Nonlinear TemperatureDistributions in Bridgesat Different Locationsin the United States

Izak C. PotgieterDirectorBruinette Kruger Stoffberg Inc.Pretoria, South Africa

William L. GambleProfessor of Civil Engineering

University of IllinoisUrbana, Illinois

A ti in the case of earthquakeloadings or wind loadings which

vary significantly throughout the conti-nental United States, a large variation inthermal loadings can be expected.Structures in some parts of the countryare more likely to experience largethermal loadings due to specific mete-orological conditions. It is not necessaryto take the severe conditions experi-enced in one part of the country intoconsideration in bridge design foranother part of the country where totallydifferent environmental conditions mayexist.

The current AASHTO Code' dealsonly marginally with temperature ef-fects in bridge design. The onl y factors

addressed are overall expansion andcontraction and no adjustments aremade for variations in weather condi-tions through the country other than adistinction between "rnoderate'* and"cold" climates, The two foreign codesthat deal extensively with temperaturedistributions through bridge super-structures are the British BS 5400' andNew Zealand Ministry of Works Spec-ifications.' No variation of design con-ditions with location are suggested bythese two codes, since they are intendedas specific design guidelines for theBritish Isles and New Zealand, respec-tively. The research on which these

`See AASH fO Specification, Section 3.16.

80

codes were based indicates that verylittle variation in thermal loadings oc-curred with a change in location in therespective countries. Ref. 4 and its de-rivative Ref. 5 are the first attempts toinclude a temperature gradient intogeneral American bridge design prac-tice.

The large variations in meteorologicalconditions in the United States neces-sitate the quantification of extremethermal loadings at representative loca-tions throughout the country accordingto which acceptable design criteria forthe American bridge codes can be de-veloped. A heat flow model was de-veloped to analyze the possible influ-ence of weather conditions.

The main emphasis in this paper isthe magnitude and effect of nonlineartemperature distributions over thedepth of a member. Appendix A con-tains a brief discussion, with illustra-tions, of the effects of a particular tem-perature distribution on a specific rec-tangular member.

HEAT FLOWCOMPUTER MODEL

In order to quantify the magnitude oftemperature differences expected inbridge decks at extreme conditions, aone-dimensional computer heat flowmodel' was used. In the development ofthe heat flow analysis, extensive use wasmade of material previously developed.Much of this work had been done inNew Zealand and Australia, with workby Priestley et al., 72 Lanigan et al.,s•'oHunt and Cook," Thurston, 12 andHirst."

The purpose of this study was not thedevelopment of new and improved heatflow models hut rather the use of thesemodels to study the variability of ther-mal effects across the continentalUnited States.

The three general processes of heattransfer are conduction, convection and

SynopsisThe effects of nonlinear tempera-

ture distributions in bridge structuresresulting from solar radiation andother related weather parameterswere investigated. A finite differenceheat flow model was developed tosimulate transient heat flow in bridges.Reasonable agreement was foundbetween computed temperature dis-tributions and experimental mea-surements for the Kishwaukee RiverBridge at Rockford, Illinois.

Meteorological data from 26 SOL-MET stations spread across the conti-nental United States were used tocompute possible extreme tempera-ture distributions in bridges located atthose stations. Temperature differ-ences approaching 5TF (32`C), thevalue contained in the New Zealandspecifications, were obtained for abare concrete deck located in thedesert Southwest United States.Moderate response was found forcoastal areas. The temperature differ-ence cited is the differential betweenthe maximum temperature during theday, at the top of the deck, and theminimum temperature. at some sec-tion low in the cross section but gen-erally not at the bottom : at the sametime.

Simplified relationships of curva-ture-depth and residual stress-depthdue to nonlinear temperature distribu-tions in box girder bridges were de-veloped for application in bridge de-sign at different locations in the UnitedStates,

radiation. All three modes are involvedin the heat transfer process which takesplace in bridges. A finite difference,heat flow model was developed whichincorporates the heat transfer variablesand weather data for the determination

PCI JOURNAL/July-August 1989 81

tf 2 497m

. 4 5 6

IC7,

I

2

in, = 25.4 mmI H = 304.8 mm k 6306m

Fig. 1. Cross section of Kishwaukee River Bridge.

of vertical temperature distributions atvarious times in a typical bridge section.

Two- and three-dimensional heat flowmodels are also possible, and may beformulated using either finite differ-ences or finite elements. For this study,there seemed to he no advantage to thefinite element approach, and the finitedifference approach was simpler to im-plement. Ref. 14 describes a studywhich used a two-dimensional heat flowmodel in a study of concrete box girderbridges. This was necessary in Re£ 14because part of the interest was thetemperature gradients through the webthicknesses and in the effects of the ori-entation of the bridge relative to anorth-south axis. Many details werenecessary for these purposes.

The one-dimensional heat flow analy-sis was adequate for this study becausethe interest was in overall response ofbridge structures to a relatively largenumber of different climatic conditions.

FIELD MEASUREMENTSON THE KISHWAUKEE

RIVER BRIDGEIn order to verify the ability of the

analytical model to predict the temper-attire distribution in a bridge deck atvarious times, given the transientweather conditions, field measurementswere performed on the KishwaukeeRiver Bridge. 15• ' 6 This is a five spansegmentally constructed box girderbridge near Rockford, Illinois. It has endspans of 170 ft (51.8 in) and three inter-ior spans of 250 ft (76.2 in) with an over-all depth of!! ft 7^ in. (3.55 m). Fig. 1shows the cross section of the bridge.

Eight Carlson strain gauges were castinto the bridge at each of three differentsections of the bridge. In the webs ofeach of those sections a gauge was lo-cated near the top, 4 ft 8 in. (1.4 m)above the soffit and just above the soffit.Strain gauges were also placed aboveand below the cavity of the box girdersection. The deck was paved with 1.75in. (45 mm) of blacktop. Apart fromstrain readings for which those gaugeswere originally intended, temperaturereadings can also be made. Temperaturereadings as well as measurements of therelated weather parameters were takenfor a 56 hour period between July 8 and10, 1982.

