105-s29 Control of Flexural Cracking in Reinforced Concrete

ACI Structural Journal/May-June 2008 301

ACI Structural Journal, V. 105, No. 3, May-June 2008.MS No. S-2006-346 received August 21, 2006, and reviewed under Institute publication

policies. Copyright © 2008, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the March-April2009 ACI Structural Journal if the discussion is received by November 1, 2008.

TECHNICAL PAPERACI STRUCTURAL JOURNAL

Excessive cracking is one of the common causes of damage inconcrete structures and results in huge annual cost to the constructionindustry. Most of the current design approaches for crack controlare empirical and based on observed crack widths in laboratoryspecimens tested under short-term loads. Most approaches fail toadequately model the increase in crack width that occurs with timedue to shrinkage. In this paper, an alternative design method forflexural crack control that overcomes many of the limitations of thecurrent code approaches is proposed. The proposed method takesinto account the time-dependent development of cracking and theincrease in crack widths with time due to shrinkage. The crackwidth calculation procedure has been shown to provide goodagreement with the measured spacing and width of cracks in avariety of slabs and beams tested in the laboratory under sustainedservice loads.

Keywords: crack control; creep; flexural cracking; reinforced concrete;serviceability; shrinkage.

INTRODUCTIONCurrent design procedures to control cracking in concrete

structures using conventional steel reinforcement are overlysimplistic and often fail to adequately account for thegradual increase in crack widths with time due to shrinkage.The bonded reinforcement in every reinforced concretebeam or slab provides restraint to shrinkage, with theconcrete compressing the reinforcement as it shrinks and thereinforcement imposing an equal and opposite tensile forceon the concrete at the level of the steel. This internalrestraining tensile force is often significant enough to causetime-dependent cracking. In addition, the connections of aconcrete member to other parts of the structure or to thefoundations also provide restraint to shrinkage. The tensilerestraining force that develops rapidly with time at therestrained ends of the member usually leads to cracking, oftenwithin days of the commencement of drying. In a restrainedflexural member, shrinkage also causes a gradual widening offlexural cracks and a gradual build-up of tension in theuncracked regions that may lead to time-dependent cracking.

Cracks occur at discrete locations in a concrete member,often under the day-to-day service loads. The width of acrack depends on the quantity, orientation, and distributionof the reinforcing steel crossing the crack. It also depends onthe deformation characteristics of the concrete and the bondbetween the concrete and the reinforcement bars at, and inthe vicinity of, the crack. A local breakdown in bond at eachcrack complicates the modeling, as does the time-dependentchange in the bond characteristics caused by dryingshrinkage and tensile creep. Great variability exists in observedcrack spacing and crack widths and accurate predictions ofbehavior are possible only at the statistical level.

Most of the current design approaches for crack controlspecified in building codes are empirical1-3 and are based onobserved crack widths in laboratory specimens tested undershort-term loads. These approaches also specify certain

detailing requirements, including maximum limits on boththe center-to-center spacing of bars and on the distance fromthe side or soffit of the member to the nearest longitudinalbar. These limits do not generally depend on any of thefactors that affect the size and location of cracks. The codesof practice1-3 also specify a minimum quantity of tensilereinforcement in those regions of the member wherecracking is likely under service loads and maximum limitsare placed on the tensile steel stress on a cracked sectiondepending on either the bar diameter or the bar spacing.2,3

The existing code approaches,1-3 however, fail to adequatelyaccount for the increase in crack width that occurs with timedue to shrinkage.

This paper outlines a design method for flexural crackcontrol that overcomes many of the limitations of the currentcode approaches. The proposed method is based on arecently developed procedure4 for the calculation of themaximum final crack spacing and crack width in a beam orslab and takes into account the time-dependent developmentof cracking and the increase in crack widths with time due toshrinkage. The crack width calculation procedure has beenshown to provide good agreement with the measured spacingand width of cracks in a variety of slabs and beams tested inthe laboratory under sustained service loads.

RESEARCH SIGNIFICANCEExcessive cracking resulting from either restrained

deformation or external loads (or both) is one of the mostcommon causes of damage in concrete structures and resultsin huge annual cost to the construction industry. Currentdesign procedures to control cracking using conventionalsteel reinforcement1-3,5 do not adequately account for thegradual increase in crack widths with time due to the effectsof shrinkage.6 This paper provides a rational method fordesigners to control flexural cracking in reinforced concretebeams and slabs and thereby improve the serviceability ofconcrete structures.

