a new formula to calculate crack spacingfor concrete plates

ACI Structural Journal/January-February 2010 43

ACI Structural Journal, V. 107, No. 1, January-February 2010.MS No. S-2008-254.R3 received February 13, 2009, and reviewed under Institute

publication policies. Copyright © 2010, American Concrete Institute. All rights reserved,including the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the November-December 2010 ACI Structural Journal if the discussion is received by July 1, 2010.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Different codes have different formulas to calculate crack spacingand crack width developed in flexural members. Most of theseformulas are based on the analysis of results of tested beams orone-way slabs. Crack control equations for beams underestimatethe crack width developed in plates and two-way slabs. It seemsthat little attention has been paid in determining the crack spacingand width in reinforced concrete plates. The behavior of reinforcedconcrete plates and two-way slabs is different from beams or one-way slabs; therefore, the methods developed for beams cannot bedirectly applied to plates and two-way slabs. In this paper, a newanalytical equation is proposed for calculating the crack spacingfor plates and two-way slabs. A special focus will be given to thickconcrete plates used for offshore and nuclear containment structures.The proposed equation takes into account the effect of steel rein-forcement in the transverse direction through the splitting bondstress. The new equation provides good estimates for crack spacingin plates and two-way slabs with different concrete covers. Eightfull-scale two-way slabs were designed and tested to examine theeffects of concrete cover and bar spacing of normal- and high-strength concrete on crack spacing. The different code expressionsare evaluated with respect to the experimental results.

Keywords: bond stress; crack spacing; plate; transverse reinforcement.

INTRODUCTIONCrack-control equations for beams underestimate the

crack width developed in plates and two-way slabs.1 Thebehavior of reinforced concrete plates and two-way slabs isdifferent from that of one-way slabs and beams; hence, themethods developed for beams cannot be directly applied toplates and two-way slabs. The expression for crack spacingis based on the beam theory in several codes, such as theCanadian offshore code CSA-S474,2 Norwegian Code NS3473E,3 and the European CEB-FIP4 model code. With theextensive use of thick concrete plates and slabs with thickconcrete covers for offshore and nuclear containmentstructures, the development of new formulas is needed topredict crack spacing and width for plates and two-wayconcrete slabs. This paper examines the different approachesand codes for estimating the crack spacing. Little attentionhas been paid in determining the crack spacing and width inreinforced concrete thick plates. A lack of available researchdata on the prediction of crack properties results in unnecessaryoverdesign of steel reinforcement to satisfy conservativecrack requirements in codes for offshore structures. Thisinvestigation presents a unique experimental work. Crackspacing of eight reinforced concrete specimens relevant tothe offshore structures had been examined. Furthermore, theresults were evaluated with regard to the available codes.

RESEARCH SIGNIFICANCEThis paper provides a rational method for designers to

calculate crack spacing for plates and two-way concrete

slabs. An accurate estimate of the crack spacing and crackwidth of thick concrete plates used for offshore and nuclearpower plant structures can result in the reduction ofsteel reinforcement. The saving of steel reinforcement tosatisfy the crack width limitations can be estimated inmillions of dollars for a single project (for example, Hiberniaoil platform). The proposed equation combines the knownbond stress effect with the contribution of splitting bondstress in the transverse direction due to the action of two-wayslabs. The proposed equation gives a good estimate for crackspacing in plates and two-way slabs with concrete covers (Cc< 2.5db). The proposed method can also be modified andused for plates with thick concrete covers (C = 2.5 – 5.0db).

PREVIOUS RESEARCHCrack width models clearly illustrate that the crack

spacing and width are functions of the distance between thereinforcing steel. Therefore, crack control can be achievedby limiting the spacing of the reinforcing steel. Maximumbar spacing can be determined by limiting the crack width toacceptable limits.

Crack spacingThe Canadian offshore code CSA-S474-042 provides the

following expression for calculating the average crackspacing. This is the same equation used by the Norwegiancode NS 3473E3

Sm = 2.0(Cc + 0.1S) + k1k2dbe′ hef b/As (1)

where Sm is the average crack spacing, mm; Cc is concretecover, mm; S is bar spacing of outer layer, mm; k1 is thecoefficient that characterizes bond properties of bars; k1 = 0.4 fordeformed bars; and k1 = 0.8 for plain bars—this is related tothe deformed rips on reinforcing bars; k2 is the coefficient toaccount for strain gradient; k2 = 0.25(ε1 + ε1)/2ε1, where ε1and ε2 are the largest and the smallest tensile strains in theeffective embedment zones; dbe′ is the bar diameter of outerlayer, mm; hef is the effective embedment thickness as thegreater of a1 + 7.5dbe′ and a2 + 7.5dbe′ but not greater than thetension zone or half slab thickness, mm; a1 and a1 are thedistances from the centers of the bars to the surface of theconcrete, mm (refer to Fig. 1);b is the width of the section,mm; and As is the area of reinforcement within the effectiveembedment thickness (mm2).

Title no. 107-S05

A New Formula to Calculate Crack Spacingfor Concrete Platesby E. Rizk and H. Marzouk

ACI Structural Journal/January-February 201044

The crack spacing provided in Eq. (1) can be divided intotwo terms. Term A is a function of concrete cover and barspacing (A = 2.0[Cc + 0.1S]). Term B relates to the type ofbar, the diameter, and type of stress (B = k1k2d ′be hef b/As).

