Modelling of extreme uniform temperature for high-speed railway bridge piers using maximum entropy and field monitoring

Abstract

Due to the atmospheric temperature and solar radiation, the effects of temperature variation in bridge structures should be considered. Such variation induces notable deformations and movements, jeopardizing the safety of bridges and high-speed trains operations. However, the temperature action is a random process and its distribution is difficult to determine. The existing methods for analyzing structural temperatures are insufficient to meet the precision requirement. Therefore, the accurate prediction of extreme structural temperatures relates to the accurate evaluation on the bridge safety. This paper proposes a robust and accurate model for predicting the extreme structural temperature of a bridge. A field experiment, spanning over 2 years, was carried out on a high-speed railway bridge; and the long-term (56-years) atmospheric temperature data was adopted. Probabilistic models for the structural temperature were established using the Maximum Entropy (MaxEnt) model and the Generalized Pareto distribution (GPD) model; and the predictions on the uniform structural temperature (T_u) with 50 and 100 years return periods are presented. Additionally, the performance between the MaxEnt model and the GPD model is compared, based on the estimates with different return levels. The results show that the MaxEnt model is more stable and is significantly robust to the variation of sample sizes; and indicates that the MaxEnt model reduces the uncertainty of outcomes and avoids the high risk of bias. The MaxEnt model has a great potential to the applications of the extreme value analysis with small sample size. It offers a wider applicability and helps solve practical problems.

Keywords

high-speed railway bridge maximum entropy uniform structural temperature probabilistic models extreme distribution

Introduction

As a critical infrastructure, high-speed railway (HSR) bridges have been rapidly constructed in China to improve its traffic system. A long-span bridge with high-piers has been identified to be a reasonable scheme and most economical in mountainous areas. The safety and stability of high piers are critical to the operation of high-speed trains. During the service life, bridge-piers are constantly subject to temperature changes due largely to the solar radiation, thus inducing thermal stresses (Li et al., 2019) and the girder deflections (Yang et al., 2018). However, the uniform extreme temperature component (T_u) of the cross section determines the expansion or contraction of a structure (Huang et al., 2018). The variation of uniform temperatures will result in significant deformations and movements on bridge piers; and this issue will be more serious with the increasing pier height. In addition, the inconsistent temperature deformation between adjacent piers has an adverse effect on the comfortability and safety of high-speed train operations. Consequently, many bridges failed to complete their service life (Murphy and Yarnold, 2018; Obrien et al., 2020; Dai et al., 2017). Therefore, it is essential to correctly predict the Tu and its effects on the life of bridge piers.

An accurate prediction on extreme temperatures is important for evaluating the bridge safety. The first step is to obtain continuous temperature monitoring data, which can be achieved by using the structural health monitoring system (Yang et al., 2020). Essentially, the time-dependent temperature data is obtained (Tao et al., 2021; Liu et al., 2019; Peiretti et al., 2014), which can be used to efficiently describe the variation of a temperature field. Then, based on the measured temperature data, it is possible to predict the extreme structural temperature (Wang et al., 2021). For example, Ding et al. (2012) analyzed the distribution characteristics and statistical features of a temperature field and then predicted the extreme temperature with a return period of 50 years. However, insufficient data is often encountered as few long-term temperature measurements spanning over 10 years are available. It is a challenging task to predict the extreme structural temperature throughout the service life based on the limited data available. Traditionally, the extrapolation method has been employed (Fu and You, 2011). To overcome the absence of long-term measured data, the relationship between the structural temperature and the atmospheric temperature has been investigated based on the 1-year observation data (Lou et al., 2018; Abid et al., 2016). Then, an extreme structural temperature predicted model can be established by using the long-term historical atmospheric temperature data.

Then, a suitable method is required to model the risk of extreme structural temperature and estimate its probability. To quantify the extreme tail risk, the statistical extreme value theory (EVT) is used to model extreme events (Stangl, 2008; Katz et al., 2002). There are two basic methods in the EVT: the block maximum (BM) approach and the peaks over threshold approach (POT). The respective representative models are the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD). Tong et al. (2002) employed the GEV model to determine the extreme structural temperature with a 50-years return period utilizing the 40-years meteorological information of Hong Kong. Zhang et al. (2021) proposed an approach to analysis the temperature difference of a steel box girder based on the 1-year observation data using the GEV model and copula function. Zhou et al. (2017) proposed a GPD-based extreme value model to feature the statistical distribution of extreme temperatures. However, the accuracy of the GEV model is unsatisfactory as only monthly or annually extreme temperatures are used instead of all measured data. Moreover, the GPD model is sensitive to the selection of the threshold value, which is a weakness of the model.

