Calculation of standard values of temperature differences in a large-span steel-concrete composite girder bridge through probabilistic statistics and cycle-life simulation analysis

Abstract

Using the monitoring temperature data from a large-span steel-concrete composite girder bridge, the spatial distribution laws of the transversal and vertical temperature differences in the steel-concrete composite girder are analyzed; then, the probabilistic statistical characteristics of the daily extreme values of temperature differences are analyzed using probability statistical method, and then cycle-life simulation is carried out to obtain the yearly maximum or minimum values in the whole bridge service life; furthermore, the standard values of positive and negative temperature differences are calculated, and finally the correlation between the standard values and the monitoring extreme values are explored using correlation analysis method. The results show that: (1) there are significant transversal and vertical temperature differences in the steel-concrete composite girder, and the maximum positive temperature difference in the transversal direction of this girder reaches 7.40°C, and the minimum negative temperature difference in the vertical direction reaches −6.80°C; (2) the generalized extreme value distribution (GEVD) has a smaller fitting error 0.0016 compared to the extreme value distribution (EVD) and the normal distribution (ND), so GEVD is selected as the optimal probability density function; (3) the maximum and minimum values of standard values for transversal temperature differences are 10.74°C and −6.38°C, respectively; the maximum and minimum values of standard values for vertical temperature differences are 4.81°C and −7.77°C, respectively; (4) there is a good linear correlation between the measured values and the standard values, with a linear correlation coefficient of 0.9044; (5) there is a big difference between current bridge codes and the calculation results by comparison, hence the actual standard values of temperature difference for steel-concrete composite girder bridges is relatively complex, and the calculation results can provide an important reference for the current bridge codes.

Keywords

bridge health monitoring steel-concrete combined girder bridge temperature gradient probability statistics standard value

Introduction

For large-span bridge structures in the whole service life, they are affected by environmental factors such as solar radiation, daily temperature differences and climate change, which may cause severe problems such as cracking and fatigue damage to bridge structures (Han et al., 2022; Huang et al., 2018, 2023; Jing et al., 2024; Li et al., 2023; Wang et al., 2021; Yang et al., 2024; Zhou et al., 2024). Therefore, the temperature field of bridge structures has a significant impact on structure damage, making it an indispensable factor in the load design of bridge structures (Keles et al., 2024; Li et al., 2023; Li et al., 2023, 2024; Shan et al., 2023). For example, Huang et al. (2018) pointed out that the changes in structural dynamic parameters resulting from damage may be masked by the change induced from variations in environmental effects, where the effect of temperature changes is deemed most important, so in structural health monitoring (SHM) of civil infrastructure systems it is of great importance to eliminate the effects of environmental changes, especially those of temperature, from the damage detection process. Accurate calculation of temperature action is of great importance for safety assessment of bridge structures.

There are many large-span bridge types, and among these bridge types the steel-concrete composite girder bridges are widely constructed in recent years because of their outstanding advantages in material utilization, economic benefit, construction convenience, durability, and seismic performance. Currently, the temperature action model of steel-concrete composite girders has been specified by the bridge design codes home and abroad (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017; Brussels: European Committee of Standardization, 2003). For example, China and USA codes specify that the temperature action of the steel components in steel-concrete composite girders is uniform, without consideration of temperature gradients (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017). Hence, current research on the temperature field of steel-concrete composite girders commonly treat the temperature field of the steel components as a uniform temperature field, without considering temperature gradients (Keles et al., 2024; Nikola et al., 2023; Wang et al., 2019, 2023; Zhu et al., 2021).

Some new research results based on the latest monitoring data show that there are temperature gradients in the steel components of steel-concrete composite girders (Huang et al., 2024a; Meng et al., 2023; Song et al., 2023; Wang et al., 2024; Xia et al., 2020; Zhou et al., 2020; Zhu et al., 2023, 2024). For example, Meng et al. (2023) analyzed a continuous steel-concrete composite railway girder bridge and found that the temperature gradient effect of the steel components has a more adverse effect on the local stress compared with the uniform temperature effect; Song et al. (2023) analyzed the temperature field data of a high-pier long-span steel-concrete composite girder bridge and found that there are certain temperature gradients in the steel components in addition to the daily and yearly temperature changes; Wang et al. (2024) analyzed the monitoring temperature field data of a steel-concrete composite girder bridge and found that the steel components has noticeable transversal and vertical temperature differences, and the transversal positive temperature difference reaches 14.75°C and the vertical negative temperature difference reaches −14.53°C; Zhu et al. (2024) analyzed the monitoring temperature field data of a railway suspension bridge with steel-concrete composite girder and found that the temperature field of the steel components have obvious non-uniform characteristics due to the influence of sun shadowing.

