224.2r

ACI 224.2R-92

(Reapproved 1997)

Cracking of Concrete Members in Direct Tension

Reported by ACI Committee 224

David Darwin*Chairman

Andrew Scanlon*Peter Gergely*SubcommitteeCo-Chairmen

Alfred G. BisharaHoward L. BoggsMerle E. BranderRoy W. CarlsonWilliam L. Clark, Jr.*

Fouad H. Fouad Milos PolvikaTony C. Liu Lewis H. Tuthill*LeRoy Lutz* Orville R. WernerEdward G. Nawy Zenon A. Zielinski

* Members of the subcommittee who prepared this report.

Committee members voting on this minor revision:

Grant T. HalvorsenChairman

Grant T. HalvorsenSecretary

Randall W. PostonSecretary

Florian BarthAlfred G. BisharaHoward L. BoggsMerle E. BranderDavid DarwinFouad H. FouadDavid W. Fowler

Peter GergelyWill HansenM. Nadim HassounWilliam LeeTony C. LiuEdward G. NawyHarry M. Palmbaum

Keith A. PashinaAndrew ScanlonErnest K. SchraderWimal SuarisLewis H. TuthillZenon A. Zielinski

This report is concerned with cracking in reinforced concrete causedprimarily by direct tension rather than bending. Causes of direct tension cracking are reviewed, and equations for predicting crack spacing andcrack width are presented. As cracking progresses with increasing load,axial stiffness decreases. Methods for estimating post-cracking axial stiffnessare discussed. The report concludes with a review of methods forcontrolling cracking caused by direct tension.

Keywords: cracking (fracturing); crack width and spacing; loads(forces); reinforced concrete; restraints;tensile stress; tension; volume change.

stiffness; strains; stresse

ACI Committee Reports, Guides, Standard Practices, andCommentaries are intended for guidance in designing, plan-ning, executing, or inspecting construction and in preparingspecifications. Reference to these documents shall not bemade in the Project Documents. If items found in thesedocuments are desired to be part of the Project Documentsthey should be phrased in mandatory language and in-corporated into the Project Documents.

.

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CONTENTS

Chapter 1-Introduction, pg. 224.2-2

Chapter 2-Causes of cracking, pg. 224.2-2 2.1-Introduction2.2-Applied loads2.3-Restraint

Chapter 3-Crack behavior and prediction equations, pg.224.2-3

3.1-Introduction3.2-Tensile strength

The 1992 revisions became effective Mar. 1, 1992. The revisions consisted ofremoving year designations of the recommended references of standards-pro-ducing organizations so that they refer to current editions.

Copyright 0 1986, American Concrete Institute.All rights reserved including rights of reproduction and use in any form or by

any means, including the making of copies by any photo process, or by any elec-tronic or mechanical device, printed, written, or oral, or recording for sound orvisual reproduction or for use in any knowledge or retrieval system or device,unless permission in writing is obtained from the copyright proprietors.

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224.2R-2 ACI COMMITTEE REPORT

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3.3-Development of cracks3.4-Crack spacing3.5-Crack width

Chapter 4-Effect of cracking on axial stiffness, pg.224.2R-6

4.1-Axial stiffness of one-dimensional members4.2-Finite element applications4.3-Summary

Chapter 5-Control of cracking caused by direct tension,pg. 224.2R-9

5.1-Introduction5.2-Control of cracking caused by applied loads5.3-Control of cracking caused by restraint of volume

change

Notation, pg. 224.23-10

Conversion factors-S1 equivalents, pg. 224.2R-11

Chapter 6-References, pg. 224.2R-116.1-Recommended references6.2-Cited references

CHAPTER l-INTRODUCTION

Because concrete is relatively weak and brittle intension, cracking is expected when significant tensilestress is induced in a member. Mild reinforcement and/orprestressing steel can be used to provide the necessarytensile strength of a tension member. However, a numberof factors must be considered in both design and con-struction to insure proper control of cracking that mayoccur.

A separate report by ACI Committee 224 (ACI 224R)covers control of cracking in concrete members in gen-eral, but contains only a brief reference to tensioncracking. This report deals specifically with cracking inmembers subjected to direct tension.

Chapter 2 reviews the primary causes of direct tensioncracking, applied loads, and restraint of volume change.Chapter 3 discusses crack mechanisms in tension mem-bers and presents methods for predicting crack spacingand width. The effect of cracking on axial stiffness isdiscussed in Chapter 4. As cracks develop, a progressivereduction in axial stiffness takes place. Methods forestimating the reduced stiffness in the post-crackingrange are presented for both one-dimensional membersand more complex systems. Chapter 5 reviews measuresthat should be taken in both design and construction tocontrol cracking in direct tension members.

