Temperature Field Boundary Conditions and Lateral

sensors

Letter Temperature Field Boundary Conditions and Lateral Temperature Gradient Eﬀect on a PC Box-Girder Bridge Based on Real-Time Solar Radiation and Spatial Temperature Monitoring

Xiao Lei 1, Xutao Fan 1, Hanwan Jiang 2,* , Kunning Zhu 3 and Hanyu Zhan 4

1 College of Civil and Transportation Engineering, Hohai University, 1 Xikang Road, Nanjing 210098, China; [email protected] (X.L.); [email protected] (X.F.) 2 Civil & Environmental Engineering Department, University of Wisconsin Platteville, Platteville, WI 53818, USA 3 China Design Group Co., Ltd., 9 Ziyun St., Nanjing 210014, China; [email protected] 4 Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, NM 88003, USA; [email protected] * Correspondence: [email protected]

Received: 29 July 2020; Accepted: 4 September 2020; Published: 15 September 2020

Abstract: Climate change could impose great influence on infrastructures. Previous studies have shown that solar radiation is one of the most important factors causing the change in temperature distribution in bridges. The current temperature distribution models developed in the past are mainly based on the meteorological data from the nearest weather station, empirical formulas, or the testing data from model tests. In this study, a five-span continuous Prestressed-concrete box-girder bridge was instrumented with pyranometers, anemometers, strain gauges, displacement gauges, and temperature sensors on the top and bottom slabs and webs to measure the solar radiation, wind speeds, strain, displacement, and surface temperatures, respectively. The continuously monitoring data between May 2019 and May 2020 was used to study the temperature distributions caused by solar radiation. A maximum positive lateral temperature gradient prediction model has been developed based on the solar radiation data analysis. Then, the solar radiation boundary condition obtained from the monitoring data and the lateral temperature gradient prediction model were utilized to compute the tensile stresses in the longitudinal and transverse directions. It was demonstrated in this study that the tensile stress caused by the lateral temperature gradient was so significant that it cannot be ignored in structural design.

Keywords: lateral temperature gradient; temperature ﬁeld; boundary conditions; solar radiation intensity; temperature monitoring

1. Introduction Most highway bridges are located in open fields and are exposed to the solar radiation, wind, rain, and other environmental changes. Those climate changes can cause substantial temperature variations in bridges, trigger cracking and bearing displacing, and therefore, endanger the serviceability, durability, and safety of the bridges. In extreme cases, the major structural elements could lose their functionality due to the severe damage causing the global structural failure [1,2]. Solar radiation plays an important role in affecting the temperature distribution in the bridge, so that relationship between the solar radiation and the temperature distribution should be properly established. Lee [3] was monitoring the temperature distribution in a five-foot-long prestressed I-shape girder with thermocouples embedded

Sensors 2020, 20, 5261; doi:10.3390/s20185261 www.mdpi.com/journal/sensors Sensors 2020, 20, 5261 2 of 15 in the flanges and the web for a year. With the continuously measured temperature variations, both vertical and lateral temperature distributions were established. A two-dimensional heat-transfer model was developed to investigate the effects of seasonal variations and girder orientations on temperature distributions. Wang et al. created a finite element model for a concrete box-girder arch bridge in thermal field caused by solar radiation based on meteorological data [4]. Abid et al. monitored air temperature, solar radiation, and wind speeds with sensors and thermocouples for more than a year. They proposed empirical equations to predict the maximum vertical and lateral temperature gradients based on the data collected [5]. The same approach was applied to concrete-encased steel girders [6]. Taysi and Abid investigated the effects of the thermal properties in a full-scale concrete box-girder segment. The individual effect of each of the thermal properties of the concrete box girder were studied with the aid of three-dimensional finite element analysis. Temperature distributions under the extreme thermal loads were compared with AASHTO’s (The American Association of State Highway and Transportation Officials) and Bridge Manual’s temperature gradient models [7]. Tian et al. conducted a numerical study on temperature effects in the train–bridge interaction system [8]. Lawson et al. used recent meteorological data from two weather stations in Nevada establishing temperature profiles for bridge superstructures and compared the results with AASHTO model [9]. Hagedorn et al. built an I-beam segment and determined vertical and transverse temperature gradients with temperature-monitoring data [10]. Rodriguez et al. implemented a dense array of thermocouples in a box-girder bridge in California and established a Finite Element model to predict bridge internal stresses under the measured temperature variations [11]. In the above-mentioned literature, either meteorological data from the nearest weather station or empirical equations were utilized to develop the temperature distribution resulting from solar radiation. However, oftentimes, the nearest weather station is still far from the bridge, so the meteorological data obtained could still deviate from the local climate conditions. Empirical equations were developed based on the given temperature data, hence they may not be applicable for a different on-site situation. Some studies used pyranometers to measure the intensity of solar radiation and thermocouples to measure the temperature distributions [5–7]. However, only the radiation intensity on the top flanges was investigated, while the effect of the flange shadow casting on the web was not discussed. In reality most box-girder bridges have tilted webs instead of vertical ones. Hence, to study the temperature distributions in box-girder bridges, it is essential to account for the tilt angle of the web and the overhang shadow casting on the web when estimating the solar radiation intensity. In this study, we instrumented a prestressed concrete box-girder bridge with pyranometers, anemometers, strain gauges, displacement gauges, and temperature sensors on the top and bottom slabs and webs to measure the solar radiation, wind speeds, strain, displacement, and surface temperatures, respectively. The aforementioned data have been collected continuously for over a year and exploited to establish the solar radiation and temperature gradient relationship. In addition, a three-dimensional finite element model was developed to simulate the temperature field in the worst scenario and analyze the corresponding stress distribution based on the real time radiation monitoring data. The novelty of the work lies in the fact that all the boundary conditions have been obtained from real-time monitoring of solar radiation, temperature, and wind speed on an in-service bridge. From the data analysis, the lateral temperature gradient effect has been studied, which has long been ignored in the bridge design practice worldwide. Finally, the finite element analysis has been performed in conjunct with the maximum lateral temperature gradient from the monitoring data. The result indicated that the tensile stresses generated by the lateral temperature gradient alone are so significant that it cannot be ignored in design.

