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Phil Bamforth 1 08 February 2010

The development of a revised unified approach for the designof reinforcement to control cracking in concrete resultingfrom restrained contraction

FINAL REPORT

Dr Phil Bamforth, Independent ConsultantDr Steve Denton, Parsons BrinckerhoffDr Jonathan Shave, Parsons Brinkerhoff

ICE Research Project 0706February 2010

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Forward

A study has been undertaken to develop a unified method for estimating crack widths

in reinforced concrete elements subject to restrained contraction. The project hasbeen supported by the Institution of Civil Engineers through their Research &Innovation Enabling Fund, The Concrete Centre and the Highways Agency.

The investigation has identified deficiencies in the current methods (BS8007 andEN1992-3) of calculation of crack widths in members subject to continuous edgerestraint resulting from assumptions which may not accurately reflect the way inwhich cracking develops in practice. In particular the role of edge restraint incontrolling cracks may not have been represented correctly.

The unified approach assumes a two stage cracking process. Initial (STAGE 1)cracking is estimated using a calculation based on the current method of EN1992-3

for end restraint only, with a modification to take account of the effect of edgerestraint in both attracting part of the load from the concrete when a crack occurs andin controlling the subsequent development of the crack. Subsequent crack growth(STAGE 2) is based on the continued contraction of the concrete relative to thereinforcement, with the assumption that higher restraint reduces the extent to whichthe crack may open.

A comparison of estimated crack widths with values observes in the field indicatesthat the proposed unified approach provides a basis for the development of a newapproach to design for reinforcement for controlling crack widths caused by restraintto contraction.

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Contents

List of Figures ............................................................................................................ 5List of Tables ............................................................................................................. 91. Introduction ...................................................................................................... 102 The current method for estimating crack width ................................................. 113 The concept behind the revised approach ........................................................ 134 Proof of the concept ......................................................................................... 15

4.1 Observed development of cracking ........................................................... 154.2 Studies by Kheder ..................................................................................... 174.3 Results of FE analysis ............................................................................... 18

5 Limitations of the current approach to design ................................................... 22 5.1 Independence of cracks ............................................................................ 225.2 Crack spacing ........................................................................................... 225.3 Stress in the reinforcement ....................................................................... 235.4 The minimum area of reinforcement .......................................................... 245.5 Strength class ........................................................................................... 245.6 Difficulty in achieving efficient conforming designs using BS8007 andEN1992 ............................................................................................................... 25

6 Development of the revised unified method of design ...................................... 266.1 Objectives ................................................................................................. 266.2 The initial simplified approach ................................................................... 266.3 The two stage process .............................................................................. 286.4 Development of expressions for Stage 1 cracking ..................................... 28

6.4.1 Cracking under end restraint according to EN1992-3 ......................... 286.4.2 Effect of element length...................................................................... 296.4.3 Application to continuous edge restraint ............................................. 32

6.5 Development of expressions for Stage 2 cracking ..................................... 337 Critical parameters for predicting crack width ................................................... 35

7.1 Tensile strength......................................................................................... 357.2 Modulus of elasticity and creep ................................................................. 38

7.2.1 Estimating the risk of cracking ............................................................ 387.2.2 Estimating (sm- cm) .......................................................................... 38

7.3 Estimating continuous edge restraint ......................................................... 387.3.1 The nature and magnitude of edge restraint ....................................... 387.3.2 Estimating the magnitude of continuous edge restraint ...................... 397.3.3 Variation in restraint with distance from the joint ................................ 41

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7.3.4 Restraint at the point of maximum restrained strain and crack width .. 428 Validation ......................................................................................................... 459. The effect of edge restraint on crack widths calculated using the current and therevised method ........................................................................................................ 4710 Combining crack widths due to early-age thermal restraint and other actions .. 48

10.1 Requirements for combining crack widths ................................................. 4810.2 Stress in the reinforcement due to early-age cracking ............................... 4810.3 Examples of combined loading .................................................................. 48

10.3.1 Cylindrical tank ................................................................................... 4910.3.2 Elements subject to bending .............................................................. 5010.3.3 The proposed method for combining loads......................................... 51

10.4 Summary................................................................................................... 5311. Conclusions ..................................................................................................... 5412. Limitations and Recommendations ................................................................... 5513. References ....................................................................................................... 56APPENDIX 1 - Derivation of the expression for the strain sm - cm used in EN1992-3for estimating crack width in a member subject to end restraint ............................... 58APPENDIX 2 - Derivation of the steel stress in a member restrained at its ends aftera single crack has relieved the stress in the uncracked section ............................... 61APPENDIX 3 - Derivation of the steel stress in a member restrained along its edge

after a single crack has relieved the stress in the uncracked section ....................... 64

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NotationAc Concrete cross sectional area

Act Area of concrete in the tensile zone used in the calculation of minimum area

of reinforcement

Ac,eff Effective area of concrete in tension surrounding the reinforcement used inthe calculation of crack spacing

Ao Cross sectional area of old (restraining) concrete section

An Cross sectional area of new (restrained) concrete section cast against Ao

As Area of reinforcement

As,min Minimum area of tensile reinforcement

B A term describing the relative load bearing capacity of the concrete and the

reinforcement in a member

++++==== 1

kkB

e

c

c Cover to reinforcement

Ecm Mean elastic modulus of concrete at 28 days

Ecm(t) Mean elastic modulus of concrete at t days

Ec,eff Mean elastic modulus of concrete effective at the time of cracking

Es Elastic modulus of reinforcement

Eo Elastic modulus of old (restraining) concrete section

En Elastic modulus of new (restrained) concrete section

fb Steel-concrete bond strength

fct,eff Mean value of tensile strength effective at the time when cracks may first

occurfctm Mean concrete strength in tension at 28 days

fctm(t) Mean value of tensile strength at time, t, if cracking is expected earlier than 28days

fky Characteristic strength of reinforcement

h Section thickness

H Height of an element

K1 Coefficient for the effect of creep on stress relaxation

k Coefficient which allows for the effect of non-uniform self-equilibrating stresswhich lead to a reduction in restraint forces

kc Coefficient which takes account of stress distribution within a sectionimmediately prior to cracking

k1 Coefficient which takes account of the bond properties of reinforcement

ka Coefficient which takes account of the uncertainty of the age at ewhichcracking occurs

kis Coefficient to account for the difference between the strength of a testspecimen and the strength in situ

kL The ratio of the natural crack spacing of unreinforced concrete to the height ofthe member

kw A coefficient applied to the maximum potential crack width achieved under

end restraint to take account of the effect of continuous edge restraint inpreventing the crack from fully opening

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L Length of an element

n Number of cracks

Leff The effective length of a member over which strain relief can occur betweencracks

R Restraint factor

Rax EN1992-1-1 notation for factor defining the external restraint

Redge Restraint factor from continuous edge restraint

Rend Restraint factor from end restraint

Rj Continuous edge restraint at the joint between new and old concrete

Rwmax Edge restraint at the point within a member at which the maximum crack widthoccurs

S Length of debonding of reinforcement at the location of a crack

Sn The natural crack spacing occurring in an edge restrained member with noreinforcement

Srm Mean crack spacing

Sr,max Maximum crack spacing, defined as the characteristic crack spacing with a5% probability of being exceeded

S0 Minimum crack spacing

T Tension force in a member

T1 Difference between the centreline peak temperature in a member and themean ambient temperature

wk Crack width defined in EN1992-1-1 with a 5% probability of being exceeded

wp The maximum potential crack width occurring under end restraint according toEN1992-3

wk1 Stage 1 crack width based on the load transferred from the concrete to thereinforcement

wk2 Stage 2 crack width based on the residual contraction of the concrete relativeto the steel after Stage 1 cracking has occurred.

c Free coefficient of thermal expansion of concrete

ct Coefficient taking account of the effect of sustained loading and otherunfavourable n situ effects

e The modular ratio

ca Autogenous shrinkage strain

cm Mean strain in concrete within the length of debonding

ctr Residual tensile strain in uncracked concrete, i.e. in areas where full bondexists between concrete and reinforcement

cr Crack-inducing strain in concrete defined as the restrained strain less theresidual strain in the concrete within the debonded zone.

free The strain which would occur if a member was completely unrestrained

r Restrained strain in concrete

r Residual contraction in concrete within the debonded zone after Stage 1cracking has occurred

s Strain in reinforcement

sm Mean strain in steel reinforcement

smax Maximum strain in reinforcement at the crack location

smaxr Maximum residual strain in reinforcement at the crack location

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ctu Ultimate strain capacity of concrete in tension (tensile strain capacity)

Microstrain = 1 x 10-6mm/mm

Percentage of steel based on the area of concrete in tension Act

p,eff Effective steel percentage based on the area Ac,eff

s Stress in the reinforcement

Bar diameter

Glossary

Crack-inducing strain The component of restrained strain that is relieved when acrack occurs and is exhibited as crack width. This is therestrained strain less the residual strain in the concrete withinthe zone of debonding of reinforcement after a crack has

occurred.