The CEB-FIP4 model code crack spacing expression isrecommending a different approach compared to other codes(CSA2 and NS3). Meanwhile, the bond effect of CEB-FIP4

is treated in a different manner. For a cracked reinforcedconcrete section, an increase in loading will result in anincrease in steel strain. This will cause an elongation of thereinforcing bar in which the bar ribs will tend to movetoward the nearest crack relative to the surrounding concrete.The stress in the steel caused by steel strain will be reduceddue to the bond stress τbk between the steel and surroundingtensile concrete. Therefore, instead of using the factor k1 toaccount for the bond effect, the CEB-FIP4 model code uses thebond stress τbk directly in the expression as shown in Eq. (2)

(2)

For stabilized cracking

(3)

The CEB-FIP4 model code provides the following expressionfor calculating the average crack spacing for stabilized cracking

(4)

where ls,max is the length over which slip between the steelreinforcement and concrete occurs (approximating crackspacing in stabilized cracking), mm. Steel and concretestrains, which occur within this length, contribute to thewidth of the crack; σs2 is the steel stress at crack, MPa; σsEis the steel stress at point of zero slip, MPa; ϕs is the bardiameter, mm; τbk is the lower fractile value of the average

ls max,

2σs2 σsE–

4τbk

----------------------ϕs=

ls max,

ϕs

3.6ρs ef,

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

Srm23---ls max,

=

bond stress, MPa; ρs,ef is the effective reinforcement ratio(As/Ac, ef); Ac, ef is the effective area of concrete in tensionlimited by slab width and height equal to the lesser of 2.5(c + φ/2) or (h – c)/3 (mm2); and Srm is the average crackspacing, mm.

A reevaluation of cracking data5 provides a new equationbased on the physical phenomenon for the determination ofthe flexural crack width of reinforced concrete members.This study shows that previous crack width equations arevalid for a relatively narrow range of covers up to 63 mm(2.5 in.). Frosch (1999) introduced this phenomenon into anew expression that was adopted by ACI 318-99.6 For crackcontrol in beams and one-way slabs, ACI 318-996 requiresthe spacing of reinforcement closest to a surface in tensionnot exceed

(5)

where fs is the calculated stress in reinforcement at serviceload = unfactored moment divided by the product of steelarea and internal moment arm. Alternatively, fs can be takenas 0.60 fy; Cc is the clear cover from the nearest surfacein tension to the flexural tension reinforcement; and S isthe center-to-center spacing of the flexural tension reinforce-ment nearest to the surface of the extreme tension face.

Crack control in plates and two-way slabsThe cracking width in plates and two-way slabs is

controlled primarily by the steel stress level and spacing ofthe reinforcement in two perpendicular directions. In addi-tion, the clear concrete cover in plates and two-way slabs isnearly constant (20 mm [0.8 in.]) for most interior structuralslabs, whereas it is a major variable in the crack controlequations for beams. Analysis of data on cracking in platesand two-way slabs1 has provided the following equation forpredicting the maximum crack width

(6)

where the terms under the square root are collectively termedthe grid index, and k is a fracture coefficient (k = 2.8 × 10–5)for uniformly loaded restrained two-way action square slabsand plates. For concentrated loads or reactions or when theratio of short to long span is less than 0.75 but larger than 0.5,a value of k = 2.1 × 10–5 is applicable. For span aspect ratiosless than 0.5, k =1.6 × 10–5; β = 1.25 (chosen to simplifycalculations, although it varies between 1.20 and 1.35); andfs is the actual average service-load stress level or 40% of thespecified yield strength fy, ksi.

Desayi and Kulkarni7 developed an approximatemethod to predict the maximum crack width in two-wayreinforced concrete slabs. The researchers calculated themaximum crack width based on an estimation of the crackspacing at any given stage of loading, which is betweenthat stage and the ultimate load. As a result of the two-wayaction of the slabs, when the stretching of bars in Direc-tion X and the concrete surrounding them is considered,the bars in the perpendicular direction can be assumed tobear against the concrete surrounding them. The spacingof cracks formed in Direction X can be calculated usingthe following formula

S 95 000,fs

------------------ 2.5Cc–=

w kβfs I=

E. Rizk is a PhD Candidate and Research Assistant at Memorial University ofNewfoundland, St. John’s, NL, Canada, and is an Assistant Lecturer at MenoufiyaUniversity, Shibin el Kom, Egypt. He received his BSc and MSc from MenoufiyaUniversity in 1999 and 2005, respectively. His research interests include crackingand minimum reinforcement of offshore structures and shear strength of two-wayconcrete slabs.

ACI member H. Marzouk is Chair of the Civil Engineering Department at RyersonUniversity, Toronto, ON, Canada. He received his MSc and PhD from the Universityof Saskatchewan, Saskatoon, SK, Canada. He is a member of ACI Committees 209,Creep and Shrinkage in Concrete, and 213, Lightweight Aggregate and Concrete. Hisresearch interests include structural and material properties of high-strength andlightweight high-strength concrete, offshore design, creep, and finite element analysis.

