Introduction

Background of the study

With the bridge manufacturing technology regardless of the improvement, the concrete structure gradually toward the thin-walled, lightweight, and other directions, in which the concrete rectangular hollow pier by virtue of excellent performance, is widely used in the field of bridge construction1,2. However, concrete rectangular hollow piers are placed in the natural environment, due to the concrete material is a poor conductor of heat, in the strong drop (rise) temperature effect, the concrete will produce a large internal temperature effect3,4, which produces a large temperature tensile stress, leading to structural cracking, which is detrimental to the safety of the structure5.During bridge design, the range of initial temperature change will be predicted according to the corresponding codes and standards, but in the actual construction process, it is often difficult to achieve the conditions considered in the design due to changes in the construction period, environment and other conditions6. For bridges in the service stage, due to the thermal expansion and contraction of concrete materials, the larger temperature action of the cross-section will lead to structural deformation, and a large number of concrete disease analyses show that many bridges cracking, breaking, and other diseases are related to temperature action7,8,9,10.

Therefore, an accurate understanding of the bridge temperature is the key to evaluating its performance during the bridge design and in-service stages11. Research on the existing concrete temperature testing found that most of the traditional temperature monitoring programs require temperature sensors to be pre-buried in the concrete structure, which has the advantages of intuition and accuracy, but it cannot be applied to the concrete structure in the service stage and the lack of internal concrete temperature information12,13,14. Compared with the traditional temperature testing program, the method of predicting the change rule of concrete structure temperature by studying the long-term change relationship between the average internal temperature of the concrete structure and the external air temperature of the concrete can grasp the range of the internal temperature change of the concrete more accurately in the absence of the internal temperature information of the concrete15,16,17.

In recent years, many organizations and scholars have conducted more research in the field of concrete temperature monitoring and temperature prediction. China's General Specification for Highway Bridge and Culvert Design for the role of the average temperature is divided into three climate zones, but due to the vastness of China, the countrywide climate varies greatly, and this treatment appears to be more rough18. British Bridge Regulations BS-5400, for the range of changes during the bridge erection, the minimum and maximum shade temperature will be increased and decreased by 2 °C respectively on the basis of the 120-year return period, for the shade temperature above mean sea level adjusted according to the altitude, for the minimum shade temperature, every 100 m higher minus 0.5 °C, for the maximum shade temperature, every 100 m higher minus 1.0 °C19. The 1st edition (1994) of the AASHTO LRFD highway bridge design specification divides the country into mild and cold regions, and gives a direct range of average bridge temperatures according to bridge materials20; the 3rd edition (2004) and later AASHTO highway bridge design specification supplements a new scheme for the role of the average temperature of bridges, which is according to the material type gives the national contour maps of the maximum and minimum effective temperatures, the old and new schemes co-exist and can be either one or the other21,22,23. Yingshu Yuan15 et al., in order to predict the temperature response of concrete in natural climatic environments, a model based on the heat conduction of concrete proposed by considering the natural weather variations induced by diurnal and seasonal temperatures, and based on the extreme value of the difference decomposition, gave a method to deal with climatic temperature data, and the results showed that the prediction model can be more effective in predicting the temperature response of concrete under natural environment.MP Dissanayaka24 et al., applied the finite element method and the finite difference method to the concrete temperature prediction model, and found that applying these two methods to predict the temperature of concrete is more effective, and emphasized the importance of cold water pipe is important for the cooling of concrete structure, but the calculation method is more complicated and cannot be applied to the long-term prediction of concrete temperature. Haoyu Zeng25 et al., studied and established a three-dimensional transient heat transfer model, calculated the model using the finite element method, and compared and analyzed the calculation results with the measured data. The results showed that: the prediction of temperature distribution in concrete by the three-dimensional model was relatively accurate, with an error of less than 4%; the surface temperature difference in July was twice as much as that in December; the wind speed had a cooling effect on the surface of the concrete, and the larger the wind speed was, the stronger the cooling effect was. Guo An26 et al. analyzed the on-site temperature monitoring data collected by the engineering examples, and obtained an empirical formula for calculating the cooling effect of the concrete, by combining The results of measured data and empirical formula was compared and analyzed, and it was found that the empirical formula had better calculation effect, and the equation of the rate of temperature rise of the heat of hydration was derived. LiBao Yui27 et al., in order to predict the trend of temperature change of mass concrete, constructed a CART prediction model based on the technology of large-data processing, and the results of the study showed that the model and the method can predict the temperature change of the mass concrete better, and the prediction accuracy is higher, and can be used for practical purposes. The prediction accuracy is high, which can provide necessary guidance for the actual project.