Crack width The crack width at the surface of the concrete. Maximumcrack widths designed to EN1992-1-1 are design targetcharacteristic values with only a 5% chance of beingexceeded.

Early-age Typically up to 7 days

Free strain The strain that would occur in the concrete if it was completelyunrestrained

Microstrain 1 x 10-6strain

Restrained strain The component of free strain which is restrained and whichgenerates stresses in the concrete

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List of FiguresFigure 1 Types of restraint dealt with by EN1992-3................................................. 11Figure 2 The strain distribution in the steel and the concrete after cracking in amember subject to direct tension............................................................................. 11Figure 3 The effect of edge restraint on crack distribution and crack size............... 13Figure 4 Early-age thermal cracking in a 500mm thick wall.................................... 15Figure 5 The relationship between date of casting and number of cracks................ 16Figure 6 The relationship between date of casting and mean crack width.............. 16Figure 7 The relationship between the length of a wall and the height of the maximumcrack width [5, 6]and restrained strain [7]................................................................ 17Figure 8 The relationship between crack spacing and restraint [from interpretation

data in refs 5 and 6]................................................................................................. 18

Figure 9 Estimated restraint using the model described in CIRIA C660.................. 19Figure 10Modelling the imposition of edge restraint................................................ 19Figure 11Strain contours from the FE model corresponding to half the crack spacing................................................................................................................................ 20Figure 12 Reduction in crack width for a fixed crack spacing corresponding to theimposition of edge restraint..................................................................................... 21Figure 13 A typical crack pattern in a 19m long wall............................................... 22Figure 14 Observed crack pattern in a base-restrained wall [5].............................. 23Figure 15 The relationship between the bar diameter, the bar spacing, the area ofreinforcement and the estimated crack width to BS8007 (300mm wall using C30/37)................................................................................................................................ 25Figure 16 Comparison of estimated and reported crack widths.............................. 27Figure 17 Comparison of estimated and reported crack widths using currentlyavailable methods................................................................................................... 27Figure 18 An element subject to end restraint only................................................. 29Figure 19The effect of length on the crack width immediately after cracking (i.e.before additional contraction).................................................................................. 30Figure 20 Crack development for R = 0.4 [NB The max estimate includes the factorct= 0.80 in the derivation while the EN1992 estimate does not]............................. 31Figure 21 Crack development for R = 0.39 [NB The max estimate includes the factorct= 0.80 in the derivation while the EN1992 estimate does not]............................. 31Figure 22 An element subject to continuous edge restraint.................................... 32Figure 23 Development of stage 2 cracking............................................................ 33Figure 24 The mean tensile strength development, fctm(t) for CEM I concreteaccording to EN1992-1-1......................................................................................... 36Figure 25 Distribution of early-age in situ tensile strength for C30/37 concrete [4].. 37Figure 26 Early thermal cracking in the walls of a box-section tunnel wall............... 39

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Figure 27 The variation in elastic modulus at early age and the influence on the ratioof En/Eo and the restraint derived using equation 11 [It is assumed in this examplethat the ratio An/Aoequals 1]................................................................................... 40Figure 28 A comparison of restraint values from ACI 207 [15] and Emborg [18]..... 41Figure 29 Comparison of estimated and measured restraints using the ACIapproach and the revised Emborg model [En/Eo= 0.7]............................................ 42Figure 30 The CIRIA C660 calculator for continuous edge restraint modified to showthe restraint at the point of maximum crack width.................................................... 43Figure 31 Estimated restraint using the CIRIA C660 calculator for a 3m high wall ofvarying length (An/Ao= 1)........................................................................................ 44Figure 32 Estimated restraint using the CIRIA C660 calculator for a 3m high wall ofvarying length (An/Ao= 1)........................................................................................ 44Figure 33 Comparison of estimated and measured crack widths assuming a length

coefficient kL= 1.5................................................................................................... 45

Figure 34 Estimated stage 1 and stage 2 crack widths and the total compared withmeasured values..................................................................................................... 46Figure 35 The effect of restraint on the estimated crack width................................ 47Figure 36 Distribution and combination of strains in a water tank........................... 49Figure 37The effect of imposed actions on early age cracking in cantilever slabs ona bridge beam......................................................................................................... 51Figure 38Concrete and steel strains adjacent to a crack, indicating influence ofimposed loading on crack width............................................................................... 52List of TablesTable 1 Expressions for (sm - cm) given in EN1992-3 for continuous edge restraint(equ.2) and end restraint (equ.3)............................................................................. 12Table 2 Input data for the probabilistic analysis for strength class C30/37.............. 37Table 3 Estimated in-situ tensile strength at early age compared with valuescalculated in accordance with EN1992-1-1.............................................................. 37

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1. IntroductionEN1992-3:2006, [1] for the design of liquid retaining and containment structures, isreplacing BS8007 [2] , to provide the method for the design of reinforcement to

control cracking resulting from restrained contraction. Two particular conditions ofrestraint are considered, end restraint and continuous edge restraint. While thesetwo conditions are dealt with in the same way in relation to the initiation of cracking(which occurs when the restrained contraction exceeds the tensile strain capacity ofthe concrete), the post cracking behaviour is dealt with very differently.

When cracking occurs due to end restraint, the load from the concrete is assumed tobe transferred entirely to the reinforcement and the resulting crack width is estimatedfrom the average strain in the reinforcement over the length over which de-bondingoccurs. In this case, increasing contraction leads to an increase in the number ofcracks but is assumed to have no effect on the maximum crack width other than thatassociated with the increase in tensile strength of the concrete as it gets older. This

condition is not dealt with in BS8007.

When cracking occurs due to continuous edge restraint the crack width is currentlyassumed to be strain limited and in direct proportion to the magnitude of therestrained contraction. In this case it is assumed that increasing restrainedcontraction leads to an increase in the crack width but has no effect on the number ofcracks, which are assumed to be independent of one another. In this case EN1992-3has adopted broadly the same approach as BS8007.

The idea for a revised approach grew initially from questioning the need to have twovery different design approaches to deal with variations of essentially the samephenomenon (i.e. restrained contraction) the only difference being the boundaryconditions. However, as the study proceeded it became apparent that many of theassumptions underlying the current design method for continuous edge restraint maynot be sufficiently robust, providing further support for the need for a unified designmethod which reflects more reliably the way in which cracking develops in practice.

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2 The current methods for estimating crack width

EN1992-3 provides methods for the design of reinforcement for crack control. Itdeals with two specific conditions of restraint as shown in Figure 1.

(a) restraint of a member at its ends (b) continuous restraint along one edge

Figure 1 Types of restraint dealt with by EN1992-3

EN1992-3 refers to EN1992-1-1 [3] (expression 7.8) for the calculation of crack width.The characteristic crack width (95 percentile) wkmay be estimated using the generalexpression;

wk= Sr,max(sm - cm) (1)

where Sr,max is the maximum crack spacingsm is the mean strain in the steel over the length Sr,maxcm is the mean strain in the concrete over the length Sr,max

The assumed strain distributions in the steel and in the concrete after cracking arebased on a member in direct tension, as illustrated in Figure 2.

sm

c

s

cm0.5ctu

= 0

ctu

smax

(sm cm)

Sr,max

Figure 2The strain distribution in the steel and the concrete after cracking in a membersubject to direct tension

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The maximum stress in the steel smaxoccurs at the location of the crack and reducesas bond with the concrete is re-established over a distance 0.5 Sr,max on either side ofthe crack.