As known, it is important to find a more suitable model to predict the extreme temperature. The Maximum Entropy (MaxEnt) method (Jaynes, 1957) can be used to establish the statistical inference of the probability density function (PDF) of extreme temperatures; and it uses all available information of the sample, described as statistical moments. By maximizing the entropy, the best probability model for the data can be identified, which is least biased on the given information. The MaxEnt method offers an enhancement way for predicting extreme temperatures robustly. Recently, the MaxEnt method has been widely used in the field of ocean engineering (Dong et al., 2013a) and hydrology (Singh et al., 2012).

In this study, the MaxEnt method is employed to predict the extreme temperature of a bridge pier, based on the measured field data. First, the short-term (2-years) temperature measurements of bridge piers are analyzed and the relationship between structural and atmospheric temperatures is established. Second, a comparison is made to assess the performance and stability of MaxEnt method. The comparison results show that: the MaxEnt method is more feasible and robust; and it is very stable with the variation of threshold values.

Research background

With the rapid development of HSRs, the movement of bridge construction is also vigorous. To span over valleys and rivers, high-piers with hollow reinforced concrete are usually used to support HSR bridges. However, the temperature gradient will develop in the concrete structure due to the poor thermal conductivity. Under the combined action of uniform temperature change and gradient temperature, temperature deformations will be induced in bridge piers, which will be more apparent with the increasing pier height. Moreover, the variation of temperatures in bridge piers significantly influences the overall deflection and deformation of the bridge.

Although the weather conditions vary at different locations, the intrinsic law between atmospheric temperature and structural temperature is the same. Therefore, the key is to find the intrinsic law. Then, the bridge pier temperature distribution and extreme temperature can be estimated using the atmospheric temperature data, obtained from local weather stations.

A field experiment was carried out to evaluate the effect of the atmospheric temperature on the temperature distribution. The tested HSR bridge pier is 39m and located at the longitude 115°20^′ east and the latitude 27°38^′ north (in Jiangxi Province, China). To monitor the temperature, 28 temperature gauges were installed in the reinforcement mat (Section 1-1, Figure 1) before pouring the concrete. The temperature gauge used is BGK-3700, which has the accuracy up to ±0.2°C and the measurement range of -30-70°C. The specific measuring points are shown in Figure 2 where the capital letters (E, S, W, N, EN, ES, WN, and WS) represent the geographical locations. For example, E1 denotes the first gauge at the east location. The monitoring period was from July 6, 2018 to July 6, 2020, being in the operation stage of the HSR. The temperature data was collected every 0.5 hour; and the remote data acquisition was realized by the wireless transmission. The loss tolerance for the data collection was less than 0.1%, ensuring the data integrity. Representative temperature monitoring data (Gauge E1) are shown in Figure 3.

Figure 1.

The structural health monitoring system of HSR bridge-pier.

Figure 2.

The locations of temperature gauges (cm).

Figure 3.

Monitoring data graph of Gauge E1.

To analyze the intrinsic relationship between structural temperature and atmospheric temperature, the observation data of the meteorological station (M57799) located in the closest vicinity to the bridge site is used, which covers a long time period of about 60 years (1964–2020) and is utilized to predict the 100-years extreme structural temperature.

Intrinsic law between structural temperature and atmospheric temperature

According to the limit state design method, the representative temperature is represented by the extreme value during the entire design life of the bridge. For example, the representative temperature for a 100-years return period is calculated according to the 100-years climatic data. To calculate the extreme value, the key is to establish a temperature probabilistic model (Xu et al., 2021).