However, current research results mainly focuses on the monitoring period, which cannot fully reflect the most unfavorable conditions throughout the entire service life, so the analysis results cannot be used as standard values for thermal action design. According to current bridge codes (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017; Brussels: European Committee of Standardization, 2003), the standard value is the most unfavorable temperature difference value in the whole bridge service life, and yearly maximum or minimum values in the whole bridge service life should be used for calculation of standard values, but the monitoring data is difficult to cover the whole bridge service life.

Therefore, this research puts forward a calculation method of standard values of temperature differences through probabilistic statistics and cycle-life simulation analysis using the monitoring data of a large-span steel-concrete composite girder bridge. First, the spatial distribution patterns of the transversal and vertical temperature differences in steel-concrete composite girder bridges are revealed; second, the probability statistical characteristics of the daily positive and negative temperature differences are studied and their cycle-life simulation in 100 years is carried out; finally, the standard values of positive and negative temperature differences are calculated and then compared with current bridge codes home and abroad. The research results have important reference significance for temperature action design.

Bridge health monitoring system

To study the spatial distribution characteristics of the temperature field of large-span steel-concrete composite girder bridges, the monitoring data of the temperature field at Hongwan Waterway Main Channel Bridge were collected using the bridge health monitoring system to analyze the temporal variation characteristics of the temperature field of the steel-concrete composite girder bridge.

The bridge is a five-span steel-concrete composite cable-stayed girder bridge in a northwest-southeast direction, which is located in Xiangzhou District, Zhuhai City. It starts from the connection line of the Hong Kong-Zhuhai-Macao Bridge and ends at the intersection of the Jiangmen-Zhuhai Expressway and the Zhuhai-Hege Expressway. The steel components are composed of rectangular and double-side I-shaped cross-sections, and the steel components is connected to the overlying concrete bridge deck through shear keys, forming a steel-concrete composite girder structure, as shown in Figure 1. To monitor the temporal variation characteristics of the steel-concrete composite girder, five temperature sensors are arranged at the top of the steel-concrete composite girder, denoted as S₁ to S₅; in addition, three temperature sensors are arranged at the bottom of steel-concrete composite girder, denoted as S₆ to S₈. The temperature sensors have a sampling frequency of 1 Hz and continuously collect data for 1 year.

Figure 1.

Temperature sensor arrangement.

The long-term monitoring law of temperature field of steel-concrete composite girder

Long-term monitoring rules of temperature values

The temperature values collected by the temperature sensor at position S_i is represented by T_{S
i}. Considering that the temperature data collected by various sensors have roughly the same changing trends, the temperature values T_S4 from September 1, 2023, to August 31, 2024 are selected. Firstly, the yearly trend of the temperature values T_S4 is plotted as shown in Figure 2, showing that the temperature trend has a distinct seasonal variation trend, with higher values in summer and lower values in winter. Furthermore, the daily trend of the temperature values T_S4 within 1 day on October 2, 2023 is plotted as shown in Figure 3. It can be seen that the temperature trend exhibits distinct diurnal variations, with higher temperatures during the day and lower temperatures at night. What should be mentioned is that during the temperature data acquisition process, some temperature data were lost, and a convolutional neural network model was employed to recover the missing data. The recovery method is available by referring to Huang et al. (2025) and Zhang et al. (2024).

Figure 2.

Time history of monitoring temperatures of T_S4.

Figure 3.

Daily trend of monitoring temperatures of T_S4.