The strain in the concrete is zero at the crack and is assumed to increase up to amaximum potential value equal to the tensile strain capacity of the concrete when fullbond with the reinforcement has been established. When the strain in the concreteexceeds this value another crack is formed.

The crack width is therefore calculated as the mean extension of the reinforcementover the length of debonding less the residual tensile strain in the concrete over thesame length.

Equation 1 is applied to both end restraint and continuous edge restraint conditionsbut the way in which (sm - cm)is estimated differs significantly for the two conditions.Expressions provided in EN1992-3 for (sm - cm)are given in Table 1.

Table 1 Expressions for (sm - cm) given in EN1992-3 for continuous edge restraint(equ.2) and end restraint (equ.3)

Symbol

Continuous edge restraint End restraint

(sm - cm) = Raxfree(equ.2)

++++==== 1

1

E

fkk0.5-(

es

eeffct,ccmsm )

(equ.3)

Expression M.3 of EN1002-3 Expression M.1 of EN1992-3

Rax Restraint

free Free contraction

fct,effEffective tensile strength of concrete at thetime of cracking

Reinforcement ratio

e Modular ratio

Es Modulus of elasticity of reinforcement

k, kc

Coefficients for stress distribution and theeffect of self-equilibrating stresses (seeEN1992-1-1, 7.3.2 Minimum reinforcementareas)

Comparing the two expressions for (sm - cm)indicates immediately the differencebetween the two approaches. No two parameters are common to both expressions.What is most surprising is that for continuous edge restraint, no account is taken ofthe load transferred from the concrete to the steel when cracking occurs (i.e. asinfluenced by the tensile strength of the concrete or the steel ratio) despite the factthat this is the only consideration when estimating the minimum area ofreinforcement for both restraint conditions.

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3 The concept behind the revised approach

The proposed unified approach is based on the assumption that the maximumpotential crack width wpis that which may occur under conditions of end restraint,

and that when continuous edge restraint is applied, wpis reduced to take account ofthe following;

a. Part of the load from the concrete is transferred into the restraining member,thus reducing the stress transferred to the reinforcement

b. The edge restraint will inhibit the extent to which a crack may open. The higherthe edge restraint, the less strain relief that may occur and hence the smallerthe crack width that may develop.

c. A new crack may be influenced by the presence or lack of existing crackswhich may determine the degree of stress relaxation between the cracks

The concept may be explained by considering the following simplified exampleshown in Figure 3 for an element in direct tension. If the length of the specimen ismarginally greater than the crack spacing then, with sufficient load, a single crackmay be expected as shown in Figure 3a. Now consider the element subject tocontinuous edge restraint and how the cracking will develop (Figure 3b).

a) Element subject to direct tension

EDGE RESTRAINT

EDGE RESTRAINT

b) Element subject to uniaxial tension and edge restraint

Figure 3 The effect of edge restraint on crack distribution and crack size

In this very simple case it is clear that the when subject to edge restraint, the singlecrack may be prevented from opening to its full potential and a number of smallercracks may develop. The edge restraint is therefore acting in a similar way toreinforcement by attracting some of the load and distributing the cracks. Thisconcept provides the basis for the revised unified approach.

In the revised approach the restraint is assumed to influence not only the magnitude

of restrained strain, and hence the risk of cracking, but also the way in which cracksare distributed. Unlike the current approach of EN1992-3 (and BS8007) whichassumes that wider individual cracks may be formed with higher restraint, the revised

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approach considers that the restraint may act to prevent cracks opening to their fullpotential and hence, while higher restraint will increase the risk of cracking and leadto a greater integrated crack width, higher restraint may also act to distribute theintegrated crack width over a larger number of smaller individual cracks.

So, while the current basis for design for control of cracks due to continuous edgerestraint asks the question What causes cracks to be as wide as they are?therevised approach poses the question What prevents cracks achieving their potentialwidth?This places an entirely different emphasis on the effect of edge restraintcompared with BS8007 and EN1992-3.

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4 Proof of the concept

The hypothesis that restraint inhibits crack opening has been examined bycomparing predicted performance with observed crack development; by investigating

theories on crack development proposed by other researchers; and by undertakingFE analysis.

4.1 Observed development of cracking

The basis for design according to EN1992-3 (and BS8007) predicts the following;

For end restraint, increasing restrained contraction will increase the number ofcracks but have no effect on individual crack widths, which are determined bythe load transferred from the concrete to the steel when a crack occurs

For continuous edge restraint, increasing the restrained contraction will cause

the individual crack widths to increase but have no effect on the number ofcracks, which are assumed to be independent.

Cracking behaviour was observed on a structure comprising a number of similarwalls, subject to continuous edge restraint as shown in Figure 4, cast over a period ofabout 4 months.

550mm

500mm

Horizontal bars20@125 both facesCover = 65mm

2650mm

500mm350 x 350mm splay

Full slab width = 26100mm

Figure 4 Early-age thermal cracking in a 500mm thick wall

The construction started in late summer when both ambient and concrete mixtemperatures were high and the T1value (i.e. the drop from the peak temperature tomean ambient temperature) was reported to be about 35oC. Construction continuedto late in the year when the ambient temperature had dropped significantly, leading to

much lower mix temperature and a T1value of about 20

o

C with reduced thermalcontraction. Results are presented in Figure 5 and 6 showing the number of cracks ineach wall and the mean crack width respectively plotted against the date of casting.

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0

2

4

6

8

10

12

14

16

18

19/08

26/08

02/09

09/09

16/09

23/09

30/09

07/10

14/10

21/10

28/10

04/11

11/11

18/11

Date (dd/mm)

Numberofc

racks

Figure 5 The relationship between date of casting and number of cracks

Figure 5 illustrates very clearly that the number of cracks reduced as the thermalcontraction reduced in the cooler months. For continuous edge restraint theassumption within the current method of design that cracks are independentsuggests that the number of cracks would be unaffected by the magnitude ofrestrained contraction, although the crack width would be expected to reduce. Asshown in Figure 6, while the mean crack width reduced marginally the reduction wasless than predicted using the methods of EN1992-3 (modified in accordance withCIRIA C660 [4]) and BS8007.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Meancrackwidth

(mm)

Date (dd/mm)

Results

EN1992-3/C660

BS8007

Linear (Results)

35oC 20oC30oC 25oC

Assumed T1 values

Linear best f itof results

Figure 6 The relationship between date of casting and mean crack width

The observed behaviour of cracking in these walls subject to continuous edgerestraint indicates therefore that the crack development (in relation to the number ofcracks) is most consistent with the performance currently expected of memberssubject to end restraint. The relationship between crack width and the magnitude ofrestrained strain is also less strong than expected based on the assumption inEN1992-3 (and BS8007) that the crack width is directly proportional to the magnitudeof restrained contraction, although the latter does appear to have some influence oncrack width.

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4.2 Studies by Kheder

Kheder [5, 6] has published extensive data for crack widths and has proposed ananalytical method for design. While it is felt that the analysis proposed by Khederdoes not capture fully the mechanics of cracking, the presentation of the

experimental results is useful in supporting the concept behind the revised approach.

For walls, Kheder has suggested that the primary crack spacing is a function not onlyof the reinforcement but also of the wall geometry and in particular the height (H). Inaddition, reported crack width data suggest that the maximum width may be likely tooccur at height which is about 0.1Labove the joint (L= length of wall). This isconsistent with values of restrained strain in uncracked walls reported by Anson et al[7] which achieved a maximum value at about the same relative height (Figure 7).

y = 0.1033x

R2= 0.8395

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20 25

Length (m)

Height(m)

Anson et al - max restrained strain

Kheder - max crack width

Figure 7 The relationship between the length of a wall and the height of the maximumcrack width [5, 6]

and restrained strain [7].

It is recognised in current methods for estimating restraint in walls that restraint is afunction of the wall height in relation to both its cross sectional area and thelength/height ratio[4]. The crack spacing (and hence crack width) and restraint are

linked, therefore, through their relationship with the wall geometry. Using datapresented by Kheder [5, 6]the relationship shown in Figure 8 has been obtained.Increased restraint reduces the crack spacing leading to an increase in the number ofcracks and hence a reduction in the width of individual cracks.

This is consistent with the hypothesis that higher edge restraint will reduce crackwidth.