Fig. 1—Effective embedment thickness (effective tension area).


(7)

where fb is the bond strength; kb is the constant to account forthe surface characteristics of the bar and the distribution ofbond stress; fbb is the bearing stress; kt is the constant toaccount for the distribution of tensile stress; and ft is thetensile strength of the concrete.

The diameter of the bars in Direction X is ϕ1 and thespacing between bars is S1. In Direction Y, the diameter is ϕ2and the spacing between bars is S2.

The researchers7 presented an equation to predict crack widthin two-way slabs, which considered the bond force on the barsand anchorage forces due to cross wires. The proposed methodgives a large coefficient of variation, compared to the morepractical equation used by Nawy and Blair.1

Marzouk and Hossin8 tested eight square full-scalespecimens to investigate the crack width and spacing ofhigh-strength concrete slabs, five high-strength concreteslabs, and three normal-strength concrete slabs. The structuralbehavior, with regard to the deformation and strengthcharacteristic of high-strength concrete slabs of variousthicknesses and different reinforcement ratios (0.40 to2.68%), were studied.

ANALYTICAL MODELThe presented theoretical model for calculating crack

spacing for two-way slabs combines the known effect ofbond stress with the splitting bond stress in the transversedirection, which is due to the action of two-way slabs.

Bond stress distributionFor a concrete section between two successive cracks in a

tensile test specimen, zero bond stresses at the two crackedsections and at the midpoint can be assumed. Variation of thebond stress between these two zero-points (between themidpoint and the nearest cracked section) were establishedby many researchers9,10 based on experimental results. Inthe present research, it is further assumed that the peak bondstress occurs at the midsection between the two zero points,with a parabolic variation. These two assumptions greatlysimplify the mathematical formulation in calculating thebond stress. The resulting bond stress distribution closelyagrees with the experimental observations.9,10 The resultingparabolic bond stress distribution between two successiveflexural cracks is shown in Fig. 2(b).

Figure 2 shows a cross section of a slab and the layout ofreinforcement in the Directions X and Y. Stretching the barsin Direction X with the concrete surrounding the bars willresult in another crack at a Distance x = smx. At the sametime, as a result of the two-way action of the slabs, stretchingthe bars in a perpendicular direction results in splittingcircumferential forces in Direction X. A sufficient bondforce is developed at this location (x = smx), which, togetherwith the splitting stresses along the transverse bars, is just largeenough to induce a maximum tensile stress equal to thetensile strength of concrete.

Longitudinal steel reinforcement (loading direction)The equilibrium forces acting on a concrete Section 1-1 in

Direction X are shown in Fig. 2(a) for a unit width of the slabin Direction Y

α1kt ftAct1

πφ1kb fb/s1( ) φ2 fbb/s2( )+-------------------------------------------------------------= (8)

The constant kt accounts for the distribution of tensilestress in Section 1-1 on the effective area of concrete Actx andfctm is the mean tensile strength value of the concrete that iscalculated according to the CEB-FIP4 model code.

The number of bars per unit width in X direction is nx, andthe peak bond strength is fbo, calculated using the CEB-FIP4

Model Code equation. The CEB-FIP4 model code(Table 3.1.1) provides the following expression forcalculating peak bond stress for confined and unconfinedconcrete for different bond conditions

(9)

For cases where failure is initiated by splitting of theconcrete (unconfined concrete), the coefficient μ is takenequal to unity and, hence, fbo is calculated as follows

(10)

23--- πdbx fbosmx nx +

contribution of transverse steel reinforcement kt fctmActx=

fbo μ fc′ MPa( )=

fbo 1.0 fc′ MPa( )=

Fig. 2—Distribution of bond stress, splitting stress andtensile stress over a section: (a) plan of two-way plate;and (b) cross section of two-way plate.

46 ACI Structural Journal/January-February 2010

It should be noted that Eq. (10) is only valid for concretecovers equal to or less than 2.5db (Cc ≤ 2.5db), Cc is the clearconcrete cover, for plates and two-way slabs with thickconcrete covers greater than the radius of the effectiveembedment zone (Cc > 2.5db), it was found that a value of0.75 for the coefficient μ will be more consistent, so Eq. (10)can be written as follows

(11)

This is due to the fact that such plates act as cross sectionsthat contain two separate materials: a reinforced concretepart and a plain concrete part.

Transverse steel reinforcement and splitting bond stress

The contribution of the transverse steel reinforcement isconsidered through splitting bond stress. Consider a concretecylindrical prism with a diameter of cs (diameter of effectiveembedment zone) containing a bar with a diameter of db, asshown in Fig. 3(c). The radial components of the forces onthe concrete, shown in Fig. 3(a) and (b), cause a pressure pon a portion of the cross section of the prism. This is equilibratedby tensile stresses in the concrete on either side of the bar. InFig. 3(a), the distribution of these stresses has been previouslyassumed to be parabolic; this assumption has been found toprovide more consistent values compared to the experimentalresults. Splitting is assumed to occur when the maximumstress is equal to the tensile strength of the concrete fctm. Forequilibrium in the transverse direction in a prism with alength equal to ly

(12)

where K is the ratio of the average tensile stress to themaximum tensile stress and equals 0.33 for the parabolicstress distribution. A rearrangement gives

