TRANSPORT and ROAD RESEARCH LABORATORY

Department of the Environment Department of Transport

TRRL LABORATORY REPORT 765

TEMPERATURE DIFFERENCES IN BRIDGES: BASIS OF DESIGN REQUIREMENTS

by

Mary Emerson, B.Sc.

Any views expressed in this Report are not necessarily those of the Department of the Environment or of the Department of Transport

Bridge Construction Division Structures Department

Transport and Road Research Laboratory Crowthorne, Berkshire

1977 ISSN 0305-1293

Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on ! st April 1996.

This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.

CONTENTS

Page

Abstract

1. Introduction

2. Definitions

2.1 Temperature distribution

2.2 Temperature difference

2.2.1 Positive and reversed temperature differences

2.2.2 Temperature difference distribution

2.2.3 Maximum temperature difference

2.3 Temperature gradient

2.4 Discussion

3. Environmental conditions

3.1 Positive temperature differences

3.2 Reversed temperature differences

3.3 Discussion

4. Influence of depth of surfacing on maximum temperature difference

4.1 Concrete decks

4.1.1 Positive temperature differences

4.1.1.1 Discussion

4.1.2 Reversed temperature differences

4.1.3 Extreme environmental conditions

4.2 Composite decks

4.2.1 Concrete areas

4.2.1.1 Positive and reversed temperature differences

4.2.2 Steel areas

4.2.3 Discussion

4.3 Steel box decks

4.3.1 Positive temperature differences

4.3.2 Reversed temperature differences

1

1

2

2

2

2

3

3

3

3

3

4

5

5

6

6

6

6

8

10

10

10

10

10

12

12

12

13

5. Co-existence of effective bridge temperatures with maximum positive and reversed temperature differences

5.1 Maximum positive temperature difference

5.2 Maximum reversed temperature difference

6. Influence of depth of surfacing on effective bridge temperature

6.1 Concrete decks

6.2 Composite and steel box decks

7. Conclusions

8. Acknowledgements

9. References

10. Appendix 1 - Effective bridge temperature

11. Appendix 2 - Modifications to the method of calculation given in LR 561

Page

13

13

13

14

14

1 5

15

15

15

35

38

© CROWN COPYRIGHT 1977 Extracts from the text may be reproduced, except for

commercial purposes, provided the source is acknowledged

TEMPERATURE DI FFERENCES IN BRIDGES: DESIGN REQUIREMENTS

BASIS OF

ABSTRACT

It is shown how the magnitude of temperature differences in both concrete bridge decks and the concrete deck slabs of composite bridge decks under various depths of surfacing may be determined by using a simple modific- ation to the method of calculation of bridge temperatures described in TRRL Report LR 561. Empirical methods are used to estimate the magnit- ude of temperature differences in the steel areas of composite decks and in

steel box decks.

Values of effective bridge temperatures likely to co-exist with maximum positive and reversed temperature differences are given, and the influence of depth of surfacing on minimum and maximum effective bridge temperatures

is discussed.

The clauses relating to temperature differences in a new British Stand- ards Institution document on bridge loading, BS 5400, are based on the information contained in this Report, and the Department of Transport Technical Memorandum No. BE1/77 is to be revised to include this

information. O

1. INTRODUCTION

When the British Standards Institution decided to revise and expand BS 153 'Specification for steel girder

bridges, Part 3A, Loads', to include composite and concrete bridges, the Transport and Road Research

Laboratory was asked to provide information on bridge temperatures and temperature differentials upon

which the temperature clauses to be i~cluded in this revised document could be based. (The revised

document is BS 5400: 'Steel, Concrete and Composite Bridges'. The temperature clauses are contained in

Part 2: Loads.) The temperature clauses in the Department of Transport Technical Memorandum (Bridges)

No. BE1/771 are to be revised to contain similar information.

The basis of the derivation of extreme values of bridge temperatures has already been published in TRRL Report LR 7442. This Report provides the basis of the information to be contained in the clauses

in both BS 5400 and BE1/77 (revised edition) which deal with temperature differences.

w e r e ;

1.

The data for steel, composite and concrete bridges, requested by the British Standards Institution,

The influence of depth of surfacing on maximum positive and reversed temperature differences t .

tDefmitions of these terms are given in Section 2. 1

. The values of effective bridge temperatures* likely to co-exist with maximum positive temperature differences.

. The values of effective bridge temperatures likely to co-exist with maximum reversed temperature differences.

. The influence of depth of surfacing on minimum and maximum effective bridge temperatures.

(This information would have been more appropriate in LR 744, but was not requested until after that Report had been written.)

The method of calculation of temperature distributions in bridge decks (required for 1 above) is given

in TRRL Report LR 5613. At the time LR 561 was written a satisfactory method of calculating temperatures

through layers of surfacing on concrete had not been found and in order that the temperatures in the concrete

could be calculated, an approximation was used to 'by-pass' the surfacing. This approximation has now been

superseded; relevant details are given in Appendix 2. The empirical method given in LR 561 for dealing with

the influence of surfacing on the temperature of the top flange plate of a steel box deck still stands.

All measurements mentioned in the Report have been obtained from site experirnents on seven

bridges in England and Wales. A description of these bridges and their instrumentation can be found in TRRL Report LR 6414.

Aspects of temperature differences not specifically related to the information contained in BS 5400 or in BE 1/77 are to be presented in another Report.

All times mentioned are G.M.T.

2. DEFINITIONS

2.1 ,Temperature distribution

A temperature distribution is the variation of temperature through any vertical section of a bridge

deck at any instant in time. (See Section 2.4.) An example is given in Figure l(a).

2.2 Temperature difference

A temperature difference at a given depth is defined as the difference between the temperature of the

surface of the deck and the temperature of the deck at the specified depth. (See Section 2.4.)

2.2.1 Positive and reversed temperature differences. A temperature difference is said to be positive if the temperature of the surface of the deck is higher than the temperature of the deck at the

specified depth. Conversely, a temperature difference is said to be reversed if the temperature of the surface

of the deck is lower than the temperature of the deck at the specified depth.

* The effective temperature of a bridge is defmed in Appendix 1.

2.2.2 Temperature difference distribution. A temperature difference distribution is the variation

of temperature difference through any vertical section of a deck at any instant in time. (See Section 2.4.)

It is derived from a temperature distribution by calculating the temperature difference at all levels through

the depth of the deck, using the temperature of the surface of the deck as the datum temperature. Figure 1 (b)

shows the temperature difference distribution which has been derived from the temperature distribution

shown in Figure 1 (a).

Distributions of temperature differences can consist wholly of positive temperature differences, wholly

of reversed temperature differences, or of a combination of both.

2 .2 .3 Maximum t empe ra tu r e di f ference. A maximum temperature difference, whether positive

or reversed, is obtained from a temperature difference distribution and is the maximum value of the

difference in temperature between the deck surface and the coldest (for maximum positive differences) or

hottest (for maximum reversed differences) area of the deck, irrespective of the depth of the coldest (or

hottest) area of the deck.

2.3 Temperature gradient

The correct definition of a gradient is a rate of change in a quantity with distance; thus a temperature

gradient is a rate of change in temperature with distance, the distance, in this case, being the depth of the

deck. Because the variation of temperature with depth, i.e. the temperature distribution, is usually non-

linear, the temperature gradient is not constant, as is shown in Figure 1 (c).

The term 'temperature gradient' is used in TRRL Reports LR 561 and LR 7025, and should have read

'temperature distribution', or 'distribution of temperature'.