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0

500

1000

1500

2000

2500

0.35 0.40 0.45 0.50 0.55 0.60

Restraint

Maximumcrackspaci

ng(mm)

Figure 8 Example of the relationship between crack spacing and restraint [frominterpretation data in refs 5 and 6]

4.3 Results of FE analysis

A series of finite element models were developed with the aim of investigating theeffect of edge restraint on crack width and crack spacing [8]. The modelling sought toinvestigate the validity of the hypothesis that increased edge restraint reduces crackwidths by distributing the restrained strain over a larger number of smaller cracks,and to provide evidence to develop the unified approach further. This was done bycomparing the crack widths obtained with edge restraint in place with the crackwidths corresponding to an end-restrained condition.

The modelling work considered the case of cracking in long walls, which have theworst restraint conditions leading to full height cracks.

Consider a long wall that cracks somewhere in the central region, with the crackextending to full height. Ignoring for now the effect of reinforcement, the crackeffectively creates two walls of shorter length. If further cracking occurs until the wallis fully cracked we may end up with a series of short walls, each with a lengthcorresponding to the crack spacing. If the crack spacing is less than two times theheight of the wall, then restraint factors given in EN1992-3 and those estimated

using the method described in CIRIA C660 (Figure 9), suggests that the strains at thetop of the wall (and therefore the crack width for a given crack spacing) are notinfluenced by the edge restraint.

This premise has been investigated further with a series of FE models. The modellingsought to investigate the effect of imposing an edge restraint on a wall that has hadthe edge restraint temporarily removed, using the principle of superposition. Theprocess is illustrated in Figure 10 and is described as follows:

1) A long wall subject to edge restraint has cracked at a certain crack spacing.The edge restraint is now temporarily removed by releasing the axial force inthe base at the crack positions. The cracks will then open up to the crack

width wpcorresponding to end-restrained conditions.

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0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Restraint

ProportionalHeig

ht

842

1

L/H3

Figure 9 Estimated restraint using the model described in CIRIA C660

Figure 10Modelling the imposition of edge restraint

wp

s

w = p-0.5w

Imposed deformation

0.5 p 0.5wp

Equivalent to end -restrained member of length s

w

s

w = -0.5w

Imposed deformation

0.5 0.5

Equivalent to end -restrained member of length s

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2) The edge restraint may then be reintroduced to a model corresponding to aportion of wall between cracks. Due to symmetry, the mid point betweencracks has zero deflection and may be considered to be fixed in the model, sothat the width of the model is half the crack spacing. The effect of edgerestraint is included by adding a stiff beam to model the presence of the baseand then pulling the base of the wall back to its original position. This is doneby imposing a deflection of 0.5wp at the base of the wall portion adjacent tothe crack.

3) The deflection at the top of the wall 0.5warising from the deformation in 2)indicates half the reduction in crack width at the top of the wall due to theedge restraint.

4) The factor by which the maximum crack width has been reduced by the edgerestraint is calculated as

k = (wp -w)/ wp (4)

An example of the output from the FE analysis is shown in Figure 11

Figure 11Strain contours from the FE model corresponding to half the crack spacing

It was expected that the primary variable affecting the factor k would be the ratio sr/H,where sris the crack spacing and His the wall height. The potential effects of thewall thickness h/Hand the Poissons ratio were also investigated using sensitivityanalyses. As expected, the wall thickness and Poissons ratio both had very littleeffect on the results and the primary effect was due to the crack spacing. This isillustrated in Figure 12.

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6 7 8

Crack spacing/Wall height

w/w

p

Figure 12 Reduction in crack width for a fixed crack spacing corresponding to theimposition of edge restraint

As shown in Figure 12, for crack spacing less than two times the wall height, the

maximum crack width (at the top of the wall) was not affected by the imposition ofedge restraint. This finding is consistent with the predicted result implied by Figure 9,i.e. that the edge restraint does not affect the crack width at the top of the wall for agiven crack spacing, when that crack spacing is less than twice the height.

These findings indicate that there does seem to be a relationship between therestraint factor, which remains higher towards the top of longer walls (compared with

shorter walls), and the reduction in crack width. The reduction in the crack width wcould theoretically by derived from an integration of the restraint factors along the topof the wall (or at any other height of interest) between crack positions. Since theserestraint factors will vary between 0 and Rax(where Raxis the restraint factor midwaybetween cracks) this would suggest that:

rfreeax sRw (5)

As the reduction in crack width wwill increase with the magnitude of restraint, thissupports the hypothesis that higher restraint will lead to smaller cracks.

In practice, cracks do not always extend to the full height of the wall and an attemptwas also made to model the behaviour of partial height cracks. However, any finiteelement model that contains an explicit crack will exhibit a singularity at the tip of thecrack, where the theoretical tensile stress becomes very large locally. In theory thistensile stress will always exceed the tensile strength, leading to extension of the

crack to the full height of the wall. Fracture energy models were attempted to predictthe pattern and gradual growth of cracks by defining material parameters for fractureenergy and initiation stress. At a local level there will always be cracks at the bottomof the wall at the interface with the base, since at this position there is a discontinuityin strain. However, only some of these will develop and grow into observable cracks.Therefore, instead of seeking to predict where cracking occurs, the analysis aimed tomodel whether the existing cracks would grow, and how far they grow. This analysiswas only partially successful in modelling the cracking behaviour. Problems withconvergence of the model had to be addressed, which then required unrealistic inputparameters to be used. However, the general behaviour of certain cracks growingand others not growing was observed but in view of the limited scale of thisprogramme modelling of partial height cracking was not pursued further.

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5 Limitations of the current approach to design

As part of the study a thorough investigation was undertaken of both the assumptionsunderlying the current approach to design for edge restraint and its application [9,10].

5.1 Independence of cracks

Both BS8007 and EN1992 assume cracks to be independent, with the crack widthbeing determined by the relief of restrained strain within the zone over whichdebonding occurs (Figure 2). Hence it is assumed that in the area beyond Sr,max,there is no strain relief as a result of the crack occurring which could lead to the crackwidth changing. Is this a reasonable assumption? With full restraint this may beacceptable but in most practical cases only partial restraint exists and strain reliefmay occur over some distance beyond Sr,max, thus influencing both the extent towhich the existing crack may open and the location of subsequent cracks. Thishypothesis is supported by observations of crack patterns, a typical example of which

is shown in Figure 13. Within the 19.5m long wall, while there were areas in whichthe crack spacing was consistent with the design, there were also large uncrackedareas spanning over 3-4m. This suggests that cracking may have resulted in stressrelief over a much larger area than assumed by the theory behind BS8007 andEN1992.

0.2 0.1

0.15

0.15

0.150.15

0.1 0.2 0.2

0.2

0.15 0.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.15

0.150.15

0.15

0.15

0.1

0.1

0.1

0.2

0.20.15 0.2

0.20.2 0.1

0.15

0.15

0.150.15

0.1 0.2 0.2

0.2

0.15 0.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.15

0.150.15

0.15

0.15

0.1

0.1

0.1

0.2

0.20.15 0.2

0.2Strain relief?Strain relief?

Figure 13 A typical crack pattern in a 19m long wall

5.2 Crack spacing

BS8007 and EN1992-3 both make the assumption that the crack spacing isdetermined by the reinforcement alone and that (accepting the normal variability) thecracks will be equally spaced and of equal size). Results published by Kheder [5]indicate that neither assumption may be correct and that there are two levels of crackspacing and crack width that should be considered. Observed cracking behaviour ofconcrete walls has indicated that primary and secondary cracks differ in both thespacing and the width as shown in Figure 14.

Analysis of the observations reported by Kheder indicate that the primary crackspacing is influenced most significantly by the element geometry (for a wall, theheight is a critical factor) with the amount of reinforcement playing a secondary role.The spacing of the secondary cracks is, however, more consistent with that which iscurrently attributable to the reinforcement.