(13)

where cs is the diameter of the effective embedment zonewhere the reinforcing bar can influence the concrete bond,which is also known as the diameter of the splitting cylinder,


Pdbly

2------------- K

cs

2----

db

2-----–⎝ ⎠

⎛ ⎞ fctmly=

P 0.33cs

db

----- 1–⎝ ⎠⎛ ⎞ fctm=

arbitrarily taking cs = (3.0 – 3.5) db; and db is the bar diameter. Fora triangular stress distribution, K equals 0.5.11

The contribution of the transverse splitting bond can beestimated by considering the equilibrium of forces acting onconcrete to the left and right of Section 1-1 (Fig. 2(a)), andthe unit width of the slab in transverse Direction Y

Contribution of transverse steel reinforcement = K(cs – dby)fsp,tly (14)

The different components of the right hand side of Eq. (14)can be estimated in the following:

The splitting bond stress fsp,t can be assumed to be equalto fctm. The diameter of the effective embedment zone cs =3.0dby. The length of the effective embedment zone ly istaken equal to the slab unit width. Therefore, Eq. (14) can bewritten as follows for a unit width in the Y direction

Contribution of transverse steel reinforcement = 0.33(3dby – dby)fctm (15)

The contribution of the splitting bond stress determinedfrom Eq. (15) can be substituted into Eq. (8), representing theequilibrium forces in Direction X to determine the crackspacing as follows

(16)

The crack spacing formed in Direction X can be estimatedas follows

(17a)

Similarly, the spacing of cracks formed in Direction Y canbe estimated as follows

(17b)

Equations (17a) and (17b) give the crack spacing inDirections X and Y, respectively, at a given stage of loading.The proposed model suggests that increasing bar diameter dbwill result in decreased crack spacing and, hence, decreasedcrack width. Also, increasing the number of bars (decreasingbar spacing) will result in decreasing crack spacing and,hence, achieving required crack control. To use the previousexpression, values of kt, Actx, Acty, fctm, and fbo must beestimated. The constant kt is a tensile stress factor thatdepends on the distribution of tensile stress on concrete areasActx and Acty. The value kt is the ratio of the average tensilestress area to the actual tensile stress area within the effectiveembedment thickness hef. For thick slabs, the tensile stressdistribution within the effective embedment thickness istrapezoidal and, hence, kt could be assumed equal to 0.67 to1.0. In the proposed expression, for plates and two-way slabshaving concrete covers of (Cc < 2.5db), tensile stress on theconcrete is assumed to be uniformly distributed and, hence,kt can be taken as equal to unity.7 For thick plates and two-way slabs with thick concrete covers that are greater than2.5db and less than 5.0db, tensile stress distribution on the

23---πdbx fbosmxnx 0.33+ 2dby( )fctm kt fctm Actx=

smxkt fctmActx 0.67– dby fctm

23---πdbx fbonx

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

smykt fctm Acty 0.67– dbx fctm

23---πdby fbony

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

Fig. 3—Stresses in circular concrete prism subjected tobond stresses: (a) parabolic stress distribution; (b) triangularstress distribution; and (c) diameter of effective embedmentzone.


concrete is assumed to be trapezoidal and, hence, kt can betaken equal to 0.67; this assumption was found to be moreconvenient. The values of Actx and Acty, which are the effectivestretched area of concrete in the X and Y direction, areassumed to be

Actx = hefxb (18a)

Acty = hefyb (18b)

where hef is effective embedment thickness (shown in Fig. 1)as the greater of a1 + 7.5d ′be and a2 + 7.5d′be but not greaterthan the tension zone or half slab thickness, mm; and b is thewidth of the section, mm.

Crack spacing for beams and one-way slabsThe proposed equation can be used to calculate the crack

spacing for beams and one-way slabs by modifying the peakbond strength fbo, according to the CEB-FIP4 model codeprovisions (Table 3.1.1). For cases where failure is initiatedby shearing of the concrete between the ribs (all other bondconditions), fbo is calculated as

(19)

The crack spacing can be estimated as

(20)

where n is the number of bars per unit width. The constant ktis a tensile stress factor, which depends on the distribution oftensile stress on concrete areas Act. In the present research, avalue of 0.67 for the coefficient kt was found to be moreconsistent for beams and one-way slabs.

EXPERIMENTAL INVESTIGATIONThe variables considered in the current investigation are

the concrete cover, slab thickness, and bar spacing fornormal- and high-strength concrete. The selected values forthe proposed experimental testing are typical for the possibleuse in Canadian offshore applications. A total of eightconcrete slabs were tested. Five normal-strength concreteslabs (NS) and three high-strength concrete slabs (HS) wereselected for the experimental investigation of the crackingbehavior study as detailed in Table 1. The slabs’ thicknessesranged from 150 to 400 mm (6 to 16 in.) and were designedto examine the effect of slab thickness on the crackingbehavior. The details of a typical test specimen are shown inFig. 4. The test slabs were classified into three series.

