014. To avoid the impact of surface covering on the box girder before and after the track slab construction, the temperature field analysis utilized monitoring data spanning a full annual cycle after the completion of track slab construction, from 00:00 on June 9, 2014 to 12:00 on June 8, 2015. The measured data of each measurement point during the full annual cycle are shown in Figure 4. Due to the use of 21 measuring points in the box girder, it is difficult to display each temperature time history curve separately. Representative measuring points in box girder are selected for display from those with similar distribution areas. Actually, the on-site acquisition device is powered by the system consisting of solar panel and storage battery. When the solar panel cannot provide electricity due to cloudy weather or other reasons, the battery serves as backup power supply equipment to provide electricity. Due to the depletion of battery energy, malfunction and maintenance of data acquisition device, temperature data is missing and discontinuous in some time periods. This is a shortcoming in the on-site data acquisition of this study, which is also a common problem encountered in the field of health monitoring and has stimulated research in data recovery (Zhang et al., 2024).

Figure 4.

Measured temperatures of box girder and track slab: (a) Temperature time history curve of measurement points in box girder and (b) temperature time history curve of measurement points in track slab.

During the entire annual cycle period, the highest measured temperature of the box girder was up to 49.2°C, which was recorded at measuring point 19 on the top surface at 15:30 on July 21. The lowest temperature was also recorded at measuring point 19, reaching 1.5°C. For the ballastless track slab, both the highest and lowest temperatures were recorded at measuring point 36, where the top and side surfaces were directly exposed to solar radiation and in direct thermal convection with the air. The highest temperature, 50.8°C, was recorded at 15:30 on July 21, while the lowest temperature, −1.7°C, was recorded at 7:30 on December 23.

Based on the measured temperature, the equivalent linear STDs for the cross-sections of box girder and track slab were calculated, as shown in Figure 5(a) and (b), respectively. To better quantify the temperature field of structure, the measured temperature field can be represented by the overall uniform temperature and equivalent STD. Due to the longitudinal fixation of the track slab, this article did not study the overall uniform temperature which causes the overall expansion and contraction, but only studied the equivalent STD that plays a major role in the vertical warping deformation of the structure. Due to the large vertical dimension of the box girder cross-section, the equivalent linear STD in the summer is significantly higher than that in the winter. The heat transfer rate is related to the length of the transmission path and the structural size (Saggion et al., 2019). With the increase of surface-volume ratio, the heat transfer efficiency increases. In addition, the heat transfer rate decreases as the transmission path increases. Compared to box girder, track slab has a larger surface-volume ratio and a shorter transmission path. Therefore, the effects of daily radiation and heat transfer are more pronounced for the ballastless track slab, resulting in less noticeable differences in the equivalent STDs between summer and winter. Therefore, although the box girder and track slab are concrete structures, the representative values of their STDs should be considered separately. A statistical analysis shows that the annual variation range of the vertical STD for the box girder cross-section is from −2.4°C to 7.8°C, with the maximum positive STD typically occurring at 15:30 and the maximum negative STD occurring at 7:00 in the morning. For the ballastless track slab, the maximum positive STD usually occurs at 13:30, and the minimum STD often occurs at 7:30 in the morning, with the annual variation range of the vertical STD being from −4.9°C to 10.9°C.

Figure 5.

Time history curves of combined STDs of the box girder and track slab: (a) Box girder and (b) track slab.

The aforementioned maximum entropy function was used to fit the distribution of STD for box girder and track slab. The data frequency histogram and the density function fitting curves are shown in Figure 6. The PDF and CDF in the legend represent probability density function and cumulative distribution function, respectively, which is also applicable to the following figures. As the PDF and CDF are dimensionless, the coordinate legends corresponding to PDF and CDF do not have unit annotations. The fitting performance was verified through a Probability-Probability (PP) plot (Petrov et al., 2013), as shown in Figure 7, where the sample points are mostly aligned along the black line from (0, 0) to (1, 1), indicating a good performance of the maximum entropy fitting model. Compared to the empirical distribution CDFs, the fitting root mean squared errors of the maximum entropy distribution CDFs are only 0.016 and 0.011, respectively, for the box girder STD and track slab STD. As shown in Figure 8, the CDF curves for the STDs related to the box girder and track slab were fitted separately, using the maximum entropy probability density function shown in equation (12). Based on the calculated CDF curves, a univariate return period analysis was conducted, the 0.99 quantile of the fitted CDF distribution represents the statistical variable’s representative value of 100-year return period. Specifically, for a 100-year return period, the representative value of the STD for box girder is 8.7°C, while for the track slab, the representative value of its STD is 11.5°C.