The above study shows that: for the average temperature role of concrete bridges, from the development trend in the norms of various countries, there is an increasing tendency to determine the range of changes in the average temperature of bridges based on the information of air temperature in various places; the main influencing factor of the average temperature of the bridge structure is the change of air temperature, and at the same time, the solar radiation also has a certain influence; based on the observation data, the study of the average temperature of the concrete structure is a function of the external air temperature, and it is proposed and verify the feasibility of predicting the range of average temperature change of hollow piers; at present, domestic and international research on the role of temperature in bridges mainly focuses on concrete girders, with relatively little research on piers, and the relevant bridge codes at home and abroad also lack the provisions on the role of temperature in piers.

Therefore, this study on the Changjiahe Special Bridge concrete rectangular hollow pier temperature distribution for long-term observation, analyzes the average temperature of the concrete hollow pier and pier outside the shade of the relationship between the air temperature, the relationship with the air temperature inside the pier, put forward and verify the average temperature of the hollow pier time-by-time value of the prediction method, the average temperature of the hollow pier of the prediction of the range of the law. Thus, in the absence of concrete internal temperature information, can be the more accurate grasp of the range of concrete temperature change, and provide a reference for such similar work.

Purpose and significance of the study

The purpose of the study in this paper is to analyze the variation of internal temperatures of concrete rectangular hollow piers when subjected to strong cooling (warming) effects in a natural environment and the temperature stress problems they may cause, which in turn may lead to concrete cracking. In bridge design, the initial temperature of concrete rectangular hollow piers is usually predicted to guide engineering practice. However, due to the influence of various factors such as construction period and environment during the actual construction process, there may be deviations between the predicted value and the actual situation, resulting in incomplete data on the internal temperature of the concrete, which in turn affects the construction and maintenance of rectangular hollow piers. Therefore, this study aims to fill this knowledge gap and provide a reliable reference for concrete temperature changes.

The research innovation of this paper lies in the observation of the rectangular hollow core concrete pier of Changjiahe Bridge for 212 days, and the collection of detailed measured temperature data. By statistically analyzing these data, we propose two methods for accurately predicting the average temperature range of the concrete rectangular hollow core pier: one is to monitor the specific value of the temperature change of the concrete hollow core pier hour by hour, and the other is the maximum and minimum values of the average temperature change during a certain period of time. These methods not only make up for the problem of insufficient information on the internal temperature of concrete, but also provide a reliable reference basis for similar projects to take corresponding measures to cope with the impact of temperature changes. In addition, these methods can also predict the average temperature change of similar bridges during the service stage, which further enhances their practicality and application value.