2. Experimental Instrumentation and Data Acquisition The bridge was built in 2002. It has ﬁve continuous spans and each span is 25 m in length. All the sensors were mounted on the fourth span from the south on upstream side. Three optical pyranometers (JMFS-1001) were installed on the top slab, upper corner on the web, and lower position on the web, SensorsSensors2020 2020, 20,, 526120, x FOR PEER REVIEW 3 of3 15of 15 Sensors 2020, 20, x FOR PEER REVIEW 3 of 15 Sensorsposition 2020, 20 on, x FOR the PEERweb, REVIEW respectively, for each bridge (Figure 1). The optical pyranometers have3 spectralof 15 position on the web, respectively, for each bridge (Figure 1). The optical pyranometers have spectral respectively,range of 0.3~3 for each μm bridgeand can (Figure detect1). the The direct optical radiation pyranometers from the have sun spectral and the range reflected of 0.3~3 radiationµm and from positionrange of on0.3~3 the μ web,m and respectively, can detect thefor eachdirect bridge radiation (Figure from 1). the The sun optical and thepyranometers reflected radiation have spectral from canthe detect other the objectives direct radiation as well from as the the radiation sun and from the reﬂected the incident radiation sunlight from with the otheran angle. objectives 58 temperature as well rangethe other of 0.3~3objectives μm and as well can asdetect the radiationthe direct from radiation the incident from the sunlight sun and with the an reflected angle. 58radiation temperature from as thesensors radiation were embedded from the incident inside or sunlight on the surface with an of angle. the box 58 girder temperature in the midspan sensors cross were section embedded (Figure sensorsthe other were objectives embedded as well inside as the or onradiation the surface from of the the incident box girder sunlight in the with midspan an angle. cross 58 section temperature (Figure inside2). Two or on anemometers the surface of for the wind box girder speed in monitoring the midspan were cross installed section on (Figure the outer2). Two side anemometers and at the bottom for sensors2). Two wereanemometers embedded for inside wind or speed on the monitoring surface of werethe box installed girder inon the the midspan outer side cross and section at the (Figurebottom windof the speed bridge, monitoring as shown were in Figure installed 3. on the outer side and at the bottom of the bridge, as shown in of2).Figure theTwo bridge,3 anemometers. as shown for in wind Figure speed 3. monitoring were installed on the outer side and at the bottom of the bridge, as shown in Figure 3.

Figure 1. Radiation sensor (optical pyranometer) positions at the midspan. Figure 1. Radiation sensor (optical pyranometer) positions at the midspan. Figure 1. Radiation sensor (optical pyranometer) positions at the midspan. Figure 1. Radiation sensor (optical pyranometer) positions at the midspan.

FigureFigure 2. Temperature 2. Temperature sensor sensor layout layout at the at the midspan. midspan. Figure 2. Temperature sensor layout at the midspan. Figure 2. Temperature sensor layout at the midspan.

Figure 3. Anemometers for wind speed monitoring positions. Figure 3. Anemometers for wind speed monitoring positions. Figure 3. Anemometers for wind speed monitoring positions. A comprehensive monitoring system was integrated with an all sealed chassis (JMBV-1164), A comprehensiveFigure monitoring 3. Anemometers system for was wind integrated speed monitoring with an positions. all sealed chassis (JMBV-1164), an an integratedA comprehensive data acquisition monitoring module system (JMZX-32A), was integrated a temperature with an all acquisition sealed chassis module (JMBV-1164), (JMWT-64RT), an integrated data acquisition module (JMZX-32A), a temperature acquisition module (JMWT-64RT), integratedandA a DTUcomprehensive data mobile acquisition internet monitoring modulemodule system (JMTX-2017).(JMZX-32A), was integrated Thea temperature system with enables an acquisition all sealed 24/7 automatic chassis module (JMBV-1164), continuous(JMWT-64RT), data an and a DTU mobile internet module (JMTX-2017). The system enables 24/7 automatic continuous data integratedandacquisition a DTU datamobile through acquisition internet wire or module module wireless (JMTX-2017). (JMZX-32A), transmission. Th a etemperature The system data enables used acquisition in 24/7 this automatic study module were continuous (JMWT-64RT), collected data from acquisition through wire or wireless transmission. The data used in this study were collected from andacquisitionMay a 2019DTU tomobilethrough May 2020internet wire at or a module sampling wireless (JMTX-2017). transmission. rate of 1 time Th perThee system hour. data enablesused in 24/7this automaticstudy were continuous collected fromdata May 2019 to May 2020 at a sampling rate of 1 time per hour. acquisitionMay 2019 to through May 2020 wire at aor sampling wireless ratetransmission. of 1 time per The hour. data used in this study were collected from May3. 2019 Solar to Radiation May 2020 andat a Heatsampling Transfer rate ofTheory 1 time per hour. 3. Solar Radiation and Heat Transfer Theory 3. Solar Radiation and Heat Transfer Theory

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3. Solar Radiation and Heat Transfer Theory The Fourier heat transfer diﬀerential equation was employed to describe the heat conduction in the concrete bridge [12–15]:

∂T ∂ ∂T ∂ ∂T ∂ ∂T ρCp = (k ) + (k ) + (k ) (1) ∂t ∂x ∂x ∂y ∂y ∂z ∂z

3 where ρ, Cp, and k are density in kg/m , speciﬁc heat in J/kg ◦C, and thermal conductivity of concrete W/m ◦C, respectively. T refers to the temperature at any point in the girder at any time t. The boundary condition on the surfaces of the box girder can be written as [12–15]:

∂T ∂T ∂T k nx + k ny + k nz + qc + qs + qre = 0 (2) ∂x ∂y ∂z where nx, ny, and nz are the direction cosines of the normal vectors; qc, qs, and qre are the convection, 2 the total solar radiation and the long-wave radiation in W/m . The solar radiation qs consists of three components including direct solar radiation qdr, solar scattered radiation qsr, and ground reﬂected radiation qgr, as shown in Equation (3).

qs = qdr + qsr + qgr (3)

3.1. Direct Solar Radiation

Direct solar radiation comes straight from the Sun. The direct solar radiation energy Im can be expressed in the form of Equation (4) [13]:

sin h I = I (4) m 0 1 p sin h + −p where h is solar altitude angle; p is atmospheric transparency coeﬃcient, and I0 is solar constant. Solar altitude angle h is generally written as Equation (5) [16].

sin(h) = cos ϕ cos δ cos τ + sin ϕ sin δ (5) where φ is the latitude of the location; δ is solar declination, and δ = 23.45 sin [360 (284 + N)/365] in which N is the day of the year; τ is the solar hour angle, which is zero at the noon, negative before noon, and positive afternoon converting each hour to 15◦. The direct solar radiation projected on the box-girder web qdr in Equation (3) is calculated as:

qdr = Im cos θ (6) where θ is the angle between the Sun rays and the normal line to the web surface and cosθ = cosβ sinh + sinβ cosh cos(γz γ) in which β is the angle between web and the horizontal direction, γ is the − surface azimuth angle, and γz is the azimuth angle of the sun. In a box-girder bridge, the overhang casts shadow on the web and the solar radiation in the shadow should be deducted from the total. The equation of shadow length is deﬁned by Equation (7) [16]:

tan h ls = lc (7) sin(90 + γ γz) sin β cos β tan h − − where lc is the length of the overhang. Sensors 2020, 20, 5261 5 of 15

3.2. Scattered Solar Radiation Solar heat scatters on the structures in all directions. Jain assumed that the scattering intensity of the horizontal plane received is linearly related with the transmission coeﬃcient and the averaged radiation. Scattering radiation Id in the atmosphere can be obtained with Equation (8) [17]:

I = (1 1.13 k )Im (8) Sensors 2020, 20, x FOR PEER REVIEW d − T 5 of 15 where kT is the transmission coeﬃcient accounting for the attenuation of solar radiation in the Id = (1 − 1.13 kT) Im (8) atmosphere, which takes a value in between 0.3~0.8 for latitudes of 30~40◦. whereThe scattered kT is thesolar transmission radiation coefficientqsr received accounting by an arbitrary for the wallattenuation is: of solar radiation in the atmosphere, which takes a value in between 0.3~0.8 for latitudes of 30~40°. The scattered solar radiation qsr receivedqsr = Ibyd ( 1an+ arbitrarycos β)/2 wall is: (9)

qsr = Id (1 + cosβ)/2 (9) 3.3. Ground Reﬂected Radiation

3.3.The Ground bottom Reflected slab of Radiation the box girder receives the short-length wave radiation reflected from the ground. The reflected solar radiation can be calculated by Equation (10) [18]: The bottom slab of the box girder receives the short-length wave radiation reflected from the ground. The reflected solar radiation can be calculated by Equation (10) [18]: qgr = ρ∗(Im + I )(1 cosβ)/2 (10) d − =+−ρβ* ()() qcosgrII m d 1/2 (10) where ρ* is the ground reflection coefficient and ρ* = 0.1 for ground reflection and ρ* = 0.2 for waterwhere reflection. ρ* is the ground reflection coefficient and ρ* = 0.1 for ground reflection and ρ* = 0.2 for water reflection. 4. Theoretical and Measured Solar Radiation Intensity 4. Theoretical and Measured Solar Radiation Intensity It is evident from the monitoring data shown in Figure4 that the solar radiation intensity on the top slabIt reached is evident maximum from the monitoring in summer data due shown to the in direct Figure solar 4 that exposure, the solar radiation while the intensity intensity on the on the top slab reached maximum in summer due to the direct solar exposure, while the intensity on the lower web reached maximum in winter. Solar radiation intensity on the lower web had maximum lower web reached maximum in winter. Solar radiation intensity on the lower web had maximum value in winter is because the top slab casts shadow on the web and the sun altitude angle gets smaller value in winter is because the top slab casts shadow on the web and the sun altitude angle gets smaller leadingleading to higher to higher solar solar radiation radiation in in winter. winter.

(a)

Figure 4. Cont.

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(b)

FigureFigure 4. 4.Solar Solar radiationradiation intensity on on different diﬀerent parts parts of of the the bridge bridge (sampling (sampling frequency: frequency: 1 time/h). 1 time /(h).a) (aSolar) Solar radiation radiation variations variations on on the the top top slab slab from from 29 29 June June 2019 2019 to to 15 15 January January 2020; 2020; (b)) solar radiation variationsvariations on on the the lower lower web web from from 29 29 June June 2019 2019 to to 15 15 January January 2020. 2020.