FLEXURAL CRACKING MODELRecently, Gilbert4 proposed a model for predicting the

maximum final crack width, w*, in reinforced concrete flexuralmembers based on the Tension Chord Model of Marti et al.7

The model was shown to provide good agreement with themeasured final spacing and width of cracks in a range ofreinforced concrete beams and slabs tested in the laboratoryunder sustained service loads for periods in excess of 400 days.6

The notation associated with the model is shown in Fig. 1.

Title no. 105-S29

Control of Flexural Cracking in Reinforced Concreteby R. Ian Gilbert

ACI Structural Journal/May-June 2008302

In the following, Gilbert’s model is used to develop asimple procedure to ensure that the final maximum flexuralcrack width in a beam or slab is less than a selectedmaximum design crack width, wmax.

Consider a segment of a singly reinforced beam of rectan-gular section subjected to an in-service bending moment Msgreater than the cracking moment Mcr. The spacing betweenthe primary cracks is s, as shown in Fig. 1(a). A typical crosssection between the cracks is shown in Fig. 1(b) and a crosssection at a primary crack is shown in Fig. 1(c). The crackedbeam is idealized as a compression chord of depth c andwidth b and a cracked tension chord consisting of the tensilereinforcement of area As surrounded by an area of tensileconcrete Act as shown in Fig. 1(d). The centroids of As andAct are assumed to coincide at a depth d below the top fiberof the section.

For the sections containing a primary crack (Fig. 1(c)),Act = 0 and the depth c and the second moment of area aboutthe centroidal axis, Icr , may be determined from a cracked

section analysis. Away from the crack, the area of theconcrete in the tension chord of Fig. 1(d) (Act) is assumed tocarry a uniform tensile stress σct that develops due to thebond stress τb that exists between the tensile steel and thesurrounding concrete.

For the tension chord, the area of concrete between thecracks, Act , may be taken as

Act = 0.5(h – c)b* (1)

where b* is the width of the section at the level of thecentroid of the tensile steel (that is, at the depth d). At eachcrack in the tension chord of Fig. 1(d), σst1 = T/As, σc = 0, and

(2)

As distance z from the crack increases, the stress in thesteel reduces due to the bond shear stress τb between the steeland the surrounding tensile concrete. For reinforced concreteunder service loads, where σst1 is less than the yield stress fy,Marti et al.7 assumed a rigid-plastic bond shear stress-sliprelationship, with τb = 2fct at all values of slip and where fctis the direct tensile strength of concrete. In reality, themagnitude of τb is affected by steel stress, concrete cover,bar spacing, transverse reinforcement (stirrups), lateralpressure, degree of compaction, and size of bar deformations.In addition, τb is likely to be reduced with time by tensilecreep and shrinkage. Experimental observations by Gilbertand Nejadi6 and others indicate that σb reduces as the stressin the reinforcement increases and, consequently, the tensilestresses in the concrete between the cracks reduces (that is,tension stiffening reduces with increasing steel stress).

Gilbert4 proposed that τb = α1α2 fct, where α1 depends onthe steel stress at the crack (and varies from 3.0 at low stresslevels to 1.0 at high stress levels); and where α2 = 1.0 forshort-term calculations and α2 = 0.5 for long-term calculations.These values of α1 and α2 where calibrated to provideagreement with the results of a detailed experimental studyof cracking in reinforced concrete beams and slabs underboth short-term and long-term sustained loads.6 To avoid thediscontinuities in α1, it is herein assumed that α1 is independentof steel stress and equal to 2.0 (as proposed by Marti et al.7).That is, for short-term calculations, the bond stress τb = 2.0fctand, for long-term calculations in the determination of thefinal maximum crack width, τb = 1.0fct.

An elevation of the tension chord is shown in Fig. 2(a) andthe stress variations in concrete and steel in the tension chordare illustrated in Fig. 2(b) and (c), respectively. Followingthe approach of Marti et al.,7 the concrete and steel tensilestresses in Fig. 2(b) and (c), where 0 < z ≤ s/2, may beexpressed as

(3)

where ρtc is the reinforcement ratio of the tension chord(= As/Act) and db is the reinforcing bar diameter. Midwaybetween cracks, at z = s/2, the stresses are

TnMs d c–( )

Icr

---------------------------As=

σstxTAs

-----4τbz

db

-----------– and σcx4τbρtcz

db

------------------==

R. Ian Gilbert is a Professor of civil engineering and an ARC Australian ProfessorialFellow in the School of Civil and Environmental Engineering at the University of NewSouth Wales, Sydney, Australia. His research interests include serviceability and thetime-dependent behavior of concrete structures.