The measurements were comparedwith the results of the computer heat

82

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TIME (HOURS)

Fig. 2. Comparison of measured and computed temperaturevariations in web of Kishwaukee River Bridge.

flow simulation. Fig. 2 shows a compari-son of measured and computed tem-perature changes at the top, middle andbottom of one of the webs, as explainedpreviously. The plotted points representmeasurements while the curves wereextracted from computed results. Simi-lar results were obtained for the othercross sections. The results were deemedof acceptable accuracy for the extensionof model simulation to other bridgeswhere hourly weather data were avail-able.

WEATHER DATACOLLECTION INCLUDING

SOLAR RADIATION

There are 89 weather stations in theUnited States where solar radiation in-tensities are recorded. Solar radiationhas been recorded on an hourly basis atonly 40 stations for varying periodsstarting in 1952." For all stations, how-ever, meteorological measurements re-lated to solar energy are available,which, in absence of radiation data, canbe used to estimate instantaneous vaI-

ues of radiation. Data on hours of sun-shine, or percentage of possible sun-shine, are also widely available.Methods for the use of these data to es-timate solar radiation can be foundelsewhere. 17,18

Typical meteorological year tapes wereprepared by the Sandia Laboratories inAlbuquerque for the years 1954 to1972. 1a Data were prepared for 26SOLMET stations and were processedas follows: Nine indices (total horizontalradiation; maximum and minimum, andmean of both dry bulb and dewpointtemperature, and the maximum andmean of wind speed) were identified ascritical. For the preparation of the tapes,the data were weighted with the solarindex as 50 percent and the rest of thefactors as 50 percent. Typical monthswere identified by their closeness tolong-term cumulative distribution func-tions.

The 26-station weather tape was usedfor this study to determine the magni-tude and distribution of temperaturesthat can be expected in typical concretebridge structures on days with extremeconditions at these locations. Each sta-

PCI JOURNAL/July-August 1989 83

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Fig. 3. Computed maximum temperature differences (°C) of 2 m (6.56 ft) deepbridge structures (unsurfaced) located at different SOLMET stations.

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Fig. 4. Computed maximum temperature differences (°C) of 2 m (6.56 ft) deepbridge structures, with 50 mm (2 in.) blacktop, located at different SOLMET stations.

84

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10rig. n. Computed maximum temperature differences (°C) of 2 m (6.56 ft) deepbridge structures, with 100 mm (4 in.) blacktop, located at different SOLMETstations.

tion is represented by 12 differentmonths in its meteorological history forwhich a total of 8760 hours of climaticdata are available. The list of names ofstations appears in Table 1 as well as inFigs. 3 to 5.

SELECTION OFEXTREME DAYS

June 21 is the longest day of the yearin the northern hemisphere and would,with absolutely clear sky conditions, ex-hibit the maximum total daily solar ra-diation of the year. Perfect clear skyconditions, however, are very rare inparts of the country, especially in someof the coastal regions. The conditionsexpected to produce maximum thermalloadings were therefore selected fromthe available data for the months of Juneand July, because those clearly containthe most favorable conditions for ex-

treme temperature differentials.The three important meteorological

parameters which govern the tempera-ture distribution in concrete bridgestructures are total solar radiation, airtemperature variation during the clayand the average wind speed. It is alsoimportant to know whether the threespecific weather parameters can exhibitextreme values simultaneously to pro-duce an absolute extreme temperaturedistribution. Emerson 2° determined thatextreme environmental conditions wereunlikely to occur on the same day inEngland.

The two most important weather fac-tors to produce extreme temperaturedistributions are very high radiation ac-companied by calm or very low windspeed. Solar radiation is the main sourceof heat energy to a bridge, and heat islost back to the surroundings by rera-diation and convection. The convectiveheat transfer coefficient is the smallest

PCI JOURNAL/Jufy-August 1989 85

Table 1. Number of days* when recorded weather data approachesthe maximum temperature difference for each station.

Extreme dayMaximum

temperature No. ofAverageStation and No. Radiation wind speed difference days

(k]/m2 1 (mis) (°C)

1 Fort Worth, TX 30943 4.8 24 52 Lake Charles, LA 28667 3.5 21 43 Columbia, MO 31021 3.0 25 54 Apalachicola, FL 29089 3.1 22 25 Miami, FL 25878 2.9 22 46 Brownsville, TX 28135 4.7 21 87 Charleston, SC 27661 2.8 24 28 Nashville, TN 29180 1.9 27 19 Dodge City, KS 34096 4.3 26 1

10 Caribou, ME 30794 1.8 23 211 Madison, W1 29367 1.6 24 112 El Paso, TX 33296 2.5 30 713 Albuquerque, NM 33818 1.8 30 114 Ely, NV 33226 3.7 30 415 Phoenix, AZ 32755 2.0 32 916 Santa Maria, CA 30924 3.2 26 317 Bismark, ND 30612 1.0 27 1IS Great Falls, M'1 31928 2.3 27 219 Medford, OR 31237 2.2 28 920 Seattle, WA 292.25 1.7 27 121 Fresno, CA 33500 2.8 31 922 Cape Hatteras, NC 29343 3.2 23 223 Sterling, VA 28378 1.0 27 124 Boston, MA 33270 5.1 22 125 New York, NY 28753 3.3 20 126 Omaha, NB 30048 2.1 26 4

''These days are arbitrarily detirnerl ms having a total solar radiation no less than97 percent of the extreme and an average wind speed of no more than 20 percent

more than for the corresponding extreme day.

under calm conditions and increasesrapidly with increased wind speed. Thedaily temperature variation has asmaller overall effect, presumably be-cause of the very small volumetric spe-cific heat of air relative to concrete. Forinstance, within the range of expectedtemperature variations of 14°F to 36°F(8°C to 20°C), the maximum temperaturedifference in a 6.56 ft (2 m) deep beamwas increased by 3.6° (2°C).

On the other hand, an increase ofwind speed from 0 to 11 mph (0 to 5mis), which is highly likely, would de-

crease the maximum temperature dif-ference in the same beam by about28.8°F (16°C). In order to determine thepossible coexistence of the extreme en-vironmental conditions on any givenday, correlation analyses were done onthe three important weather parameters.

Correlation AnalysisTable 2 shows the correlation coeffi-

cients for the different variables, As canbe expected, there is a relatively goodcorrelation between the total daily ra-

B6

Table 2. Summary of correlation coefficients between the threegoverning weather parameters for June and July at 26 weather stations.