Long-term monitoring rules of temperature difference values

The temperature difference between any two measuring points S_i and S_j is denoted as T_Si,Sj, which is T_Si minus T_Sj. The transversal temperature differences T_S1,S2, T_S2,S3, T_S3,S4, T_S4,S5, T_S6,S7, T_S7,S8 and the vertical temperature differences T_S1,S6, T_S3,S7, T_S5,S8 are calculated, and their maximum and minimum values are shown in Table 1. It can be seen that there are significant temperature differences between measuring points. For example, the maximum positive transversal temperature difference between measuring points S₃ and S₄ can reach 7.40°C, and the maximum negative vertical temperature difference between measuring points S₁ and S₆ can reach −6.80°C. This is different from current bridge codes which only consider uniform temperature action on the steel part of the steel-concrete composite bridge (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017; Brussels: European Committee of Standardization, 2003).

Table 1.

Extreme values of temperature differences of hongwan waterway bridge (°C).

Temperature differences		Minimum value	Maximum value
Transversal mode	T _S1,S2	−1.32	4.91
	T _S3,S4	−0.69	7.40
	T _S6,S7	0.57	6.08
	T _S2,S3	−5.77	2.98
	T _S4,S5	−4.20	−1.77
	T _S7,S8	−4.10	1.00
Vertical mode	T _S1,S6	−6.80	0.59
	T _S3,S7	−4.89	4.41
	T _S5,S8	−6.54	2.53

With regard to the maximum transversal temperature difference T_S3,S4 and the maximum vertical temperature difference T_S1,S6, their time histories from September 1st, 2023 to August 31st, 2024 are further plotted in Figure 4(a) and (b). It can be seen that the time histories show a uniform random characteristic without obvious seasonal variation features; the transversal temperature difference at T_S3,S4 is significantly higher in positive values than in negative values, and the vertical temperature difference at T_S1,S6 is significantly higher in negative values than in positive values. This further confirms the existence of significant transversal and vertical temperature differences between measurement points, suggesting that temperature differences should be considered in the design of steel-concrete composite girder bridges.

Figure 4.

Time histories of temperature differences (a) T_S3,S4 and (b) T_S1,S6.

Probabilistic statistical characteristics and cycle-life simulation analysis

The monitoring period of measured temperature values is relatively short, which cannot fully reflect the most unfavorable conditions throughout the entire service life, so the analysis of probabilistic statistical characteristics and cycle-life simulation is necessarily carried out to evaluate the temperature values in the whole bridge service life.

Probabilistic statistical characteristics analysis

According to the monitoring rules of temperature difference values above, the time histories show a uniform random characteristic, which is suitable to be used for probabilistic statistical characteristics analysis. The key point is to determine the probability density function of temperature difference values. The temperature difference values are divided into positive temperature differences and negative temperature differences. The daily maximum and minimum temperature differences are selected for analysis, corresponding to positive and negative temperature differences respectively. Several possible probability density functions are discussed and compared to determine the best probability density function to fit the probability density value with least errors.

The possible probability density functions are Normal Distribution Function (ND), Generalized Extreme Value Distribution Function (GEVD) and Extreme Value Distribution Function (EVD), and their equations are written as follows:

ND : f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{{(x - μ)}^{2}}{2 σ^{2}}}

(1)

where x denotes the positive or negative temperature difference, f(x) denotes the probability density function of x, μ denotes the mean value and σ denotes the standard deviation.

GEVD : f (x) = \frac{1}{β} {[1 + ξ (\frac{x - μ}{β})]}^{- \frac{1}{ξ} - 1} e^{- {[1 + ξ (\frac{x - μ}{β})]}^{- \frac{1}{ξ}}}

(2)

where x denotes the positive or negative temperature difference, f(x) denotes the probability density function of x, μ denotes the location parameter, β denotes the scale parameter and ξ denotes the shape parameter.

EVD : f (x) = \frac{1}{β} e^{- \frac{x - μ}{β} - e^{- \frac{x - μ}{β}}}

(3)

where x denotes the positive or negative temperature difference, f(x) denotes the probability density function of x, μ denotes the location parameter and β denotes the scale parameter.