Fig. 1—Cracked reinforced concrete beam and idealizedtension chord model.1

Fig. 2—Tension chord—actions and stresses.7


(4)

The maximum crack spacing immediately after loadings = smax occurs when σc2 = fct, and from Eq. (4)

(5)

where τb = 2.0fct. The minimum spacing is half themaximum value, that is, smin = smax/2.

The instantaneous crack width wi is the difference betweenthe elongation of the tensile steel over the length s and theelongation of the concrete between the cracks and is given by

(6)

Under sustained load, additional cracks occur betweenwidely spaced cracks (usually when 0.67smax < s ≤ smax).The additional cracks are probably due to the combinedeffect of tensile creep rupture and shrinkage. As a consequence,the number of cracks increases and the maximum crackspacing reduces with time. The final maximum crackspacing s* is only approximately 2/3 of that given by Eq. (5),but the final minimum crack spacing remains approximately1/2 of the value given by Eq. (5).

As previously mentioned, experimental observations indicatethat τb decreases with time, probably as a result of shrinkage-induced slip and tensile creep. Hence, the stress in the tensileconcrete between the cracks gradually reduces. Further,although creep and shrinkage will cause a small increase inthe resultant tensile force T in the real beam and a slightreduction in the internal lever arm,8 this effect is relativelysmall and is ignored in the tension chord model presentedherein. The final crack width is the elongation of the steelover the distance between the cracks minus the extension ofthe concrete caused by σcx plus the shortening of theconcrete between the cracks due to shrinkage. For a finalmaximum crack spacing of s*, the final maximum crackwidth is

(7)

where εsh is the shrinkage strain in the tensile concrete (–ve);n = Es/Ee; Ee is the effective modulus given by Ee = Ec/(1 + ϕcc);Ec and Es are the elastic modulus of the concrete and theelastic modulus of steel, respectively; and ϕcc is the creepcoefficient of the concrete.

A good estimate of the final maximum crack width isgiven by Eq. (7), if s* is the maximum crack spacing after alltime-dependent cracking has taken place, that is, s* =0.67smax. If smax is given by Eq. (5), s* may be taken as

(8)

By rearranging Eq. (7), the steel stress on a cracked sectioncorresponding to a particular crack width w* is given by

σst2TAs

-----2τbs

db

-----------– and σc22τbρtcs

db

------------------==

smaxfct db

2τbρtc

---------------=

wis

Es

----- TAs

-----τbs

db

------- 1 nρtc+( )–=

w* s*Es

----- TAs

-----τbs*

db

----------- 1 nρtc+( ) εsh– Es–=

s*db

6.0ρtc

--------------=

(9)

By substituting Eq. (1) and (8) into Eq. (9) and selecting amaximum desired crack width in a particular structure w*,the maximum permissible tensile steel stress can be obtained.

COMPARISON WITH TEST DATAA total of 12 simply-supported beams and one-way slabs

were subjected to constant sustained service loads for aperiod of 400 days by Gilbert and Nejadi.6 Full details of thetest program and test results are available elsewhere.6 Eachspecimen was prismatic, with a rectangular cross section(b = 250 mm [9.8 in.] and d = 300 mm [11.8 in.] for the sixbeams and b = 400 mm [15.8 in.] and d = 130 mm [5.1 in.]for the six one-way slabs) and a span of 3500 mm (138 in.),and was carefully monitored throughout the test to record thetime-dependent deformation, together with the gradualdevelopment of cracking and the gradual increase in crackwidths with time. The parameters varied in the tests were theshape of the section b/d, the number of reinforcing bars, thespacing between bars sb, the concrete cover ct, and thesustained load level.

Details of the 12 specimens are provided in Table 1. Allspecimens were cast from the same batch of concrete and allthe tests commenced when the specimens were 14 days old.The measured elastic modulus, compressive strength, andtensile strength of the concrete at the age of first loadingwere Ec = 22,820 MPa (3310 ksi), fc = 18.3 MPa (2650 psi),and fct = 2.00 MPa (290 psi) and the measured creep coefficientand shrinkage strain associated with the 400-day period ofsustained loading were ϕcc = 1.71 and εsh = –0.000825.