The EVT and MaxEnt methods are usually used to describe the probability of extreme events. The temperature probabilistic model is obtained by fitting the long-term historical data of structural temperatures. However, most bridge temperature measurements were taken within 10 years. To obtain the extreme temperature, one alternative is to establish the intrinsic law between the structural temperature and the atmospheric temperature. Then, the structural temperature can be predicted according to the intrinsic law and atmospheric temperature. Obviously, the accuracy of the extreme value model is affected by the intrinsic law. Some researchers have tried to simulate the structural temperature based on the atmospheric temperature and solar radiation using the heat-transfer analysis (Xia et al., 2013). It needs to establish a finite element (FE) model conforming to complex physical models for the heat-transfer analysis. These physical models are necessary to correlate loads with structural reactions and to detect correct boundary conditions. The input parameters for such models are often difficult to determine, which needs to be estimated. Therefore, the heat-transfer analysis requires costly computations to obtain a structural temperature model during the pier’s service life since physical models are required. There are also some simple calculation methods. In the General Specifications for Design of Highway Bridges and Culvers of China (JTG D60-2015) (Ministry of Transport of P.R. China, 2015), the intrinsic law is expressed as

T_{u, \max} = 24.14 + (T_{t, \max} - 20) / 1.40,20 {}^{\circ}C \leq T_{t, \max} \leq 45 {}^{\circ}C

(1)

T_{u, \min} = 1.17 + T_{t, \min} \times 0.63, - 50 {}^{\circ}C \leq T_{t, \min} \leq - 2 {}^{\circ}C

(2)

where T_u,max and T_u,min are respectively daily maximum and minimum uniform structural temperatures; and T_t,max and T_t,min are respectively the corresponding daily maximum and minimum average value of atmospheric temperature. The Euro code (Eurocode 1) (European Committee for Standardization, 2003) states the following law:

T_{u, \max} = T_{t, \max} + 2,30 {}^{\circ}C \leq T_{t, \max} \leq 50 {}^{\circ}C

(3)

T_{u, \min} = T_{t, \min} + 8, - 50 {}^{\circ}C \leq T_{t, \min} \leq 0 {}^{\circ}C

(4)

As mentioned above, T_u,max needs to be known in advance. The uniform temperature T_u over the cross section governs the expansion or contraction of a structure. In the experiment, it is estimated as (European Committee for Standardization, 2003),

T_{u} = \frac{\sum T_{i} A_{i}}{\sum A_{i}}

(5)

where T_i is the temperature of gauge i, and A_i is the area surrounding gauge i. The cross-section is divided into eight parts by eight directional temperature gauges and the section is then divided into several small areas according to the depth of the measuring points. There are five gauges and three gauges in this section with small areas divided by the midline of adjacent measuring points as shown in Figure 4.

Figure 4.

Typical cross-section of a partition (cm).

Equations (1)–(4) have been used in some other studies (Wang et al., 2020). However, their application is limited as the climate conditions in different areas are not the same. An improved linear relationship formula is hence proposed (Lou et al., 2018), which uses the statistical regression method to establish their intrinsic law based on the measured structural and atmospheric temperatures (T_u and T_t).

In this study, the atmospheric temperature data from the M57799 meteorological station is employed, while the measured structural temperature is used. Figure 5 shows the monitoring data of Gauge E1 (T_E1), uniform temperature (T_u), and atmospheric temperature (T_t). It can be seen that the change trends of the three temperature spectra are similar, and only phase separation exists between the values. Accordingly, the daily maximum and minimum uniform temperature T_u,max and T_u,min were found to be well linearly correlated to the daily maximum and minimum average value of atmospheric temperature T_t,max and T_t,min, respectively, as shown in Figure 6(a) and 6(b). The coefficient of determination (R-square) values of the proposed formulas for T_u,max and T_u,min were 0.961 and 0.922, respectively. The achieved prediction accuracy of uniform temperature is sufficient for practical purposes. The formulas of the T_u,max and T_u,min of the bridge piers are expressed as equations (6) and (7).

Figure 5.

Records of uniform structural temperature, atmospheric temperature and Gauge E1 temperature for the period from July 6, 2018 to July 6, 2020.

Figure 6.

The intrinsic law between uniform structural temperature and atmospheric temperature.

The intrinsic laws proposed in this study were compared with those provided in the China Bridge Design Specification (JTG D60-2015) and Euro code (Eurocode 1) as shown in Figure 7(a) and 7(b). As seen, the daily maximum uniform temperature is slightly higher than the corresponding atmospheric temperature in the three methods and the maximum uniform temperature curves obtained by various methods are close. Figure 7(a) shows that the proposed regression curve is close to the Euro code values at high temperatures. For the minimum uniform temperature shown in Figure 7(b), the proposed regression curve is lower than Chinese and Euro codes curves, implying that the lower temperature limit given by the Chinese and Euro codes is conservative. Therefore, an intrinsic law between structural temperature and atmospheric temperature can be established according to local conditions, which is suitable for predicting the extreme uniform temperature of structures.

\begin{array}{l} T_{u, \max} = 1.198 + 1.038 T_{t, \max} \\ R^{2} = 0.961 \end{array}

(6)

\begin{array}{l} T_{u, \min} = 2.017 + 1.059 T_{t, \min} \\ R^{2} = 0.922 \end{array}

(7)

Figure 7.