Change rule of the average temperature of the concrete hollow pier

Introduction to the dependent works

Changjiahe Bridge, 2687 m long, for the separation of a double-span six-lane curved bridge design, single deck width of 16.65 m, consisting of the main bridge and approach bridge. The upper part of the main bridge is (75 m + 2 × 140 m + 75 m) prestressed concrete continuous rigid structure, the lower piers are thin-walled hollow piers, pile foundation, the highest pier height of 90 m, the upper part of the approach bridge is prestressed concrete small box girder, a total of 30 m box girder 585, 40 m box girder 120, the lower piers are columnar piers and hollow piers, abutments are columnar platforms, piers foundation is pile foundation. The temperature test selects the representative rectangular hollow pier of the whole bridge, the section length is 7.50 m, width is 3.00 m, wall thickness is 50 cm, 20 × 20 cm chamfer is set at the corner of the pier, the maximum height of the pier is about 60 m, the latitude of the location of the pier is 35° 10′ 48ʺ N, the longitude is 109° 0′ 9ʺ E, and the elevation is about 1110 m, and the direction of the pier is northeast-southwest, and the section long edge shows northeast-southwest direction, and the section long edge shows northeast-southwest direction, and the section long edge shows northeast-southwest direction, and the section long edge shows northeast-southwest direction, and the section long edge shows northeast-southwest direction, and the section long edge shows northeast-southwest direction, and the section long edge shows northeast—northwest direction. The longitudinal direction of the abutment is northeast-southwest, and the angle between the central axis of the longitudinal direction of the cross-section and the north direction is about 62 º, as shown in Fig. 1a.

Figure 1
figure 1

Structure of abutment and arrangement of measurement points. (a) Schematic structure of a concrete rectangular hollow pier. (b) Arrangement of temperature measurement points in the cross-section of the abutment (unit: cm).

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Change rule of the average temperature of the concrete hollow pier

It has been shown that the temperature change along the height direction of the concrete abutment is not significant, and the effect of different heights on the temperature field can be neglected28,29,30. Therefore, in this study, only one test section was buried when the measurement point was arranged, and the height from the ground was about 12 m. Because the concrete rectangular hollow pier is affected by the air temperature and sunshine, it will produce a more obvious temperature gradient in the direction of the pier thickness, and the temperature difference between the outside air temperature, the air temperature inside the pier and the concrete temperature is more obvious, and the concrete temperature shows the distribution law shown in Fig. 3. Therefore, when arranging the measurement points, the influence of the rate temperature gradient on the average temperature is needed31. 6 temperature measurement points were set up in the east, south, west, and north of the test section, located in the center of the wall, in the direction of "outside the pier → inside the pier", and numbered east (1) ~ east (6), south (1) ~ south (6), west (1) ~ west (6), and north (1) ~ north (6), respectively, taking into account that Pier outside the surface temperature gradient is larger, the measurement point arrangement of the outer side of the denser, each wall 6 measurement points from the outer surface of the distance of 3, 8, 18, 28, 39 and 48 cm respectively. box inside, outside each set a temperature measurement point, which is outside the box for the shade temperature measurement point, numbered in turn (25), (26); test section of the northwest, southwest, and southeast of the three corners of the corner of the setting of the temperature of the two measurement points, the outside measurement point 3 cm from the outer surface, the inner measuring point is 2 cm from the inner surface, numbered (27), (28), (29), (30), (31) and (32), the measurement point layout is shown in Fig. 1b.

The temperature measuring element adopts JMT-36B semiconductor temperature sensor produced by China Changsha Jinmao High-Tech Industrial Co., Ltd, with sensitivity of 0.1 °C and measuring range of − 30 °C–120 °C; the data acquisition instrument adopts JMZX-32A general automation comprehensive test system, with 32 acquisition channels, which is powered by solar panels, and the data are automatically collected and stored with DTU cell phone Internet module, can realize the data of the Internet transmission, download. Temperature measurement components and acquisition system are shown in Fig. 2, taking into account the standby capacity of the system, and the comprehensive determination of the sampling time interval of 1 h.

Figure 2
figure 2

Temperature measuring element and acquisition system. (a) Temperature sensors, (b) Automated acquisition systems.