2.4 Discussion

Except during fairly prolonged periods of heavily overcast or wet weather, temperature differences will always exist within the deck of a bridge. Their magnitude will depend on such factors as the type of constr-

uction (i.e. steel, composite or concrete), the time of day, the time of year, the depth of construction, the

depth of surfacing and, in the case of some concrete structures, the weather conditions of the previous one

or two days.

Temperature differences are considered in a vertical direction only (i.e. through the depth of the deck)

and do NOT include surfacing temperatures. In other directions (e.g. transverse), a slightly different

analytical approach is required, which is not relevant here. As temperature differences are derived from

temperature distributions it therefore follows that all temperature distributions are also considered in a

vertical direction only, and do not include surfacing temperatures.

3. ENVIRONMENTAL CONDITIONS

The combinations of environmental conditions necessary to cause large positive or reversed temperature

differences are complex, varied and more difficult to defme than those which give rise to extreme values of

minimum and maximum effective bridge temperatures 2'3- For example, consider a solid concrete deck 1.0m

deep with a vertical distribution of temperature as shown in Figure 2. The temperatures within the top

0.5m (approximately) of the deck are controlled mainly by the incident solar radiation - the greater the

3

amount of radiation, the larger the resultant temperature differences. From this approximate depth to about 0 .3m above the soffit, the deck temperatures are the result of the radiation and shade temperature levels of

the two previous days 6, and within the bottom 0.3m, the deck temperatures depend mainly upon the shade

temperature at the time and the amount of heat reflected or re-radiated from the ground beneath the bridge.

Further complications arise in that the temperature distributions in concrete areas of composite decks

and the temperatures of top flange plates of steel box decks are also influenced mainly by solar radiation,

very. high values of which can occur at any time between May and early August, whereas the temperature

distributions in steel areas of composite decks and the temperatures of soffits of steel box decks are influenced

mainly by the shade temperature. Between May and early August the shade temperature can vary over

almost the full annual range, making it impossible to define an unique set of environmental conditions which

will give rise to a large positive temperature difference. It is also impossible, at present, to assign a return period to the occurrences of large positive temperature differences.

A similar situation exists under re-radiation conditions when considering temperature distributions

caused by cooling of the deck surface, resulting in reversed temperature differences, for these can occur at any time of the day, night or year.

Analysis of twelve years of measurements has shown that:

. The largest measured values of both positive and reversed temperature differences in steel, composite and concrete bridges occur more than once a year.

. The largest measured positive temperature differences occur between May and August. The thne of

day at which they occur depends upon the type of construction and the depth of surfacing; for a

steel box deck it can be as early as 1200 hours and for a concrete deck, as late as 1900 hours. (The

greatest depth of surfacing beneath which deck temperatures have been measured is i02mm.)

3. The largest measured reversed temperature differences occur at any time of the day, night or year.

4. It is possible for a large positive and a large reversed temperature difference to occur within the deck of a bridge in less than 24 hours.

The various combinations of environmental conditions upon which the sets of calculations of positive

and reversed temperature differences are based are described below. In each case other combinations of

environmental conditions will exist which will give rise to the same values of positive and reversed temperature differences.

3.1 Positive temperature differences

The environmental conditions upon which the calculation of maximum positive temperature differences have been based are:

. For concrete decks and concrete areas of composite decks, a summer day with a total of total radiation

on a horizontal surface of 7500Wh/m 2, a range of shade temperature of 15°C and a wind speed of approximately 8km[hr.

4

. For steel box decks, a summer day with a total of total radiation on a horizontal surface of 8000Wh/m 2,

a range of shade temperature.of 20°C and a wind speed of approximately 8km/hr.

3.2 Reversed temperature differences

The environmental conditions upon which the calculation of maximum reversed temperature

differences have been based are:

1. For concrete decks and concrete areas of composite decks, a winter night with the standard night-

time re-radiation of -110W/m 2 3,5, an overnight range of shade temperature of 15°C and a wind

speed of approximately 4km/hr.

2. For steel box decks, a winter night with the standard night-time re-radiation of -110W/m 2, an over-

night range of shade temperature of 20°C and a wind speed of approximately 4km/hr.

3.3

1.

Discussion

The reasons why the environmental conditions given in Sections 3.1 and 3.2 are different for concrete

and steel are: (i) the starting conditions on which the calculation of concrete and steel temperatures are based

are different (see Appendix 2 and LR 561) and (ii) the thermal properties of concrete and steel are very different; this influences their response

to variations in radiation and shade temperature.

. The phrase 'total radiation' is a standard meteorological term used to describe the 'type' of radiation.

(For example, it is also possible to measure direct or diffuse radiation.) The phrases 'a day with a total

of total radiation', or 'a daily total of total radiation' describe the 'whole amount', or, in other words,

the total of total radiation measured on that day.

. In the environmental conditions described in Sections 3.1 and 3.2, the actual values of the shade

temperature do not matter. The effect of increasing the level of the shade temperature by, say, 10°C,

but keeping the range at 15°C is to increase all the calculated temperatures by 10°C; thus the

temperature differences remain the same. This effect is to be discussed in more detail in another Report,

and is also mentioned, in a slightly different context in LRs 702 and 7607.

. The environmental conditions described in Sections 3.1 and 3.2 are not extreme. A daffy total of

total radiation on a horizontal surface can be as high as 8500Wh/m 2 and a daffy (or overnight) range

of shade temperature as high as 22°C. It is thought that the value of -110W/m 2 used t~or the night-

time re-radiation is close to the maximum (though it is interesting to note that Hunt and Cooke 8

used a value of -69.4 BThU/hr ft 2 (= -219W/m2), which is considerably higher than other published

values9,10).

However, if combinations of these extreme environmental conditions are used in the calculations, they

result in values of temperature differences which have never been measured (see Section 4.1.3). This

gives added confirmation to the various statements made in LRs 744 and 78311 that it is unlikely that

combinations of extreme environmental conditions will occur on thesame day.

5. Because the environmental conditions upon which the calculations of temperature differences have been based are not extreme, the calculated temperature differences are not classed as extreme.

4. INFLUENCE OF DEPTH OF SURFACING ON MAXIMUM TEMPERATURE DIFFERENCE

4.1 Concrete decks

4.1.1 Positive temperature differences. Using the environmental conditions described in Section 3.1, and amending the method of calculation given in LR 561 (details of the amendments are given in Appendix 2), temperature difference distributions were calculated for depths of construction of 0.2m, 0.4m, 0.6m, 0.8m, 1.0m and 1.5m, each with the following surfacing conditions:

1. waterproofing layer only,

2. 50mm surfacing, 3. 100mm surfacing,

4. 150mm surfacing,

5. 200mm surfacing and

6. unsurfaced.

For each calculation the temperature difference distribution which contained the maximum positive

temperature difference was plotted out, and the results are shown in Figures 3 to 8 (inclusive). The values

of the maximum positive temperature differences and the times at which they occurred are given in Table 1

For comparison some of the largest measured values of positive temperature differences (for depths of construction greater than 1.0m) are:

57mm surfacing : 16°C

64mmsurfacing : 17°C

90mm surfacing : 14°C

92mmsurfacing : 15°C

102mm surfacing : 13°C

(Medway Bridge cantilever span),

(Adur Bridge slip road),

(Mancunlan Way),

(Hammersmith Flyover),

(Coldra Viaduct).

The largest positive temperature difference measured through the 0.23m thick deck slab of the Medway

Bridge viaduct span, with 57mm of surfacing, was approximately 17°C. The agreement between these measured temperature differences and the calculated values given in Table 1 is good.