The first group (Series I) was designed to investigate theeffect of concrete cover and bar spacing on the crack width.The group was made of two slabs designated as NS1 (h =150 mm [6 in.]) and NS2 (h = 200 mm [8 in.]). The slabs hadthe same concrete cover (Cc = 40 mm [1.6 in.]), withdifferent bar spacing. The effect of concrete strength oncrack spacing was not considered because it has a smallinfluence on the crack spacing in Series I. The second group(Series II) was designed to investigate the effect of concretecover, concrete strength, and corresponding change in steelratio on the average crack spacing. The slabs of this group


smkt fctAct

23---πdb fbon----------------------=

had different slab thicknesses (250 and 300 mm [10 and 12 in.]),different concrete cover thicknesses (60 and 70 mm [2.4 and2.75 in.]), and different bar sizes (15M and 25 M [No. 5 andNo. 8]) but with the same big bar spacing of 368 mm (14.5 in.).

All the specimens of Series I and Series II were designedto fail in flexure, as recommended by Marzouk andHussein.12 The third group (Series III), however, wasdesigned to investigate the effect of other modes of failure oncrack spacing. The specimens in this series were designed tofail under pure punching failure mode, as discussed byOsman et al.13 The third group includes two thick, heavy,reinforced specimens HS3 (h = 350 mm [14 in.]) and NS5(h = 400 mm [16 in.]) with the same 70 mm (2.75 in.) thickconcrete cover and heavy reinforcement ratio, which is atypical practice for offshore structures.

Test procedureA typical cross section of the tested specimen is shown in

Fig. 4. The tested slab was placed in the frame in a verticalposition. The test slabs were simply supported along all fouredges with the corners free to lift. The specimen was initiallyloaded up to 10% of the ultimate load. Then, crack gaugeswere installed using epoxy glue on the tension surface of theslab and left for 1 hour to enable the epoxy to dry. The loadwas released and then reapplied at a selected load incrementof 44.0 kN (10 kips). The tested slabs were carefullyinspected at each load step. The cracks were marked manuallyafter mapping all the cracks on the specimen. Crack mapping of

Table 1—Details of test specimens

SeriesNo.

Slabno.*

Com-pressivestrength fc′ , MPa

Barsize,mm

Barspacing s,

mm

Concretecover Cc,

mm

Slab thickness,

mmDepth,

mm

Steel ratio ρ,%

INS1 44.7 10 210 45 150 105.0 0.48

NS2 50.2 15 240 40 200 152.5 0.54

II

NS3 35.0 15 368 60 250 182.5 0.35

HS1 70.0 15 368 60 250 182.5 0.35

NS4 40.0 25 368 70 300 217.5 0.73

HS2 64.7 25 368 70 300 217.5 0.73

IIIHS3 65.4 35 289 70 350 262.5 1.44

NS5 40.0 35 217 70 400 312.5 1.58*NS is normal-strength slabs; and HS is high-strength slabs.Note: 1 MPa = 145 psi; 1 mm = 0.0394 in.

Fig. 4—Details of typical test specimen HSC2 (thickness300 mm [12 in.]).


the specimen was depicted by means of photographs at eachstage of loading throughout the experiment. These photographswere inserted in a computer aided AutoCAD software draftingpackage on a two-dimensional grid with a scale of one to one.Cracks were retraced on the computer using AutoCAD, tools,and the spacing was measured and averaged using the software.It was found that for all the specimens, the first crack forms alongthe reinforcing bar and passes through the slab center or close tothe slab center. The second crack forms along theperpendicular reinforcing bar in the other direction.

Test resultsThe first crack of each specimen was visually inspected

and the corresponding load was recorded as the first crackload. The yield steel strain was recorded at a value of2000 με, which produced a stress in the steel reinforcing barequal to 400 MPa (58 ksi). The yield strain was measured ata location 150 mm (6 in.) from the center of the slab. Thevalue of 2000 με was suggested based on experimentalobservations of the stress-strain curve of a single reinforcing bar.

In all tested slabs, the initial observed cracks were firstformed tangentially under the edge of the column stub,

followed by radial cracking extending from the column edgetoward the edge of the slab.

For the slabs failing in flexure (NS1, NS2, NS3, and HS1),the crack pattern observed prior to punching consisted of onetangential crack, roughly at the column outline, followed byradial cracking extending from the column. In all slabs,flexure yield lines were well developed. This failure can beclassified as flexure failure. For the slabs failing by flexure-punching, the crack pattern observed prior to punchingconsisted of almost no tangential crack; radial cracking extendingfrom the column was the most dominant crack pattern.Failure patterns of the tested slabs are shown in Fig. 5 to 7.

It was noticed that increasing the concrete cover resultedin increased crack spacing. Test results of Series II indicatedthat increasing the concrete cover from 60 to 70 mm (2.4 to2.75 in.) increased the crack spacing from 245 to 261 mm(9.7 to 10.3 in.). The test results of Series I (Specimens NS1 andNS2) and Series II indicated that as the bar spacing is increasedfrom 210 to 240 mm (8.3 to 9.5 in.), the crack spacing increasedfrom 201 to 221 mm (8 to 8.7 in.), respectively.

Series II included two specimens (NS3 and HS1) reinforcedwith low steel reinforcement ratio 0.35%, and two specimens(NS4 and HS2) reinforced with medium steel reinforcementratio 0.7% to fail under flexure mode. All specimens had thesame bar spacing 368 mm (14.5 in.). It is interesting to pointout that the average crack spacing almost equal to 253 mm(10 in.) was much smaller than the bar spacing.