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Measured average temperature change rule of the concrete hollow pier

Calculation of average temperature of the concrete hollow pier

Calculating the average temperature of a moment of concrete rectangular cross-section is a more reasonable approach based on the cross-section area represented by each measurement point, using the weighted average method of calculation. For rectangular hollow piers, one can first find out the average temperature of each wall, ignoring the wall along the length direction of the temperature difference, approximate along the wall thickness direction of the temperature weighted average as the average temperature of the wall. The temperature distribution of the wall along the wall thickness direction is nonlinear, the temperature of the measurement points C1 ~ C6 at a certain moment is t1 ~ t6, respectively, and the temperature values are plotted in the thickness direction to obtain the actual temperature distribution curve, as shown in Fig. 3. Segmented folding lines are used instead of the actual curve distribution, and the average temperature \(\overline{{T}_{j}}\) of each wall is shown as Eq. (1).

$$\begin{array}{c}\overline{{T}_{j}}=\frac{{\sum }_{i=1}^{6}{t}_{i}{h}_{i}}{h},\end{array}$$
(1)

where C1 ~ C6 for the concrete rectangular hollow pier wall measurement point number; t1 ~ t6 for the measurement point C1 ~ C6 to monitor the temperature of the corresponding moment.

Figure 3
figure 3

Graphical representation of wall weighted average temperature calculation.

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After finding the average temperature \(\overline{{T}_{j}}\) of each wall, it is then weighted and averaged with the weight being the length L_j of each wall, and the average temperature ¯T of the abutment is shown in Eq. (2).

$$\begin{array}{c}\overline{T}=\frac{{\sum }_{j=1}^{4}\overline{{T}_{j}}\cdot {L}_{j}}{{\sum }_{j=1}^{4}{L}_{j}},\end{array}$$
(2)

Analysis of the change rule of shade temperature outside the concrete hollow pier

The hollow pier average temperature change rule and the pier outside the temperature change are closely related, to study the hollow pier average temperature change rule, first of all to figure out the pier outside the temperature change rule. Through the bridge site at the shade temperature long-term observation, found that the outside shade temperature change can be regarded as a different cycle change and the superposition of random change. There are three main cyclic changes, annual cycle change, daily cycle change and step up and down change. As can be seen in Fig. 4, in the long term, the temperature shows year-to-year cyclic changes, and the annual changes in temperature can be approximated by a sinusoidal curve, with the average monthly temperature generally reaching its lowest value in December or January and its highest value in July or August. As can be seen from Fig. 5, in addition to the yearly cycle, the air temperature also exhibits a day-by-day cyclic variation, and the daily variation of the air temperature can also be approximated by a sinusoidal curve, with the air temperature generally reaching its highest value around 14:00 p.m. and its lowest value around 5:00 a.m. The mean air temperature is generally around 14:00 p.m. and its lowest value is around 5:00 a.m. As can be seen in Fig. 6, the average air temperature does not change linearly when warming from winter to summer or cooling from summer to winter but shows a stepwise increase or decrease, and the period of each step is about 10 days. From winter → summer, the temperature shows a step increase, each step of the change process is roughly" a sharp decline (length of about 2 ~ 3 days) → rapid warming (length of about 7 days)", the final temperature is higher than the highest temperature before the last step cooling; from summer → winter, the temperature shows a step decrease, each step of the change process is roughly" a sharp decline (length of about 2 ~ 3 days) → rapid warming (length of about 7 days)". The process is roughly as follows: "Sharp decline (duration about 2 ~ 3 days) → slow warming (duration about 3 ~ 4 days) → slow cooling (duration about 3 ~ 4 days)", and the final temperature is lower than the maximum temperature before the previous step.

Figure 4
figure 4

Annual cycle of temperature.

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Figure 5
figure 5

Daily cycle of temperature.

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Figure 6
figure 6

Step change in air temperature.