Comparison of calculation methods for uniform structural temperature.

Methods for distribution function

This section introduces the MaxEnt method and a numerical algorithm. Then, a brief overview of the traditional GPD method is given.

Maximum entropy method

The MaxEnt method is used to characterize the probability distribution functions even when the given data is limited. The entropy has been proposed in the field of thermodynamics and then introduced into the information theory described by Shannon (1948). It is defined as a measure of uncertainty in the process of system information transmission. The principle of MaxEnt argues that the probability distribution, that maximizes information entropy, is the best estimate. That is, the least biased distribution estimate is the probability distribution with the MaxEnt. The entropy for probability distribution S(f) is given by

S (f) = - \int_{D} \ln (f (x)) f (x) d x

(8)

where ln represents the natural logarithm and f(x) is a probability distribution function. According to the principle of MaxEnt method, the distribution f(x) that maximizes the entropy S(f) is the least biased estimation. Then, the problem can be transformed into an optimization problem subjected to the following constraints:

\int_{D} f (x) d x = 1, f (x) \geq 0

(9)

\int_{D} g_{i} (x) f (x) d x = m_{i}, i = 1,2, \cdot \cdot \cdot, r

(10)

where D is the integration interval and g_i(x) is the arbitrary basis function with statistical moments m_i. Then, by introducing the Lagrangian function and corresponding multipliers λ_i, the following expression is obtained,

\begin{array}{l} L (f, λ) = S (f) + (λ_{0} + 1) [\int_{D} f (x) d x - 1] + \\ \sum_{i = 1}^{r} λ_{i} [\int_{D} g_{i} (x) f (x) d x - m_{i}] \end{array}

(11)

Seeking the MaxEnt is essentially to maximize this Lagrangian function under the conditions of equations (9) and (10).

\frac{\partial L (f, λ)}{\partial f} = \int_{D} (- \ln f (x) + λ_{0} + \sum_{i = 1}^{r} λ_{i} g_{i} (x)) d x = 0

(12)

After some preliminary calculations, the following expression is obtained:

f (x) = \exp [λ_{0} + \sum_{i = 1}^{r} λ_{i} g_{i} (x)]

(13)

Further, assuming that g_i(x)=xⁱ satisfies the given constraints, then, the expression of the MaxEent probability density function is then

f (x) = \exp (λ_{0} + \sum_{i = 1}^{r} λ_{i} x^{i}), i = 1,2, \cdot \cdot \cdot, r

(14)

whose distribution function is

F (x) = \int_{D} \exp (λ_{0} + \sum_{i = 1}^{r} λ_{i} x^{i}) d x, i = 1,2, \cdot \cdot \cdot, r

(15)

Equation (14) is the general formula of the probability density function (PDF). The probability density function f(x) is obtained by solving the Lagrangian multipliers from the nonlinear system with (r+1) equations. The analytical solution of λ_i can be obtained when r is small (r ≤ 3), which is discussed in Gotovac et al. (2009). However, it is hard to obtain when r > 3 (Dong et al., 2013b). Therefore, a number of numerical optimization techniques, such as the iterative scaling, gradient descend of Abramov (2007, 2009), Newton iterative method, and BFGS procedure of Berger et al. (1996), are proposed.

In this study, the improved Newton iterative method (Rockinger and Jondeau, 2002), which is an efficient and accurate estimation technique, is used. The procedure is briefly described as follows.