Total radiation Total radiation Temperature variationVs Vs vs

temperature variation average wind speed average wind speed

Mean 0.65 0.03 -0.20Standarddeviation 0.13 0.2{) 0.21

diation and temperature variation. Thisindicated that large temperature varia-tions will accompany high daily radia-tion. Lower correlations were invariablyfound for coastal stations with a higherincidence of cloud cover. No correlationexists between average wind speed andeither of the other two parameters.

Climatical Data forThermal Analyses

Due to the complexity of the me-teorological system, it is not justified tosimpl y combine extreme climatical con-ditions in order to determine hypotheti-cal maximum temperature differencesthrough a concrete structure. Provisionfor such a hypothetical condition canlead to overdesign and unnecessarilyincrease construction costs. An attemptwas therefore made to determine realis-tic extreme conditions.

Data for two actual days recorded ateach weather station during June andJuly were used to calculate the corre-sponding temperature distributions.The first day represents the day with thehighest total solar radiation during thetypical meteorological year. The secondday was selected from those during thetypical year for which at least 95 percentof the radiation of the first day was re-corded. The main priority for the choiceof the second day was a combination ofvery low average wind speed and hightotal radiation. In making the selection,very little attention was given to tern-perature variation during the day.

Evaluation of MaximumRadiation Values

The method by Threlkeld' $ and Duf-fie and Beckman19 is generally used insolar engineering for the theoreticaldetermination of instantaneous beam aswell as diffuse solar radiation. In orderto evaluate the measured maximum ra-diation values, the solar radiation mea-surements at the different SOLMETstations were compared with the theo-retical clear sky distribution on June 21.On average, theoretical total values are2.5 percent less than those obtainedfrom the hourly data. A larger deviationwas obtained at coastal areas where ab-solutely clear sky conditions are rare.

When comparing the statistical aver-age daily radiation at the different sta-tions with the corresponding maximumreported values, it is evident that themeasured values are well above aver-age. The recorded radiation values aretherefore considered suitable to repre-sent the upper limit of radiation at therespective stations.

DATA FORTHERMAL ANALYSES

Thermal loading parameters that arespecifically relevant to the summer ex-tremes at the different SOLMET sta-tions were deduced from the availablemeteorological data. Also, the attenua-tion of the temperatures through thestructural concrete from the upper sur-face and specifically the influence of the

PCI JOURNALJuly-August 1989 87

Table 3. Thermal properties of concrete used inthermal analyses.

Property Concrete Bitumen Air

1.3Density (k}/m 3) 2480 2300Specific heat(J/kg°C) 922 840 1006Conductivity (Win °C) 1.384 0.744 ft 023Absorptivity 0.7 0.9

Surface Heat TransferHeat transfer on the top surface:h t = 13.5 3.88v (W1mt°C)where v = wind speed (mis)For the bottom exposed surface a value of 0.45 of the topsurface heat transfer was used. This is based on results oftests by Emerson.25For inside surfaces of flanges and partially protected lowersurfaces, a value of 0.2 of top surface heat transfer wasused.

thickness of existing bituminous sur-facing were determined. Finally, de-formation and stress criteria, which canbe used as guidance for the incorpora-tion of thermal loading into the bridgedesign process, were developed fromthe resulting analyses.

In the theoretical process of thermalanalysis, where only limited data areavailable, the values of parameters haveto be reliable. Assumptions made in thetheoretical analysis are discussed in thefollowing categories: climatical data,thermal properties of structural materialand starting conditions in the structure.

Climatical DataThe ambient conditions as measured at

the 26 SOLMET stations were used forthe determination of the variation inthermal loading throughout the UnitedStates. The time was recorded at all thestations according to the solar time sys-tem, which is a more standardized ap-proach than the regular time zone sys-tem. Wind speed was measured about33 ft (10 m) above ground. To allow forthe probable reduced exposure ofbridges at heights which would nor-mally be between fi and 26 ft (2 and 6 m)

above ground and possible partial pro-tection by the surroundings, the windspeed values used in the thermal analy-ses were reduced to 80 percent of thoserecorded. All the other weather datawere used exactly as recorded.

Thermal Properties ofStructural Material

A range of thermal properties is re-ported in the literature. Priestley2' per-formed a parametric study on the influ-ence of different thermal properties ofconcrete on the response of bridgestructures. The influence of changes inthese properties within the range of ex-perimentally determined values wasfound to be small. One exception, how-ever, is the absorptivity of the exposedupper surface. This factor will be dis-cussed in more detail later in this paper.The values for different thermal prop-erties and variables that were used inthe thermal analyses are presented inTable 3.

Starting Temperature DistributionA slightly higher temperature of 3 to

5°F' (2 to 3°C) normally exists at the

88

upper parts of the deck compared to thelower parts at the minimum temperaturedistribution during the night. For theseanalyses, however, a uniform startingtemperature distribution was assumed.The discrepancy between measured andassumed starting temperatures can bediminished by repeating the thermalanalysis for the same day, using the dis-tribution determined for the end of theinitial period as the starting distributionof the second analysis.

THERMAL ANALYSES FOR26 SOLMET STATIONS

The linear heat flow model describedearlier was used for a variety of thermalanalyses. In four series of analyses, con-crete structures of different depths wereanalyzed with plain concrete tippersurfaces, and with 2 in. or 4 in. (50 minor 100 mm) thick bitumen blacktops, re-spectively, under the specific weatherconditions at each of the 26 SOLMETstations. In the first series, the weatherdata of two different days (describedearlier) were used to compute the ten3-perature distribution through a 6.56 ft (2m) deep structure.

The day which gave rise to the largertemperature difference through thestrictures at each of the stations wasused for the analysis of structural con-crete of 3.28 ft (1.0 m) thickness in thesecond series, 1.64 ft (0.5 m) thickness inthe third series, and a structure with anair cavity in the fourth series. The thick-ness of the deck slab was 7.87 in. (0.200m), the hot-torn slab 5.91 in. (0.150 m)and the overall thickness was 3.28 ft (1.0m) in the last case.

RESULTS OFTHERMAL ANALYSES OF

SOLID STRUCTURESThe comparatively low conductivity

of concrete does not allow significant

heat penetration from the top to depthsof 3 ft (1 m) or snore. Minor temperatureincreases are also experienced in thebottom 8 in. (0.2 m) of the structure.