CHAPTER 2-CAUSES OF CRACKING

2.1-Introduction
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Concrete members and structures that transmit loadsprimarily by direct tension rather than bending includebins and silos, tanks, shells, ties of arches, roof andbridge trusses, and braced frames and towers. Memberssuch as floor and roof slabs, walls, and tunnel linings mayalso be subjected to direct tension as a result of therestraint of volume change. In many instances, crackingmay be attributed to a combination of stresses due toapplied load and restraint of volume change. In the fol-lowing sections, the effects of applied loads and restraintof volume change are discussed in relation to the for-mation of direct tension cracks.

2.2-Applied loadsAxial forces caused by applied loads can usually be

obtained by standard analysis procedures, particularly ifthe structure is statically determinate. If the structure isstatically indeterminate, the member forces are affectedby changes in stiffness due to cracking. Methods for est-imating the effect of cracking on axial stiffness arepresented in Chapter 4.

Cracking occurs when the concrete tensile stress in amember reaches the tensile strength. The load carried bythe concrete before cracking is transferred to the rein-forcement crossing the crack. For a symmetrical member,the force in the member at cracking is

in which

A, = gross area

ft'

= steel area= tensile strength of concrete

n = the ratio of modulus of elasticity of the steelto that of concrete

p = reinforcing ratio = ASIA,

After cracking, if the applied force remains un-changed, the steel stress at a crack is

fs= f =($ - 1 +rJ)fi (2.2)

For n = 10, fi’ = 500 psi (3.45 MPa). Table 2.1 givesthe steel stress after cracking for a range of steel ratiosp, assuming that the yield strength of the steel& has notbeen exceeded.

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Table 2.1-Steel stress after cracking for various steelratios

Ll1- - l + nD

LT*ksi (MPa)

*Assumes f, < f,.

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CRACKING OF CONCRETE MEMBERS IN DIRECT TENSION 224.2R-3

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For low steel ratios, depending on the grade of steel,yielding occurs immediately after cracking if the force inthe member remains the same. The force in the crackedmember at steel yield is A&

2.3-RestraintWhen volume change due to drying shrinkage, thermal

contraction, or another cause is restrained, tensilestresses develop and often lead to cracking. The restraintmay be provided by stiff supports or reinforcing bars.Restraint may also be provided by other parts of themember when volume change takes place at differentrates within a member. For example, tensile stressesoccur when drying takes place more rapidly at the ex-terior than in the interior of a member. A detaileddiscussion of cracking related to drying shrinkage andtemperature effects is given in ACI 224R for concretestructures in general.

Axial forces due to restraint may occur not only intension members but also in flexural members such asfloor and roof slabs. Unanticipated cracking due to axialrestraint may lead to undesirable structural behavior suchas excessive deflection of floor slabs1 and reduction inbuckling capacity of shell structures.2 Both are directresults of the reduced flexural stiffness caused byrestraint cracking. In addition, the formation of cracksdue to restraint can lead to leaking and unsightly con-ditions when water can penetrate the cracks, as inparking structures.

Cracking due to restraint causes a reduction in axialstiffness, which in turn leads to a reduction (or relax-ation) of the restraint force in the member. Therefore,the high level stresses indicated in Table 2.1 for smallsteel ratios may not develop if the cracking is due to

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restraint. This point is demonstrated in Tam and Scan-lon’s numerical analysis of time-dependent restraint forcedue to drying shrinkage.3

CHAPTER 3-CRACK BEHAVIOR ANDPREDICTION EQUATIONS

3.1-IntroductionThis chapter reviews the basic behavior of reinforced

concrete elements subjected to direct tension. Methodsfor determining tensile strength of plain concrete arediscussed and the effect of reinforcement on devel-opment of cracks and crack geometry is examined.

3.2-Tensile strengthMethods to determine tensile strength of plain con-

crete can be classified into one of the following cate-gories: 1) direct tension, 2) flexural tension, and 3) in-direct tension . .44 . Because of difficulties associated withapplying a pure tensile force to a plain concrete spec-imen, there are no standard tests for direct tension.Following ASTM C 292 and C 78 the modulus of rup-ture, a measure of tensile strength, can be obtained bytesting a plain concrete beam in flexure. An indirectmeasure of direct tensile strength is obtained from thesplitting test (described in ASTM C 496). As indicated inReference 4, tensile strength measured from the flexuretest is usually 40 to 80 percent higher than that measuredfrom the splitting test.

Representative values of tensile strength obtainedfrom tests and measures of variability are shown inTables 3.1 and 3.2.