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Primary cracks

Figure 14 Observed crack pattern in a base-restrained wall [5]

5.3 Stress in the reinforcementFor continuous edge restraint, the stress in the steel after cracking is not taken intoaccount in estimating crack width in the methods of either BS8007 of EN1992-3.This may be demonstrated in two ways as follows.

a) Within the method of EN1992-3 for continuous edge restraint, consider theeffect of an increase in the area of reinforcement Asby increasing the number ofbars. According to EN1992-1-1, this affects only the crack spacing Sr,max. Thecrack-inducing strain cr = (sm- cm) is determined solely by the restrained strainin the concrete prior to cracking Raxfreehence no account is taken of an increasein Asin reducing the steel strain after cracking. The mean strain in the steel aftercracking is therefore assumed to be entirely independent of the area ofreinforcement. This seems to be in contradiction to the mechanism of crackingand despite the fact that an essential part of the design process is to achieve aminimum area of steel to maintain the steel stress below its yield value whencracking occurs.

b) Consider the magnitude of the crack inducing strain estimated for edgerestraint (sm- cm) using EN1992-3 in a severe case scenario assuming thefollowing;

Restraint Rax= 0.5 (including creep effects) T1= 60

oC, for strength class C40/50 requiring a high cement content

c= 13/o

C using aggregate with a high coefficient of thermalexpansion

Autogenous shrinkage ca = 22 at 3 days for C40/50

In this case the crack-inducing strain using expression M3 in Annex M ofEN1992-3 will be;

(sm- cm) = Rax (cT1+ ca) (6)

Hence, (sm- cm) = 0.5 (13 x 60 + 22) = 401

If the strain and stress in the steel is assumed to vary linearly from its maximum

value at the point of the crack to the point at which full bond to the concrete is re-

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established, then the maximum strain in the steel may be estimated using theexpression;

smax= 2(sm- cm) = 802 (7)

Hence the maximum stress in the steel,

s= Essmax = 200,000 x 801 x 10-6=160MPa (8)

The characteristic yield strength of the reinforcement fyk= 500 MPa, hence evenin this severe case the steel stress is only about 32% of fyk. This being the case,the method seems inconsistent with the current requirements to provide As,min,which appears over-conservative for an edge restrained member at early age.

5.4 The minimum area of reinforcement

Both BS8007 and EN1992 require a minimum area of reinforcement calculated on

the basis of a requirement to achieve steel stresses below the yield strength of thesteel. Based on expression 7.1 of EN1992-1-1, As,min may be calculated using theexpression,

As,min = kck Act(fct,eff/ fyk) (9)

Where kck Act is the effective area of concrete in tension (the coefficients kand kcrespectively allow for the effect of non-uniform self-equilibratingstresses and of the stress distribution with in the section SeeSection 6.4.1 equ.12)

fct,eff is the effective tensile strength of the concrete at the time of

crackingfyk is the characteristic yield strength of the reinforcement

However, as shown in Section 5.3, even under the most extreme conditions, themagnitude of restrained strain is unlikely to be sufficient to achieve more than about0.32 fyk. Furthermore, it is expected that the restraining element will attract some ofthe load from the concrete in the restrained element when the crack occurs. It maybe possible therefore to allow As,minto be less than kck Act(fct,eff/fyk) (EN1992-1-1,expression 7.1) if it can be demonstrated that the load transfer to the restrainingelement causes a reduction in the stress in the steel. The form of the reductionfactor needs to be evaluated. However, if the edge restraint is Redge, then an initialsuggestion is that A

s,minneed only be required to exceed (1-R

edge) k

ck A

ct(f

ct,eff/f

yk); i.e.

when the restraint is higher and the restraining member is carrying more load, alower value of As,minwould be acceptable.

5.5 Strength class

Within the approach of BS8007 and EN1992 for edge restraint, the strength class isonly taken into account in relation to the difference in temperature change associatedwith the different cement contents. No account is taken of the strength of theconcrete in relation to the stress transferred to the steel when cracking occurs.

As the revised unified method is based on the potential crack width for an elementsubject to end restraint, the strength class is taken into account directly for the edge

restraint condition.

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5.6 Difficulty in achieving efficient conforming designs using BS8007and EN1992

In carrying out the calculations using BS8007 it was observed that for someconditions, crack widths of 0.2mm and 0.3mm could not be achieved effectively

without exceeding either the reinforcement spacing of 5 (c + /2) or by using As 5(c + /2)

Non-conforming zone

As < As,min

Figure 15 The relationship between the bar diameter, the bar spacing, the area ofreinforcement and the estimated crack width to BS8007 (300mm wall using C30/37)

To achieve a crack width of 0.2mm or greater requires either that the minimum steelratio is not met (in which case the method is assumed not to apply) or that large barsat excessive spacing are used. To conform in relation to both the minimum area ofreinforcement and bar spacing leads to much smaller cracks than permitted.

Hence, no practical advantage may be achieved in this example if the allowablecrack width is greater than about 0.15mm. A 0.15mm design crack width may beachieved with 12@190,As= 595mm

2. This is only marginally above As,min(=520mm2)

and provides little scope for reducing Asby permitting wider cracks. Indeed, toachieve wider cracks requires the use of larger bars at impractically wider spacingwhile maintaining Asat about 600mm

2. Under such conditions it would make moresense to simply design to achieve the smaller crack width than necessary. Thissuggests a flaw in the current approach and applies to both BS8007 and EN1992,particularly for thinner sections.

In practice, where the structural reinforcement is checked to determine its adequacyfor early thermal crack control, it is likely that the expected crack widths wouldtherefore have been significantly less than 0.2mm. Thus, situations are likely to havearisen for which a 0.2mm crack would be considered to be acceptable, while at thesame time being greater than would have been predicted by the method of BS8007.

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6 Development of the revised unified method of design

6.1 Objectives

The revised unified method of design was developed with the following aims.

To achieve a single method of design for dealing with both continuousedge restraint and end restraint

To represent as closely as possible the mechanism of cracking and post-cracking behaviour

6.2 The initial simplified approach

The revised method of design principally involves a change in the way in which thecrack width is calculated. An initial approach was developed using the method ofcalculation for the end restraint condition (equ.3) to estimate the maximum potential

crack width wpand then to apply a crack reduction factor kwto take account of theeffect of edge restraint in preventing crack opening. Hence,

wk= kwwp. (10)

The original concept considered that the crack reduction factor kwcould berepresented by 1-Redgewhere Redgerepresents continuous edge restraint hence

wk= wp (1 - Redge). (11)

The FE studies that were carried out (see Section 4.3) indicated that there doesseem to be a relationship between increasing edge restraint and reduction in crack

width, supporting the basis for the revised approach for estimating crack widths.However, the FE study also indicated that prediction of crack widths based on wk=wp (1 Redge) was over simplistic and that factors other than restraint should beconsidered.

In parallel with the FE analysis, case studies were investigated [9] in which the crackwidth was initially estimated using equ.11. Recognising that the revised approachmay lead to a reduction in minimum steel requirements by acknowledging that undercontinuous edge restraint at least some of the load from the concrete is transferred tothe restraining element immediately after cracking, elements with reinforcementratios lower than normally acceptable (0.2%) were included [5]. The resultsdemonstrated that while the simple expression wk= wp (1 - Redge) reliably predicted

crack widths in many normal circumstances, it became progressively less reliable atlow steel ratios as shown in Figure 16.

It was observed, however, that current methods for estimating crack widths, whilebeing limited to elements with steel in excess of the minimum requirement, seem tobe able to predict cracks widths in the order of those reported in elements with lowsteel ratios as shown in Figure 17. Despite the methods of BS8007, CIRIA 91,EN1992 and CIRIA C660 each requiring a minimum area of reinforcementappreciably greater than 0.2% each of these methods led to calculated crack widthsat least of the same order as those reported in base-restrained walls with this lowsteel ratio.

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low steel

ratio (0.2%)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Measured crack width (mm )

Estimatedcrackw

idth(mm)

Figure 16 Comparison of estimated and reported crack widths

0.4845

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Estimatedcrackwidth(mm)

Measured crack width (mm)

Kheder method

R = 0.7223

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7



BS8007

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7



CIRIA C660

R = 0.622

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7



EN1992-3

R = 0.7279

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7



CIRIA 91

Figure 17 Comparison of estimated and reported crack widths using currentlyavailable methods

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Recognising that the simple version of the revised approach was markedly lessreliable than existing techniques over the wide range of conditions investigated, afurther development of the revised approach was proposed. Having observed inpractice that the magnitude of contraction did have some effect on the crack width(although less than expected assuming proportionality Section 4.1) a two stagecracking process was investigated.