Substituting equation (14) into (10) leads to

\exp (- λ_{0} - \sum_{i = 1}^{r} λ_{i} m_{i}) = \int_{D} \exp [\sum_{i = 1}^{r} λ_{i} (g_{i} (x) - m_{i})] d x

(16)

Defining a function Q(λ₁,…,λ_r) as

Q (λ_{1}, \cdot \cdot \cdot, λ_{r}) = \exp (- λ_{0} - \sum_{i = 1}^{r} λ_{i} m_{i})

(17)

then equation (14) can be rewritten as

f (x) = \exp [\sum_{i = 1}^{r} λ_{i} (g_{i} (x) - m_{i})] / Q (λ_{1}, \cdot \cdot \cdot, λ_{r})

(18)

Substituting equation (9) into (10) gives,

\int_{D} (g_{i} (x) - m_{i}) f (x) d x = 0

(19)

According to equation (17), the following expression can be obtained:

\frac{\partial Q}{\partial λ_{i}} = \int_{D} (g_{i} (x) - m_{i}) \exp [\sum_{i = 1}^{r} λ_{i} (g_{i} (x) - m_{i})] d x

(20)

Substituting equation (20) into equation (16) yields

j_{i} = \frac{\partial Q}{\partial λ_{i}} = \exp (- λ_{0} - \sum_{i = 1}^{r} λ_{i} m_{i}) \int_{D} (g_{i} (x) - m_{i}) f (x) d x

(21)

Considering j_i (the gradient of Q(λ₁,…,λ_r)) equal to 0, the following hessian matrix of Q(λ₁,…,λ_r) can be obtained,

J_{i j} = \frac{\partial^{2} Q}{\partial λ_{i} \partial_{j}} = \exp (- λ_{0} - \sum_{i = 1}^{r} λ_{i} m_{i}) \int_{D} (g_{i} (x) - m_{i}) (g_{j} (x) - m_{j}) f (x) d x

(22)

According to the above conditions, the minimum Q(λ₁,…,λ_r) in the domain D has a unique solution. Then, the optimization problem is transformed into the problem of finding the minimum Q(λ₁,…,λ_r) in the domain D. By adopting a numerical optimization technique, the iterative equation becomes,

λ^{(k)} = λ^{(k - 1)} - {(J^{(k)})}^{- 1} j^{(k)}

(23)

The calculation procedure consists of four steps: (i) selecting the domain D and initial vector λ₀; (ii) finding the corresponding Q(λ₁,…,λ_r) (17), j_i (21) and J_ij (22) using the Gauss-Legendre quadrature (Lether, 1978); (iii)updating the vector of Lagrange multipliers $λ^{(k)} = λ^{(k - 1)} + δ^{(k)}$ considering $δ^{(k)} = - {(J^{(k)})}^{- 1} j^{(k)}$ ; and (iv) stopping the iteration when a required accuracy is meet, or returning to Step (iii).

Extreme value theory (EVT) method

In statistics, the EVT method is usually used when inferring the possibility of the occurrence of extreme events (Coles et al., 2001). Its primary purpose is to describe the tail of the distribution of random variables. For the temperature action, the EVT method can be used to predict the extreme temperature that may occur during the life of the bridge. For this purpose, the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD) are proposed. The GEV commonly uses block maximal values to fit the distribution, while the data fitted by the GPD model is generated by the peaks over threshold (POT) method. The block maxima (BM)are created by dividing the analysis period into several parts with the same size and then choosing the maximum observation of each new part. According to Caires (2016), with peaks over the threshold (POT) samples of two or more observations per block on average, the estimates are more accurate than the corresponding block maxima (BM) estimates and with more than 200 years of data the accuracies of the two approaches are similar and good. In the case of bridge-pier temperatures, the blocks are chosen for a period of 1 year and the block maxima are the annual maxima. The number of POT samples is in thousands, while the number of BM samples is only in dozens. POT is more efficient than BM under this circumstance, with the number of exceedances larger than the number of blocks. Modeling only the annual maxima is an inappropriate approach to the extreme value analysis when the measured extreme data is limited. Therefore, the GEV distribution is not suitable for such applications (Zhou et al., 2017; Xu et al., 2021). This study mainly introduces the GPD with its distribution function defined as

H (x) = {\begin{cases} 1 - {(1 + k \frac{x - u}{σ})}^{- 1 / k}, k \neq 0 \\ 1 - \exp (- \frac{x - u}{σ}), k = 0 \end{cases}

(24)

where u is the threshold value; k and σ represent respectively the shape parameter and scale parameter. To assess the structural risks caused by extreme events, the estimation of return level (X_p) is commonly used, which can be written as

X_{P} = u + \frac{σ}{k} [{(p_{u} / T_{p})}^{k} - 1]

(25)

where Tp=1/(pn_y) is the return period and n_y is an average annual number of observations in the random variable X, and p_u is the exceedance rate of threshold which can be estimated with the following expression:

p_{u} = \frac{n {X > u}}{n}

(26)

where n is the number of samples and n{X > u} is the number of samples exceeding the threshold u.