With regard to the parameter values of the three probability density functions, the maximum likelihood estimation method is used to obtain the optimal values. For convenience, the MATLAB Probability Density Fitting Toolbox can be used for quick solutions. In order to compare the fitting effect of the three probability density functions to determine the best probability density function with least fitting errors, the fitting errors error are calculated by the measured cumulative probability values v_measured minus the fitted cumulative probability values v_fitted as follows:

e r r o r = \frac{\sqrt{\sum_{i = 1}^{N} {(v_{measured, i} - v_{fitted, i})}^{2}}}{N}

(4)

where N denotes the number of the temperature difference values. What should be mentioned is that cumulative probability values not probability density values are used because the measured probability density values are not easily calculated but the measured cumulative probability values are easily calculated as follows:

v_{measured} (x_{T}) = \frac{n (E (X \leq x_{T}))}{N}

(5)

where v_measured (x_T) denotes the measured cumulative probability values of temperature difference x_T, n (E (X ≤ x_T)) denotes the number n of temperature differences X which satisfies the event X ≤ x_T. The fitted cumulative probability values v_fitted are calculated as follows:

v_{fitted} (x_{T}) = \int_{- \infty}^{x_{T}} f (x) d x

(6)

where v_fitted (x_T) denotes the fitted cumulative probability values of temperature difference x_T.

According to the method above, the probability density values of daily maximum and minimum temperature differences (denoted by T_{Si,Sj,maximum} and T_{Si,Sj,minimum} respectively) including transversal temperature differences T_S1,S2, T_S2,S3, T_S3,S4, T_S4,S5, T_S6,S7, T_S7,S8 and vertical temperature differences T_S1,S6, T_S3,S7, T_S5,S8 are fitted by the three probability density functions ND, GEVD and EVD, as shown in Figure 5.

Figure 5.

The fitted probability density functions of daily extreme temperature differences (a) T_{S1,S2,maximum}, (b) T_S1,S2,minimum, (c) T_S2,S3,maximum, (d) T_S2,S3,minimum, (e) T_S2,S4,maximum, (f) T_S3,S4,minimum, (g) T_S4,S5,maximum, (h) T_S4,S5,minimum, (i) T_S6,S7,maximum, (j) T_S6,S7,minimum, (k) T_S7,S8,maximum, (l) T_S7,S8,minimum, (m) T_S1,S6,maximum, (n) T_S1,S6,minimum, (o) T_S3,S7,maximum, (p) T_S3,S7,minimum, (q) T_S5,S8,maximum, and (r) T_S5,S8,minimum.

The fitting errors of the three probability density functions ND, GEVD and EVD for each positive or negative temperature differences are calculated using equation(4) as shown in Table 2. For convenience of error comparison, the average fitting errors of the three probability density functions are further calculated in Table 2. It can be seen that GEVD has the smallest average fitting errors 0.0016 compared with the other probability density functions, so in this study GEVD is selected as the optimal probability density functions of positive or negative temperature differences.

Table 2.

The fitting errors of three probability density functions.

Temperature differences	Fitting errors
Temperature differences	ND	GEVD	EVD
T _{S1,S2,maximum}	0.0047	0.0019	0.0093
T _{S1,S2,minimum}	0.0031	0.0042	0.0069
T _{S2,S3,maximum}	0.0020	0.0024	0.0010
T _{S2,S3,minimum}	0.0040	0.0015	0.0010
T _{S3,S4,maximum}	0.0056	0.0019	0.0053
T _{S3,S4,minimum}	0.0009	0.0014	0.0035
T _{S4,S5,maximum}	0.0022	0.0016	0.0035
T _{S4,S5,minimum}	0.0013	0.0009	0.0048
T _{S6,S7,maximum}	0.0011	0.0017	0.0023
T _{S6,S7,minimum}	0.0014	0.0016	0.0015
T _{S7,S8,maximum}	0.0014	0.0012	0.0015
T _{S7,S8,minimum}	0.0015	0.0014	0.0021
T _{S1,S6,maximum}	0.0012	0.0012	0.0044
T _{S1,S6,minimum}	0.0006	0.0009	0.0021
T _{S3,S7,maximum}	0.0019	0.0013	0.0044
T _{S3,S7,minimum}	0.0021	0.0009	0.0005
T _{S5,S8,maximum}	0.0015	0.0008	0.0034
T _{S5,S8,minimum}	0.0010	0.0021	0.0044
Average fitting errors	0.0021	0.0016	0.0037