Series III included two specimens (HS3 and NS5) reinforcedwith heavy steel reinforcement ratios 1.44 and 1.58% withthe same 70 mm (2.75 in.) thick concrete cover. Test resultsrevealed that crack control (crack width) can still be achievedby limiting the spacing of the reinforcing steel despite usingthick concrete cover. The crack width can be calculated bymultiplying the crack spacing by the steel strain. The steelstrain can be determined at any loading by determining theneutral axis and assuming linear stress distribution.

DISCUSSIONVerification of proposed model

A total of 12 simply-supported beams and one-way slabswere subjected to constant sustained service loads for aperiod of 400 days by Gilbert and Nejadi.14 The parametersvaried in the tests were the shape of the section b/h, thenumber of reinforcing bars, the spacing between bars s, theconcrete cover Cc, and the sustained load level. A comparisonbetween beam Series 1 and 2 (Table 2) demonstrates thatincreasing the clear concrete cover increases the averagecrack spacing. This is because the crack spacing srm isinversely proportional to the effective reinforcement ratioρeff. Increasing the bottom cover increases the effectivetension area of the concrete and decreases the effective rein-forcement ratio, which results in a larger crack spacing. Also,increasing the tensile reinforcement area decreases crackspacing and reduces crack width (because crack spacing isinversely proportional to the effective reinforcement ratio).

Frosch et al.15 tested 10 one-way slabs to determine theeffects of bar spacing and epoxy coating thickness on crackwidth and spacing. The primary variables evaluated in thestudy were the spacing of the reinforcement and the epoxycoating thickness. The parameters varied in the tests were thereinforcing bars type, the spacing between bars s, and thesustained load level. The measured crack width and spacingwere also compared to calculated crack width and spacing.Major conclusions derived from this investigation include:

Fig. 5—Crack patterns of Series I: (a) NS1 (thickness150 mm [6 in.]); and (b) NS2 (thickness 200 mm [8 in.]).


spacing of reinforcement significantly affected the width andspacing of cracks; and as the reinforcement spacingdecreased, the spacing of primary cracks decreased and thenumber of primary cracks increased.

To verify the validity of the new proposed model, themodel was applied to predict the average crack spacing ofnormalweight concrete test slabs reported in the literature(that is, tests other than those of the authors). The resultsindicate that a very good correlation exists between theoreticaland measured average crack spacing values and betweentheoretical and calculated average spacing values usingCSA-S474-04 and NS-92 codes were very close to theexperiments with approximately 5% error. In this paper, themodel has been applied to 30 tests to predict the average crackspacing of beams and one- and two-way concrete slabs. Thegeometry of test slabs, the analysis, and the results are shownin Tables 2 to 4 and include 12 test results of Gilbert and

Nejadi;14 10 test results of Frosch et al.,15 with differentconcrete covers and different bar spacing; and 8 test results ofMarzouk and Hossin,8 with different concrete strengths,different concrete covers, and different bar spacing.

For the proposed model, the overall average theory/testratio was 1.028 with a standard deviation of 0.149, givingstrong support to the ability of the proposed model to evaluatethe average crack spacing in tested slabs. It is also worthemphasizing that the slabs analyzed and presented in Tables2 to 4 cover many variables that influence crack spacing,such as concrete strength, bar spacing, and concrete cover.Bearing this in mind, as well as the fact that the tests themselvesare one-to-one scale models of the prototype and the inevitablescatter of test results in concrete behavior, the theoreticalmodel developed herein is an excellent representation of thephysical behavior of tested specimens.

Fig. 6—Crack patterns of Series II: (a) NS3 (thickness 250 mm [10 in.]); (b) HS1 (thickness 250 mm [10 in.]); (c) NS4(thickness 300 mm [12 in.]); and (d) HS2 (thickness 300 mm [12 in.]).


Tables 2 to 4 show a comparison between the calculatedvalues of crack spacing with the measured experimentalvalues reported by different researchers.8,14,15 Analysis ofthe results given in Tables 2 to 4 indicates that the newproposed model provides good estimates for crack spacing inslabs having small and thick concrete covers.

Figures 8 and 9 show a comparison in the calculations forcrack spacing between the new proposed model and differentcodes with the measured experimental values by Marzoukand Hossin.8 Figures 8 and 9 indicate that the crack spacingvalues estimated using the presented model, CSA-04, NS-92,and EC2-04 codes were very close to the experimentswith approximately 5 to 9% error. The codes’ expressionsfor crack spacing are based on the beam theory, whereasthe presented model is rational because it is based on two-way action.