The geographical variation of themaximum temperature differencesthrough the 6.56 ft (2 m) structure withan exposed concrete top surface, 2 and 4in. (50 and 100 mm) blacktop, respec-tively, are presented in Figs. 3 to 5. Thethree sets of anal y ses show similarvariations of values with location andshow a decrease in temperature differ-ences with an increase in blacktopthickness.

Analysis of the geographical presen-tation of the results can Iead to the iden-tification of regions with similar thermalloadings. The stations representing thedesert and semidesert parts of the coun-try, e.g., western Texas, New Mexico,Arizona, Nevada and eastern California,exhibited the highest temperature dif-ferences between the top and bottomparts of the bridge structures. The val-ues obtained for structures with noblacktop in this region (Fig. 3) are be-tween 54 and 58°F (30 and 32°C). On theother hand, coastal weather parametersmoderate the differences along the coastline. The frequent occurrence of highwind speeds at Dodge City, Kansas,where the highest one day radiation ofall the SOLMET stations was recorded,moderated the calculated temperaturedifference of an unsurfaced bridgestructure at 47°F (26°C).

The climatical data for the stations inthe Midwest produced very similar re-sults with temperature differencesranging from 43 to 47°F (24 to 26°C).Nashville, 'Tennessee, with a maximumtemperature difference of 49°F (27°C),can also be included in this region.Thermal analyses for all the stationsalong the Gulf Coast, including Miami,yielded temperature differences of 38 to40°F (21 or 22°C). The recorded solar ra-diation at these stations were medium tolow. The fact that this part of the countryhas the lowest atmospheric clearness

PCI JOURNAL/July-August 1989 89

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--- Brownsville, TXPhoenix, AZ

------- New Zealand Code

0 5 10 15 20 25 30

35

TEMPERATURE (°C)

Fig. 6. Maximum temperature distributions in 2 m (6.56 ft) deepbridge structures (unsurfaced).

iE

2

.4

number supports the results. The con-stant Gulf Coast winds of between 4.5and 13 mph (2 and 6 mis) resulted in thereduction of maximum temperaturedifferences.

Table I indicates the number of daysin the typical year that would approachthe maximum calculated temperaturedifferences for each station. These dayswere arbitrarily defined as having a totalsolar radiation of at least 97 percent ofthe day which caused the maximumcondition together with an average windspeed of no more than 20 percent higherthan the average wind speed of the ref-erence day. For the vast majority of thestations, the extreme conditions werevery seldom approached during Juneand July.

Weather data for the stations atBrownsville, Texas; El Paso, Texas;Phoenix, Arizona; Medford, Oregon;and Fresno, California, show a probableincidence of 7 to 9 days during the 2month period with conditions favorable

for large temperature differences.Brownsville is the only station of thisgroup for which the computed maxi-mum temperature difference was lowerthan 50°F (28°C) for the concrete struc-ture with no blacktop. At the other sta-tions, extreme conditions were ap-proached between 1 and 5 days per year.Distress, in the form of unwantedcracking, is more likely to occur with re-peated high thermal loadings at thefirst-mentioned stations.

SENSITIVITY ANALYSESAs stated earlier, the important

weather parameters influencing tem-perature differences are radiation andwind speed, while the ambient temper-ature variation has a smaller influence.Two sets of sensitivity analyses wereperformed on a 6.56 ft (2 m) deep bridgeusing the weather data of Stations 4 and13. The results indicate that a 10 percentincrease in the total radiation would in-

90

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-------- 9rownsv i 1 1e, TXPhoenix, AZ

- - New Zealand Code

F1.4

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00 5 10 15 20 25

TEMPERATURE (°C)30 35

Fig. 7. Maximum temperature distributions in 2 m (6.56 ft) deepbridge structures, with 50 mm (2 in.) blacktop.

crease the maximum temperature dif-ference by about 5.4°F (3°C).

The total radiation received by thestructure is directly related to the ab-sorptivity of the top surface. A variationin this parameter of up to 20 percent canexist for the same structure dependingon the top surface. If the wind speed in-creases by 2.2 mph (1 m /s), the temper-ature difference is reduced by about2.7°F (1.5°C). Average wind speed dif-ferences of up to 6.7 mph (3 m/s) arequite common on days with very hightotal radiation. When different days withextreme conditions are compared, thetotal ambient temperature changes com-pare within 2 to 7°F (1 to 4°C) and therelated temperature differences in bridgestructures correspond within 2°F.

VERTICAL TEMPERATUREDISTRIBUTIONS

The wide variety of thermal condi-tions which exist throughout the United

States was reflected in the maximumtemperature difference results obtainedfor webs. The variation in temperaturedistributions under these conditions andespecially the bridge response to thesetemperature loadings are important inbridge analysis. In order to determinethe vertical temperature distribution ina typical web structure, the 6.56 ft (2 m)deep rectangular section described pre-viously was analyzed.

The SOLMET weather stations atBrownsville,'1'exas, and Phoenix, Arizona,represent the lower and upper limits ofthe computed maximum temperaturedifferences (Figs. 3 to 5). The tempera-ture distributions associated with theweather data of the other stations fallmainly within the bounds set by thesetwo stations. The temperature distribu-tions obtained from the weather data ofthese two stations are illustrated in Figs.6 to S. The graphs represent the thermalanalyses for the structures with an ex-posed concrete upper surface, 2 in. (50

PCI JOURNALIJuIy-August 1989 91

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Brownsville. TXPhoenix, AZNew Zealand Code

0 5 10 15 20 25 30 35

TEMPERATURE (°C)

Fig. 8. Maximum temperature distributions in 2 m (6.56 ft) deepbridge structures, with 100 mm (4 in.) blacktop.

mm) bitumen blacktop and 4 in. (100mm) blacktop, respectively, in compari-son with the design temperature dis-tribution as specified by the New Zea-land Ministry of Works. 3 This tempera-ture distribution was developed aftermodel tests as well as extensive analysesof various prototype bridges in NewZealand. (The small temperature dis-tribution function of the New Zealandcurve for the bottom of the section isomitted for clarity.)

AIthough the temperature distribu-tions for the two weather stations do notdiffer significantly, the horizontal sep-aration in the top 8 in. (0.2 m) of thestructure is between 7 and 18°F (4 and10°C) for the unpaved structures. Dif-ferences of this magnitude in tempera-ture distribution can cause substantialdifferences in bridge response.