Cycle-life simulation analysis

One important part of temperature field analysis is to calculate the standard values of temperature differences for thermal effect analysis. According to current bridge codes (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017; Brussels: European Committee of Standardization, 2003), the standard value is the most unfavorable temperature difference value in the whole bridge service life with the exceedance probability 2%. What should be mentioned is that during calculation of standard values the yearly maximum or minimum values in the whole bridge service life need to be known in advance, but the monitoring period for the temperature data is actually not long enough (e.g. the monitoring period of this bridge is only 1 year), so cycle-life simulation analysis is necessary to obtain the the yearly maximum or minimum values in the whole bridge service life.

In this research, a uniform random sampling method is employed to simulate the yearly maximum or minimum values in 100 years for this bridge by virtual of cumulative distribution functions of GEVD above. In detail, the daily maximum or minimum values in 100 years are firstly simulated using the cumulative distribution functions of GEVD above, and then the yearly maximum or minimum values are selected from the daily maximum or minimum values.

Hence, the key step is how to simulated the daily maximum or minimum values in 100 years. First, the number of simulated daily maximum or minimum values in 100 years is calculated, which is 36,500; then, uniform random sampling in the interval (0,1) is carried out 36,500 times to obtain the sampling values; third, each sampling value is treated as the cumulative probability value p_i, so the corresponding daily maximum or minimum values T_S,i are calculated as follows:

p_{i} = \int_{- \infty}^{T_{s, i}} f (x) d x

(7)

where f(x) denotes the probability density function GEVD, and its integration is the cumulative distribution function. What should be mentioned is that T_S,i is the upper limit of integration, which can be directly calculated using the inverse GEVD function in MATLAB software. Finally, the daily maximum or minimum values of temperature differences in 100 years are simulated as shown in Figure 6.

Figure 6.

Cycle-life simulation results (a) T_S1,S2,maximum, (b) T_S1,S2,minimum, (c)T_S2,S3,maximum, (d)T_S2,S3,minimum, (e)T_S3,S4,maximum, (f)T_S3,S4,minimum, (g) T_S4,S5,maximum, (h) T_S4,S5,minimum, (i) T_S6,S7,maximum, (j) T_S6,S7,minimum, (k) T_S7,S8,maximum, (l) T_S7,S8,minimum, (m) T_S1,S6,maximum, (n) TS1,S6,minimum, (o)T_S3,S7,maximum, (p) T_S3,S7,minimum,(q) T_S5,S8,maximum, and (r)T_S5,S8,minimum.

Standard values of temperature differences of steel-concrete composite girders

Calculation method of standard values

According to current bridge codes (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017; Brussels: European Committee of Standardization, 2003), the yearly maximum and minimum temperature difference values T_max and T_min should not exceed the positive and negative standard values S_p and S_n in the whole bridge service life, respectively:

T_{\max} \leq S_{p}

(8a)

T_{\min} \geq S_{n}

(8b)

According to the cycle-life simulation results above, the number of T_max and T_min is 100, so these T_max and T_min should satisfy the following inequalities:

{T_{\max} (1), T_{\max} (2), . . ., T_{\max} (100)} \leq S_{p}

(9a)

{T_{\min} (1), T_{\min} (2), . . ., T_{\min} (100)} \geq S_{n}

(9b)

where T_max(i) and T_min(i) denotes the T_max and T_min in the ith year, respectively; i = 1,2,…,100. According to the definition of standard values, the occurrence probability P of the inequalities which are treated as random events is 0.98, which are expressed as follows:

P ({T_{\max} (1), T_{\max} (2), . . ., T_{\max} (100)} \leq S_{p}) = 0.98

(10a)

P ({T_{\min} (1), T_{\min} (2), . . ., T_{\min} (100)}) \geq S_{n}) = 0.98

(10b)

Consider that the values of T_max and T_min in each year are random and independent which follow the same probability density function, so equations (10a) and (10b) are further written as follows:

\int_{- \infty}^{S_{p}} f_{p} (T_{\max}) d T_{\max} = \sqrt[100]{0.98}

(11a)

\int_{S_{n}}^{+ \infty} f_{p} (T_{\min}) d T_{\min} = \sqrt[100]{0.98}

(11b)

where f_p (T_max) or f_p (T_min) denotes the probability density function of T_max or T_min respectively, which can be determined using the cycle-life simulation data above. What should be mentioned is that S_p and S_n are the upper or lower limit of integration, which can be directly calculated using the inverse probability density function in MATLAB software.