4.1.1.1 Discussion. It can be seen from any of Figures 3 to 8 that for depths of construction greater

than about 0.5m the maximum temperature difference remains constant from approximately 0.5m to within

about 0.3m of the soffit. This does not happen in practice - it is caused by the theoretical linear temperature

distribution used as the 0800 hour starting condition for the calculations 3. In other words, for decks with

a depth of construction greater than about 0.5m, the temperature of the deck between about 0.5m to within

about 0.3m of the soffit remains at the 0800 hour starting temperature because the heat input to the upper

and lower surfaces does not penetrate to these levels. When a measured temperature distribution at 0800

hours is used as the starting condition, the resultant temperature difference distributions are much more representative of what happens in practice. An example of this is shown in Figure 9.

6

, . . I

0

0 ~J

| • | o | , |

0 0 0 0 0 0 ~.~ ~ ('~") 0 t"~") 0 (''~ ('~'~

~ '

o ~ ,

0

0 0 0 0 0 0

0 0,..w

n m m m m u m

~ N ~ m ~ m d ~ ~ ~ ~ ~ ~ ~ .

0

o

7

The reason why actual temperature distributions at 0800 hours are not used as the starting condition is that they are different every day. Some suggestions about the use of a more realistic starting condition than the linear temperature distribution used at present are to be discussed in another Report.

The temperature difference distributions shown in Figures 3 to 8 have been calculated for solid concrete slabs only, because:

. both measurements and theoretical calculations 5 have shown that temperature distributions in box constructions differ very little from those in solid slabs,

2. measurements of temperature distributions in the Coldra Viaduct, a beam and slab construction (though admittedly almost ceUular4), differ very little from those in solid slabs and

. the slight variations in the shapes of the temperature difference distributions in different forms of

construction are of less consequence than the inaccuracy of the shapes of the temperature difference distributions calculated from a linear starting condition.

4 .1.2 Reversed t empe ra tu r e differences. Using the environmental conditions described in Section 3.2, the method of calculation given in LR 561 and the starting conditions described in Appendix 2,

temperature difference distributions were calculated for the same depths of construction and surfacing as

described in Section 4.1.1. The distributions which contain the maximum reversed temperature differences are shown in Figures 10 to 14 (inclusive). For this set of calculations the unsurfaced condition is the same

as the waterproofed condition. (The reason for this is that for the re-radiation conditions under discussion,

the emissivities of surfaced and unsurfaced concrete are the same. Variations in the emissivity and absorpt-

ivity of a surface are to be discussed in another Report.) The values of the maximum reversed temperature

differences and the times at which they occurred are given in Table 2. Figures 13 and 14 show that for

depths of surfacing of 150mm and 200mm the soffit temperature is sometimes lower than that of the deck surface.

For comparison some of the largest measured values of reversed temperature differences (for depths of construction greater than 1.0m) are:

57mm surfacing

64mm surfacing

90mm surfacing

92mm surfacing

102mm surfacing

: 8°C (Medway Bridge cantilever span),

: 9°C (Adur Bridge slip road), : 8°C (Mancunian Way),

: 8°C (Hammersmith Flyover),

: 5°C (Coldra Viaduct).

In the analysis of measured temperature differences, values less than 5°C were not logged. The largest

reversed temperature difference measured through the 0.23m deck slab of the Medway Bridge viaduct span,

with 57mm of surfacing, was less than 5°C. The agreement between these measured temperature differences and the calculated values given in Table 2 is good.

The discussion contained in Section 4.1.1.1 applies equally to the heat loss conditions which result in

reversed temperature differences, except that the theoretical linear starting condition for the calculations is taken at 1600 hours 3 (see also Appendix 2).

8

I L l - , I

I--

0

o o ~ ~ ~: "~

~ 0 0 0 0 0 0 ~ o o o ~ , ~ , ~ ,

.~ 0 0 : 0 0 0

° ° °

~ 1 ° = ~ ° ,~ o ~

~ o ~ o0 ~o ~ ~o ~ ~ . , ~ . ~ ~ • ° ° °

~ ~'-~ ~'~ '~" 0 ,-~ i ~ ~'~ ~'~

° ~ . . ~

t ,~ o o o o

o ~

~ ~ o ~ o ~ ~ - ~ o

~.~ o

c5

I

0

g

4 .1 .3 Extreme environmental conditions. For interest, both positive and reversed temperature

difference distributions in a 1.0m deep solid concrete slab with 100mm of surfacing were calculated using

the combinations of extreme environmental conditions described in paragraph 4 of Section 3.3. The

distributions which contained the maximum positive and reversed temperature differences are shown in

Figures 15 (a) and (b) respectively. The values of these maximum temperature differences are 16.5°C and

10.5°C respectively, compared with the values of 13.3°C (Table 1) and 8.1°C (Table 2), for more normal environmental conditions. Values as large as these have never been measured.

4.2 Composite decks

4.2.1 Concrete areas

4.2.1.1 Positive and reversed temperature differences. Using the environmental conditions

described in Sections 3.1 and 3.2, and the method of calculation given in LR 561, amended as described in

Appendix 2, temperature difference distributions were calculated for depths of concrete deck of 0.2mand

0.3m with the same surfacing conditions as described in Section 4.1.1. The temperature difference

distributions which contain the maximum positive and reversed temperature differences are shown in

Figures 16 (a) and (b) respectively, and the values of these maximum positive and reversed differences, and

the times at which they occurred, are given in Tables 3 and 4 respectively. Maximum measured values of

positive and reversed temperature differences, beneath 58mm of surfacing are 12°C and 3°C respectively.

The agreement between these values and the calculated values given in Tables 3 and 4 is good.

it can be seen from Figure 16 (and also from Figures 3 to 8 and 10 to 14) that for these smaller

depths of construction (up to about 0.6m) the heat gain (or loss) is sufficient to penetrate the full depth

of the deck and the theoretical linear temperature distribution used as the starting condition for the calculations 3 is no longer evident.

The shapes of the temperature distributions shown in Figures 16 (a) and (b) are controlled mainly by

the solar radiation and re-radiation respectively and are almost independent of the shade temperature. (The

effects of varying the radiation and shade temperature are to be discussed in another Report.)

The discussion contained in paragraphs 3, 4 and 5 of Section 3.3 is also relevant to the concrete areas of composite decks.

4.2.2 Steel areas. The steel areas of composite decks are usually shaded from direct solar radiation by

the concrete deck slab (except during the early morning, or late evening, when the sun is below the level of

the concrete). Similarly, during cooling conditions, it is the concrete deck which-loses most heat by re-

radiation. Because of this 'protection' afforded by the concrete, the temperatures of the steel components

of the deck, below about 0.5m from the soffit of the concrete, are controlled by the shade temperature.

(Measurements have shown that any influence the temperature of the concrete has on the temperature of

the steel is lost within approximately the top 0.5m of the steel.) Empirical methods of determining the

temperature distribution through the depth of the steel are given in LR 561.

J

(Note: The derivation of the distribution of temperature through the depth of the steel is necessarily

empirical, for the flow of heat through the steel is not one-dimensional (because of sideways heat losses or gains) and does not, therefore, lend itself to linear heat flow analysis.)

10

. . . I

I-

°...~

. e o

0 0

o

° ~ ,

~ o

~ ~ °

0 ° ~ ~

o

~o ~o~ o

~ 0 0 " ~

o

- . 1

0

o

0

0

° ~

N - o

o

~ ~ o ~ o~ o

¢5 d

o

11

4.2.3 Discussion. It is stated in Section 3 that very high values of solar radiation can occur at any time

between May and early August, during which months the shade temperature can vary over almost the full

annual range. Thus, although the maximum positive temperature differences within the concrete may be predicted with some confidence, they can co-exist with a variety of temperature difference distributions through the depth of the steel.