Comparison of experimental versus theoretical estimates of crack spacing

Table 5 shows a comparison between the calculated valuesof crack spacing with the measured experimental values. Forbar spacing greater than 300 mm (12 in.), the Norwegiancode NS 3474 E3 and the Canadian offshore code CSA-S474-042

overestimates the average crack spacing by approximately33%. In general, the calculated average crack spacing was

Table 2—Comparison between calculated crack spacing values using code formulas with measured experimental values for test specimens by Gilbert and Nejadi14

Slab no.Concrete cover

Cc , mm Height h, mmBar spacing s,

mm fc′ , MPa NS/CSA, mm CEB, mmNew proposed

model, mmExperimentalresults, mm

Beam 1-a 40 348 150 36 179 190 173 192

Beam 1-b 40 348 150 36 179 190 173 186

Beam 2-a 25 333 180 36 151 190 157 149

Beam 2-b 25 333 180 36 151 190 157 163

Beam 3-a 25 333 90 36 105 127 105 109

Beam 3-b 25 333 90 36 105 127 105 104

Slab 1-a 25 161 308 36 184 178 177 131

Slab 1-b 25 161 308 36 184 178 177 128

Slab 2-a 25 161 154 36 124 119 118 92

Slab 2-b 25 161 154 36 124 119 118 131

Slab 3-a 25 161 103 36 100 89 88 89

Slab 3-b 25 161 103 36 100 89 88 117*NS is normal-strength slabs; and HS is high-strength slabs.Note: 1 MPa = 145 psi; 1 mm = 0.0394 in.

Fig. 7—Crack patterns of Series III: (a) HS3 (thickness350 mm [14 in.]); and (b) NS5 (thickness 400 mm [16 in.]).

Fig. 8—Comparison of crack spacing equations at 150 mm(6 in.) bar spacing.


higher than test results, and as both the concrete cover andbar spacing increased, the crack spacing increased theoreticallyand experimentally. For bar spacing less than 250 mm (10 in.),the CEB-FIB4 model code underestimates the averagecrack spacing by approximately 31%, compared to the onemeasured during testing.

SUMMARY AND CONCLUSIONSMost of the available expressions for estimating the crack

spacing and width are based on beams and test results forone-way slabs. The behavior of reinforced concrete platesand two-way slabs is different from the behavior of the one-waybeams. A new analytical expression is recommended for platesand two-way slabs with longitudinal and transversereinforcements. The proposed method takes into considerationthe effects of steel bond in the loading direction and the

Table 4—Comparison between calculated crack spacing values using code formulas with measured experimental values for test specimens by Frosch et al.15


Cc , mm Height h, mmBar spacing s,



B-6 46 203 152 46.6 167 118 136 175B-9 46 203 229 44.4 213 177 211 229B-12 46 203 305 44.5 260 236 282 249B-18 46 203 457 47.4 352 355 411 310E12-6 46 203 152 46.7 167 118 136 170E12-9 46 203 229 46.4 213 177 206 226E12-12 46 203 305 45.7 260 236 278 257E12-18 46 203 457 46.8 352 355 414 338E6-9 46 203 229 46.1 213 177 207 203

E18-9 46 203 229 45.9 213 177 207 188Note: 1 MPa = 145 psi; 1 mm = 0.0394 in.

Table 3—Comparison between calculated crack spacing values using code formulas with measured experimental values for test specimens by Marzouk and Hossin8


Cc , mmSlab thickness,

mmBar spacing s,



NSC1 30 200 150 35.0 125 77 126 134HSC1 50 200 150 68.5 165 68 187 171HSC2 60 200 150 70.0 186 63 188 185HSC3 30 200 200 66.7 146 100 182 163HSC4 30 200 250 61.2 167 125 184 172HSC5 30 150 100 70.0 107 56 111 120NSC2 30 200 240 33.0 204 204 225 223NSC3 40 150 240 34.0 228 182 230 239

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

Table 5—Comparison between calculated crack spacing values using code formulas with measured experimental values

Slab no.Concrete

cover Cc , mmSlab thickness,

mmBar spacing s,



NS1 45 150 210 44.7 211 137 248 201NS2 40 200 240 50.2 216 176 234 221NS3 60 250 368 35.0 341 279 320 245HS1 60 250 368 35.0 341 279 361 263NS4 70 300 368 70.0 331 225 273 261HS2 70 300 368 64.7 331 225 304 246HS3 70 350 289 65.4 276 160 273 264NS5 70 400 217 40.0 252 145 226 250

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

Fig. 9—Comparison of crack spacing equations at 250 mm(10 in.) bar spacing.

ACI Structural Journal/January-February 201052

contribution of the splitting bond stresses for the transverse steel.The main conclusions can be summarized as follows:• The proposed method gives a good estimate for crack

spacing in plates and two-way slabs with concrete coversequal to (Cc < 2.5db).

• The proposed method can be used for thick concretecovers (Cc = 2.5 – 5.0db), plates, and two-way slabsafter reducing one third of the tensile stress constant kt.For two-way slabs with concrete covers larger than5.0db, however, it can be speculated that the crack spacingbehaves randomly. This is due to the fact that such slabsact as cross sections that contain two separate materials.

• For bar spacing greater than 300 mm (12 in.), theNorwegian code NS 3474 E3 and the Canadian offshorecode CSA-S474-042 overestimates the average crackspacing by approximately 33%. In general, the calculatedaverage crack spacing was higher than the measuredone from test results.

• For bar spacing less than 250 mm (10 in.), the CEB-FIB4

model code estimated average crack spacing smallerthan the one measured during testing.

• The analytical and experimental investigations revealedthat the average crack spacing is increased as the bar spacingor the concrete cover is increased for the specimens with lowreinforcement ratio that fail in flexure.