With regard to f_p (T_max) or f_p (T_min), 100 yearly maximum or minimum values are firstly selected from the 100-year cycle-life simulated temperature difference data; then, the probability density values are calculated using through probabilistic and statistical analysis; finally, GEVD is used to fit the probability density values to determine the parameter values as shown in Figure 7. What should be mentioned is that GEVD is selected as the probability density function of T_max or T_min; the reason is that if the best probability density function of daily maximum or minimum values is GEVD as analyzed above, the best probability density function of yearly maximum or minimum values is also GEVD according to the probabilistic and statistical theory.

Figure 7.

The fitted probability density functions of yearly extreme temperature differences (a) T_S1,S2,maximum, (b) T_S1,S2,minimum, (c) T_S2,S3,maximum,(d) T_S2,S3,minimum, (e) T_S3,S4,maximum, (f) T_S3,S4,minimum, (g) T_S4,S5,maximum, (h) T_S4,S5,minimum, (i) T_S6,S7,maximum, (j) T_S6,S7,minimum, (k)T_S7,S8,maximum, (l)T_S7,S8,minimum, (m) T_S1,S6,maximum, (n)T_S1,S6,minimum, (o) T_S3,S7,maximum(p) T_S3,S7,minimum, (q) T_S5,S8,maximum, and (r) T_S5,S8,minimum.

Calculation results

According to equations (11a) and (11b) the calculation results of standard values of positive and negative temperature differences S_p and S_n are shown in Table 3. Furthermore, the spatial distribution of the standard values of temperature differences is shown in Figure 8. What should be mentioned is that the S_p of T_S4,S5, T_S1,S6 and the S_n of T_S6,S7 are not provided, and the reason is that the calculated values of S_p are negative and the calculated value of S_n is positive.

Table 3.

Standard values T_N and T_P of temperature differences (°C).

Temperature differences		S _n	S _p	Maximum value	Minimum value	China code	USA code	European code
Transversal mode	T _S1,S2	−1.52	5.42	10.74	−6.38	0	0	±15
	T _S2,S3	−6.38	3.10
	T _S3,S4	−1.03	10.74
	T _S4,S5	−4.38	\
	T _S6,S7	\	7.19
	T _S7,S8	−4.78	1.66
Vertical mode	T _S1,S6	−7.77	\	4.81	−7.77	0	0	±15
	T _S3,S7	−5.16	4.70
	T _S5,S8	−6.65	4.81

Figure 8.

The spatial distribution of the standard values of temperature differences.

It can be seen that the maximum and minimum values of standard values for transversal temperature differences are 10.74°C and –6.38°C, respectively; the maximum and minimum values of standard values for vertical temperature differences are 4.81°C and –7.77°C, respectively. Currently, the China and USA bridge codes treat the standard values as 0°C, and the European bridge code treats the standard values as ±15°C, which are different from the calculation results by comparison (Ministry of Transport of the People’s Republic of China, 2020; American Association of State Highway and Transportation Officials, 2017; Brussels: European Committee of Standardization, 2003). Hence, the actual standard values of temperature difference for steel-concrete composite girder bridges is relatively complex, and the calculation results can provide an important reference for the current bridge codes.

Furthermore, there is a one-to-one correspondence between the monitoring values and the standard values for temperature differences. For example, the maximum value of monitoring positive temperature difference for T_S1,S2 is 4.91°C and its corresponding standard value is 5.42°C. Hence, the correlation scatter plot between the monitoring values and the standard values of the temperature differences for the Hongwan Waterway Bridge is shown in Figure 9. It can be seen that there is a good linear correlation between the measured values and the standard values, with a linear correlation coefficient of 0.9044. A linear function is used to fit the linear correlation which is expressed as S = 1.18 M + 0.39, where M denotes the monitoring value and S denotes the standard value.