The measured and predicted final maximum crack widthsare compared in Table 2. The mean value of predicted-to-measured final maximum crack widths (w*/wmax) is 1.54 andthe coefficient of variation is 21.4%. Considering thevariability of cracking in concrete and the requirement forconservatism in design-oriented equations such as Eq. (7), theagreement with test data is considered to be entirely satisfactory.

MAXIMUM STEEL STRESS FOR CRACK CONTROLThe model outlined in the previous section is herein used

to examine the effects of various parameters on themaximum tensile stress permitted in the main longitudinal

fstw*Es

s*-------------

τbs*

db

-----------+ 1 nρtc+( ) εsh+ Es=

Table 1—Details of test specimens6

Beam db, mm No. of bars As, mm2 ct, mm s, mm fst, MPa

B1-a 16 2 400 40 150 227

B1-b 16 2 400 40 150 155

B2-a 16 2 400 25 180 226

B2-b 16 2 400 25 180 153

B3-a 16 3 600 25 90 214

B3-b 16 3 600 25 90 129

Slab db, mm No. of bars As, mm2 ct , mm s, mm fst, MPa

S1-a 12 2 226 25 308 252

S1-b 12 2 226 25 308 195

S2-a 12 3 339 25 154 247

S2-b 12 3 339 25 154 171

S3-a 12 4 452 25 103 216

S3-b 12 4 452 25 103 159

Note: 1 MPa = 145 psi; 1 mm = 0.0394 in.

304 ACI Structural Journal/May-June 2008

tensile reinforcement if the maximum crack width is to belimited to a preselected value w*. The maximum permittedsteel stress is determined using Eq. (9).

Crack control in reinforced concrete slabConsider a one-way reinforced concrete slab of thickness

h containing a single layer of longitudinal tensile bars ofdiameter db at a bar spacing sb. The clear cover to thereinforcement from the tension face is ct. The area oftensile reinforcement per unit width of the slab is As and itis located at an effective depth of d (= h – ct – db /2) from thecompressive face of the slab. The characteristic compressivestrength of the concrete is fc′ . Unless noted otherwise, the slabdimensions and material properties are taken as h = 200 mm(8 in.); db = 12 mm (0.5 in.); ct = 20 mm (0.79 in.); w* =0.35 mm (0.0138 in.); fc′ = 32 MPa (4640 psi); Ec = 28,600 MPa(4140 ksi); ϕcc = 2.5; εsh = –0.0006; fct = 2.04 MPa (296 psi);and Es = 200 GPa (29,000 ksi).

In Fig. 3, the effect of bar diameter on the maximumpermissible steel stress is shown. For a given bar spacing, anincrease in bar diameter results in an increase in As and anincrease in the maximum steel stress required to produce acrack width of 0.35 mm (0.0138 in.). Of course, under aparticular in-service sustained moment, an increase in bardiameter results in an increase in As and a reduction incrack width.

The effect of varying the slab thickness on the maximumpermissible steel stress for a slab containing 12 mm (0.5 in.)diameter tensile bars is shown in Fig. 4. The slab depth has amarked influence on the maximum steel stress required toproduce a maximum particular crack width, with themaximum steel stress increasing as the slab depth decreases.

Figure 5 shows the effect of changing the bar diameter, butat the same time adjusting the bar spacing so that the area oftensile reinforcement remains constant. For a given reinforce-ment ratio (As/bd), if the bar diameter is reduced (that is,smaller diameter bars at closer centers are used), themaximum steel stress required to limit the maximum finalcrack width increases. In this case, the slab thickness was200 mm (8 in.) and the maximum final crack width was 0.35 mm(0.0138 in.). Of course, under a particular in-servicesustained moment, using smaller diameter bars at closercenters will result in a reduction in crack width.