• Test results revealed that crack control can still beachieved by limiting the spacing of the reinforcingsteel, despite using thick concrete covers.

ACKNOWLEDGMENTSThe authors are grateful to the Natural Sciences and Engineering

Research Council of Canada (NSERC) for providing the funds for theproject. The authors would also like to thank M. Hossin for providing histest data. Sincere thanks are due to M. Curtis, S. Organ, D. Pike, and thetechnical staff of the Structural Engineering Laboratory, Memorial University ofNewfoundland, for their assistance during the preparation of the specimensand during testing. Sincere thanks are extended to Capital Ready Mix Ltd.,Newfoundland, for providing the concrete for this project.

NOTATIONAc,ef = area of concrete symmetric with reinforcing steel divided by

number of barsAs = area of reinforcement within the effective embedment thicknessb = width of sectionCc = clear cover from nearest surface in tension to flexural tension

reinforcementcs = diameter of effective embedment zone where reinforcing bar

can influence concrete bondd = effective depth to centroid of tensile reinforcementd ′c be = equivalent bar diameter of outer layer of barsfbo = maximum bond strengthf ′c = uniaxial compressive strength of concrete (cylinder strength)fctm = mean value of concrete tensile strength at time that crack formsfsp.t = splitting bond stressfy = yield stress of steelh = section heighthef = effective embedment thicknessk1 = coefficient that characterizes bond properties of bars

k2 = coefficient to account for strain gradientkt = tensile stress factorls,max = length over which slip between steel and concrete occurs; steel

and concrete strains, which occur within this length, contribute towidth of crack

S = center-to-center spacing of flexural tension reinforcementnearest to surface of extreme tension face

Smx = crack spacing for cracks normal to x reinforcementSmy = crack spacing for cracks normal to y reinforcementSrm = average stabilized crack spacingρs,ef = effective reinforcement ratio; and equals area of steel considered

divided by area of effective zone where concrete can influencecrack widths

ρs2 = reinforcement stress at crack locationρsE = steel stress at point of zero slipτbk = lower fractile value of average bond stress

REFERENCES1. Nawy, E. G., “Crack Control in Reinforced Concrete Structures,” ACI

JOURNAL, Proceedings V. 65, No. 10, Oct. 1968, pp. 825-836.2. Canadian Standards Association (CSA), “Offshore Concrete Structures,”

CSA-S474-04, Mississauga, ON, Canada, 2004, 68 pp.3. Norwegian Standard, NS 3473 E (English translation), “Concrete

Structures, Design Rules,” Norwegian Council for Building Standardization,Oslo, Norway, 1992, 79 pp.

4. Comité Euro-International du Béton-Fédération de la Précontrainte(CEB-FIP), Model Code 1990, Bulletin D’Information, No. 203-305,Lausanne, Switzerland, 462 pp.

5. Frosch, R., “Another Look at Cracking and Crack Control inReinforced Concrete,” ACI Structural Journal, V. 96, No. 3, May-June1999, pp. 437-442.

6. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318M-99) and Commentary (ACI 318RM-99),” AmericanConcrete Institute, Farmington Hills, MI, 1999, 369 pp.

7. Desayi, P., and Kulkarni, A. B., “Determination of Maximum CrackWidth in Two-Way Reinforced Concrete Slabs,” ICE Proceedings, Department ofCivil Engineering, Indian Institute of Science, Bangalore, India, V. 61,No. 2, June 1976, pp. 343-349.

8. Marzouk, H., and Hossin, M., “Crack Analysis of Reinforced ConcreteTwo-Way Slabs,” Research Report RCS01, Faculty of Engineering andApplied Science, Memorial University of Newfoundland, St. John’s, NL,Canada, 2007, 159 pp.

9. Jiang. D.; Shah, S. P.; and Andonian, A., “Study of the Transfer ofTensile Forces by Bond, ACI Structural Journal, V. 81, No. 3, May-June1984, pp. 251-259.

10. Kankam, C., “Relationship of Bond Stress, Steel Stress, and Slip inReinforced Concrete,” Journal of Structural Engineering, AmericanSociety of Civil Engineers, V. 123, No. 1, 1997, pp. 79-85.

11. MacGregor, J., and Bartlett, F., Reinforced Concrete: Mechanics andDesign, first Canadian edition, Prentice Hall, Scarborough, ON, Canada, 2000,1042 pp.

12. Marzouk, H., and Hussein, A., “Experimental Investigation on theBehavior of High-Strength Concrete Slabs,” ACI Structural Journal, V. 88,No. 6, Nov.-Dec. 1991, pp. 701-713.

13. Osman, M.; Marzouk, H.; and Helmy, S., “Behaviour of High-Strength Lightweight Concrete Slabs under Punching Loads,” ACIStructural Journal, V. 97, No. 3, May-June 2000, pp. 492- 498.

14. 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.

15. Frosch, R.; Blackman, D.; and Radabaugh, R., “Investigation ofBridge Deck Cracking in Various Bridge Superstructure Systems,” FHWA/IN/JTRP Report No. C-36-56YY, File No. 7-4-50, School of CivilEngineering, Purdue University, West Lafayette, IN, 2003, 286 pp.

a new formula to calculate crack spacingfor concrete plates

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