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Change rule of the average temperature of the concrete hollow pier

In order to investigate the degree of difference between the average concrete hollow pier temperature and air temperature, the meteorological 4-point average air temperature is utilized instead of the 24-h average air temperature. Using the above method to calculate the daily average pier temperature and daily average air temperature, can be obtained as shown in Fig. 7, a typical air temperature step change stage of the average pier temperature and average air temperature relationship graph, hollow pier average time-by-time temperature and air temperature changes in the relationship as shown in Fig. 9. As can be seen in Fig. 7, the average temperature of the concrete hollow piers and the same as the air temperature, but also show a day-by-day cycle and step change. As can be seen from Figs. 7 and 9, when the time-by-time cycle changes, the hollow pier's average temperature extremes appear to lag behind the air temperature, about 7:00 a.m. or so the lowest, lagging behind the air temperature for about 2 h, about 19:00 p.m. or so the highest, lagging behind the air temperature for about 5 h; the hollow pier average temperature with the fluctuations of the air temperature fluctuations, but the magnitude of the daily change is much smaller than the air temperature; due to the thermal inertia of the concrete, the average pier temperature is generally higher than the average daily temperature, the average temperature. When the temperature drops sharply, the difference between the two increases, when the temperature rises sharply, the difference between the two decreases, when the temperature warming rate is too fast, there may also be the case of daily average air temperature over the daily average pier temperature. The difference between all daily average pier temperatures and daily average air temperatures during the observation period is statistically obtained as shown in the scatter plot in Fig. 8, from which it can be seen that the average difference during the observation period is 1.6 °C (Fig. 8).

Figure 7
figure 7

Daily mean pier temperature vs. daily mean air temperature.

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Figure 8
figure 8

Difference between daily mean dun temperature and daily mean air temperature during the observation period.

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Figure 9
figure 9

Variation of mean temperature of hollow piers in relation to air temperature.

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Statistical analysis of the correlation between the mean temperature of concrete hollow piers and air temperature

Relationship of mean hollow pier temperature to mean daily air temperature and mean daily in-pier air temperature

Adopt the calculation method of Article 2.1 to get the time-by-time value of the average hollow pier temperature, and adopt the calculation method described in Article 2.3 to get the daily average air temperature. Take the daily maximum and minimum hollow pier average temperature and the daily average air temperature of the day respectively, analyze the correlation between the two, and get the correlation graph as shown in Figs. 10 and 11. As can be seen from Figs. 10 and 11, the linear correlation coefficients between the daily maximum and minimum hollow pier average temperatures and the daily average air temperature are R = 0.980 and R = 0.973, respectively, which are highly linearly correlated. The air temperature in the narrow sense refers to the shade temperature outside the pier, and the air temperature in the broad sense includes the air temperature inside the pier in addition to that. Separately take the daily average air temperature, the daily average air temperature inside the pier, and the daily average hollow pier temperature comparison analysis, analysis of the correlation between the two, the correlation obtained as shown in Figs. 12, and 13. From Figs. 12, 13 can be seen, the average daily air temperature, the average daily air temperature within the pier, and the average daily hollow pier temperature between the linear correlation coefficient were R = 0.980, R = 0.998, are highly linear correlation; the average daily hollow pier temperature and the average daily air temperature within the pier of the linear proportionality between the 1.030, and therefore can be taken as an approximation of the average daily air temperature within the pier is equal to the average daily air temperature.

Figure 10
figure 10

Daily maximum hollow pier average temperature vs. daily average air temperature.

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Figure 11
figure 11

Average daily minimum hollow pier temperature vs. daily average air temperature.

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Figure 12
figure 12

Daily average off-pier air temperature vs. daily average hollow pier temperature.

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Figure 13
figure 13

Daily average in-pier air temperature vs. daily average hollow pier temperature.

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Relationship between daily variability of mean hollow pier temperature and daily variability of air temperature

According to the results of the previous study, when the average temperature of the hollow pier changes day by day like the air temperature, the variation of the average temperature of the hollow pier is much smaller than that of the air temperature. As shown in Fig. 14, for a peak, corresponding to the left and right two valley points, and "valley—peak" and "peak—valley" absolute values is not equal, so in order to simplify the calculation, define the daily air temperature, hollow pier temperature change is much smaller than the air temperature. Therefore, in order to simplify the calculation, the definition of daily air temperature, hollow pier temperature variation for the average of the two, that is, the daily variation of air temperature as shown in Eq. (3); hollow pier temperature daily variation as shown in Eq. (4).