FromFrom the the aforementioned aforementioned solarsolar radiationradiation theory,theory, thethe theoreticaltheoretical solarsolar radiationradiation intensity was calculatedcalculated and and compared compared withwith thethe measured intensit intensity,y, as as shown shown in in Figure Figure 5.5 .During During the the day day of of31 31December December 2019, 2019 the, the measured measured solar solar radiation radiation intensities intensities on the on thetop topslab slab and andupper upper portion portion of the of web the webwere were smaller smaller than than the theoretical the theoretical ones, ones, while while the measured the measured solar solarradiation radiation intensities intensities on the onlower the lowerportion portion of the of web the were web close were to close the theoretical to the theoretical values. In values. addition, In addition, note that notein Figure that 5a,c in Figure there 5wasa,c therea sudden was a drawdown sudden drawdown around 2 aroundpm, which 2 pm, may which be attributed may be attributedto the temporary to the temporary cloudiness. cloudiness. The other Thereason other for reason the underestimation for the underestimation of the theoretica of the theoreticall equations equations may ascribe may ascribe to the to ignorance the ignorance of the of theatmosphere atmosphere counter counter radiation radiation and and earth earth surface surface radiation radiation at night. at night.

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(a) (b)

(c)

FigureFigure 5. 5. ComparisonComparison between between measured measured and and theoretical theoretical solar solar radiation radiation intensities. intensities. ( (aa)) Solar Solar radiation radiation onon the the top top slab; slab; ( (bb)) solar solar radiation radiation on on the the upper upper web; ( c) solar radiation on the lower web.

5. Solar Radiation and Temperature Gradient Relationship 5. Solar Radiation and Temperature Gradient Relationship TheThe box-girder box-girder surface surface temperature temperature increases increases ra rapidlypidly under under the the solar solar radiation radiation forming forming a a large large temperaturetemperature gradient. gradient. The The maximu maximumm daily daily temperature temperature gradients gradients were were obtained obtained from from maximum temperaturetemperature during during the the day day at at #12, #12, #13, #13, and and #14 #14 sens sensoror subtracted subtracted by the by lowest the lowest temperature temperature at #15, at #16,#15, #17, #16, #20, #17, #21, #20, #26, #21, #27, #26, #39, #27, #40, #39, #46, #40, #47, #46, #48, #47, #49, #48, #50, #49, and #50, #51. and It #51. was Itfound was foundthat the that daily the maximumdaily maximum lateral lateraltemperature temperature gradient gradient was correl was correlatedated with daily with dailymaximum maximum solar solarradiation. radiation. The PearsonThe Pearson correlation correlation coefficient coeffi cientfor each for location each location on the on box the girder box girder was calculated, was calculated, as shown as shown in Table in 1.Table 1. Table 1. Pearson correlation coefficients between daily maximum temperature gradient and daily Table 1. Pearson correlation coefficients between daily maximum temperature gradient and daily maximum solar radiation. maximum solar radiation. Item Daily Maximum Solar Radiation Item Daily Maximum Solar Radiation Position Top slab (s1) Upper web (s2) Lower web (s3) Position Top slab (s1) Upper web (s2) Lower web (s3) PearsonPearson correlation correlation coefficients coefficients 0.124 0.124 0.276 0.276 0.879 0.879

ItIt isis indicatedindicated in in Table Table1 that 1 thethat correlation the correlation between between maximum maximum daily positive daily lateral positive temperature lateral temperaturegradient and gradient maximum and daily maximum solar radiationdaily solar on radiation the lower on web the islower the highest.web is the In highest. Figure6 ,In a Figure linear 6,regression a linear regression was utilized was to utilized best ﬁt theto best data fit described the data indescribed Equation in (11).Equation (11). 𝑇 = 𝐼 + (11) + max-la = 0.01377 max-lw +0.4838 T max-la 0.01377Imax-lw 0.4838 (11) where 𝑇 max-la is the maximum daily positive lateral temperature gradient in °C and 𝐼max-lw is the T+ I maximumwhere max-la dailyis solar the maximum radiation dailyintensity positive on the lateral lower temperature web. The standard gradient deviation in ◦C and ofmax-lw 𝑇 max-lais d the is T+ equalmaximum to 2.196 daily °C. solar The dashed radiation line intensity in Figure on 6 the represents lower web. the The𝑇 max-la standard+2𝑑, and deviation it is just of abovemax-la all dthe is + + temperatureequal to 2.196 gradient◦C. The data. dashed Theline dashed in Figure line is6 therepresents envelope the forT themax-la temperature2d, and gradient it is just data above and all can the be used for maximum daily positive lateral temperature gradient prediction through the given maximum daily solar radiation intensity on the lower web (Equation (12)).

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temperature gradient data. The dashed line is the envelope for the temperature gradient data and can be used for maximum daily positive lateral temperature gradient prediction through the given maximum daily solar radiation intensity on the lower web (Equation (12)). Sensors 2020, 20, x FOR PEER REVIEW 8 of 15 Sensors 2020, 20, x FOR PEER REVIEW 8 of 15 + + = + T max-la 2d 0.01377Imax-lw 4.8752 (12) 𝑇 max-la +2𝑑=0.01377𝐼max-lw + 4.8752 (12) 𝑇 max-la +2𝑑=0.01377𝐼max-lw + 4.8752 (12)