The parameters of the GPD can be achieved by many techniques such as the maximum likelihood (ML) method (Coles et al., 2001), the probability weighted moments method (Hosking et al., 1985) and L-moments method (Pandey et al., 2001). This study adopts the maximum likelihood (ML) method.

Application of the models

This study was carried out based on the observed data (T_t) of the meteorological station (M57799) located in the closest vicinity to the bridge site, taken from 1964 to 2020. Then, with the proposed model, the extreme T_u, can be obtained by the historical data T_t and the intrinsic law.

Results of the GPD model

The initial step of the POT approach is to determine the threshold range to determine the sample size for an analysis. Selecting an appropriate threshold is the key to achieve high estimation accuracy. The threshold should be high enough to support the asymptotic basis for the GPD model and low enough in order to avoid high variance. The mean residual life method (Anagnostopoulou et al., 2012) is applied to determine the threshold range. In this study, the selected threshold range is 32–38°C.

As an example, the diagnostic plots for the daily maximum atmospheric temperature (T_t,max) dataset with the threshold value of 36°C are shown in Figure 8, including the probability, quantile, the return level plots as well as the probability density distribution function. The information represented by these graphs is an indication of good fit, as discussed in Coles (2001). Regarding the probability and quantile plots, each set of plotted points is nearly linear, which supports the validity of the fitted model. In addition, the return level plot (with 95% confidence interval) provides a satisfactory representation of the empirical estimates. Moreover, the density distribution plot is consistent with the histogram of the data. Consequently, Figure 8 reveals that the GPD distribution fits satisfactorily the POT sample. The diagnostic plots of other threshold ranges have a similar pattern and a similar level of good fit.

Figure 8.

Diagnostic plots for the GPD model with the threshold value of 36°C.

Since the focus of this study is the applicability of the MaxEnt method to describe the extreme temperature behavior, these results will be used to compare with the extreme prediction generated by the MaxEnt method.

Results of the MaxEnt model

Different from the GPD model, the MaxEnt model is a non-parametric method. The numerical optimization algorithm based on a finite domain is needed when employing the MaxEnt model. Besides, the POT samples and the threshold range used in the MaxEnt model are the same as those in the GPD model. Then, when the POT sample is selected, the lower boundary of the defined domain is naturally determined. The upper boundary has no pre-defined value and can be an arbitrary number. However, it should be a moderate one. If it is too low, the PDF fitted by the model will be the truncated in the high tail region. While, the PDF will be expanded if the upper bound is too high. Therefore, the upper boundary needs to select an appropriate value to ensure the accuracy of the model.

To obtain the appropriate value for the upper boundary, the MaxEnt model is tested with the lower boundary fixed at 36°C and the upper boundary varying from 45°C to 65°C. Variations of return level with upper boundary of the MaxEnt model are shown in Figure 9. As seen, the return levels increase with an increasing upper boundary. 55°C is used as an ideal value for the upper boundary of the MaxEnt model domain since the high quantiles tend to be constant and do not rise again when the upper boundary exceeds 55°C. Therefore, it is used as a reasonable upper boundary in this study.

Figure 9.

Variations of return levels with upper boundary of the MaxEnt model with the lower boundary at 36°C and the upper boundary varying from 45°C to 65°C.

After determining the upper and lower boundaries, the appropriate order of sample moments should be selected as constraints. Variations of return level with order of moments (constrains) are shown in Figure 10. To ensure that all important characteristics of the distribution are captured, the sample moments from order one to order four are used. In Figure 10, the achieved accuracy of the 100-years return level is 0.1°C, which is sufficient for practical purposes. Increasing the order of moments higher than 4, the impact on accuracy is insignificant. Therefore, the sample moments up to order four is adopted in all POT samples.

Figure 10.

Variations of return level with order of moments with the lower boundary at 36°C and the upper boundary at 55°C.