The computed temperature distribu-tions have a notably larger gradient thanthe curve of the New Zealand Code.The maximum temperature distributionis a function of the blacktop thickness.

The blacktop serves as insulation and,due to thermal lag, the maximum teTn-perature distribution occurs at a laterstage in the concrete structure when it iscovered with a thicker pavement.Maximum temperature distributionswere generally recorded at 1400, 1530,and 1700 hours local solar time for theunsurfaced structure, 2 and 4 in. (50 and100 mm) blacktop structures, respec-tively.

The computed temperature distribu-tion for Phoenix, Arizona, closely re-sembles the corresponding curve of theNew Zealand Code for the structurewith 2 in. (50 mm) of blacktop. For theunsurfaced structure, the computedcurve is slightly below that of the NewZealand Code, while for the structurewith the 4 in. (100 mm) bitumen pave-ment, the computed temperatures areslightly above the Code values.

The bottom portion of the New Zea-land curve is not shown in these figuresbecause of the clutter it would create. Itappears in Fig. Al, showing a lower

92

surface temperature 2.7°F (1.5°C) higherthan the web temperature 8 in. (0.2 m)above the soffit. The computed soffittemperature differences, however,ranged between 3.6 and 5.4°F (2 and3°C) above the web temperatures. Theconstant temperature region in the webwas reached at about the same 8 in. (0.2m) height above the soffit, The temper-ature differentials for a bridge withstructural depth of 20 in. (0.5 m) do notreach the maximum values obtained forthe deeper structures and the computedsoffit temperatures for this structure arealso slightly lower.

INDUCED CURVATURESOF BRIDGE SECTIONS

Temperature loadings do not inducethe same response in all types ofbridges. The response is mainly depen-dent on the type of cross section of asuperstructure, the span configuration,the degree of fixity and the bridge ma-terial.

In this section, an attempt will bemade to generalize the response in typi-cal box girder cross sections dependingon their depths and to obtain a generalthermal curvature for the typical sec-tions depending on the specific thermalloading. A variety of bridge cross sec-tions have been analyzed elsewhere,'but only the general box girder crosssection is discussed here.

Constant depth box girder bridges areused for longer span bridges with spanlengths normally in the range of 78to 250 f (24 to 75 in). A wide range ofcross sections are used in practice and,although some basic dimensional limitsare set by the AASHTO Code,' the dif-ferent structural requirements as well asaesthetic considerations make it difficultto generalize cross sections. However,similar thermal curvature responseswere found for different structures withthe same section depth. The response of18 existing box girder bridges were

computed using the thermal analysisprogram.' The bridges studied includedthe Kishwaukee River Bridge,`5.1e theexperimental segmental bridge at PerinState, 2' and others from various pub-lished and unpublished sources.

The temperature distributions fromBrownsville, Texas (Station 6), andPhoenix, Arizona (Station 15), were usedFor the analyses, since these stationsproduced the lowest and highest tem-perature gradients, respectively, of the26 stations considered. Provision wasmade for the fact that at extreme condi-tions, temperatures in the slab above theair cavity are a few degrees higher thanthose at the same height in the web sec-tions and that the temperature gradientin the slab is also smaller.s•"

For each of the box girder bridges,.only the most typical section was ana-lyzed. The thermal curvatures for the 18bridges computed from the temperaturedistributions at Stations 6 and 15 areshown in Figs. 9 and 10. Only the plot-ted points for the unsurfaced bridge areshown.

The relatively small scatter of pointsmade it possible to represent the valueswith a third order polynomial. The pres-entation of the data points by poly-noinials can be slightly improved by di-viding the depth range into two sets, 3 to8 ft, and 8 ft and above (1 to 2.5 m and2.5 m and above), for two individual setsof equations.

In order to compute the top and hot-torn flange stresses of bridge decks, val-ues of residual stresses must also beknown. The residual stress is the ther-mally induced stress at the soffit of astatically determinate member Isee Fig.A2(d)1. A linear Function depending onthe deck depth was found to yield ac-ceptable results. The residual stressesshould not be taken to be particularlyaccurate because they were derivedfrom a one-dimensional heat flow analy-sis applied to a two-dimensional crosssection, but the stresses should properlyindicate trends.

PCI JOURNAL/July-August 1989 93

240STA7I0N 6 - UNSVRFACED

220 ------ STATION 6 - 50mm BLACKTOP

200 ------- STATION 6 - 100mm BLACKTOP

180160

0140120

wX 100I- 80

'•ter60

v 40

20

01.5

2. 2.5 3. 3.5SECTION DEPTH (m)

Fig. 9. Temperature induced curvatures for box girders subjected toStation 6 weather conditions.

The different curvature and residualstress polynomials are as follows:

Station 6 (SI units)UnsurfacedOr= 374-358d+130dE-16.2d3fr _ - 1.3+0.074d50 mm blacktop:fir= 307- 287d + 103d2 - 12.8d3fr = - 1.2 + 0.045d100 mm blacktop:Or= 198-184d+66.4d2-8.2d1f,. _ -0.92 +0.015d

Station 15Unsurfaced:q5r= 638-5854+208d2-25.4d-f,. _ -2.1+0.18c!50 mm blacktop:

r= 522 - 455 4 + 157 th - 18.8 d 3

fr = - 2.0+0.12d100 mm blacktop:fir = 347 - 294 d + 100 d 2 - 11.9dfr = - 1.6 +0.09d

Note: I radlm = 0.3048 rad/ft; 1 ksi - 6.895 %fPa6.895 Nlmm l : ! m = 3.281 ft = 39.37 in.

whereOr = curvature due to temperature

differences (rad/m x 10S)f, = residual stress (MPa) at bottom

of section, determinate memberd = girder depth (in)

Station 6 (Imperial units, convertedfrom SI units)Unsurf~ced:Or = 114-33.31) +3.68i)0.1401)f, =-189+3.271)2 in. blacktop:OT - 93.6 -- 26.71) + 2.921) 2 - f1.110!)J' _ -174+1.99!)4 in. blacktop:OT = 60.4- 17.11) +1.881)2-0.0711)3fr = - 133+0.661)

Station 15UnsurfacedOr = 194 - 54.31) + 5.891) 2 - 0.219 !) 3

fr = -304+7.961)2 in. blacktop:4>T- 159- 42.31) +4.451)2-0.16203fr = - 290+5.301)4 in. blacktop:der = 106 - 27.31) + 2.831) 2 - 0.1031)'

94

240220

-. 200p 180

160O140

`^ 120LU 100

80

rr 60U 40

200

STATION 15 - UNSURFACED------ STATION 15 - 5O BLACKTOP•--•--- STATION 15 - 10omm BLACKTOP

1.5 2. 2.5 3. 3.5SECTION DEPTH (m)

Fig. 10. Temperature induced curvatures for box girders subjected toStation 15 weather conditions.