Figure 9.

The correlation between measured extreme values and standard values.

Conclusion

Using the monitoring data of a large-span steel-concrete composite girder bridge, this research puts forward a calculation method of standard values of temperature differences in the whole service life is put forward by combination of probabilistic statistical characteristics analysis and life-cycle simulation analysis. The main conclusions are drawn as follows:

(1) The monitoring results show that there are significant temperature differences between measuring points, where the maximum positive transversal temperature difference between measuring points S₃ and S₄ can reach 7.40°C, and the maximum negative vertical temperature difference between measuring points S₁ and S₆ can reach –6.80°C. This is different from current bridge codes which only consider uniform temperature action on the steel part of the steel-concrete composite bridge;

(2) ND, GEVD and EVD are discussed and compared to determine the best probability density function to fit the probability density value with least errors. GEVD has the smallest average fitting errors 0.0016 compared with the other probability density functions, so in this study GEVD is selected as the optimal probability density functions of positive or negative temperature differences. Furthermore, cycle-life simulation method is put forward to obtain the yearly maximum or minimum values in the whole bridge service life using GEVD;

(3) The maximum and minimum values of standard values for transversal temperature differences are 10.74°C and –6.38°C, respectively; the maximum and minimum values of standard values for vertical temperature differences are 4.81°C and –7.77°C, respectively. There is a good linear correlation between the measured values and the standard values, with a linear correlation coefficient of 0.9044, so a linear function is used to fit the linear correlation which is expressed as S = 1.18 M + 0.39;

(4) Currently, the China and USA bridge codes treat the standard values as 0°C, and the European bridge code treats the standard values as ±15°C, which are different from the calculation results by comparison. Hence, the actual standard values of temperature difference for steel-concrete composite girder bridges is relatively complex, and the calculation results can provide an important reference for the current bridge codes.

Future work

This research put forwards a calculation method of standard values of temperature differences in the whole service life, which can provide a reference for thermal action design to evaluate the temperature effect of large-span steel-concrete composite girder bridges. What should be mentioned is that the monitoring results of T_s1,s6, T_s3,s7, and T_s5,s8 reveals the big temperature differences, but the number of temperature sensors may be not enough to reflect the whole vertical temperature gradients (Huang et al., 2024b). Hence, in the future more temperature sensors will be installed to obtain a more detailed temperature gradient.

Footnotes

ORCID iD

Gaoxin Wang

Funding

China Railway Group Limited Science and Technology Research and Development Program (2022 - Key - 44).

Declaration of conflicting interests

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

Data Availability Statements

The data that support the findings of this study are available from the corresponding author upon reasonable request.*

References

American Association of State Highway and Transportation Officials (2017) AASHTO LRFD Bridge Design Specifications: SI Units 10^th Edition 2017. AASHTO.

Brussels: European Committee of Standardization (2003) Eurocode 1: Actions on Structures-Part 1-5 General actions-thermal Actions: EN 1991-1-5. England The European Union Per Regulation.

Han

Zhao

Zhang

, et al. (2022) Performance assessment of railway multispan steel truss bridge bearing by thermal excitation. Journal of Civil Structural Health Monitoring 12: 163–178.

Huang

Gül

Zhu

(2018) Vibration-based structural damage identification under varying temperature effects. Journal of Aerospace Engineering 31(3): 04018014.

Huang

Cai

, et al. (2023) Comparison of the corrugated steel web composite box-girder and traditional girders regarding temperature field under solar radiation. Engineering Structures 291: 116419.

Huang

Zhang

Luo

, et al. (2024a) Construction and experimental verification of an efficient numerical model of one-dimensional temperature field for large-span cable-stayed bridge with steel-concrete composite girders. Engineering Mechanics 41(S1): 197–205.

Huang

Zhang

, et al. (2024b) Nonlinear modeling of temperature-induced bearing displacement of long-span single-pier rigid frame bridge based on DCNN-LSTM. Case Studies in Thermal Engineering 53(2024): 103897.

Huang

Wan

Zhu

(2025) Reconstruction of structural acceleration response based on CNN-BiGRU with squeeze-and-excitation under environmental temperature effects. Journal of Civil Structural Health Monitoring 15: 985–1003.