This problem of providing one maximum positive temperature difference distribution for the full depth of the deck, for each depth of surfacing, is made more complicated by the fact that the times of

occurrence of maximum positive temperature differences in the concrete deck slab vary according to the

thickness of the surfacing (see Table 3), thus increasing the variety of temperature difference distributions

which could exist through the depth of the steel. Because of this no theoretical maximum positive temper- ature differences, through the full depth of a bridge deck, are presented.

A temperature difference distribution, through the full depth of the deck of the Adur Bridge main

viaduct, which contains a maximum measured positive temperature difference is shown in Figure 17 (a).

The same problems exist when considering theoretical reversed temperature differences and, for the

same reasons, no theoretical maximum reversed temperature differences through the full depth of a bridge deck are presented.

A temperature difference distribution, through the full depth of the deck of the Adur Bridge main

viaduct, which contains one of the largest measured reversed temperature differences in the concrete deck, reverting to positive temperature differences in the steel, is shown in Figure 17 (b).

4.3 Steel box decks

4.3.1 Positive t empera tu re differences. Using the environmental conditions described in Section 3.1

and the combination of the theoretical and empirical methods of deriving deck temperatures given in LR 561, temperature difference distributions were calculated for an unsurfaced deck.

Because the thermal properties of surfacing and steel are dissimilar, it is not, at present, possible to

calculate temperatures through a surfacing/steel system, and an empirical method 3 has been used to derive the temperature difference distributions under 20mm and 40mm of surfacing. (See also Appendix 2.)

The distributions which contained the maximum positive temperature differences are shown in Figure 18 (a).

The values of these maximum positive temperature differences, and the times at which they occurred are given in Table 5:

TABLE 5

Unsurfaced

Max. temp. diff. (°C)

Time (h)

20mm surfacing

Max. temp. diff. (°C)

Time (h)

40mm surfacing

Max. temp. Time diff. (°C) (h)

30 Noon* 27 1300 t 24 1400 +

(Note: * - calculated time; t _ extrapolated time (see Appendix 2); + - time derived from measurements.)

For comparison, maximum measured positive temperature differences are 31 °C, unsurfaced, and 24°C under 38mm of surfacing.

12

In the above analysis it is assumed that the thickness of the top flange plate is 11.4mm (the only value

for which measured temperatures are available) and that the depth of the steel box exceeds 0.6m. Temperature

difference distributions in different depths of box can be derived by assuming that the temperature difference

increases linearly from its value at 0.6m to 30°C, 27°C or 24°C (depending on the thickness of the surfacing)

at the soffit.

4.3.2 Reversed temperature differences. Using the environmental conditions described in Section 3.2

and the combination of theoretical and empirical methods of deriving deck temperatures given in LR 561, tem-

perature difference distributions were calculated for an unsurfaced deck.

Measurements have shown that the effect of 38mm of surfacing is to reduce the temperature of the top

flange plate by about 2°C, the steel being warmer than the surfacing. It has been assumed that the effect

of 40mm of surfacing is the same, and the temperature distributions containing the maximum reversed

temperature differences of 8°C, unsurfaced and 6°C, surfaced, are shown in Figure 18 (b). "They both occur

at about 0600 hours. Maximum measured values are 7°C, unsurfaced and 6°C under 38mm of surfacing.

The discussion contained in paragraphs 3, 4 and 5 of Section 3.3 is also relevant to positive and reversed

temperature differences in steel box decks.

5. CO-EXISTENCE OF EFFECTIVE BRIDGE TEMPERATURES WITH MAXIMUM POSITIVE AND REVERSED TEMPERATURE DIFFERENCES

5.1 Maximum positive temperature difference

Analysis of measurements has Shown that the lowest effective bridge temperatures likely to co-exist with

maximum positive temperature differences are:

(i) 15°C for concrete and composite decks and

(ii) 25°C for steel box decks.

Thus for a concrete or composite bridge it is possible for a maximum positive temperature difference to co-

exist with any effective bridge temperature between 15°C and the extreme maximum values given in LR 744.

(Although it is thought unlikely that a maximum positive temperature difference will co-exist with an

extreme maximum value of effective bridge temperature, as the former usually occurs earlier in the day and

will therefore have decreased from its maximum value by the time the maximum effective bridge temperature

is reached, the possibility of coincidence cannot be disregarded.)

Similar observations apply to steel box bridges, in which it is possible for a maximum positive temperature

difference to co-exist with any effective bridge temperature between 25°C and the extreme maximum values

given in LR 744.

5.2 Maximum reversed temperature difference

Maximum reversed temperature differences in concrete, composite or steel box bridges can occur at any

time of the day, night or year and may be assumed to co-exist with any effective bridge temperature within

the full thermal range given in LR 744, with the following reservation.

13

It is extremely unlikely that a maximum reversed temperature difference can exist between about

1000 hours and midnight on (or after) a hot sunny day when conditions are such that an extreme maximum

effective bridge temperature will occur. It is therefore suggested that no maximum reversed temperature difference is likely to exist:

(i) in a concrete deck, within 2°C of the values of maximum effective bridge temperature given in column A of Table 4 in LR 744;

(ii) in a composite deck, within 4°C of the values of maximum effective bridge temperature given in column A of Table 4 in LR 744;

(ii) in a steel box deck, within 8°C of the values of maximum effective bridge temperature given in column A of Table 4 in LR 744.

It should be noted that the above values are based on judgement and not on experimental or theoretical evidence.

(Note: The information contained in Sections 5.1 and 5.2 relating to composite and steel box bridges

has been derivedfrom measurements from one bridge of each type only, and may not be very accurate.)

6. INFLUENCE OF DEPTH OF SURFACING ON EFFECTIVE BRIDGE TEMPERATURE

It has been shown elsewhere 2,12 that the effective temperature of a bridge is influenced by the shape of the

cross-section of its deck, and it is not possible to investigate the influence of the depth of surfacing without

taking this into consideration. (The shape of the cross-section is 'described' by the ratio of the area of the cross-section of the deck to its width, in units of m2/m.)

6.1 Concrete decks

Using the method described in Appendix 2 of LR 744, the minimum and,maximum effective temperatures

of the simple bridge cross-sections shown in Figure 19 were calculated using the environmental conditions

described in Sections 3.2 and 3.1 respectively and the surfacing conditions described in Section 4.1.1. (No

calculations are included for box decks as it was shown in LR 744 that they followed the pattern of behaviour of solid slab decks.)

Taking the minimum and maximum effective temperatures of each cross-section under 100mm of

surfacing as the datum temperatures, the variations in minimum and maximum effective bridge temperatures

of the various cross-sections under other depths of surfacing were calculated. These variations are shown in Figures 19 (a) and (b) respectively.

In view of all the approximations necessarily incorporated in order to predict a single extreme minimum

or maximum effective bridge temperature for every conceivable form of construction and shape of cross-

section, as was done to derive the temperatures given in Tables 2 and 4 of LR 744, the superimposition on

these rather imprecise temperatures of a precise ref'mement to allow for the depth of surfacing would seem unjustiffmble.

14

6.2 Composite and steel box decks

The views expressed in the above paragraph apply equally to composite and steel box decks. However,

the variations in minimum and maximum effective bridge temperatures under different depths of surfacing

were calculated, or derived empirically, and the results for a composite deck are shown in Figure 201

(Referring to Figure 20 - measurements have shown that, provided the cross-sectional area of steel is not

greater than about 25 per cent of the total cross-sectional area of the deck, its temperature influences the

effective temperature of the deck by less than 2°C. Steel temperatures have therefore not been included in

the Figure.)