Table 2—Measured6 and predicted maximumcrack widths at 400 days

BeamMaximum steel stress fst , MPa

Maximum crack width, mm

Ratiow*/wmax

Measuredwmax

Predicted w*(Eq. (7))

B1-a 226 0.38 0.383 1.01

B1-b 154 0.18 0.304 1.69

B2-a 225 0.36 0.382 1.06

B2-b 153 0.18 0.303 1.68

B3-a 213 0.28 0.239 0.85

B3-b 128 0.13 0.180 1.39

SlabMaximum steel stress fst , MPa

Maximum crack width, mm

Ratiow*/wmax

Measuredwmax

Predictedw* (Eq. (7))

S1-a 252 0.25 0.412 1.65

S1-b 195 0.20 0.347 1.73

S2-a 247 0.23 0.272 1.18

S2-b 171 0.18 0.216 1.20

S3-a 216 0.20 0.183 0.92

S3-b 159 0.15 0.154 1.03

Note: Mean (w*/wmax) = 1.54; and coefficient of variation (w*/wmax) = 21.4. 1 MPa =145 psi; 1 mm = 0.0394 in.

Fig. 3—Effect of bar diameter on maximum permissiblesteel stress.

Fig. 4—Effect of section depth on maximum permissiblesteel stress.


For a slab containing 12 mm (0.5 in.) diameter bars, theeffect of varying the maximum desired crack width w* isshown in Fig. 6. As the permissible crack width increases,the maximum permissible tensile steel stress also increases.For exposure classifications where crack widths have noinfluence on durability, the selection of a maximum desiredfinal crack width w* of 0.30 to 0.35 mm (0.012 to 0.014 in.)will generally be acceptable from the point of view ofaesthetics and the cracks will not detract from the appearance ofthe structure. Where the crack will not be visible andaesthetics is not important, a wider crack may be acceptable—perhaps up to 0.55 mm (0.022 in.). Where durability is anissue, the maximum desired final crack may be as low as0.15 mm (0.006 in.) in aggressive environments but notgreater than 0.3 mm (0.012 in.).

The effect of variations in the final shrinkage strain on themaximum permissible steel stress is shown in Fig. 7 for a200 mm (8 in.) thick slab containing 12 mm (0.5 in.) diameterbars. As expected, as the final shrinkage strain increases, themaximum permitted tensile steel stress decreases. Of course,under particular in-service conditions, an increase in thefinal shrinkage strain results in wider cracks.

The effect of varying the concrete strength on themaximum permissible tensile steel stress is shown in Fig. 8for a 200 mm (8 in.) thick slab containing 12 mm (0.5 in.)diameter bars. It is assumed herein that the concrete strengthonly affects the tensile strength, the elastic modulus, andthe creep coefficient. In all cases, the final shrinkage wasεsh = –0.0006. Clearly, the concrete strength does notsignificantly affect the maximum tensile steel stress requiredfor crack control.

Design exampleConsider a 150 mm (5.91 in.) thick, simply-supported one-

way slab located inside a building. With appropriate regard for

durability, the concrete strength is selected to be fc′ = 32 MPa(4640 psi) and the cover to the tensile reinforcement is takento be 20 mm (0.79 in.). The final shrinkage strain is taken to

Fig. 5—Effect of bar diameter and reinforcement ratio (As/bd)on maximum steel stress.

Fig. 7—Effect of final shrinkage strain on maximumpermissible steel stress.

Fig. 6—Effect of maximum final crack width on maximumsteel stress.

306 ACI Structural Journal/May-June 2008

be εsh = –0.0006. Other relevant material properties are Ec =28,600 MPa (4140 ksi); n = Es/Ec = 7.00; ϕcc = 2.5; fct =2.04 MPa (296 psi); and Es = 200 GPa (29,000 ksi). Theeffective modulus is therefore Ee = Ec/(1 + ϕcc) = 8170 MPa(1180 ksi) and the effective modular ratio n = Es/Ee = 24.5.The tensile face of the slab is to be exposed and the maximumfinal crack width is to be limited to w* = 0.3 mm (0.0118 in.).

After completing the design for strength and deflectioncontrol, the required minimum area of tensile steel is 650 mm2/m(0.307 in.2/ft). Under the full service loads, the maximum in-service sustained moment at midspan is 20.0 kN·m (14.7 kip·ft).The designer must select the bar diameter and bar spacing sothat the requirements for crack control are also satisfied.

Case 1—Use 10 mm (0.394 in.) bars at 120 mm (4.72 in.)centers, that is, As = 655 mm2/m (0.309 in.2/ft) at d = 125 mm(4.92 in.).

Referring to Fig. 1, elastic analysis of the cracked sectiongives c = 29.6 mm (1.16 in.) and Icr = 50.3 × 106 mm4

(121.0 in.4). The maximum in-service tensile steel stress onthe fully-cracked section at midspan is calculated usingEq. (2) and is σst = T/As = 7.00 × 20 × 106 × (125 – 29.6)/50.3 × 106 = 265 MPa (38.4 ksi).