$$\begin{array}{c}\Delta T=\frac{\Delta {T}_{1}+\Delta {T}_{2}}{2},\end{array}$$
(3)

where ΔT is the daily variation of temperature; \(\Delta {T}_{1}\) is the absolute value of temperature "valley—peak"; \(\Delta {T}_{2}\) is the absolute value of temperature "peak—valley".

$$\begin{array}{c}\Delta t=\frac{\Delta {t}_{1}+\Delta {t}_{2}}{2},\end{array}$$
(4)

where Δt is the daily variation of the average temperature of the hollow pier; \(\Delta {t}_{1}\) is the absolute value of the "valley—peak" of the average temperature of the hollow pier; \(\Delta {t}_{2}\) is the absolute value of the "peak—valley" of the average temperature of the hollow pier.

Figure 14
figure 14

Schematic illustration of the daily variation of air temperature and the daily variation of hollow pier temperature.

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According to this definition, the daily variation of hollow pier temperature and the daily variation of air temperature in the observation period of 212 days are calculated, and the correlation between the two is analyzed, and the correlation graph shown in Fig. 15 is obtained. As seen in Fig. 15, the daily variation of the average hollow pier temperature and the daily variation of air temperature are approximately linear, with a correlation coefficient of R = 0.899, and the two are highly linearly correlated, from which the daily variation of air temperature can be predicted by the daily variation of the hollow pier temperature.

Figure 15
figure 15

Daily variation of air temperature vs. daily variation of hollow pier temperature.

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Prediction of the average temperature of concrete rectangular hollow piers

Hour-by-hour value prediction and validation of hollow pier mean temperature

For concrete rectangular hollow piers, an initial bridge temperature is usually predicted at the time of design, but in actual construction, it is often difficult to achieve the condition considered in the design due to changes in conditions such as construction period and environment. Therefore, it is hoped that the average temperature of the bridge can be predicted more accurately in order to grasp the degree of difference from the design-anticipated condition so that appropriate remedial measures can be taken.

According to the study in Sect. "Change rule of the average temperature of the concrete hollow pier”, the average concrete hollow pier temperature reaches its lowest value around 7:00 a.m. and its highest value around 19:00 p.m., with good symmetry. Assuming that the average temperature of the concrete hollow pier is a folded line change, the time-by-time value of the average temperature of the hollow pier can be predicted according to the changes in air temperature inside and outside the pier, and the specific realization process is:

  1. (1)

    the pier temperature for 24 h of continuous testing, get the daily average pier temperature; from the analysis of Sect. “Calculation of average temperature of the concrete hollow pier” it can be seen that the daily average hollow pier temperature and the daily average pier temperature linear proportionality between the pier temperature of 1.030, so it can be approximated to take the daily average hollow pier temperature is equal to the daily average pier temperature, can be taken the day 13 h of the average temperature of the concrete hollow pier for the average pier temperature, that is, point A in Fig. 17 Take the average temperature of the hollow pier at 13:00 on the next day as the average daily air temperature in the pier on the next day, i.e. point B in Fig. 17; take the average temperature of the hollow pier at 1:00 on the next day as the average of the two moments of A and B, i.e. point C in Fig. 17;

  2. (2)

    Continuous testing of the shade temperature outside the pier to get the temperature rise \(\Delta {T}_{1}\) and temperature drop \(\Delta {T}_{2}\), according to the study in Sect. "Analysis of the change rule of shade temperature outside the concrete hollow pier", the daily variation of the hollow pier temperature and the daily variation of the air temperature is approximately linear, the introduction of α for the daily variation of the temperature of the hollow pier and the daily variation of the air temperature than the value of the value of the ratio (as in Eq. (5)), will be the paper's actual 212 days of observation of the α take the arithmetic mean \(\overline{\alpha }\) is 0.21, as shown in Fig. 16 below. Equation (6) is shown, then the hollow pier temperature \({T}_{D}\) at 19:00 is Eq. (7), i.e. point D in Fig. 17;