No variations in the minimum effective temperature of a steel box bridge for different depths of

surfacing are given - because none exist (within the accuracy which the method of derivation allows).

Measurements suggest that the maximum effective temperature of a steel box bridge with 40mm of surfacing

will be between 3°C and 4°C higher than that of an unsurfaced bridge, and with 20mm of surfacing, about

2°C higher than that of an unsurfaced bridge.

7. CONCLUSIONS

1. It has been shown that the magnitude of temperature differences in both concrete bridge decks and the

concrete deck slabs of composite bridge decks under various depths of surfacing may be determined

by the use of a simple modification to the method of calculation of bridge temperatures described in

LR 561.

2. By comparing calculated and measured temperature differences for various combinations o f shade

temperature and radiation, confirmation is obtained of the statements made in LRs 744 and 783

that extreme values of these two environmental conditions are unlikely to occur on the same day.

. It is suggested that to modify the effective temperature of a bridge to allow for the depth of the

surfacing, without taking into account the shape of the cross-section of the deck is art unjustifiable

refinement.

8. ACKNOWLEDGEMENTS

This Report was prepared in the Bridge Construction Division (Division Head: Mr W I J Price) of the

Structures Department of TRRL.

9. REFERENCES

1. Department of Transport. Technical Memorandum (Bridges) BE 1/77: Standard Highway Loadings,

1977.

. EMERSON, MARY. Extreme values of bridge temperatures for design purposes. Department of the Environment, TRRL Report LR 744, Crowthorne, 1976 (Transport and Road Research Laboratory).

. EMERSON, MARY. The calculation of the distribution of temperature in bridges. Department of the Environment, TRRL Report LR 561, Crowthorne, 1973 (Transport and Road Research Laboratory).

15

4. MORTLOCK, J D. The instrumentation of bridges for the measurement of temperature and

movement: Department of the Environment, TRRL Report LR 641, Crowthorne, 1974 (Transport and Road Research Laboratory).

. JONES, M R. Bridge temperatures calculated by a computer program. Department of the Environment, TRRL Report LR 702, Crowthorne, 1976 (Transport and Road Research Laboratory).

. EMERSON, MARY. Bridge temperatures estimated from the shade temperature. Department o f the Environment, TRRL Report LR 696, Crowthome, 1976 (Transport and Road Research Laboratory).

. JONES, M R. Calculated deck plate temperatures for a steel box bridge. Department of the Environment Department o f Transport, TRRL Report LR 760, Crowthorne, 1977 (Transport and Road Research Laboratory).

. HUNT, B and N COOKE. Thermal calculations for bridge design. Proc. A.S.C.E., 10, ST9, pp 1763- 1782, 1975.

9. BILLINGTON, N S. Thermal properties of buildings: London Cleaver-Hume Press, 1952.

10. WILLIAMSON, P J. The estimation of heat outputs for road heating installations. Ministry of Transport, RRL Report LR 77, Crowthorne, 1967 (Road Research Laboratory).

11. EMERSON, MARY. Temperatures in bridges during the hot summer of 1976. Department of the Environment Department of Transport, TRRL Report LR 783, Crowthome, 1977 (Transport and Road Research Laboratory).

12. BLACK, W, D S MOSS and MARY EMERSON. Bridge temperatures derived from measurement of movement. Department of the Environment, TRRL Report LR 748, Crowthome, 1976 (Transport and Road Research Laboratory).

1 6

0

0.2

A 0 . 4 - E

0.6 a

0.8

1.0 -

m

(a) Temperature distribution ~, measured in the Adur Bridge

slip road at 1630hrs on 1st June 1971.

I i ~ I I I I I I I

14 16 18 20 22 24 26 28 30 32 Temperature (°C)

34

A

E t "

E3

0.2

0.4

0.6

0.8 B

1.0 ~-

20

Z / Surfacing "

ence distribution I ~ I derlived fr lmFigi l(a) I I

I 18 16 14 12 10 8 6 4 2 0

Temperature difference (°C)

0.2

0.4

a 0.6

0.8

1.0

14

- ~ / J Temperature i / ~ ~ gradients_

(c) Temperature gradients _ \ ~ . . - I i I I I I I I I

16 18 20 22 24 26 28 30 32 34 Temperature (°C)

Fig. 1 EXAMPLES OF A TEMPERATURE DISTRIBUTION, A TEMPERATURE DIFFERENCE DISTRIBUTION AND TEMPERATURE GRADIENTS

0 ~,-

c 0 0 o +a

r,g ~ '~ ~ e-- e.-

0=50 c- 0 ~- 0 o 0

~ N .9

0

o~-~ Q . e-

E E ~

~ ¢" ¢" e'~

¢'~ e_ .~, e..

I I

• ¢.D

1.0

0

¢n

I -

Z

o ~

~ g g N ~ F-'"

5,," ~ ~ o <

. . I 1.1.1

m Z

e ~

• . N eq Z W

U J

I N

0 d d d d d d d d d

(w) 4}.dao

. 0

0

f ~

E e!.

0 0

-t-

I -

0

E =. -"1

E

0

E ~.--H

\

/

/ '

/

I i I i

(W) UO!~OnJ~SUO:) ~0 q:~da(]

+

E

I I I I I

t ~

0

0 .

c.O

r ~.

0 ~4

U W

W I= W OC U Z 0 u

Z 0

r . I--

o

Z UJ

,<

a .

w

._~ U.

r ~

r ~

L e 3 , r ~

I E

• \

\ \ 1

\ \ 1

I C ~

<

E ~ =. -,,.I I E o

d

I ~

\ / /

/ /

J I I I I I l I !

6 d d d d d 6 d

(u J) uo!~.onJ~suoo ~.o 4~.da 0

I I

E LO --

/ I I

0

¢.0

8 C

7- E o;

t~ 1,-

m-

p . .

W

I -

Z 0

I

Z

I-

p-

Z W

k~ M.

UJ

I - - ,< IIC

O .

._~ L.6

÷

-t-

i 0.1

0.2

0.3

0.4

I \

%.

"~ Soff i t at 0.2m k

0.4m

±

0.5

~ 0.6

0.7

O u 0 .8 "5

o. 0.9

1.0

0.6m

3_

0 . 8 m

3_

1.0m

1

1.1

1.2

1.3

The merging of curves

A t A

B C

1 . 4 m

1.5

15 14 13 12 11 10

1 .5m

I I I I ~ I I I 9 8 7 6 5 4 3

Temperature difference (°C)

Fig. 5 TEMPERATURE DIFFERENCE DISTRIBUTIONS -- CONCRETE DECKS

I 2

-t-

0

.,...~E E E E E

d-"] d-"i . : -"1

\ \

\ "\

f J

J

/ / J. L L I _ _ ~

o ~ o d d o ~ o o .~

(w} uo!,~0nzlsuoo J.o q:~da(]

I,-.-

J ,1. L

o I

Z 0 I--

=O

~C

== z u j

~- L~ lk -

= - - z ~ ~ 0

._~

. ° o

\ \ \ \ ~ - ,

~.'. ~-~ ,'.'..~ ~,'-', \

\ 0 \

\ \ \ \ \ \

\ \ \ \ \ \ x 'x \ \ \ " - ?

I I I I I I ~ ] I I I

o c5 d d o c~ d o c:; ,-:

(LU) uo!13nJ~.Suoo j o q:~da 0

;illll

I I I I

0

_ ~ z

m

~ z 0. W ¢ ~ E m ~

k-

O.

k- lID

._~ _ _ 0 U.