ACI 209R suggests the following expressions to esti-

Table 3.1-Variability of concrete tensile strength: Typical results5

Mean 1 Standard deviation 1 Coefficient

I strength,I

within batches,psi psi I

ofvariation,

Type of test (MPa) (MPa)

Splitting test 405 (2.8) 20 (0.14)Direct tensile test 275 (1.9) 19 (0.13)Modulus of rupture 605 (4.2) 36 (0.25)Compression cube test 5980 (42) 207 (1.45)

percent

5

:31/2

Table 3.2-Relation between compressive strength and tensile strengths of concrete6

Compressivestrength

of cylinders,psi (MPa)

1000 (6.9)2000 (13.8)3000 (20.7)4000 (27.6)5000 (34.5)6000 (41.4)7000 (48.2)8000 (55.1)9000 (62.0)

Modulus of rupture*to compressive

strength

0.230.190.160.150.140.130.120.120.11

Strength ratioDirect tensile

strength tocompressive

strength

0.110.100.090.090.080.080.070.070.07

Direct tensilestrength tomodulus of

rupture*

0.480.530.570.590.590.600.610.620.63

*Determined under third-point loading.

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mate tensile strength as a function of compressivestrength

modulus of rupture: f, = gr [w, (fc’])]“’ (3.1)

direct tensile strength: fi’ = gt [wc Oc,‘)]” (3.2)

where

;::= unit weight of concrete (lb/ft3)= compressive strength of concrete (psi)

gr = 0.60 to 1.00 (0.012 to 0.021 for wc inkg/m3 and& in MPa)

gt = 0.33 (0.0069)

Both the flexure and splitting tests result in a suddenfailure of the test specimen, indicating the brittle natureof plain concrete in tension. However, if the deformationof the specimen is controlled in a test, a significantdescending branch of the tensile stress-strain diagram canbe developed beyond the strain corresponding to maxi-mum tensile stress. Evans and Marathe7 illustrated thisbehavior on specimens loaded in direct tension in a test-ing machine modified to control deformation. Fig. 3.1

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500mix w/c age

(4)_ 1:1:2 0.45 65 days

(5)_1::4 0.60 270 "

(6)_1:3:6 0.90 70 "

I I I I I800 1200 1600 2000 2400

longitudinal tensile strain x 106

Fig. 3.1-Tensile stress-strain diagrams for concrete7

(includes unloading portion)

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shows tensile stress-strain curves that include unloadingbeyond the maximum tensile stress. More recent work byPetersson8 shows that the descending branch of the curveis controlled primarily by localized deformatiou acrossindividual cracks, indicating that there are large dif-ferences between the average strain (Fig. 3.1) and localstrains.

3.3-Development of cracksWhen a reinforced concrete member is subjected to

tension, two types of cracks eventually form (Fig. 3.2).
One type is the visible crack that shows at the surface ofthe concrete, while the other type does not progress tothe concrete surface. Broms9 called cracks of the firsttype primary cracks and those of the second type se-condary cracks.
Each of the two types of cracks has a different geo-metry. The primary or external cracks are widest at thesurface of the concrete and narrowest at the surface ofreinforcing bars.10-12” The difference in crack width be-tween the concrete surface and the reinforcing bar issmall at low tension levels (just after crack formation),and increases as the tension level increases; therefore,the crack width at the reinforcing bar increases moreslowly than the width at the concrete surface with anincrease in load. The deformations on the reinforcingbars tend to control the crack width by limiting the slipbetween the concrete and the steel.

The secondary, or internal, cracks increase in widthwith distance away from the reinforcement before nar-rowing and closing prior to reaching the concrete surface.More detail on internal crack formation is presented inReference 13.

Because of the variability in tensile strength along thelength of a tension member, cracks do not all form at thepyright American Concrete Institute

same stress level. Clark and Spiers14 estimated that thefirst major crack forms at about 90 percent of the aver-age concrete tensile strength and the last major crack atabout 110 percent of the average tensile strength. Soma-yaji and Shah15 used a bilinear stress-strain diagram forconcrete in tension to model the formation of cracksalong the member at increasing load levels. They as-sumed that the tensile strength beyond first cracking wasa function of the strain gradient in the concrete along thelength of the bar.

Induced tensile stresses caused by restrained concreteshrinkage affect the amount of cracking that is visible ata given tensile force. This has been made apparent bytensile tests conducted to compare the performance ofType I cement and Type K

16cement in concrete specimens.shrinkage-compensating)

Specimens placed underthe same conditions of environment and loading hadmarkedly different cracking behavior.

When specimens made with Type I cement had fullydeveloped external cracks, the specimens made with TypeK cement exhibited fewer and narrower external cracks.The Type K specimens exhibited first cracking at a higherload than the Type I specimens, and in some tests novisible cracking was evident in Type K specimens.

The compressive stress induced in the concrete by therestrained expansion of the Type K cement was appar-ently responsible for increasing the loads both at firstcracking and at which cracking was fully developed. Thus,efforts to compensate for concrete shrinkage also appearto help reduce cracking.