.fr = – 2.Z + 3.981)where

OT = curvature due to temperaturedifferences (rad/ft x 10 6)

Jr residual stress (psi) at bottom ofsection, determinate member

1) = girder depth (ft)

APPLICATION TOPRETENSIONED BRIDGES

The temperature differentials whichare computed and observed in concretebridges lead to significant stresses anddeformations. The temperature relatedstresses are often of the same generalmagnitude as the allowable service loadstresses, as the stresses shown in Appen-dix A demonstrate.

However, it is suggested that thebridge design profession move very slow-ly in incorporating thermal stress calcula-tions and stresses into the design process.While it is clear that thermal stresses anddeformations have damaged a few struc-tures, 2a it appears that these are a rela-tively few cases in a very large bridge

population.One result of the restraint of thermal

deformations is the development of posi-tive moments at interior supports (seeAppendices A and B). The compositepretensioned I-girder bridge which ismade continuous by means of deck rein-forcement and compression concrete castbetween the girder ends would appear tobe especially vulnerable to damage.There is no dead load negative moment atthe interior supports, and there may beno positive moment connection of anykind. Moments maybe induced as a resultof creep and shrinkage strains, but thesemoments can he either positive or nega-tive, depending on many variables.24

Thus, the thermal deformations wouldbe expected to cause significant positivemoments and, consequently, cracking atinterior supports, but this damage doesnot appear to have occurred in this typeof bridge. If the thermal deformationscause significant positive moment crack-ing at the interior supports, the forces aretemporarily relieved. However, if thecracks are of any size, they will not closevery well when the thermal deformationsare removed, and an accumulation of

PCI JOURNALIJuly-August 1989 85

local crushing damage would be ex-pected. This does not appear to havebeen reported.

The recommendation that changes beintroduced very cautiously is based onthe observation that very little damage

has occurred in bridges which can betraced directly to thermal stresses anddeformations.

It is clear that thermally induced crack-ing has damaged some post-tensionedbox girder bridges

Theoretical methods are available togenerate values for weather parametersthat can be used for the successful ther-mal analysis of bridge structures at anylocation in the United States for whichonly limited weather data exist. A widerange of thermal loadings will be pro-duced due to the large variation ofclimatical parameters. It is evident thatthe combination of weather conditionswhich will result in an extreme tempera-ture distribution in bridge structureswould be clear days with high solar radia-tion and very low wind speeds.

A large range in air temperature alsoincreases the differential temperature inthe bridge, but this is a smaller factor.The data for the 26 SOLMET stations

were successfully used to compute prob-able temperature distributions in bridgeslocated at different parts of the UnitedStates. The geographical region at high-est risk for stress due to thermal loadingswas identified as the desert Southwest. Inturn, the effects of nonlinear tempera-ture distributions in bridges along thecoast are much less important in design.

The analysis of bridge structures forthermal loadings is appreciably simpli-fied by the use of the curvature-depthand the residual stress-depth relation-ships. If the magnitude of thermal stress-es or the size of the project warrantsmore in-depth beat flow analyses, theequations would give acceptable ac-curacy for preliminary design purposes.

REFERENCES

1. American Association of State Highwayand Transportation Officials (AASHTO),Standard Specifications far HighwayBridges, 13th Edition, 1983.

2. British Standard BS 5400, Steel, Concreteand Composite Bridges, Part 2. BritishStandards Institute, 1978.

3. New Zealand Ministry of Works andDevelopment, Highway Bridge IJesignBrief, Issue C, Office of the Chief CivilDesign Engineer, Wellington, New Zea-land, 1973, 51 pp., Amended in August1976.

4. Imbsen, R. A., Vandershaf, D. E., Scham-ber, R. A., and Nutt, R. V., "Thermal Ef-fects in Concrete Bridge Superstruc-tures," National Cooperative HighwayResearch Program Report 276, Transpor-tation Research Board, Washington, D.C.,

September I985, 99 pp.5. AASHTO, "Guide, Specifications—Ther-

mal Effects in Concrete Bridge Super-structures," in preparation.

6. Potgieter, I. C., and Gamble, W. L.,"Response of Highway Bridges to Non-linear Temperature Distributions," CivilEngineering Studies, Structural ResearchSeries 505, University of Illinois, 1983,219 pp.

7. Priestley, M. J. N., "Design of ConcreteBridges for Temperature Gradients," ACIJournal, V. 75, No. 5, May 1978, pp. 209-•217.

S. Cooke, N.. Priestley, M. J. N., and Thur-ston, S. J., "Analysis and Desigr. of Partial-ly Prestressed Concrete Bridges UnderThermal Loading," PCI JOURNAL, V. 29,No. 3, May-June 1984, pp. 94-114.

96

9. Lanigan, A. G., "The Temperature Re-sponse of Concrete Box Girder Bridges,"PhD Thesis and Report No. 94, School ofEngineering, University of Auckland,Auckland, New Zealand, September1973.

10. Bryant, A, H., Buckle, I. C., and Lanigan,A. G., "Prediction of Temperatures in BoxGirder Bridges," Proceedings, 7th Aus-tralian Road Research Board Conference,V. 7, Part 7, 1974, pp. 296- 308.

11. Hunt, B.. and Cooke, N., "Thermal Cal-culations for Bridge Design," Proceed-ings, ASCE, Journal Structural Division, V.1Ot, No. ST9, September 1975, pp.1763-1781. (See also Discussion, V. 102,No, ST6, June 1976, pp. 1277-1279, V.103, No. ST4, April 1977, p. 919.)

12. Thurston, S. J., "Thermal Stresses in Con-crete Structures," PhD Thesis and Re-search Report 78-21, Department of CivilEngineering, University of Canterbury,Christchurch, New Zealand, November1978.