Additionally, to obtain 95% confidence intervals for estimating the return levels generated by the MaxEnt model, the bootstrap technique (Efron, 1979) is used. For each threshold, i.e., POT sample, 100 bootstrap samples were generated based on the original sample, i.e., resampling with replacement. The MaxEnt model have been employed on them and the 95% confidence interval can be calculated. Then, the result can be used as a quantitative description of the uncertainty for MaxEent model, which will be further compared with that generated by the GPD model.

The probability density and return level plots are presented in Figures 11 and 12 respectively to illustrate how the MaxEnt model fits with the data. Figure 11 shows that the density estimate is consistent with the histogram of the data, indicating that the POT samples are supporting the fitted MaxEnt model. The return level is consistent with the empirical estimates. Therefore, the return level plot and the corresponding 95% confidence interval (Figure 12) can be directly compared with the return level plot of the GPD model (as shown in Figure 8). As seen from Figure 12, the MaxEnt model has a similar level of fitness and the same shape pattern as previously observed in the GPD model. However, The MaxEnt model has a significantly smaller 95% confidence interval, indicating smaller uncertainty.

Figure 11.

Probability density plot of the MaxEnt model for the dataset (T_t,max).

Figure 12.

Return level plots of the MaxEnt and the GPD model for the dataset with a threshold at 36°C and the 95% confidence intervals (CIs).

Robustness comparison of the MaxEnt and GPD models

Table 1 summarizes the results of the MaxEnt and GPD models, including the threshold, the size of POT samples under the analyzed threshold range, and the 100-years return levels with the 95% confidence intervals of the GPD and MaxEnt models. As indicated in Table 1, the GPD model has relatively broader 95% confidence interval for higher threshold values.

Table 1.

Threshold, number of threshold exceedances and 100-years return levels with 95% confidence intervals obtained by the GPD and MaxEnt models.

Data set	Threshold u (°C)	No. of exceedances n_u	$T_{t, \max}^{100 y r}$ in the GPD mode (°C)		$T_{t, \max}^{100 y r}$ in the MaxEnt mode (°C)
Data set	Threshold u (°C)	No. of exceedances n_u	Predicted value	GPD-95%CI	Predicted value	Bootstrap-95%CI
M57799	32	4584	40.83	(40.75,41.87)	41.10	(40.96,41.24)
	33	3788	40.92	(40.81,41.90)	41.11	(40.94,41.20)
	34	2874	40.99	(40.70,41.98)	41.13	(40.89,41.27)
	35	1978	41.10	(40.71,42.08)	41.15	(41.04,41.34)
	36	1215	41.32	(40.74,42.27)	41.32	(41.06,41.45)
	37	623	41.64	(40.77,42.56)	41.63	(41.46,41.78)
	38	251	41.80	(40.72,42.72)	41.82	(41.70,41.92)

CI: confidence intervals.

Compared with the results obtained by the GPD model, the uncertainty gap of the MaxEnt model is considerably smaller; and similar conclusion is found in the remaining combinations of POT sample and threshold value. As for each of the analyzed thresholds and POT samples, the deviation between the return levels obtained by the MaxEnt model to be used on the bootstrap samples and the return levels obtained by the MaxEnt model to be employed on the original samples, is small enough to meet the following requirements: the return levels estimated from the original samples fall within 95% confidence interval around the corresponding bootstrap means. It indicates that the calculation results of the MaxEnt model are accurate and stable in both the bootstrap samples and the original samples, and the MaxEnt produces reliable tail estimates with less variability. In addition, the 100-years return levels predicted by the MaxEnt model remain basically consistent, as the number of samples(n_u) decreases. The MaxEnt model is not sensitive to the sample size, thus expanding its application scope. In general, the results generated by the MaxEnt model are in good agreement with those obtained by the GPD model; while the MaxEnt model is more stable and is robust to the variation of sample sizes.

The appropriate threshold is a critical factor for the accuracy and robustness of the GPD model. However, it has no significant effect on the MaxEnt model. A further analysis of the sensitivity for the models (MaxEnt and GPD) was made. Figure 13 shows the prediction of the extreme temperature related to 100-years return level and the corresponding 95% confidence interval. As seen, the prediction of extreme temperature increases with the increasing threshold; and the prediction results of both models are nearly the same.

Figure 13.

Estimates of 100-years return levels obtained by the MaxEnt and GPD models and the 95% confidence intervals (CIs).