The area of concrete in the tension chord is obtained usingEq. (1) and is Act = 60,200 mm2 (93.3 in.2). The reinforcementratio of the tension chord is ρtc = As/Act = 0.0109. With thefinal bond stress taken as τb = 1.0fct = 2.04 MPa (296 psi) andthe maximum final crack spacing obtained from Eq. (8) ass* = 10/(6.0 × 0.0109) = 153 mm (6.04 in.), the maximumpermissible steel stress required for crack control is obtainedusing Eq. (9).

fst0.3 200 000,×

153---------------------------------- 2.04 153×

10------------------------- 1 24.5 0.0109×+( ) –+=

0.0006 × 200,000 = 310 MPa (45.0 ksi)

The actual stress at the crack σst = 265 MPa (38.4 ksi) isless than fst = 310 MPa (45.0 ksi) and, therefore, cracking iseasily controlled.


For this section, c = 29.6 mm (1.17 in.) and Icr = 50.1 ×106 mm4 (121 in.4). The maximum in-service tensile steelstress on the fully-cracked section at midspan is σst = T/As =263 MPa (38.1 ksi). The area of concrete in the tension chordis Act = 60,200 mm2 (93.3 in.2). The reinforcement ratio ofthe tension chord is ρtc = As/Act = 0.0111. With τb = 1.0 fct =2.04 MPa (296 psi) and s* = 12/(6.0 × 0.0111) = 181 mm(7.13 in.), the maximum permissible steel stress required forcrack control (obtained using Eq. (9)) is fst = 251 MPa(36.4 ksi), which is just less than the actual maximum stressat the crack σst = 263 MPa (38.1 ksi). Therefore, the finalmaximum crack width may just exceed the desiredmaximum of 0.3 mm (0.012 in).


For this section, c = 29.5 mm (1.16 in.) and Icr = 48.7 ×106 mm4 (117.0 in.4). The maximum in-service tensile steelstress on the fully-cracked section at midspan is σst = T/As =266 MPa (38.6 ksi). The area of concrete in the tensionchord is Act = 60,270 mm2 (93.42 in.2) and the reinforcementratio of the tension chord is ρtc = As/Act = 0.0111. With τb =1.0fct = 2.04 MPa (296 psi) and s* = 16/(6.0 × 0.0111) =240 mm (9.45 in.), the maximum permissible steel stressrequired for crack control is fst = 169 MPa (24.5 ksi) (Eq. (9)),which is much less than the actual steel stress due to thesustained moment of σst = 266 MPa (38.6 ksi). Therefore,crack control is not adequate and the maximum final crackwidth will exceed 0.3 mm (0.012 in.).

By contrast, the procedures for crack control specified byACI 318-05,1 Eurocode 2,2 AS3600,3 and Gergely and Lutz5

all suggest that cracking is adequately controlled in all threeof the aforementioned cases. Each of these methods does notadequately account for the time-dependent increase in crackwidths due to shrinkage. It is not surprising that excessivelywide cracks are a common serviceability problem in manyreinforced concrete structures throughout the world.

CONCLUDING REMARKSThe procedure outlined previously provides a simple and

reliable approach to crack control and has been proposed forinclusion in the next edition of the “Australian Standard forConcrete Structures,” AS3600. In the design for crackcontrol at the serviceability limit state, the designer mustselect the maximum desired final crack width in the structureand then ensure that the tensile steel stress on the crackedsection under the sustained service load is less than themaximum value fst given by Eq. (9). The approach has beenshown to provide good agreement with measured final crackwidths in beam and slab specimens under sustained serviceloads for a period of 400 days.

Sensible detailing should always be specified for crackcontrol. For example, the distance from the side or soffit of abeam to the center of the nearest longitudinal bar should notexceed approximately 100 mm (4.0 in.) and the center-to-

Fig. 8—Effect of concrete strength on maximum permissiblesteel stress.


center spacing of bars near a tension face of a beam or slabshould not exceed approximately 300 mm (12 in.).4

For exposure classifications where crack widths have noinfluence on durability, the selection of a maximum desiredfinal crack width w* of 0.3 to 0.35 mm (0.012 to 0.014 in.)will generally be acceptable from the point of view ofaesthetics and the cracks will not detract from the appearance ofthe structure. Where the crack will not be visible andaesthetics is not important, a wider crack may be acceptable—perhaps 0.5 to 0.6 mm (0.02 to 0.025 in.). Where durabilityis an issue, the maximum desired final crack may be as lowas 0.15 mm (0.006 in.) in aggressive environments but notgreater than 0.3 mm (0.012 in.).