-I-

0

E~ 6~ o

\ \ / \

k

I I I I I

o d o d o

III E ~ ~o

E E

0

///// !

E

/

0

f~

0

__ tO

c~ d o d . . . . . .

(w) uo!1onJlsuoo ~o 4:~da(]

"0

E

v U 1.14

I-- W

Z 0 (,.)

I

Z 0 r--

i -

0 Z I.W

14. I.L

5

I - -

I.U a .

+

+

\ \ \ \

\ \

\ \

\ \

\ \ \ \ \ \ \ \

\ \ \ \ \ \ \ \ \ \ \ \

\ \ \ \

C

E~ E

\ \ \ \

\ \ \ \ " N \

\ \ \ \

\ \ \ \ \ \ \ \

\ \ \ \

\ N

\ \

\ \

, I I I

d d o

I Igl

(tu) Ll~da 0

0

g 0 0 . _ C ) . _

~_ ~ "- .~ " -~.

~ ~ ~E ~ o o. o ~ . -~ . -

~" ~ ~ 3 ®w o

~ N ~ $~._~

I f

d d d d o.

I ~ . CO

~0

L~

0 CO

E

LO

F~

o

o ~

0 o

Z 0 I -

n~

u J

w

u J

u ~ - - I - - I

I

uJ

W

0

Z 0

Q.

0 0

+

I

0 o

..~

d d I I I

(w) uo!lonJ:~suoo ~.o q~.da 0

d c~ d d o . . . . . . I I I I I I I I I

a

\

"-, \ _

/ \ \

'\ / \

I \ I \

I \ I \

I \ I \

/ •

/'

e~

E E - "- ~ -*'1 " - 4

E E E

1 I

I-- ~ m

E

,r-

c~

I'N

r-- ~

E

W C~

g4 I-- MJ CC

Z 0

I o~ Z 0 1- 012

CC I-"

Z UJ

W

Q.

0

ig

+

÷

0

\ \ \ \ ' ~

\ E // ~'~ 7 , . \ \ \~ \ \ \ \ \\ \ \'~ ,

\ \ \ ' ~ i I m

\\\\

I I I I

f ~

\ \

\ \

\

\ \

(tu) uo!:~onJlsuoo JO q:l.da O

¢n (D r--. o0 o~.

I I I I I

E E E

0 0 ,~-

I I I I I

. 11 II

,< m

E

0

O0

p,~ A

U

m ~

E

0

t

Z 0

L~

Ou z

I-- ,<

W

N

U.

o o 6 o

\ \

\ \

\ \

% \

f

C CO

"0

o / m !

O3 \ \ \ \

I 0

I

( w ) uo!:~onJ~suoo Jo 41da O

d c~ d d o I I l I I

CI

\ \

E E E E

I I I I I

o l

E l--

0

0 3

:'5

O. E

¢'3

0

I

Z 0 I -

ra

I - ¢/'J

0 uJ ,,=~-

U . W

~, ,u Z 0

I -

U.I ¢t.

g.I I--

M.

-I-

+

O o o d o

Ca

I e~

\

\ \, \

\ \ \ ',

\ \

, \ \ ',

~- E E "

( w ) uo!~.OrlJ~.Suo3 Jo q~dac]

I I I I I

\ E E E

I I ] co

. 111 -

< m - - e~

~ E

I

Z 0 I -

k-

~ w I--

E ~

W

I -

o o 6 o o \ \ \ \ \ ~ I I 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q \ \ \ \ \ \ ~ \ \ \ \ \ \

\ N \ \ \ \

\ \ \ ~ \ N

X % \ ~ \ N ~ \ \ \ x \ \ \ \ \ \ %

\ \ ~ \ \ \

\ \ ~ . \ \

~ g \ x

\ \ \ \ \ \

~ N N N \ \ / N \ N \ \ \ N '\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N

\ \ \ \ \ \ E

, \ \ \ % x I m

(uJ) uo!~.onJ~suo:) ~.o qJ, d e Q

~ . r-, ~ o~ o . ~ ~ ~. ~.. u~ 6 o 6 o 6 . . . . .

I I I I I ] I I I I

\

0 0

t :

,< E °-H

I r~

I E

CO I

r-. Z 0 l -

c~ QQ

l--

uJ ~, ~ fO

~ Z LU W ,r',

0"] U.

2 • -- OC

0. 2E UJ

0 {.--

t ' )

CN

+

E .o

2 E Q

Q

0.1

0.2 S 0.3

0.4 I 0.5

0.6

0.7

0.8 ~ 0.9

1.0 i 17 16 15 14 13 12

100mm surfacing

(a) Positive temperature difference distribution

I I I I I I I I I I I 11 10 9 8 7 6 5 4 3 2 1

Temperature difference (°C)

100mm surfacing 0

0.1

0.2

9 (b) Reversed temperature difference

I d i s t r i b ~ = ~ f J I ~ L I ~ I I

1 2 3 4 5 6 7 8 9 10 Temperature difference { °C)

0.7

0.8

0.9

1.0 11

0.3

0.4 g 0

0.5 C @

0.6 "6 . C

Fig. 15 T E M P E R A T U R E D I F F E R E N C E D I S T R I B U T I O N S C A L C U L A T E D U S I N G E X T R E M E ENVIRONMENTAL CONDITIONS (CONCRETE DECK)

A

E t -

O

2 t -

O o

¢-

E3

0 R

0.1 -

0.2 -

0.3

0.4

0.5

0.6

0.7

0.8

0.9 -

1 . 0 -

17 16 15 14

100mm surfacing

(a) Positive temperature difference distr ibution

13 12 11 10 9 8 7 6

Temperature difference ( °C)

5 4 3 2 1 0

! !00mm surfacing

(b) Reversed temperature difference distribution

0 1 2 3 4 5 6 7 8 9

Temperature difference ( °C)

0

- 0.1

0.2

0.3

0.4

0.5

0.6 Jto. ~ 0 . 8

0.9

1.0 10 11

A

E ¢-

O o~

(.~

2 t -

O r J

N - O t -

E~ E3

Fig. 15 TEMPERATURE DIFFERENCE DISTRIBUTIONS CALCULATED USING EXTREME ENVIRONMENTAL CONDITIONS (CONCRETE DECK)

÷

\ \ \ \

l £

~0

CO

03

c:3

7-

(N

Lo

(D

r ~

C0 ,r-

03

C)

CN

\ \

\ \

\ \

J t*q

d o o (t.u) qels >pap (w) qels >~oap

a,taJouoo jo 4:~da0 asa~ouoo jo q~da0

O

¢N

t.¢3

(.O

GO

~3

o ,e.- ~

~J

"" i. ~ o O-

t..-

L~

(.O ,p-

CO

Ce ¢.-

O

F~

¢-N

eq ¢N

o

( w ) q e l s ) 1 3 a p

a ~ . a J o u o o J.o Ll :~da 0

~7. ( '4 o ' )

o o (5

2

Q.

E

E o ~

N

('..u) qels >loap .~ a:~a,~ouoo ~,o q:~da G

o o d 0o

.=

"O ~ r

E I -

(-N

O

6/3

¢..) I.g

UJ I - -

o a .

o CJ

M. o

.- I ¢O

( . ) UJ GI

gJ I - - I.M

Z O

I

Z O

I ' -

I - -

5 UJ ¢J Z

l.LI I.I. I.l=

5 I.LI

I-- ¢¢ CC UJ Q.