13. Hirst, M. J. S - , "Thermal Loading of Con-crete Bridges," Proceedings, InternationalConference on Short and Medium SpanBridges, Toronto, 1982, V. 1, pp. 115-124.

14, klbadry, M. M., and Ghali, A., -Tempera-ture Variations in Concrete Bridges,"Journal of Structural Engineering, V. 20,No, 10, October 1983, p. 2355-2374.

15. Russell, H. C., Sliiu, K. -N., Gamble, W.L., and Marshall, V., "Measured and Cal-culated Time-Dependent Deformations ina Post-Tensioned Concrete Box GirderBridge," Proceedings, International Con-ference on Short and Medium SpanBridges." Toronto, 1982, V. 1, pp. 439-450.

16. Marshall, V., and Gamble, W. L., "TimeDependent Deformations in SegmentalPrestressed Concrete Bridges," Civil En-

gineering Studies, Structural ResearchSeries No. 495, UILU-ENG-81-2014,University of Illinois at Champaign-Ur-bana, Urbana, IL, October 1981, 242 p.

17. Duffie, J. A., and Beckman, W. A., SolarEnergy Thermal Processes, John Wile y &Sons, New York, NY, 1980.

I8. Threlkeld, J. L., Thermal EnvironmentalEngineering, Prentice Hall, Inc., Engle-wood Cliffs, NJ, 1970.

19. Typical Meteorological Year Tapes (TMY)SOLMET, Sandia Laboratories, Albuquer-que, NM.

20. Emerson, M., "Extreme Values of BridgeTemperatures for Design Purposes,"TRRL Report LR744, Department of En-vironment, England, 1976.

21. Priestley, M. J. N., "Design ThermalGradients for Concrete Bridges," NewZealand Engineering, V. 31, No. 9, Sep-tember 1976, pp. 213-219,

22. Hoffman, D. C., McClure, R. M., andWest, H. II., " Temperature Study of anExperimental Segmental Bridge," PCIJOURNAL, V. 28, No, 2, March-April1983, pp - 78-97.

23. Podulny, Walter, Jr., "The Cause ofCracking in Post-Tensioned Concrete BoxGirder Bridges and Retrofit Procedures,"PCI JOURNAL, V. 30, No. 2, March-April1985, pp. 82-139.

24. Mossiossian, V., and Gamble, W. L.,"Time Dependent Behavior of Noncom-positc and Composite Prestressed Con-crete Structures Under Field andLaboratory Conditions," Civil Engineer-ing Studies, Structural Research SeriesNo. 385, University of Illinois at Urbana-Champaign, Urbana, IL, May 1972, 517pp.

25. Emerson, M., "The Calculation of the Dis-tribution of Temperature in Bridges,"TRRL Report LR561, Department of En-vironment, England. 1973.

NOTE: Discussion of this article is invited. Please submityour comments to PCI Headquarters by April 1, 1990.

PCI JOURNAL/July-August 1989 97

APPENDIX A --- EXAMPLE OF TEMPERATUREINDUCED STRESS CALCULATIONS

(EXAMPLE IN SI UNITS)

A simple example is used to show theanalysis steps and the general results ofapplying a nonlinear temperature dis-tribution to an elastic, untracked con-crete member. A rectangular section isused. More typical sections such as T-sections can be handled, but the amountof arithmetic increases significantly andthe general pattern of response remainsthe same.

Fig. Al shows the section, and thetemperature distribution prescribed bythe New Zealand Ministry of Works andDevelopment." The fifth order curve hasbeen shown to be a reasonable approxi-mation of measured distributions, andthe 32C differential is appropriate forNew Zealand. This temperature dis-tribution produces stresses and defor-mations in the member which can besuperimposed with those from othercauses, within the limits of elasticity,These temperature stresses can be com-puted by a series of steps, with the coin-ponents added to produce total stresses.

Step 1 — Compute stresses assumingdeformations caused by temperature in-crease are completely restrained. Thestress at any point is then:

fr = T(y) a^Er

whereT(y) = temperature change at section

considereda, = coefficient of thermal expan-

sion of material

Note: The metric (SI) values of E, and C t were pur-Ixasely chosen to be round whole numbers in orderto simplify the arithmetic so that the main attentioncould be focused on the process. The values arereasonable for concrete.Conversion !actors: I in. = 25,4 mm; E. = 25 kN/uun 2 — 3133 x 10 psi; 1 ksi = 6.895 MPa; C, = to x1U-I1°C = 5.56 x 10-6!•F; ¢. = 103.2 x 10 6 radlm2.621 x IU D rad/in.; 1°C = 1.8°F differential.

= Young's modulus of materialFig. A2(b) shows the stresses resulting

in a 1700 mm deep section from thetemperature distribution shown in Fig.Al, assuming the case of complete re-straint of thermal strains. The top fiberstress is –8.0 N/mm z. The stresses havebeen resolved into the two force com-ponents also shown in Fig. A2(b).

Step 2 — The stresses are, of course,not fully restrained, and this is takeninto account by applying the two com-pression forces as tension forces actingat the same locations. Stresses fromthese two forces are computed using thefamiliar equation:

_f =P A +l– Mull

and the resulting stress distribution isshown in Fig. A2-(c). This distribution islinear.

Step 3 —'The stresses shown in Figs.A2(b) and (c) are then added, giving thedistribution shown in Fig. A2(d). Thisfinal stress distribution is in internalequilibrium, with the stresses produc-ing neither net moment nor net thrust.

If the member is statically detenni-nate, there are no other forces to be con-sidered, although there are also curva-tures. If the original temperature dis-tribution had been linear, the stresscomponents would have completelycanceled out, although there still wouldhave been curvatures.

Step 4 — The curvature induced canbe found by converting the stressesshown in Fig. A2(c) to strain and divid-ing the algebraic difference between topand bottom fiber strains by the sectiondepth, or by dividing the net momentcaused by the forces shown in Fig. A2(b)by El. For this particular example, thecurvature is:

98

fc; N/mm2

0 -5 -80

171.4

600mN/ m

mm

+3.14 0

+

b mm

b OT +32°C

5th Order Curve

AT ( y ) = AT (I O ^5

1 200 mm

h

h —1400 mm

Linear200 mm

*-I 4+ AT = + 1.5° C

(a) Section (b) Temperature Distribution FromNew Zealand MWD

Fig. Al. Rectangular section and nonlinear temperature distribution.