    $$\begin{array}{c}\alpha =\frac{\Delta t}{\Delta T},\end{array}$$
    (5)
    $$\begin{array}{c}\overline{\alpha }=\frac{{\sum }_{i=1}^{n}{\alpha }_{i}}{n} n=212,\end{array}$$
    (6)

where α is the ratio of daily variation of hollow pier temperature and daily variation of air temperature; \(\Delta T\) is the daily variation of air temperature; \(\Delta t\) is the daily variation of average temperature of hollow pier; \(\overline{\alpha }\) is the arithmetic mean of the ratio of daily variation of hollow pier temperature and daily variation of air temperature in the observation interval.

$$\begin{array}{c}{T}_{D}=\frac{1}{2}\left({T}_{A}+{T}_{C}\right)+\frac{\overline{\alpha }}{4}\left(\Delta {T}_{1}+\Delta {T}_{2}\right),\end{array}$$
(7)

where: \(\overline{\alpha }\) is the arithmetic mean of the ratio of the daily variation of the hollow pier temperature and the daily variation of the air temperature in the observation interval; \({T}_{A}\) is the average temperature of the concrete hollow pier at 13:00 on the same day; \({T}_{C}\) is the average temperature of the hollow pier at 13:00 on the next day; \(\Delta {T}_{1}\) is the absolute value of the temperature rise on the same day; \(\Delta {T}_{2}\) is the absolute value of the temperature drop on the same day.

Figure 16
figure 16

Ratio of daily variation of hollow pier temperature to daily variation of air temperature.

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Figure 17
figure 17

Predicted computational schema for daily average abutment temperatures.

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  1. (3)

    Because A and D, D and C between the concrete hollow pier temperature is a linear change, the use of linear interpolation method, can be obtained in the middle of each moment of the concrete hollow pier temperature;

  2. (4)

    Using the same method, the concrete hollow pier temperature values of other moments can be obtained sequentially.

Using the above method, the predicted value of the hollow pier temperature between May 4 and May 19 during the observation period was obtained, and the comparison with the measured value is shown in Fig. 18, with a high degree of curve coincidence and a small difference, which shows that the prediction method has a high degree of accuracy. According to the above method, the predicted value of concrete box girder temperature can be obtained by only testing the shade temperature inside and outside the bridge pier at the same time.

Figure 18
figure 18

Predicted and measured values of abutment temperature.

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Prediction and validation of the range of variation in the mean temperature of hollow piers

For the prediction of the range of change in the average temperature of concrete rectangular hollow piers, according to the research and analysis in Sects. “Calculation of average temperature of the concrete hollow pier”, “Analysis of the change rule of shade temperature outside the concrete hollow pier”, to determine the range of change in the average temperature of hollow piers during a longer period of time, there can be the following two methods:

  1. (1)

    One is to directly calculate the maximum and minimum values of the hollow pier mean temperature by using the daily average off-pier air temperature during that time period according to Relationship Eqs. (8) and (9) identified in Figs. 10 and 11;

  2. (2)

    The other is based on Fig. 12 to determine the relationship Eq. (10), through the time period of the daily average pier outside the air temperature, calculated the day of the average hollow pier temperature, and then based on Fig. 15 to determine the relationship Eq. (11), the use of daily variation of air temperature to calculate the daily variation of the average temperature of the hollow pier, if you can not get to know the day of the accurate daily variation of the air temperature, according to the results of the previous study, the maximum, the minimum average hollow pier temperature can be taken as the average daily variation of air temperature of the month of July, January, and the minimum average air temperature of the month of July. If the average daily variation of air temperature is not known on the day, according to the results of the previous study, the maximum and minimum hollow pier average temperature can be taken as the average daily variation of air temperature in July and January respectively, then the maximum and minimum hollow pier average temperature will be shown in Eqs. (12) and (13).