18.0 18.0

16.0 16.0

14.0 ） 14.0 ） ℃ ℃ （ 12.0 （ 12.0

10.0 10.0

temperature 8.0 temperature 8.0

Lateral maximum daily gradient gradient daily maximum Lateral 6.0

4.0 4.0

2.0 2.0

0.0 0.0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Maximum daily radiation intensity of lower web（w/m2) Maximum daily radiation intensity of lower web（w/m2) FigureFigure 6. 6. RelationRelation between between maximum maximum daily daily positive positive lateral lateral temperature temperature gradient gradient and and maximum maximum daily daily Figure 6. Relation between maximum daily positive lateral temperature gradient and maximum daily solarsolar radiation radiation intensity intensity on on the the lower lower web. web. solar radiation intensity on the lower web. 6.6. Simulation Simulation of of Temperature Temperature Field Field with with Finite Finite Element Element Method Method 6. Simulation of Temperature Field with Finite Element Method 6.1. Finite Element Model 6.1. Finite Element Model 6.1. Finite Element Model A five-span continuous bridge model was created with ANSYS software. Since the bridge is A five-span continuous bridge model was created with ANSYS software. Since the bridge is symmetricA five-span in centerline, continuous half of bridge the five model spans (2.5was spans) created were with modeled ANSYS in ANSYS.software. The Since three-dimensional the bridge is symmetric in centerline, half of the five spans (2.5 spans) were modeled in ANSYS. The three- symmetricthermal element in centerline, Solid 90 washalf selectedof the five for thespans structural (2.5 spans) analysis. were The modeled 3D model in ANSYS. has 49,848 The elements three- dimensional thermal element Solid 90 was selected for the structural analysis. The 3D model has dimensionaland 205,400 nodes,thermal as element shown in Solid Figure 907 .was selected for the structural analysis. The 3D model has 49,848 elements and 205,400 nodes, as shown in Figure 7. 49,848 elements and 205,400 nodes, as shown in Figure 7.

Figure 7. The 3D Finite Element model of the PC box girder. Figure 7. The 3D Finite Element model of the PC box girder. Figure 7. The 3D Finite Element model of the PC box girder. 6.2. Parameters Used in Finite Element Modeling 6.2. Parameters Used in Finite Element Modeling 6.2.1. The Thermo-Physical Properties of the Box Girder 6.2.1. The Thermo-Physical Properties of the Box Girder Note that the asphalt pavement was built into the FE model together with the PC box girder. Note that the asphalt pavement was built into the FE model together with the PC box girder. The thermo-physical properties for both of the asphalt and concrete are listed in Table 2. The thermo-physical properties for both of the asphalt and concrete are listed in Table 2.

Sensors 2020, 20, x FOR PEER REVIEW 9 of 15 Sensors 2020, 20, 5261 9 of 15 Table 2. The thermo-physical properties of the box girder and its asphalt pavement.

6.2. Parameters Used inDensity Finite Element Specific Modeling Capacity Heat Thermal Conductivity Absorption Material (kg/m3) (J/kg °C) (W/m °C) Rate 6.2.1. TheConcrete Thermo-Physical 2500 Properties of 880 the Box Girder 2.5 0.4 Asphalt concrete 1700 1000 1.403 0.8 Note that the asphalt pavement was built into the FE model together with the PC box girder. The thermo-physical properties for both of the asphalt and concrete are listed in Table2. 6.2.2. Boundary Conditions It was Tableobserved 2. The during thermo-physical the one-year properties long of the monitoring, box girder and maximum its asphalt pavement.daily positive lateral temperature gradient varied with solar altitude angle h. In summer, solar altitude angle h is larger Speciﬁc Capacity Heat Thermal Conductivity Absorption than inMaterial winter, so thatDensity the solar (kg radiation/m3) duration in the day is shorter leading to a smaller maximum (J/kg ◦C) (W/m ◦C) Rate daily positive lateral temperature gradient. It is evident in Figure 8 that December 31st witnessed the Concrete 2500 880 2.5 0.4 largestAsphalt maximum concrete daily positive 1700 lateral temperature 1000 gradient in the year. 1.403 0.8 Taking the monitoring data of the wind speed, solar radiation, and temperature within and out of the box girder on 30–31 December 2019 (Figure 9), it was observed that the ambient temperature 6.2.2. Boundary Conditions on the top of the box girder and radiation intensity almost doubled on December 31 compared with that ofIt wasDecember observed 30. during As the theambient one-year temperature long monitoring, increased maximum on December daily positive31, the outer lateral surface temperature of the gradientweb had varied been exposed with solar to altitude solar radiation angle h. Inlonger summer, in winter solaraltitude than the angle other h istime larger during than inthe winter, year. soMeanwhile, that the solar the radiationtemperature duration within in the the box day girder is shorter had leading less fluctuation to a smaller and maximum stayed low. daily Hence, positive the lateralcombining temperature effects of gradient.the ambient It temperature is evident in and Figure solar8 radiation that December increase, 31st and witnessed the not much the changed largest maximuminbox temperature, daily positive the maximum lateral temperature temperature gradient gradie innt thein the year. year, was generated in those days.

16.0 Transverse maximum daily gradient temperature 14.0

12.0 ）

℃ 10.0 （ 8.0

6.0

Temperature 4.0

2.0

0.0 2019/5/1 2019/7/20 2019/10/8 2019/12/27 2020/3/16 2020/6/4 Time (days)

Figure 8.8. Maximum daily positive lateral temperature gradient variations.variations.

Taking the monitoring data of the wind speed, solar radiation, and temperature within and out of the box girder on 30–31 December 2019 (Figure9), it was observed that the ambient temperature on the top of the box girder and radiation intensity almost doubled on December 31 compared with that of December 30. As the ambient temperature increased on December 31, the outer surface of the web had been exposed to solar radiation longer in winter than the other time during the year. Meanwhile, the temperature within the box girder had less ﬂuctuation and stayed low. Hence, the combining eﬀects of the ambient temperature and solar radiation increase, and the not much changed inbox temperature, the maximum temperature gradient in the year, was generated in those days.