MJ I - -

¢JD t.,-

c~ LE

4-

(LU) 4~doCl

0 '~" CO ~ CO 0 ~t

d d ~ ~ e,i e,i

o d

:._c~ ~ ' - 8 : ¢ . ) : l

i i - o ~ ,

E ~ ~ ~ 0

. _~ o _ . ~ . = z -

- - C 0 . ~ ~D " 0 c', I,M • ~ m -- o -- 70 I.I. <~

~ ~- ~ ~ ~ 0 ~- ~ 0 ~

N~ °~ I ~o j ~

g m

N I x , 7 ~

• - 2 CD ¢ N , - -

0

d d ,.- ,.- cq ¢'q

( w ) 4~.de O

0.1

E

c 0.2 O

o D 4-J

£ ~ 0.3 E 0 U

~ o.4 e-

a 0.5

0.6

Unsurfaced

20 mm surfacing

40 mm surfacing

/ - Top flange plate

//// , /

, / ! , /

/ /

(a)// / So.,t / I / I = / I t

35 30 25 20 15 10 5

Temperature difference (°C)

" 0.1

0.2

0.3

0.4

0.6

(b) I I

0 5 10

Temperature difference PC)

A

E c -

O 4-J

t -

O

"5 ¢ -

E E3

Fig. 18 T E M P E R A T U R E D I F F E R E N C E D I S T R I B U T I O N S - STEEL BOX DECK

E %

,:5

-8

"~ (5 E ,~

O0

~D >

O

6 aJ " -

O Z

E

%

. _ 0

• ~ b "6 b _ .~_,~ ~e

z ~ E ' n o

E

%

0

8 o

Z

E

(D

> ,

t - O

> ,

. B N.- / / /

A . Q

,~- ¢0 i'~ ,-- 0 "T

(0o) aJnleJadwa~ aBp!Jq a^!~.oa~a wnLu!xeuJ u! uo!~,e!Je/~

C~ I

O3

~ o ~

3

(Oo) aJnleJadwa~ a6p!Jq aA!].oajja

m n w ! u ! w u! uo!:le!Je A

('N

I

0 0

0

E E

t -

r-

E

0

0

0 0

0

,r,.- A

E E

t -

O ~

121 0

0

v

L i i

LU I-- i i i n "

C J

Z

O o

I 1J.I n "

.I-- < n,-

I ,U a, .

i i i

I-. I ,U

n,- ¢n

i i i

> I-. t,j i i i

U, . i i

I.,U

z O (..9 z

,< i l 1=

0

O. I.U a t l

0

¢j z

U.

Z

._~ i1

f- 0

e-

0 0

f f f

/ /

/ /

; / / /

A

v

I I I I

I I I I I

0 0

(Oo) aJn),eJadLua~, aBpuq aA!lOa~.~.a uJnLu!xeuJ u! uo!},eueA

I I I I

E

d o

E -HI il-- --I~ ~

l L 6

Q)

0

0 z

\ \

~' . _ \~x'x,

~, o ~ - \ -~

, , ,',% (0o) aJn),eJadwe),

efip!Jq aA!:l.:)aJ.J.a LunLu!u!uJ U! uo!~.epeA

0

E E

E

e"

__ O LO

0

0 0

0

E E

8g

g a

v co W a

o a .

0 ~J

I U.I

U.I O.

W

a

I,LI >

I- i i i I 1 i i i i i

Z O (.9 z

0

z I,- O. W a

U.

0 i11

Z W

,...1 U . z

0

._~ U .

10. APPENDIX 1

EFFECTIVE BRIDGE TEMPERATURE

The effective temperature of a bridge is defined as the temperature which governs the longitudinal movement

of the deck. It is derived from the sum of the products o f areas between isotherms and their mean temperatures,

divided by the total area o f cross-section o f the deck. In practice it is difficult to locate isotherms exactly

and an approximation can be made by dividing the cross-section into areas, the mean temperatures o f which

can be determined by measurement. The method of calculation is given below.

The change in temperature and change in length o f any unconstrained material are connected by the

equation L 1 = Loa0

where L 1 = change in length

L o = original length

c~ = coefficient o f expansion of the material

0 = change in temperature.

(0 o = original temperature)

Consider two adjacent sections within the bridge, each of unit length, one o f concrete and one o f steel. Let

the area of the concrete section be A 1 and Youngs modulus E 1 , and let the area o f the steel section be A 2

and Youngs modulus E 2. Assume that both the steel and concrete sections are subjected to the same change

in temperature, 0. If the two sections were free to expand independently, a unit length o f the concrete

section would expand by an amount a I 0 (since L 0 = 1 and t~ 1 is the coefficient o f thermal expansion of

the concrete), and a unit length of the steel section would expand by an amount a20 (L 0 = 1 and a 2 is the

coefficient o f thermal expansion of the steel). But the two sections are bonded together and must therefore

expand by the same amount. Let this amount be y. If there is no bending of the sections (see discussion below)

then the compressive strain in the concrete section = (al 0 - y ) , assuming a I > a 2, and the tensile strain in the

steel section = - ( y - a 2 0 ) = (a20-Y).

stress in concrete = E~(a l0-Y) and

stress in steel = E2(a20-y) .

.. loadin concrete = A1El(Otl0-Y) and

loadin steel = A2Ea(a20-Y).

Since the loads in the two sections balance (the system is not constrained), the total load is zero.

i.e. AiEl(Ct l0-Y) + A2E2(ct20-y) = 0 . . . . . . . . . . . . . . . . . . (1)

A1ElOtl + A2E20t 2 ) or y = 0. . . . . . . . . . . . . . . . . . . (2)

A l E 1 + A2E 2 (AiElCtl + A2E2°t2~ Comparing equation (2) with L 1 = ct0, it can be seen that the expression _ _ _ ~ is equivalent

\ AlE1 + A 2 E 2 ]

to the 'mean' coefficient o f expansion of the non-homogeneous section. Let this be a m.

3 5

Assume now that the concrete section is subjected to a change in temperature of 0 l, and the steel

section to a change in temperature of 02; equation (1) becomes:

or

A1EI(txI01-y) + A2E2(~202-Y) = 0

I. A l E ] + A2E 2 /

. . . . . . . . . . . . . . . . (3)

. . . . . . . . . . . . . . . . (4)

le t y = Otm0e, where 0 e is the change in effective bridge temperature, then, from equations (2) and (4):

or

( AIEI0tl/~IA2E2Ot202 ) = ( A1E10tl + A2E2°~ 2 . . _ _ ?

\ AlE1 + A2E2 ~k AlE 1 + A2E 2

( Ai El OtlO l + A2E20t20 2 .~ 0 e . . . . . . . .

A1EI°tl + A2E2tx 2 /

0 e

. . . . . . . . . . . . . . . . ( 5 )

If E 2 = PE1, where p is the modular ratio of the E values of concrete and steel, then equation (5) becomes:

( AI~101 + A2P~202_~ . . . . . . . . . . . . . . . . 0 e . . . . . . ( 6 )

Al0t 1 + A2PtX 2 ]

If the bridge is of homogeneous construction, then E 1 = E 2 = E, and Otl=0t2=~, and equation (6) reduces to:

\ A,+A2 / ~--" (Oe +00) m Ik' (01 +O°)+k2 (02 +0°)1 =~101 +A202~+00" (AI +A2) AI +A 2 ) ( 7 )

From equations (6) and (7) it can be sden that, if the cross-section of the bridge is split up into areas, the

temperatures of which are known, then the effective temperature of the bridge can be calculated by summing

the products of the areas and their mean temperatures, and dividing by the total area of the cross-section of

the bridge. (In the case of the composite construction it is necessary to know the values of the coefficients

of expansion, and the ratio of the E values of the materials.)