    $$\begin{array}{c}{f}_{max}=0.942{x}_{max}+3.777,\end{array}$$
    (8)
    $$\begin{array}{c}{f}_{min}=0.925{x}_{min}+1.606,\end{array}$$
    (9)

where \({f}_{max}\) is the maximum value of the average temperature of the hollow pier; \({x}_{max}\) is the maximum value of the average daily air temperature; \({f}_{min}\) is the minimum value of the average temperature of the hollow pier; \({x}_{min}\) is the minimum value of the average daily air temperature.

$$\begin{array}{c}{f}_{A}=0.931{x}_{A}+2.774,\end{array}$$
(10)
$$\begin{array}{c}{f}_{D}=0.208{x}_{D}+0.009,\end{array}$$
(11)
$$\begin{array}{c}{f}_{max}={f}_{A}+0.5{f}_{D},\end{array}$$
(12)
$$\begin{array}{c}{f}_{min}={f}_{A}-0.5{f}_{D},\end{array}$$
(13)

where \({f}_{A}\) is the daily mean hollow pier temperature of the day; \({f}_{D}\) is the daily variation of the mean hollow pier temperature; \({x}_{A}\) is the maximum and minimum values of the daily mean air temperature; and \({x}_{D}\) is the daily variation of the air temperature.

Adopting the above two methods, the maximum and minimum average temperatures of the test piers were calculated within 7 days of the observation period of 7 days, for example, and the calculation results are shown in Table 1. As can be seen from Table 1, the average of the calculation results of Method 1 and Method 2 is very close to the measured value, and the relative error is only − 7.3–7.5%, the maximum and minimum values of the average temperature of the hollow piers are higher than that of the average daily air temperature, which is consistent with the heat storage properties of the concrete structure (Supplementary Information).

Table 1 Range of average temperature changes at test abutments during the observation period (in °C).
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Conclusion

This paper focuses on the average temperature action of concrete rectangular hollow piers, and the following main results and conclusions are obtained through the statistical analysis of the long-term observation data of the test piers:

  1. (1)

    The change of external shade temperature can be regarded as the superposition of different cyclic changes and random changes. There are three main types of cyclic variations, annual cyclic variations, daily cyclic variations, and step-up or step-down, and each step has a period of about 10 days;

  2. (2)

    The average temperature change law of concrete hollow pier and air temperature, also presents day-by-day cycle and step change; and the hollow pier average temperature extreme value time lags behind the air temperature, about 7:00 a.m. or so the lowest, about 19:00 p.m. or so the highest; hollow pier average temperature fluctuations with the temperature fluctuations and fluctuations, but the magnitude of the daily change is much smaller than the air temperature, the vast majority of cases of daily average pier temperature is higher than the daily average air temperature, and the observation period of the average air temperature, and the daily average pier temperature is higher than the daily average air temperature. Average air temperature and the average difference during the observation period is 1.6 °C;

  3. (3)

    daily maximum, daily minimum hollow pier average temperature, and daily average air temperature between the linear correlation coefficient were R = 0.980, R = 0.973, are highly linear correlation; daily average air temperature, daily average pier air temperature and daily average hollow pier temperature between the linear correlation coefficient were R = 0.980, R = 0.998, are highly linear correlation;

  4. (4)

    Hollow pier average temperature and air temperature as the same day-by-day cycle changes, the average hollow pier temperature change is much smaller than the air temperature; hollow pier average temperature daily variation and air temperature daily variation is approximately linear relationship, correlation coefficient R = 0.899, the two are highly linearly correlated;

  5. (5)

    Assuming that the average temperature of the concrete hollow pier is a folding line change, a method of predicting the average temperature of the concrete hollow pier hour by hour is proposed, and the predicted value is compared with the measured value, and it is found that the predicted value matches the measured value better, which indicates that the method has a high prediction accuracy;

  6. (6)

    Two methods are given to predict the range of changes in the average temperature of the test piers, and the results of the two are averaged and compared with the measured values, and it is found that the results are very close to each other, and the relative error is only − 7.3–7.5%, which indicates that the method has a high prediction accuracy.