6.3 The two stage process

Within the proposed two stage process, the first stage is assumed to cause the crackto open instantaneously to a value wk1when part of the load is transferred from theconcrete to the steel. During the second stage the crack opens by a further wk2 asthe concrete is assumed to continue to contract relative to the reinforcement.

Hence the full crack width wk= wk1+ wk2

Stage 1 crack width wk1is estimated using an expression based on the currentmethod for end restraint with a modification to take account of the effect of edgerestraint in both attracting load and preventing crack opening. The revision also takesaccount of the relative lengths of the cracked zone S(over which it is assumed thatdebonding occurs) and the uncracked zone (which exhibits strain relief immediatelyafter cracking).

Stage 2 cracking wk2considers how the residual contraction of the concrete withinthe cracked zone relative to the reinforcement contributes to the continued openingof the crack. Again, restraint is assumed to prohibit contraction and hence the extentto which a crack may open during stage 2.

6.4 Development of expressions for Stage 1 cracking

6.4.1 Cracking under end restraint according to EN1992-3

Stage 1 cracking is estimated using a revision to the expression for end restraintgiven in EN1992-3. The end restraint expression provided in EN1992-3 (M1) is asfollows;

++++==== 1

1

E

fkk0.5-

es

eeffct,ccmsm (12)

wherek kc are coefficients, defined in EN1992-1-1 which take account of the

stress distribution in the concrete and self-equilibrating effectsfct.eff is the mean design tensile strength of the concrete at the time of

cracking, =ctfctm(t). The [authors] recommended value of ct is 0.8

Es is the modulus of elasticity of the reinforcement.

e is the modular ratio Es/Ec,effand Ec,effis the modulus of elasticity of theconcrete at the time of cracking.Ec,eff(t)=Ecm(t)

is the ratio As/Act based on the full section thickness [Note that differsfrom p,effwhich is used to calculate the crack width for the edge-restrained condition. This is to take account of the full area in tensionprior to cracking]

As is the (total) area of reinforcement

Act is the gross section in tension

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The expression in EN1992-3 is based on an element loaded in direct tension andassumes that the load from the concrete that is transferred to the steel is maintainedfully after cracking. Derivation of this equation from first principles indicates that anerror has been introduced in equation 12 (See Appendix 1) which should be asfollows:

1

kk

E

,f0.5)(

e

c

s

effcte

cmsm

+= (13)

With regard to estimated values of (sm - cm) the error in the equation makes verylittle difference (in the order of 1%), with the revised expression giving the marginallyhigher values. The difference is greatest with lower concrete strength classes andwith higher reinforcement ratios but even in very extreme cases the difference isunlikely to exceed 2%. For practical purposes expression M1 of EN1992-3 istherefore acceptable, but for correctness the revised equation 13 is used in the

development of the revised method.

6.4.2 Effect of element length

In practice, when the load is generated by restrained contraction, as shown in Figure18, the stress transferred to the steel immediately after cracking cannot be sustainedas stress relief occurs in the uncracked zone.

L

S

L ctu

Crack

Figure 18 An element subject to end restraint only

Theoretically, the load transferred to the steel immediately after cracking can only bemaintained if the element is infinitely long. For the condition shown in Figure 18 themean residual strain in the steel after cracking and strain relief may be calculatedusing the expression;

1)(B/L)(S0.511)(B0.5

maxr,

ctusmr

++++

++++==== (14)

where

++++==== 1

kkB

e

c (15)

The development of equation 14 is given in Appendix 2.

In equation 14, as L, then (Sr,max/ L) 0 and smr0.5 ctu (B +1)

The crack inducing strain smr cm = sm 0.5 ctu= 0.5 ctu (B +1)- 0.5 ctu= 0.5 ctu B

Asctu= fct,eff/Ec,effand the modular ratioe= Es/Ec,eff, thenctu= efct,eff/ Es, and

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smr cm = 0.5 ctu B = 1

kk

E

,f0.5

e

c

s

effcte

++++ = sm cm (16)

Thus for conditions of end restraint, expression 13 represents only the extreme caseof the general equation 14 when L . For most practical conditions the length ofthe element will be a limiting factor in determining sm cmand the crack widthimmediately after cracking will be proportional to smr 0.5ctr, wherectris theresidual tensile strain in the uncracked concrete, i.e. outside the debonded zone,immediately after cracking (see Figure A3.2) This is estimated using the expression;

1)(BS0.5L

BL0.50.5

ctu

ctrsmr

+= (17)

According to EN1992-3, sm - cm= 0.5 ctuB and

( ) ( )

+

=

+

=

kk

L

S0.51

1)(BL

S0.51

0.5

e

c

3EN1992cmsm3EN1992cmsm

ctrsmr (18)

Hence the effect of element length may be accounted for by estimating the maximumcrack width for end restraint using expression M1 of EN1992-3 and dividing the resultby the factor [1 + (S/L)(k kc/e)]

Consider the example of a 300mm thick element with 20mm bars at 150mm centresand 40mm cover. The concrete is C30/37. Using expression 13, the crack inducing

strain is estimated to be 412 m. This would apply for an infinitely long element. Theeffect of element length has been estimated and is shown in Figure 19. For a typicallength of about 10m, the estimated crack inducing strain immediately after cracking isonly about 55% of the limiting value. The crack width would be similarly reducedcompared with the value estimated using EN1992-1-1.

Figure 19The effect of length on the crack width immediately after cracking (i.e. beforeadditional contraction)

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As contraction continues after the first crack, the stress in the steel increases and thefirst crack widens until a second crack develops, and so on until contraction iscomplete. An example of how the crack responds to continuing contraction is shownin Figure 20 for a 15m long wall with an assumed free contraction of 300 microstrainand R= 0.4. In this case only three cracks are formed. When cracking first developsthe crack width opens instantaneously to 0.34mm, but then reduces immediately to0.26mm as strain relief occurs in the uncracked concrete (i.e. outside the debondedzone). As contraction continues the first crack opens progressively up to 0.34mmuntil the second crack occurs, when it reduces to 0.27mm. The reduction in crackwidth after the second crack has developed is less than after development of the firstcrack as there is less uncracked concrete over which strain relief can occur. Thecrack width again increases up to 0.34mm but reduces to 0.28mm after the thirdcrack. As the restraint of 0.4 was not sufficient to cause the fourth crack, furthercontraction leads to a final crack width of about 0.32mm

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Crackwidth(mm)

Crack no.

ct = 1.0 EN1992-3

ct = 0.8 revised method

Figure 20 Crack development for R = 0.4 [NB The max estimate includes the factor ct= 0.80 in the derivation while the EN1992 estimate does not]

Had the restraint be marginally lower (0.36) the third crack would not have developedand the estimated mean crack width for the two cracks already developed would be0.34mm as shown in Figure 21.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Crackwidth(mm)

Crack no.

ct = 1.0 EN1992-3

ct = 0.8 revised method

Figure 21 Crack development for R = 0.36 [NB The max estimate includes the factor ct= 0.80 in the derivation while the EN1992 estimate does not]

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If edge restraint is high, the length over which strain relaxation occurs (i.e. the zoneof influence of the crack) will be less than if the restraint is very low.

Hence it is assumed that L,eff

= Sn/ R

edge= k

LH / R

edgewhere the length coefficient

1 < kL< 2 and equ.19 may be modified as follows:

++++

++++====

)R(1

1B0.51

Hk

RS-1

1))BR-[(10.5

edgeL

edge

edgectu

smr (20)

The stage 1 crack width is then estimated using the expression

+

=

)R(1

1B0.51

Hk

RS-1

)BR-(10.5

edgeL

edge

edgectu

cmsm (21)

This assumes that, after the crack has occurred, there is a residual tensile strain inthe concrete equal to half the strain capacity of the concrete.

6.5 Development of expressions for Stage 2 cracking

Stage 2 cracking occurs as the concrete continues to contract after the developmentof the crack. The steel in the cracked zone is maintained under stress by contractionof the concrete outside the cracked zone and despite cooling with the concrete will

be restrained from contracting between points of zero displacement between thecracks. Hence the concrete within the cracked zone is assumed to contract relativeto the steel, causing the crack to grow as shown in Figure 23.