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20.0 Average temperature in the box girder

15.0 Ambient temperature under the box girder Ambient temperature on the box girder ）

℃ 10.0 （ 5.0

0.0 Temperature -5.0

-10.0 12/30/2019 12/31/2019 01/01/2020 Time (hours) (a)

800 Radiation on the top 700 Upper web radiation 600 lower web radiation 500

400

300

200 Radiation intensity (w/m2) intensity Radiation 100

0 12/30/2019 12/31/2019 01/01/2020 Time (hours) (b)

6 wind speed on the top 5 wind speed on the bottom

） 4

2 Wind speed(m/s Wind

0 12/30/2019 12/31/2019 01/01/2020 Time (hours) (c)

FigureFigure 9. 9.Variations Variations on on the the boundaries boundari ofes the of boxthe girder.box girder. (a) Temperature (a) Temperature variations variations within andwithin outside and ofoutside the box of girder; the box (b girder;) solar ( radiationb) solar radiation intensity intensity variations variations on the top on andthe top web and surface web ofsurface the box of the girder box ongirder 30–31 on December 30–31 December 2019; ( c2019;) wind (c)speed wind speed variations variations on the on top the and top web and surfaceweb surface of the of box the girderbox girder on 30–31on 30–31 December December 2019. 2019.

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6.2.3. Comparison between the FE Analysis and Experimental Results 6.2.3. Comparison between the FE Analysis and Experimental Results Figure 10 gives the comparison between the measured temperatures on 31 December 2019 and the simulatedFigure 10 temperatures gives the comparison in FE analysis between for nine the measured temperature temperatures test points. on The 31 nine December test points 2019 were and groupedthe simulated into three temperatures with each in consisting FE analysis one for point nine temperatureon the top slab, test points.one on Thethe web, nine testand pointsone on were the bottomgrouped slab. into It three can withbe seen each from consisting the comparison one point onin theFigure top slab,10 that one the on differences the web, and between one on the measuredbottom slab. and It simulated can be seen temperatures from the comparison are less than in Figure 2 °C 10except that thethose diff onerences the top between slab. The the maximum measured measuredand simulated and temperaturessimulated temperature are less than difference 2 ◦C except on the those top on slab the reached top slab. 4.0 The °C. maximum Overall, the measured 3D FE modeland simulated can be considered temperature suitable difference for onsimulation the top slab of thermal reached conductivity, 4.0 ◦C. Overall, heat the convection, 3D FE model ambient can be temperatures,considered suitable and solar for simulationradiation in of the thermal box girder conductivity, [6,8,11,14]. heat The convection, difference between ambient temperatures,the measured and solarFE simulated radiation temperature in the box girder values [6, 8may,11,14 be]. attribut The diffederence to the between assumptions the measured made in and FE FEanalysis simulated that thetemperature heat thermal values conductivity may be attributed and the absorption to the assumptions rate are uniform made inin FEthe analysis structure. that However, the heat in thermal really, thoseconductivity parameters and the could absorption not berate evenly are uniformdistribute ind the in structure. heterogeneous However, materials in really, like those concrete parameters and asphalt.could not be evenly distributed in heterogeneous materials like concrete and asphalt.

16.0 P1-EXP 14.0 P1-FE 12.0 P12-EXP 10.0 P12-FE ） P31-EXP ℃ 8.0 （ 6.0 4.0 2.0

Temperture 0.0 -2.0 -4.0 12/31/2019 12/31/2019 01/01/2020 Time（hours） (a)

16.0 P2-EXP 14.0 P2-FE 12.0 P13-EXP 10.0 P13-FE ） P30-EXP ℃ 8.0 （ 6.0 4.0 2.0

Temperture 0.0 -2.0 -4.0 12/31/2019 12/31/2019 01/01/2020 Time（hours） (b)

Figure 10. Cont.

Sensors 2020, 20, x FOR PEER REVIEW 12 of 15

16.0 Sensors 2020, 20, 5261 P3-EXP 12 of 15 Sensors 2020, 20, x14.0 FOR PEER REVIEWP3-FE 12 of 15 12.0 P14-EXP 10.0 P14-FE ） 16.0 P29-EXP ℃ 8.0 P3-EXP