Jing

Shan

Zhang

, et al. (2024) Typhoon- and temperature-induced responses of a cable-stayed bridge. Advances in Structural Engineering 27(16): 2773–2789.

10.

Keles

Artar

Erguen

(2024) Investigation of temperature effect on the optimal weight design of steel truss bridges using cuckoo search algorithm. Structures 59: 105819.

11.

Chen

Zhou

, et al. (2023) Thermal behaviors of bridges—a literature review. Advances in Structural Engineering 26(6): 985–1010.

12.

Chen

Zhou

, et al. (2023) Thermal behaviors of bridges—a literature review. Advances in Structural Engineering 26(6): 985–1010.

13.

Nie

Zhang

, et al. (2024) Temperature field of long-span concrete box girder bridges in cold regions: testing and analysis. Structures 61: 105969.

14.

Meng

Guo

Jiang

(2023) Thermal effect on continuous steel truss girder rail-cum-road bridge with long spans and units after completion. World Bridges 51(04): 77–84, (In Chinese).

15.

Ministry of Transport of the People's Republic of China (2020) Design Code for Highway cable-stayed Bridges: JTG/T3365-01-2020. People’s Communications Press (In Chinese).

16.

Nikola

Marija

Maja

, et al. (2023) Parametric study of additional temperature stresses in continuously welded rails on steel truss railway bridges. Buildings 13(9): 1–22.

17.

Shan

Xia

, et al. (2023) Temperature behavior of cable-stayed bridges. Part I—Global 3D temperature distribution by integrating heat-transfer analysis and field monitoring data. Advances in Structural Engineering 26(9): 1579–1599.

18.

Song

Yan

Zhang

, et al. (2023) Temperature induced deformation laws of high pier and long span deck continuous steel truss bridge. Railway Engineering 63(04): 42–46 (In Chinese).

19.

Wang

Ding

Liu

(2019) The monitoring of temperature differences between steel truss members in long-span truss bridges compared with bridge design codes. Advances in Structural Engineering 22(6): 1453–1466.

20.

Wang

Zhou

Ding

, et al. (2021) Long-term monitoring of temperature differences in a steel truss bridge with two-layer decks compared with bridge codes: case study. Journal of Bridge Engineering 26(3): 05020013.

21.

Wang

Zhang

, et al. (2023) Temperature response of double-layer steel truss bridge girders. Buildings 13(11): 2889.

22.

Wang

Zhu

, et al. (2024) Long-term monitoring and standard value calculation of temperature gradients of long-span steel truss bridges. China Journal of Highway and Transport 2024: 1–15 (In Chinese).

23.

Xia

Wan

, et al. (2020) Condition analysis of expansion joints of a long-span suspension bridge through metamodel-based model updating considering thermal effect. Structural Control and Health Monitoring 27(5): e2521.

24.

Yang

Qin

Yang

, et al. (2024) Study on fire resistance of suspension bridge structure based on thermal mechanical coupling. Highway Engineer 49(1): 53–59 (In Chinese).

25.

Zhang

Huang

Wan

, et al. (2024) Missing measurement data recovery methods in structural health monitoring: the state, challenges and case study. Measurement 231(2024): 114528.

26.

Zhou

Xia

Chen

, et al. (2020) Analytical solution to temperature-induced deformation of suspension bridges. Mechanical Systems and Signal Processing 139: 106568.

27.

Zhou

Wanyan

(2024) Review on convective and radiation heat transfer between bridges and external environments. Advances in Structural Engineering 27(8): 1285–1302.

28.

Zhu

Wang

, et al. (2021) Mapping temperature contours for a long-span steel truss arch bridge based on field monitoring data. Journal of Civil Structural Health Monitoring 11(3): 725–743.

29.

Zhu

Zhang

Zhou

, et al. (2023) Thermal effect analysis of a steel truss cable-stayed bridge with two-layer decks under sunlight. Advances in Structural Engineering 27(1): 119–133.

30.

Zhu

Guo

Sun

, et al. (2024) Non-uniform spatial-temporal field of railway suspension bridge and detailed analysis of its effect. Engineering Mechanics 41(12): 1–13 (In Chinese).