The first stage of cracking will have occurred when the tensile strain capacity of theconcrete ctuwas exceeded. For restraint Redgeprior to cracking and taking intoaccount creep (factor K1), the contraction required to cause cracking will have beenctu/RedgeK1. Hence stage 2 cracking will be proportional to the residual contraction.

Hence res = free - ctu/RedgeK1. (22)

wk1

Sr,max

0.5 wk20.5 wk2

Cracked zone

Figure 23 Development of stage 2 cracking

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The contraction in the cracked zone will be inhibited by the restraint local to thecrack. This may be estimated by assuming that the restraint to contraction of theconcrete is zero at the crack (the concrete having debonded locally) and that it buildsup linearly to its pre-cracked value beyond the zone of cracking. The averagerestraint within the zone of cracking is therefore 0.5 R

edge. The additional movement

at the crack wk2will therefore be proportional to 1 0.5 Redge

Hence wk2may be calculated using the equation,

)KR

(K)0.5R(1Sw

1edge

ctufree1edgemaxr,k2 ==== (23)

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7 Critical parameters for predicting crack widthAs demonstrated in Table 1, the parameters which are currently assumed to controlthe restrained strain (sm- cm) and hence the crack width are entirely different for

edge restraint and end restraint. According to EN1992-3 and BS8007 only the freecontraction and the restraint influence (sm- cm) while for end restraint, the tensilestrength of the concrete and its modulus of elasticity (combining to give the tensilestrain capacity) are also dominant together with the area of reinforcement. Providedthat restraint is of sufficient magnitude to cause cracking, it is assumed by EN1992-3to have no subsequent effect on the width of cracks caused by end restraint. As theunified approach is based largely on the EN1992-3 approach for end restraintconsideration must be given therefore to appropriate values for these parameters.

7.1 Tensile strength

The tensile strength fct,eff is assumed to be the effective tensile stress at the time ofcracking and (

sm

cm)is directly proportional to this value. It is important therefore

that an appropriate value of fct,eff is adopted in the design process

As the first crack is most likely to occur at the weakest location it may be mostappropriate to use the lower 5 percentile value fct0.05in the calculation of the crack-inducing strain sm- cm. However, later cracks will occur when the strength hasincreased (in time) and in areas where there is higher strength than at the (weakest)location of the first crack. Furthermore, to balance forces, the stress in the steel mustbe equal at each crack location, assuming that the steel ratio is constant along thelength of the member. Hence, assuming that the length of debonding S0is constant,all cracks would be expected to be equal width and to be determined by the in situtensile strength at the location of and immediately prior to the most recent crack.

Any variation in crack width would be expected to be largely due to variations in S0asvariations in the cross sectional area and the elastic modulus of the steel would beexpected to be small.

According to EN1992-1-1 [2], fct,effis the mean value of tensile strength of theconcrete effective at the time of cracking and fct,eff = fctmor lower fctm(t)if cracking isexpected earlier than 28 days. Expressions for estimating fctmand fctm(t)are given inEN1992-1-1 and values derived using these relationships for a range of concretestrength classes using CEM I (42.5 N) are shown in Figure 24.

EN1992-1-1 also notes that The development of tensile strength with time is stronglyinfluenced by curing and drying conditions as well as the dimensions of the structural

membersand that where the development of tensile strength is important testingshould be carried out. The expressions provided by EN1992-1-1 must therefore beapplied with caution.

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C20/25

C25/30

C35/45

C40/50C45/55

C50/60C55/65

C60/75

C30/37

0

1

2

3

4

5

6

1 10 100 1000Time (days)

Meantensilestrength,

fctm(M

Pa)

Figure 24 The mean tensile strength development, fctm(t) for CEM I concrete accordingto EN1992-1-1

The value of tensile strength used in estimating crack width should represent a safevalue of tensile strength at the time at which cracking is expected, i.e. the in situtensile strength with a suitable margin. A study was carried out as part of the revisionto CIRIA 91 [12], published as CIRIA C660 [4], to determine the probability of thedesign value of fc,effbeing exceeded in situ. The following factors were considered:

The coefficient of variation of 18% that may occur in tensile testspecimens of the same strength class (defined in EN1992-1-1 by the 5%and 95% fractiles of 30% fctm)

In situcompaction and curing (including thermal effects) This wasbased on a Concrete Society study [13] which measured variations incore compressive strength.

Sustained loading which reduces the stress at which cracking occurscompared with the stress achieved when the load is applied quickly.

Age at cracking which may differ from the assumed age at cracking.

To establish a safe value for design a probabilistic analysis was undertaken. Inputdata for the probabilistic analysis (described in Appendix 10 of CIRIA C660 [4]) aregiven in Table 2 for class C30/37 concrete. The results, in the form of a histogram,are shown in Figure 25, for the early age tensile strength. It is clear that there is apotentially wide range of values of in situ tensile strength. Estimated values of in situtensile strength (5% fractile, mean and 95% fractile) are given in Table 3 comparedwith the design values derived from the expressions in EN1992-1-1. It can be seenthat the estimated in situ values are lower than the EN1992-1-1 design values byvarying degrees. The EN1992-1-1 values are, on average, 2.67, 1.64 and 1.13 timesthe 5% fractile, mean and 95% fractile values.

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Table 2 Input data for the probabilistic analysis for strength class C30/37

Input parameters Distribution Min Mean Max SD

fctm(EN1992-1-1) 28-day MPa NORMAL 2.90 0.53

fctm(EN1992-1-1) 3-day MPa NORMAL 1.73 0.32

In situcompaction and curing kis NORMAL 0.90 0.082

Sustained loading ct PERT 0.6 0.7 0.8

[Early] Age factor (relative to 3days)

ka PERT 0.54 1.00 1.28

[Long term] Age factor relativeto 28 days

ka PERT 0.90 1.00 1.10

Distribution for Early-age tensile

strength/K10

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

Mean=1.059666

0.4 0.9 1.4 1.9 2.4

1.41.4

0.4 0.9 1.4 1.9 2.4

5% 82.87% 12.13%.6512 1.38

Mean=1.059666

Figure 25 Distribution of early-age in situ tensile strength for C30/37 concrete [4]

Table 3 Estimated in-situ tensile strength at early age compared with valuescalculated in accordance with EN1992-1-1

Strengthclass

Estimated in-situ values EN1992-1-1

fcm(3) MPa

0.8 fcm(3)MPa5% Mean 95%

C20/25 0.49 0.81 1.17 1.32 1.06

C25/30 0.57 0.94 1.36 1.53 1.22

C30/37 0.65 1.06 1.54 1.73 1.38

C35/45 0.72 1.17 1.71 1.92 1.54

C40/50 0.79 1.28 1.85 2.10 1.68

C45/55 0.86 1.39 2.02 2.27 1.82

C50/60 0.91 1.49 2.16 2.44 1.95

C55/67 0.95 1.54 2.23 2.51 2.01

C60/75 0.97 1.58 2.31 2.62 2.10

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But what is an appropriate value to be used in design for control of cracking? Usingtoo high a value will lead to an unnecessarily high volume of reinforcement whileassuming too low a value leads to the risk exceeding the allowable cracks width.Design should err on the side of caution but not excessively so for a serviceabilitylimit state.

Based on the probabilistic analysis for the C30/37 concrete with the distribution ofestimated in situ tensile shown in Figure 23, a value of 0.8 fctm(t) represents a valueclose to the upper 90 percentile (with only a 12% chance of the estimated valuebeing exceeded). As cracks are most likely to form at the weaker, rather than thestronger parts of the element, this value is therefore considered to represent a safevalue for design.

7.2 Modulus of elasticity and creep

The modulus of elasticity of the concrete at the time of cracking Ec,effis used toestimate both the modular ratio and, in association with the tensile strength fct.eff, the

tensile strain capacity of the concrete. The latter is used in both the estimation of therisk of cracking and the subsequent crack width.

It must be recognised that different values of modulus of elasticity apply in relation tothe estimation of the risk of cracking and the calculation of the crack inducing strain(sm -cm).