Figure 6.

Probability function fitting curves of the STDs of box girder and track slab: (a) Box girder and (b) track slab.

Figure 7.

PP plots of the fitted CDFs for the STDs of box girder and track slab: (a) Box girder and (b) track slab.

Figure 8.

CDF curves of the STDs of box girder and track slab: (a) Box girder and (b) track slab.

Joint probability distribution of the STDs under simultaneous occurrence

When establishing the STD joint distribution model, it is essential to consider both scenarios where the STDs of the box girder and track slab independently reach their extreme values. Additionally, it should extract both box girder STD and track slab STD that appear simultaneously to fully characterize their co-occurrence correlation. This study establishes a coupled dataset for STDs of track slab and box girder through a bidirectional extreme value triggering mechanism, with the specific methodology outlined as follows: Using daily maximum STDs of one structural component as independent triggering conditions, synchronous STD observations from the other component are systematically collected at corresponding timestamps, thereby forming two asymmetric bivariate datasets. The first dataset prioritizes the daily maximum STDs of box girder as the dominant variable while capturing concurrent track slab STDs at the same timestamps; the second dataset centers on the STDs of track slab as the pivotal variable while simultaneously recording the synchronized box girder STDs. This dual-path data acquisition strategy systematically captures asynchronous response patterns and latent coupling relationships between two critical structural components during extreme temperature events, thereby establishing theoretical foundations for calibrating temperature load combination coefficients in probabilistic limit state design. The proposed approach innovatively implements a dual-extreme triggering strategy, providing novel perspective for probabilistic limit state analysis of multi-structural systems.

Therefore, the structural temperature data samples are divided into two groups, the sample group of track slab maximum STD (the daily maximum STD of track slab and the corresponding box girder STD at the same time) and the sample group of box girder maximum STD (the daily maximum STD of box girder and the corresponding slab track STD at that time). The derived two-dimensional maximum entropy function is used to construct the joint probability distribution model for the STDs, as shown in Figure 9. As the measured data perfectly conforms to the maximum entropy marginal distribution, demonstrating that the two-dimensional probability distribution obtained from the maximum entropy model can accurately fit the global distribution characteristics of the measured data. The final two-dimensional joint PDF surface and CDF surface are shown in Figures 10 and 11, respectively.

Figure 9.

Marginal distribution fitting performances based on the maximum entropy model: (a) Marginal distribution fitting of box girder STD and (b) marginal distribution fitting of track slab STD.

Figure 10.

Joint PDF and CDF when the box girder STD is determined as control variable: (a) Joint PDF surface with box girder STD as control variable and (b) joint CDF surface with box girder STD as control variable.

Figure 11.

Joint PDF and CDF when the track slab STD is determined as control variable: (a) Joint PDF surface with track slab STD as control variable and (b) joint CDF surface with track slab STD as control variable.

The calculation of design representative values is a critical component of extreme load risk analysis. In the aforementioned determination of univariate representative values, the exceedance extreme value for multi-year return periods was used. In the case of bivariate variables, two-dimensional return period contour maps are required to determine the combined representative values. Based on the joint return period contour maps for the STDs (Figure 12), it can be seen that for a given joint return period, there are various combinations of bivariate variables (box girder STD and track slab STD). In this case, the exceedance extreme value of the box girder STD for that return period can be obtained firstly using the univariate probability density function, after which the simultaneous representative value of the track slab STD is determined based on the joint return period contour lines.

Figure 12.

Contour plot of the joint return period of STDs related to box girder and track slab: (a) Contour plot of the joint return period with box girder STD as control variable and (b) contour plot of the joint return period with track slab STD as control variable.