ACKNOWLEDGMENTSThe support of the Australian Research Council through an ARC

Discovery Grant and an ARC Australian Professorial Fellowship isgratefully acknowledged.

NOTATIONAct = area of concrete in tension chord, mm2 (in.2)As = area of tensile reinforcement, mm2 (in.2)b = width of compression chord, mm (in.)b* width of section at level of centroid of tensile reinforcement,

mm (in.)c = depth of compression chord or compression zone, mm (in.)ct = clear cover to tensile reinforcement, mm (in.)d = effective depth to centroid of tensile reinforcement, mm (in.)db = bar diameter, mm (in.)Ec = elastic modulus of concrete, MPa (ksi)Ee = effective modulus of concrete, MPa (ksi)Es = elastic modulus of steel reinforcement, MPa (ksi)fc′ = characteristic compressive (cylinder) strength of concrete,

MPa (psi)fct = direct tensile strength of concretefst = stress in tensile steel at crack, MPa (ksi)fy = yield stress of steel reinforcement, MPa (ksi)h = overall depth or thickness of beam or slab, mm (in.)Icr = second moment of area of cracked transformed section, mm4 (in.4)Mcr = cracking moment, kN·m (kip·ft)Ms = in-service bending moment, kN·m (kip·ft)n = modular ratio (Es/Ec)n = effective modular ratio (Es/Ee)

s = crack spacing, mm (in.)s* = final crack spacing, mm (in.)sb = center-to-center spacing between bars, mm (in.)T = total tensile force in tension chord, kN (kips)w* = maximum final crack width, mm (in.)wi = maximum initial crack spacing (at first loading), mm (in.)z = distance along tension chord, mm (in.)εsh = shrinkage strain of concreteϕcc = creep coefficient of concreteρtc = reinforcement ratio of tension chord (Ast /Act)σc = stress in concrete, MPa (psi)σc2 = tensile stress in concrete in tension chord midway between cracks,

MPa (psi)σct = uniform average tensile stress in concrete in tension chord,

MPa (psi)σst1 = stress in reinforcement in tension chord at crack, MPa (ksi)σst2 = stress in reinforcement in tension chord midway between cracks,

MPa (ksi)τb = average bond stress, MPa (psi)

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, MI, 2005, 430 pp.

2. BS EN 1992-1-1:2004, “Eurocode 2: Design of Concrete Structures—Part 1-1: General Rules and Rules for Buildings,” European Committee forStandardization, CEN, Brussels, 2004, 224 pp.

3. Standards Australia Committee BD-002, “Australian Standard forConcrete Structures (AS3600-2001),” Standards Australia, Sydney,Australia, 2001, 176 pp.

4. Gilbert, R. I., “Cracking and Crack Control in Reinforced ConcreteStructures Subjected to Long-Term Loads and Shrinkage,” 18th AustralianConference on the Mechanics of Structures & Materials (ASMSM18), V. 2,A. J. Deeks and H. Hao, eds., the Netherlands, 2004, pp. 803-809.

5. Gergely, P., and Lutz, L. A., “Maximum Crack Width in ReinforcedConcrete Flexural Members,” Causes, Mechanism, and Control ofCracking in Concrete, SP-20, American Concrete Institute, FarmingtonHills, MI, 1968, 244 pp.

6. Gilbert, R. I., and Nejadi, S., “An Experimental Study of FlexuralCracking in Reinforced Concrete Members under Sustained Loads,”UNICIV Report No. R-435, School of Civil and Environmental Engineering,University of New South Wales, Sydney, Australia, 2004, 59 pp.

7. Marti, P.; Alvarez, M.; Kaufmann, W.; and Sigrist, V., “Tension ChordModel for Structural Concrete,” Structural Engineering International, Apr.1998, pp. 287-298.

8. Gilbert, R. I., Time Effects in Concrete Structures, Elsevier SciencePublishers, Amsterdam, the Netherlands, 1988, 321 pp.

Documents

105-s29 Control of Flexural Cracking in Reinforced Concrete