（ 14.0 6.0 P3-FE 12.0 P14-EXP 4.0 10.0 P14-FE ） 2.0 P29-EXP ℃ 8.0 Temperture

（ 0.0 6.0 -2.0 4.0 -4.0 2.0 12/31/2019 12/31/2019 01/01/2020 Temperture 0.0 Time（hours） -2.0 (c) -4.0 Figure 10. Measured12/31/2019 and simulated temperature 12/31/2019 values comparisons on 31 December 01/01/2020 2019. (a) Time（hours） Measured and simulated temperature values at the test point 1, 12, and 31 on 31 December 2019; (b) Measured and simulated temperature values at the test point 2, 13, and 30 on 31 December 2019; (c) (c) Measured and simulated temperature values at the test point 3, 14, and 29 on 31 December 2019. Figure 10.10.Measured Measured and and simulated simulated temperature temperature values valu comparisonses comparisons on 31 December on 31 December 2019. (a) Measured2019. (a) 6.2.4. StressesandMeasured simulated Caused and temperaturesimulated by the Spatial temperature values Temperature at the values test pointat Gradientsthe 1,test 12, point and in 31 1,FE 12, on Analysis 31and December 31 on 31 December 2019; (b) Measured 2019; (b) and simulated temperature values at the test point 2, 13, and 30 on 31 December 2019; (c) Measured SolarMeasured radiation and simulatedcauses differential temperature temperature values at the distributions test point 2, 13, that and result 30 on 31in Decembera vertical 2019; (y axis (c) in and simulated temperature values at the test point 3, 14, and 29 on 31 December 2019. Figure Measured7) and lateral and simulated (x axis in temperature Figure 7) valuestemperature at the test gradient point 3, 14,in theand box29 on girder. 31 December In bridge 2019. design practice,6.2.4. Stresses only the Caused vertical by temperature the Spatial Temperature gradient is Gradientsconsidered in to FE be Analysis attributed to significant stress 6.2.4. Stresses Caused by the Spatial Temperature Gradients in FE Analysis changes, and the lateral temperature gradient effect is generally ignored. In this paper, the lateral Solar radiation causes differential temperature distributions that result in a vertical (y axis in temperatureSolar radiationgradient causesin the differentialx axis direction temperature is computed. distributions Then, thatthe maximumresult in a verticalpositive (ylateral axis in Figure7) and lateral ( x axis in Figure7) temperature gradient in the box girder. In bridge design practice, temperatureFigure 7) and gradient lateral was (x axis input in inFigure the FE7) temperaturemodel, and thegradient corresponding in the box normal girder. stressIn bridge in z designaxis only the vertical temperature gradient is considered to be attributed to significant stress changes, directionpractice, and only the the transverse vertical stresstemperature in y axis gradient direct have is considered been calculated, to be attributas showned into Figures significant 11–13. stress and the lateral temperature gradient effect is generally ignored. In this paper, the lateral temperature changes,Figure and11 shows the lateral the temperature temperature distribution gradient effect at the is time generally of 15:00 ignored. on 31 December In this paper, 2019 when the lateral the gradient in the x axis direction is computed. Then, the maximum positive lateral temperature gradient lateraltemperature temperature gradient gradient in the reached x axis maximum. direction Itis is computed. clear that theTh temperatureen, the maximum on the positivetop and rightlateral was input in the FE model, and the corresponding normal stress in z axis direction and the transverse webtemperature surface reached gradient maximum was input 21 °Cin, thewhile FE the model, temperatures and the oncorresponding the bottom surfacenormal and stress inside in z the axis stress in y axis direct have been calculated, as shown in Figures 11–13. boxdirection remain andrelatively the transverse low. stress in y axis direct have been calculated, as shown in Figures 11–13. Figure 11 shows the temperature distribution at the time of 15:00 on 31 December 2019 when the lateral temperature gradient reached maximum. It is clear that the temperature on the top and right web surface reached maximum 21 °C, while the temperatures on the bottom surface and inside the box remain relatively low.

FigureFigure 11. 11. SimulatedSimulated temperature temperature distribution distribution in in the the box girder atat 15:0015:00 on on 31 31 December December 2019 2019 (unit: (unit:◦C). °C).

Figure 11. Simulated temperature distribution in the box girder at 15:00 on 31 December 2019 (unit: °C).

Sensors 2020, 20, x FOR PEER REVIEW 13 of 15 Sensors 2020, 20, x FOR PEER REVIEW 13 of 15 The positive lateral temperature gradient shown in Figure 11 was exerted onto the FE model for structuralThe analysis positive and lateral the temperaturelongitudinal gradient and transverse shown in stressFigure distributions11 was exerted due onto to the the FE lateral model for temperaturestructural gradient analysis were and obtained. the longitudinal Figure 12 and shows transverse that the stresslongitudinal distributions normal duestress to reached the lateral maximumtemperature 2.65 MPa gradient on the were left webobtained. in compression Figure 12 andshows 1.42 that MPa the in longitudinal tension on the normal other stresswebs andreached bottommaximum slabs. The 2.65 longitudinal MPa on the normalleft web tensile in compression stress caused and by 1.42 lateral MPa temperature in tension on gradient the other alone webs is and significantbottom andslabs. large The enoughlongitudinal to create normal cracks tensile in stressconcrete. caused The bytransverse lateral temperature normal stress gradient was alsoalone is calculated,significant as shown and large in Figure enough 13. Theto create bottom cracks edge ofin theconcrete. top slab The experienced transverse maximum normal stress transverse was also tensilecalculated, stress 1.21 as shown MPa, inwhile Figure the 13. all The the bottom webs andedge bottom of the top slabs slab were experienced in compression, maximum and transverse the maximumtensile compressivestress 1.21 MPa, stress whilewas 0.69 the MPa. all the It is webs indicated and thatbottom the maximum slabs were transverse in compression, tensile stress and the is comparablemaximum compressiveto the longitudinal stress was one. 0.69 Hence, MPa. Itth ise indilateralcated temperature that the maximum gradient transverse effects cannot tensile be stress ignoredisSensors comparable in 2020structural, 20, 5261 to design. the longitudinal one. Hence, the lateral temperature gradient effects cannot13 ofbe 15 ignored in structural design.

Figure 12. Longitudinal normal stress distribution under maximum positive lateral temperature Figure 12. Longitudinal normal stress distribution under maximum positive lateral temperature gradientFigure (unit: 12. Pa). Longitudinal normal stress distribution under maximum positive lateral temperature gradient (unit: Pa). gradient (unit: Pa).