Based on the analysis of the univariate extreme values of the STDs in the structure, it is evident that when the box girder STD is determined as the control factor, its exceedance representative value for the 100-year return period is 8.7°C. At this point, the combined STD for the track slab, as shown in the joint return period contour plot, is 10.3°C. Conversely, when the track slab STD is determined as the control factor, the exceedance representative value for the track slab is 11.5°C, with the corresponding combined STD for the box girder being 7.5°C. The joint representative values for the STDs of box girder and track slab, along with the corresponding combination coefficients, were obtained, as shown in Table 1. However, the specific results were obtained based on on-site measurement points, which were arranged in the track slab and box girder that have a longitudinal orientation of 9.5° to the north-south axis. For girder-track systems in other locations, meteorological environments, and solar radiation angles, thermodynamic numerical simulations can be used to analyze the time-varying temperature fields (Cai et al., 2022; Hu et al., 2023; Lin et al., 2016; Yang et al., 2017), and then the proposed method can be used to study the combined STDs related to the track slab and box girder.

Table 1.

Joint representative values for the STDs with a 100-year return period.

Control factor	Univariate representative value of 100-year return period	Simultaneous occurrence of box girder STD	Simultaneous occurrence of track slab STD	Combination coefficients of same percentile values
STD of box girder T_EG	8.7°C	—	10.3°C	1.0T_EG + 0.89T_ET
STD of track slab T_ET	11.5°C	7.5°C	—	1.0T_ET + 0.86T_EG

Conclusions

Based on the principle of curvature equivalence in sectional flexural deformation, this paper establishes an equivalent linear STD model for structures based on spatial multi-temperature measurements, and analyzes the time-varying characteristics of vertical STD in a box girder-track slab structural system. Subsequently, using measured temperature from a standard concrete box girder and track slab, a two-dimensional joint probability distribution model of the box girder STD and track slab STD was constructed using the maximum entropy method. This model was employed to analyze the combined STDs of the box girder-track slab structural system. The main conclusions are as follows:

(1) The linear STDs of the box girder-track slab structural system exhibit characteristics of being high in summer and low in winter. Due to the large vertical sectional dimension of the box girder, the equivalent linear STD in summer is significantly greater than that in winter. In contrast, the ballastless track slab, with a thickness of only 0.26 m, experiences more pronounced daily heat transfer effects, resulting in nearly identical equivalent STDs in both summer and winter. Therefore, although the box girder and track slab are both concrete structures, the STD representative value for each of the two structures should be considered separately due to the differences in structure size and shielding effect.

(2) To eliminate distribution assumptions during extreme value fitting, an unbiased estimation method based on the maximum entropy principle is proposed to fit extreme STD value. Using measured data samples as objective constraints, probability distribution functions for the equivalent STDs of the box girder and track slab are constructed, respectively. The PP plot tests and marginal distribution mapping comparisons demonstrate that the maximum entropy model provides an excellent fitting performance, enabling objective and unbiased estimation of STD representative extreme for the structure.

(3) There is temporal asynchrony in the daily extreme values of the equivalent linear STDs of different structures. To address this issue, a method based on two-dimensional maximum entropy is proposed to quantify the joint probability distribution of the box girder STD and track slab STD. The joint distribution contour plot and combination coefficients for different joint return periods can be generated. Engineering application has demonstrated that the STDs in the box girder and track slab do not reach their extremes simultaneously, and the combination coefficient corresponding to multi-year return periods can quantitatively characterizes the time-lag effect of the spatial temperature fields in different structures.

(4) Compared to the traditional extreme value analysis models, GEV, bivariate Gumbel model and bivariate Lognormal model, etc., the proposed method based on maximum entropy principle can provide a more unbiased probability distribution function without making any distribution assumptions in advance. Moreover, the maximum entropy model is not sensitive to the choice of threshold, offering greater stability and lower variability. The proposed method can determine more realistic combined STDs of HSR bridge and ballastless track system, thus enhancing the precision of structural temperature effect analysis and train driving safety assessment.

Footnotes

ORCID iDs

Hao Ge

Fen Wang

Yonghui An

Yanqun Xu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the National Key Research and Development Program of China (2022YFC3005302), Guangxi Science and Technology Base and Talent Project (AA23026011), National Natural Science Foundation of China (U23A20662), and Guangxi Natural Science Foundation (2025GXNSFBA069337).

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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