Moreover, as noted, the distinctive difference between the MaxEnt model and the GPD model is the uncertainty gap (displayed with the size of the 95% confidence interval envelope). The MaxEnt model produces smaller 95% confidence intervals across the whole threshold range, indicating that a greater stability is achieved compared with the GPD model.

Lastly, it can be stated that the MaxEnt model can predict the estimates with high quantiles as accurately as the GPD model and is more robust within the selected threshold range. Therefore, it is considered as a more suitable model than the GPD model.

Prediction of extreme temperatures on bridge piers

On the basis of the intrinsic law, the extremes of T_u can be calculated by the historical atmospheric temperature data. According to the above analysis, the performance of the MaxEnt model is deemed as a more suitable method for predicting the extreme temperature with a high return period, tested on the atmospheric temperature data obtained from observation data of meteorological station (M57799) for a time period of 56 years (1964–2020). Therefore, the MaxEnt model is also applied to predict the minimum uniform temperature ( $T_{u, \min}^{50 / 100 y r}$ ). The daily minimum average value of atmospheric temperature (T_t,min) is analyzed according to the calculation method of the MaxEnt model. In this study, the threshold is set as two and the order is set as the first four moments. Figure 14 shows that the density distribution plot is consistent with the histogram of the data. As seen in Figure 15, the drawn scatter points are basically straight lines, which validates the MaxEnt model. As a result, the MaxEnt model satisfactorily fits the POT sample.

Figure 14.

Probability density plot of the MaxEnt model for the dataset (T_t,min).

Figure 15.

Probability plot of the MaxEnt model.

Based on the MaxEnt model, the prediction of T_u could be acquired by adopting different return levels. Taking return periods of 50 and 100 years as examples, the predicted extreme values of uniform structural temperature are shown in Table 2. For this bridge-pier, the maximum uniform temperatures are 43.49°C and 44.09°C and the minimum uniform temperature are −1.73°C and −2.48°C within the 50-years and 100-years return period, respectively.

Table 2.

The predicted extreme values of atmospheric temperature and uniform structural temperature with 50 and 100 return periods.

$T_{t, \max}^{50 y r} / T_{t, \min}^{50 y r}$ (°C)	$T_{t, \max}^{100 y r} / T_{t, \min}^{100 y r}$ (°C)	$T_{u, \max}^{50 y r} / T_{u, \min}^{50 y r}$ (°C)	$T_{u, \max}^{100 y r} / T_{u, \min}^{100 y r}$ (°C)
40.18	41.32	43.49	44.09
−3.61	−4.33	−1.73	−2.48

Conclusions

This paper proposes a robust and accurate model for predicting the extreme structural temperatures of bridge-piers. A field experiment, spanning over 2 years, was carried out and the long-term (56-years) atmospheric temperature dataset was adopted. Predictions on the design T_u of the pier using the MaxEnt and GPD models are presented; and the performance of the models is discussed when producing the estimates with different return levels. Base on this study, the following conclusions can be drawn:

(1) From the case study, the prediction results predicted by the MaxEnt model are in good agreement with those obtained by the GPD model. However, as the threshold increases from 32 to 38, the prediction results by the GPD model increase significantly, showing the instability of the method. The maximum uniform temperature and the minimum uniform temperature are 43.49°C/44.09°C and −1.73°C/-2.48°C within the 50-years/100-years return period, respectively.

(2) The advantage of MaxEnt model over the GPD model is that the MaxEnt model is more robust in selecting a threshold level. The prediction results show that a greater stability is achieved compared with the GPD model; and indicates that the MaxEnt model reduces the uncertainty of outcomes and avoids the high risk of bias.

(3) The return levels predicted by the MaxEnt model remain basically consistent as the number of samples(n_u) decreases. Therefore, the MaxEnt model is robust to the variation of sample sizes, which expands its application scope. It can be concluded that the MaxEnt model has a great potential to the applications of the extreme value analysis with small sample size. It is well suited for broad applications and helps solve practical problems. All extreme values of structural temperature proposed for bridge-piers are uniform temperature. The differential temperature between the outer faces of a bridge pier is also important to bridge’s responses. Future work will focus on identifying the general differential temperature and uniform temperature actions of bridge piers, and the probable load combination.

ORCID iD

Fen Wang https://orcid.org/0000-0002-2296-7998

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by the Project of Science and Technology Research and Development Program of China Railway Corporation (No. 2017G006-N), to which the authors are very grateful.

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