0 -486

1200

1700mm

300

to) Section

—135Et= 25 kN/mm=

G1- 10x10%°C

37.5 N/mm

667mm-1.25 —1.625

(b) Fully Restrained (c) Stresses At (d) Net StressStresses and Forces Release of

Restraint

Fig. A2. Temperature stresses in determinate member.

PCI JOURNAL/July-August 1989 99

__T1.7 m

25.0 m 2 5.O m

(a) Example Two-Span Structure

$r=103.2 x10Rod/m

(b) Curvature Due To Temperature Gradient

8 T = 32.25 mm

(c) Unrestrained Deformation of Beam

M = q5 T IE1)1.5

(d) Moments Due To Restraining Forces AtCentral Support

Fig. A3. Temperature induced moments in two-span continuous beam.

Or = 103.2 * 10-e rad/m

Step 5 — The top side of the memberis the heated side, so the member bowsor hogs up. The upward deflection forthis prismatic member is:

ST = Chr (Span)2/8

If the member is statically indetermi-nate, this tendency to camber upwardproduces additional stresses and forcesin the member. Fig. A3 shows an as-sumed two-span continuous member,

with spans of 25 m, along with the cur-vature due to the temperature gradient.If the structure is made determinate byremoving the central support, it will befound that the member cambers up32.25 mm away from the central support,or at least would if it were weightless.

Step 6 — A force is required to pushthe beam back down to the central sup-port, and this causes the moment dia-grain shown in Fig. A3(d). The momentis independent of the length of the

N /mm2b, mm

0 -4.86 0 -3.29 0 -815

'700

+1,78 +

-1.625 +329 0 +167

(a) Section (bI Stress In (c1 Continuity Id) Total Temperature StressesDeterminate Stress At Central SupportBeam At Support

Fig. A4. Temperature stresses at interior support, two-span beam.

member. This moment diagram issuperposed with those from the otherdead and live loads in a determination oftotal stress. The 1.5 factor in the equa-tion for the moment due to the restrain-ing force, dr (E1)1.5, is specific to atwo-span beam and will be smaller forstructures with more spans.

Step 7 — Fig. A4 completes the su-perposition process to find the totaltemperature induced stress at the cen-tral support. Fig. A4(b) shows the stressdistribution in the determinate beam,and Fig. A4(c) shows the stress distribu-tion at the central support caused by thecontinuity moment. Fig. A4(d) shows

the total stresses over the depth of thesection at the central support.

It will be noted that these stressesgenerally are in opposition to thosecaused by the gravity load forces, butthat they are generally additive tostresses caused by post-tensioning. At amidspan section, the continuity stresseswould be half those shown here, butthey are additive to the gravity loadstresses. It is thus not always clearwhich section might he hurt worse bytemperature induced stresses, but it isclear that these elastic stresses may be amajor fraction of the usual service loadallowable stresses,

PCI JOURNAL/July-August 1989 101

APPENDIX B - EXAMPLE CALCULATION OFTHERMAL FORCES IN BOX GIRDER

(EXAMPLE IN IMPERIAL UNITS)

Consider the cross section of theKishwaukee River Bridge, shown in Fig.1. Ignore the fact that the bottom slab ofthe box is thicker near the piers so that aprismatic section can he considered.With the minimum bottom slab thick-ness, the member has the followingproperties:1 = 32,005,000 in.'A = 11,595 in.2

c.= 51.216 in.;c,= 88.534 in.Total depth = 139.75 in. = 11.646 ft =1)

The concrete had E, = 4500 ksi at 6months, for specimens stored outdoorswith the bridge.

Consider a three-span bridge, withend spans of 170 R and an interior spanof 250 ft, as shown in Fig. B1(a). There isa 2 in. asphalt topping.

If the bridge is in the desert South-

Metric (S I) conversion factors: 1 in, = 25.4 ,,nn; t ft= 0.3048 m; 1 ksi – 6.895 N/mi = 6.895 MPa; 1 x10 -e radlin. = 39.37x [ 0^" radi n; 1 kip-in. = 0,11298kN-m.

west, the Station 15 (Phoenix) data canbe used to compute the thermal curva-ture and residual stress. The curvaturecan also be obtained from Fig. 10.

^r = 10-[159 -42.31) +4.45D 2 -0.162 D ^' I rad/ft

= 10 -6 [159 - 42.3*11,646 +4.45* 11.646 2 - 0.162 * 11.646 114.039* 10 -e radlft = 1.170* 10 _*radlin.

Jr - 290 + 5.30 D = 290 +5.30*11.646 = -228 psi

The curvature corresponds to a nega-tive moment, producing upward deflec-tion, or hogging. If the same approachwere used as was shown in Appendix A,it would be found that the 590 ft longmember would camber upward 0.6109 ftat the middle of the interior span, and bysome lesser amount at the interior sup-ports. Also, the force required to restrainthe movement at the interior supports tozero could he found.

A different approach will be used inthis example. The application of a pos-

17 0' 250 170

(a) Arrangement of Spans

M = +194.793 k

(b) Thermally Induced Moments

Fig. B1. Example of moments in three-span bridge.

102

itive moment equal to 0T*EI, to bothends of a member will exactly counter-act the thermal curvature, resulting in astraight member. This is in effect a fixedend moment, and the most efficient wayof dealing with this is to make themember determinate by breaking it intothree separate spans.

Each span then has fixed end mo-ments equal to:

4 r*EI e = 1.170* 10 -' rad/in. *4500ksi*32,005,000 in.4

= 168,506 kip-in.A moment distribution solution leads

to M – 194,793 kip-in. at the interiorsupports, and the moments are dis-tributed along the length of the structureas shown in Fig. B1. The support mo-

ments are 1.156 times the fixed endmoments, and the 1.156 factor is specificto the span lengths considered.

This is a positive moment, so it pro-duces a bottom fiber tension of:fib = 194,793 kip-in.*88.534 in./

32,005,000 in.4

_ +0.539 ksiThe residual stress was –0.228 psi

(compression), leaving a net bottomfiber tension stress of +0,311 psi ten-s ion.

As noted earlier, the residual stressshould not he assumed to be too precise,but the conclusion remains that there isa significant bottom fiber tensionthroughout the interior span as a conse-quence to the thermal loading.

PCI JOURNALJJuIy-August 1989 103