7.2.1 Estimating the risk of cracking

Cracking will occur when the restrained strainr = Raxfree >ctu. It has been shownthat ctumay be estimated from form the tensile strength fctmand the modulus ofelasticity in compression Ecm[14] and ctu=. fctm/Ecm. In this case the tensile strain

that occurs is affected by creep under the sustained loading and in the expression ctu=. fct,eff/Ec,eff, where fct,effandEc,eff, are effective in situ values. Under sustainedloading the creep effective effective modulus applies and Ec,eff = Ecm(t)/K1whereEcm(t) is the mean value of modulus of elasticity at age of cracking t and K1is thecreep coefficient with a recommended value of 0.65 [4].

7.2.2 Estimating (sm- cm)

When a crack occurs the load transfer from the concrete to the steel isinstantaneous. When using equ.18 to estimate (sm - cm) the value of Ec,eff = Ecm(t),i.e. there is no effect of creep. Hence the modular ratio e= Es/ Ecm(t).

7.3 Estimating continuous edge restraint

7.3.1 The nature and magnitude of edge restraint

Within the revised approach the influence of restraint on crack width differssignificantly from that assumed by the current methods of EN1992-3 and BS8007.Using the current methods, a safe (worst case) is achieved by assuming the highestpossible restraint. However, within the revised approach continuous edge restraint isassumed to act in a similar way to the reinforcement, with higher restraint limiting theextent to which cracks may open and leading to a greater number of finer cracks. Itis important therefore that the restraint is estimated with a reasonable degree ofaccuracy.

The most recognisable form of edge restraint occurs when a wall is cast on a rigidfoundation. A typical example is shown in Figure 26 for the walls of a box-sectiontunnel. The classic crack pattern can be observed.

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Contr

action

Contr

action

Restra

int

Restra

int

Contr

action

Contr

action

Restra

int

Restra

int

Contr

action

Contr

action

Restra

int

Restra

int

Figure 26 Early thermal cracking in the walls of a box-section tunnel wall

Under these conditions, values of restraint are typically in the range from 0.3 to 0.7and it is tempting to assume an average value of 0.5 as previously proposed byCIRIA 91 and BS8007 and now by EN 1992-3. However, a difference in restraint of0.1 from this mean value will affect the level of restrained strain by 20% and wherethe crack width is critical and/or when permitted by the element geometry it isrecommended that a more rigorous assessment of the restraint is undertaken. EN1992-3 provides the option to calculate restraint factors from knowledge of thestiffness of the element considered and the elements attached to it.

As cracking resulting from edge restraint is a common occurrence considerableresearch has been undertaken to enable restraints to be predicted and therefore toprovide a more reliable estimate of the risk and extent of cracking.

7.3.2 Estimating the magnitude of continuous edge restraint

It is apparent that the restraint offered by old concrete against which a new elementis cast must be influenced the relative size and stiffness of the new element and theold. The stiffness of an element is defined by its elastic modulus and its geometry.For new elements cast against existing concrete with continuous restraint along oneedge, ACI 207.2R-73 [15] provides a method for estimating restraint based on therelative cross-sectional areas, the relative modulus of elasticity of the new and old

elements and the distance from the joint. This has been described in CIRIA 135 [16].Restraint at the joint is calculated using the equation,

o

n

o

nj

E

A

A1

1R

++++

==== (24)

where An= cross sectional area (c.s.a) of the new (restrained) pour

Ao= c.s.a. of the old (restraining) concrete

En = modulus of elasticity of the new pour concrete

Eo = modulus of elasticity of the old concrete

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While the geometry remains constant for both new and old concrete, the same doesnot apply to the elastic modulus. For the old concrete it is unlikely that the modulusEowill change significantly during the period of an early age heat cycle of concretecast against it and it is reasonable, therefore, to assume that Eoremains constant inany calculations. However, the elastic modulus of the new section will be changingrapidly over the first few days and this will therefore influence the degree to whichdeformation is restrained by the older and stiffer concrete.

Figure 25 illustrates the change in elastic modulus during an early age heat cycle [17]and its influence on the ratio Eo/Enand the restraint (assuming Ao/An =1).

Immediately after casting, when the new concrete is relatively soft, the ratio of Eo/Enis low and hence the restraint is high. As the elastic modulus increases, the ratio ofEo/Enincreases and the restraint reduces. CIRIA 135 recommends that when usingthe ACI approach to estimate restraint the ratio Eo/Enis assumed to be in the range0.7 to 0.8. As shown in the example in Figure 27, this represents the value at about

48 hours, after which time the restraint reduces to a value of about 0.55. The higherend of this range applies for thinner elements which cool and crack soonest. Thickerelements which cool more slowly develop greater stiffness in relation to therestraining element and Eo/Enis at the lower end of the range.

0

5

10

15

20

25

30

0 24 48 72 96 120 144 168

Hours

Elasticmodulus(GPa)

Restraint

En/Eo

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168

Hours

Restraint

-En/Eo

Figure 27 The variation in elastic modulus at early age and the influence on the ratioof En/Eo and the restraint derived using equation 11 [It is assumed in this examplethat the ratio An/Aoequals 1]

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7.3.3 Variation in restraint with distance from the joint

The value of restraint calculated using equ.24 is the value at the joint between thenew and the old concrete but, depending on the pour geometry, and in particular thelength/height ratio of the element, the restraint will reduce with distance from the joint

to varying degrees as shown in Figure 28. Since the development of the ACI data,further investigations have been carried out into restraint and Emborg [18] hasproposed revised estimates of restraint for elements of different L/Hratio. Theserevised values are also shown in Figure 28. In Figure 29, restraint estimated usingEmborgs curves are compared with the original ACI curves and recorded restraint inwalls. For elements with low L/H ratios, the values are every similar, but there aresignificant differences for pours with high L/H ratios.

ACI values

1

2

3 4 5 6 8 10 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.70.8

0.9

1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Restraint

Proportionalheight

Revised values (Emborg 2003)

1

1.4

2

3

45 6 7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Restraint

Proportio

nalHeight

Figure 28 A comparison of restraint values from ACI 207 [15] and Emborg [18]

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Estimation of restraintWall on a rigid base

Cells for input data5

Dimensions Wall Base

Length 12 m

Height 3 m Width 3 m

L/H 4

Thickness 0.5 m Thickness 0.5 m

c.s.a. A n 1.5 m2 c.s.a A o 1.50 m

2

L/H 4

Ratio of areas A n/A o 1.00

Ratio of moduli n o 0.70

Restraint factor at joint Rj 0.59

Height of maximum restraint 1.20 m

Restraint at maximum crack width Rwmax 0.45

Max crack width

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Restraint

Height(m)

Figure 30 The CIRIA C660 calculator for continuous edge restraint modified to showthe restraint at the point of maximum crack width

Estimates of the restraint profile and the height at which the maximum crack width isestimated to occur are shown in Figure 31 for a 3m high wall of varying length andassuming that the ratio An/Ao= 1. The results indicate that while the length of thewall influences the height at which the maximum crack width occurs, it has very littleeffect on the magnitude of restraint at that height. In each case the value of Rwmax

0.45.

The influence of varying the ratio of areas An/Aois shown in Figure 32. At the heightof maximum crack width the restraint varies within the range from about 0.3 to 0.6.However, the ratio Rwmax/ Rjremains approximately constant at about 0.78.

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0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Restraint

ProportionalHeight

842

1

L/H

Height of

maximum

crack width

Figure 31 Estimated restraint using the CIRIA C660 calculator for a 3m high wall ofvarying length (An/Ao= 1)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.5 1.0 1.5 2.0 2.5

Ratio of areas (An/Ao)

Restraint

Restraint at joint Rj

Restraint at height of

maximum crack width,Rwmax

Figure 32 Estimated restraint using the CIRIA C660 calculator for a 3m high wall ofvarying length (An/Ao= 1)

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8 ValidationTo establish the validity of the revised approach, predicted crack widths arecompared with observed cracking. Details of the structures investigated and the

assumptions used in the calculated are provide in details in reference 19.

The total crack width has been estimated from the sum of wk1and wk2and the resultsare shown in Figure 33 compared with reported crack widths for the case studiesinvestigated. The best fit curve, passing through the origin, is achieved by assumingthat the coefficient for effective length kLused in equ.18 has a value of 1.5. This isconsistent with observed values of primary crack spacing in under reinforcedsections and close to the value of 1.3 recommended by EN1992-1-1 for under-reinforced sections. While there is considerable scatter of the results the best fitrelationship is close to the li

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