The equality on the right hand side of the -----sign in equation (7) is intended to show that, if0 o is some datum temperature (e.g. the daily minimum effective bridge temperature), then the effective bridge temper-

ature at any time is obtained by adding to the datum temperature the change in temperature between the

time of the datum temperature and the required time.

36

In the above analysis it has been assumed that no thermal bending takes place, and the change in

length of the bridge has been considered in the longitudinal direction only. As the effective bridge temper-

ature is the temperature which controls the longitudinal movement of the bridge at the position of the

neutral axis, both thermal bending and lateral movement will have a second order effect on the calculation

of this temperature. This is particularly true of the multi-span continuous bridges, on which the research described in this Report is based, since significant longitudinal bending will only occur in the end spans. The

combined effect of bending and lateral movement will be considered in more detail in future research

concerned more particularly with thermal bending and thermal stresses.

37

11. APPENDIX 2

MODIFICATIONS TO THE METHOD OF CALCULATION GIVEN IN LR 561

For ease of cross-referencing, the main paragraph headings refer to the relevant Sections in the main text of the Report.

11.1 Sections 3.3 and 4.1.1

The temperature difference distributions shown in Figures 3 to 8 have been derived from temperature

distributions calculated using the method described in LR 561, with the amendments discussed in Sections

11.1.1 and 11.1.2.

11.1.1 Thermal properties of surfacing and concrete. It is well known that the thermal conduct- ivity, (k), density, (P) and specific heat, (c), of both concrete and surfacing materials can have a wide range

of values. For example, the thermal conductivity of concrete depends upon: the mix proportions, the type

of cement, the type of aggregate, the moisture content, the air content and the compaction. A set of values

for structural concrete (described in LR 561 as 'average values') is:

k = 1.4W/m°C

2400kg/m3 1 giving the diffusivity, K, - k - 0.6 x 10"6m2/sec,

c 960J/kg°C ] pc

and a set of 'average values' for surfacing material is:

k = 0.9W/m°C

P 2300kg/m3 f giving K = 0.5 x 10 -6 m2/sec.

c 840J/kg°C

(Hunt and Cooke 8 use a value of c = 1675J/kg°C for surfacing - a much higher value than any found

elsewhere.)

As these two sets of values were reasonably similar it was felt that, with the possible overlap of the

various ranges of values, it was probably accurate enough to assume that the thermal properties of concrete

and surfacing were similar enough for a surfacing/concrete layered system to be regarded as a homogeneous

concrete system. Thus, instead of increasing the value of the surface heat transfer coefficient, ht, to

46W/m2°C, to allow for the surfacing, it was kept at 23W/m2°C (the 'correct value'). The value of the

absorption coefficient, r, was reduced from 0.9 to 0.85, and bridge temperatures were calculated under

these conditions. The values of these calculated temperatures agreed well with measured values, suggesting

that the assumption of a homogeneous system was valid. Unfortunately, due to lack of data on surfacing

temperatures, it is not possible to say how accurate the temperatures in the surfacing are - except to

observe that they seem to be 'reasonable'.

11.1.2 Starting conditions. The starting conditions were amended as follows:

1. For depths of construction (includingthe surfacing) up to, and including 0.25m, the correct shade

temperature (derived from the shade temperature variation) was used as the starting condition at 0700 hours 2,6.

38

. For depths of construction (including the surfacing) from 0.30m to 0.45m, the starting temperature was

taken as the 0800 hour shade temperature (derived from the shade temperature variation) + 2°C.

. For depths of construction (including the surfacing) greater than 0.5m the starting condition was

taken as the 0800 hour shade temperature (derived as in 2 above) + 4°C.

The reason for 'staggering' the starting conditions in this way is because the more massive a construction

is, the more likely that it will be warmer than the shade temperature at 0800 hours (under the environmental

conditions described in Section 3.1). This has been confirmed by measurements 6. The influence of the

starting conditions on the calculated temperatures is to be discussed more fully in another Report.

11.2 Section 4.1.2

The starting condition used, for all depths of construction, under all depths of surfacing is as described in LR 561, i.e. the starting temperature is taken as the value of the shade temperature at 1600 hours. No

allowance is made for massiveness of construction, for, under the environmental conditions described in

Section 3.2, it is assumed that the above starting condition is correct for any depth of construction.

11.3 Table 2

The computer program which calculates night-time temperatures ends the calculation at 0845 hours,

because by this time the solar radiation is taking over and the program has not yet been amended to

accept the 'cross-over' from out-going night-time re-radiation to incoming solar radiation. At 0845 hours

the reversed temperature differences in the more massive depths of construction, under 200mm of surfacing,

had not reached a maximum value, but they were increasing so slowly that the values given in Table 2 ~ e

not likely to be in error by more than l°C. (In practice, by 0845 hours the solar radiation would be taking

over, and the reversed temperature differences would be decreasing as the surface temperature increased.)

11.4 Section 4.2.1.1

The amendments to the method of calculation given in LR 561, and the reasons for these amendments

are as described in Sections 11.1.1 and 11.1.2.

11.5 Section 4.3.1

Top flange plate temperatures of steel box decks under 20mm of surfacing have been derived by

extrapolating between the temperatures calculated, using the method described in LR 561, for unsurfaced

steel and steel under 40mm of surfacing. (It has been assumed that the effect of 40mm of surfacing is the

same as the effect of 38mm of surfacing, as measurements are only available for the latter depth.)

39

(2008) Dd443233 1,800 11/77 H P L t d S o ' t o n G1915 PRINTED IN ENGLAND

ABSTRACT

Temperature differences in bridges: basis of design requirements: MARY EMERSON, BSc: Department of the Environment Department of Transport, TRRL Laboratory Report 765: Crowthorne, 1977 (Transport and Road Research Laboratory). It is shown how the magni- tude of temperature differences in both concrete bridge decks and the concrete deck slabs of composite bridge decks under various depths of surfacing may be determined by using a simple modification to the method of calculation of bridge temperatures described in TRRL Report LR 561. Empirical methods are used to estimate the magnitude of temperature diff- erences in the steel areas of composite decks and in steel box decks.

Values of effective bridge temperatures likely to co-exist with maximum positive and reversed temperature differences are given, and the influence of depth of surfacing on mini- mum and maximum effective bridge temperatures is discussed.

The clauses relating to temperature differences in a new British Standards Institution document on bridge loading, BS 5400, are based on the information contained in thisRepo- rt, and the Department of Transport Technical Memorandum No. BE 1/77 is to be revised to include this information.

ISSN 0305-1293

ABSTRACT

Temperature differences in bridges: basis of design requirements: MARY EMERSON, BSc: Department of the Environment Department of Transport, TRRL Laboratory Report 765: Crowthorne, 1977 (Transport and Road Research Laboratory). It is shown how the magni- tude of temperature differences in both concrete bridge decks and the concrete deck slabs of composite bridge decks under various depths of surfacing may be determined by using a simple modification to the method of calculation of bridge temperatures described irt TRRL Report LR 561. Empirical methods are used to estimate the magnitude of temperature diff- erences in the steel areas of composite decks and in steel box decks.

Values of effective bridge temperatures likely to co-exist with maximum positive and reversed temperature differences are given, and the influence of depth of surfacing on mini- mum and maximum effective bridge temperatures is discussed.

The clauses relating to temperature differences in a new British Standards Institution document on bridge loading, BS 5400, are based on the information contained in thisRepo- rt, and the Department of Transport Technical Memorandum No. BE1/77 is to be revised to include this information.

ISSN 0305-1293