Figure 13. Transverse normal stress distribution under maximum positive lateral temperature gradient Figure 13. Transverse normal stress distribution under maximum positive lateral temperature (unit: Pa). gradientFigure (unit: 13. Pa). Transverse normal stress distribution under maximum positive lateral temperature gradientFigure 11 (unit: shows Pa). the temperature distribution at the time of 15:00 on 31 December 2019 when the 7. Conclusionslateral temperature gradient reached maximum. It is clear that the temperature on the top and right 7. Conclusions webThe surfaceauthors reached performed maximum an experimental 21 ◦C, while study the temperatureson temperature on effects the bottom caused surface by solar and insideradiation the box on aremain 5-spanThe relatively authorscontinuous performed low. PC box-girder an experimental bridge. Sensorsstudy on measuring temperature solar effects radiation, caused temperatures,by solar radiation strain,on winda The5-span speed positive continuous and lateral displacements temperaturePC box-girder were gradient bridge.installed shownSensors at various in measuring Figure cross 11 was sections.solar exerted radiation, The onto continuously temperatures, the FE model acquiredstrain,for structural data wind from speed analysis May and 2019 anddisplacements to the May longitudinal 2020 were was utilized andinstal transverseled to at determine various stress cross the distributions dailysections. lateral The due maximum continuously to the lateral positiveacquiredtemperature temperature data gradient from gradient May were 2019for obtained. the to PCMay box-girder Figure2020 was 12 bridge. showsutilized It that wasto thedetermine found longitudinal from the the daily normaldata lateral analysis stress maximum that reached the positivePearson’smaximum temperature correlation 2.65 MPa gradient oncoefficient the left for webthebetween PC in box-girder compression the daily bridge. maximum and 1.42It was MPa foundpositive in tensionfrom lateral the on datatemperature the analysis other webs that gradienttheand Pearson’son bottom the lower slabs. correlation web The and longitudinal thecoefficient daily maximum normalbetween tensilelateral the dailysolar stress radiationmaximum caused byintensity positive lateral reached temperature lateral maximum temperature gradient valuegradientalone of 0.879. is significanton Hence, the lower the and web solar large and radiation enough the daily on to maximum createthe we cracksb can lateral be in concrete.consideredsolar radiation The as transversethe intensity key factor reached normal that maximum stresscause was the dailyvaluealso calculated,maximum of 0.879. Hence, positive as shown the lateral solar in Figure temperatureradiation 13. on The gradient.the bottom web can Then, edge be ofconsidereda prediction the top slabas equation the experienced key forfactor the maximumthat daily cause maximumthetransverse daily positive maximum tensile lateral stress positive 1.21temperature MPa, lateral while temperature gradient the all thewas gradient. webs developed and Then, bottom using a slabsprediction solar were radiation inequation compression, intensityfor the and daily the measuredmaximummaximum on the compressivepositive lower web. lateral stressMeanwhile, temperature was 0.69 the MPa. compar gradient It isison indicated was made developed between that the maximum theusing simulated solar transverse radiation solar radiation tensile intensity stress intensitiesmeasuredis comparable and on the the tomeasured thelower longitudinal web. values Meanwhile, indicated one. Hence, the that compar the there lateral isonwere temperature made differences between gradient between the simulated e fftheects simulated cannot solar be radiation and ignored measuredintensitiesin structural values. and design. It the may measured be attributed values to indicated the fact that that the there uniform were differencesheat thermal between transfer the conductivity simulated and and measuredabsorption values. rate are It mayused be in attributedthe model. to On the the fact other that thehand, uniform a linear heat elastic thermal material transfer model conductivity was utilizedand7. Conclusions inabsorption the FE analysis rate are for used simplification, in the model. which On may the beother attributed hand, ato linear causing elastic the difference.material model In the was future,utilized anThe improved in authors the FE FE analysis performed model for accounting ansimplification, experimental for the which non-linear study may on temperature bematerial attributed properties, e toffects causing caused cracking, the bydifference. solarand stress radiation In the re-distributionfuture,on a 5-span an improved will continuous be developed FE PCmodel box-girder to accounting generate bridge. more for the Sensors realistic non-linear measuring stress material distribution solar properties, radiation, result. In temperatures,cracking, addition, and it isstress strain, suggestedre-distributionwind speedthat actual and will displacements monitoring be developed data were to be generateinstalled used to establish atmore various realistic the cross temperature stress sections. distribution Thefield continuously boundary result. In conditions addition, acquired datait is whensuggestedfrom possible. May that 2019 actual to May monitoring 2020 was utilized data be toused determine to establish the dailythe temperature lateral maximum field boundary positive temperature conditions whengradient possible. for the PC box-girder bridge. It was found from the data analysis that the Pearson’s correlation

coeﬃcient between the daily maximum positive lateral temperature gradient on the lower web and the

daily maximum lateral solar radiation intensity reached maximum value of 0.879. Hence, the solar radiation on the web can be considered as the key factor that cause the daily maximum positive lateral temperature gradient. Then, a prediction equation for the daily maximum positive lateral temperature gradient was developed using solar radiation intensity measured on the lower web. Meanwhile, the comparison made between the simulated solar radiation intensities and the measured values indicated that there were differences between the simulated and measured values. It may be attributed to the fact that the uniform heat thermal transfer conductivity and absorption rate are used in the model. On the other hand, a linear elastic material model was utilized in the FE analysis for simplification, which may be attributed to causing the difference. In the future, an improved FE model Sensors 2020, 20, 5261 14 of 15 accounting for the non-linear material properties, cracking, and stress re-distribution will be developed to generate more realistic stress distribution result. In addition, it is suggested that actual monitoring data be used to establish the temperature field boundary conditions when possible. It was discovered that the daily maximum positive lateral temperature gradient took place at 15:00 on 31 December 2019 over the one-year monitoring. Entering the obtained daily maximum positive lateral temperature gradient into the FE model, the maximum longitudinal tensile stresses obtained were 1.42 MPa on the inner and farther webs and bottom slabs, and the maximum transverse ones were 1.21 MPa at the bottom of the top slab. The study demonstrated that the positive lateral temperature gradient effects were so significant that they should be taken into account in structural design. Our future work will focus on developing an advanced three-dimensional solar radiation model accounting for modifications based on field testing data and comparing it with the vertical temperature gradient model in AASHTO bridge design specification.

Author Contributions: Conceptualization, X.L.; methodology, X.L.; formal analysis, X.F.; investigation, K.Z.; data curation, X.L. and X.F.; writing—original draft preparation, H.Z., H.J.; writing—review and editing, X.L.; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China (No. 51108152). Acknowledgments: We would like to acknowledge the reviewers who provided us constructive suggestions that helped us improve our work. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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