3.4-Crack spacingAs a result of the formation of cracks in a tension

member, a new stress pattern develops between thecracks. The formation of additional primary cracks con-tinues as the stress increases until the crack spacing isapproximately twice the cover thickness as measured tothe center of the reinforcing bar.”

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TFig. 3.2-Primary and secondary cracks in a reinforcedconcrete tension member

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oro

There is, of course, a considerable variation in thespacing of external cracks. The variability in the tensilestrength of the concrete, the bond integrity of the bar,and the proximity of previous primary cracks, which tendto decrease the local tensile stress in the concrete, arethe main cause of this variation in crack spacing. For thenormal range of concrete covers, 1.25 to 3 in. (30 to 75mm), the average crack spacing will not reach the limit-ing value of twice the cover until the reinforcement stressreaches 20 to 30 ksi (138 to 200 MPa).11

The expected value of the maximum crack spacing isabout twice that of the average crack spacing.11 That is,the maximum crack spacing is equal to about four timesthe concrete cover thickness. This range of crack spacingis more than 20 percent greater than observed for flex-ural members.

The number of visible cracks can be reduced at agiven tensile force by simply increasing the concretecover. With large cover, a larger percentage of the crackswill remain as internal cracks at a given level of tensileforce. However, as will be discussed in Section 3.5, in-creased cover does result in wider visible cracks.

3.5-Crack widthThe maximum crack width may be estimated by mul-

tiplying the maximum crack spacing (4 times concretepyright American Concrete Institute
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cover) at high steel stress by the average strain in thereinforcement. When tensile members with more thanone reinforcing bar are considered,” the actual concretecover is not the most appropriate variable. Instead, aneffective concrete cover t, is used. t, is defined as afunction of the reinforcement spacing, as well as theconcrete cover measured to the center of the reinfor-cement. 11 The greater the reinforcement spacing, thegreater will be the crack width. This is reflected as anincreased effective cover. Based on the work of Bromsand Lutz,11 the effective concrete cover is

2 ,

(3.3)

in which d, = distance from center of bar to extremetension fiber, in., and s = bar spacing, in.

The variable t, is similar to the variable 3/o used inthe Gergely-Lutz crack width expression for flexuralmembers,17 in which A = area of concrete symmetricwith reinforcing steel divided by number of bars (in.2).Using t,, it is possible to express the maximum crackwidth in a form similar to the Gergely-Lutz expression.

Due to the larger variability in crack width in tensionmembers, the maximum crack width in direct tension isexpected to be larger than the maximum crack width inflexure at the same steel stress.

The larger crack width in tensile members may be dueto the lack of crack restraint provided by the compressionzone in flexural members. The stress gradient in a flex-ural member causes cracks to initiate at the most highlystressed location and to develop more gradually than ina tensile member that is uniformly stressed.

The expression for the maximum tensile crack widthdeveloped by Broms and Lutz” is

Wm a x = 4 ES t, = O.l38f, t, x 10-3 (3.4)

Using the definition of t, given in Eq. (3.3), Wmax may beexpressed as

Wmnn = 0.138 f,d,I-7-T

1 + -& 2 x 1O-3 (3.5)c

p; yyainet,; 3/C for a single layer of reinforcement2s/d, see Fig. 3.3), which is approximately equal

to 1.35d, Jl+(s/4d,)2 for S/d, between 1 and 2. Thus,for tensile cracking

Wmax = 0.10 f, “p x 10-3(3.6)

Eq. (3.6) can be used to predict the probable maximumcrack width in fully cracked tensile members. As withflexural members, there is a large variability in themaximum crack width. One should fully expect the maxi-mum crack width to be 30 percent larger or smaller than

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2d x 2sc = 2d$

2

Fig. 3.3-3p parameter in terms of bar spacing

the value obtained from Eq. (3.6).The maximum flexural crack width expression17

Wma* = O.O76fif, “&& 10-3 (3.7)

in which B = ratio of distance between neutral axis andtension face to distance between neutral axis and cen-troid of reinforcing steel = 1.20 in beams may be usedtocompare the crack widths obtained in flexure and ten-sion.

Using a value of @ = 1.20, the coefficient 0.076fl inEq. (3.7) becomes 0.091 compared to a coefficient of 0.10in Eq. (3.6) for tensile cracks. This indicates that for thesame section and steel stress f, the maximum tensilecrack will be about 10 percent wider than the maximumflexural crack.

The flexural crack width expression in Eq. (3.7), with13 = 1.2, is used in the following form in ACI 318

Z =f,3@ (3.8)

A maximum value of z = 175 kips/in. (30.6 MN/m) ispermitted for interior exposure, corresponding to alimiting crack width of 0.016 in. (0.41 mm). ACI 318limits the value of z to 145 kips/in. (25.4 MN/m) forexterior exposure, corresponding to a crack width of0.013 in. (0.33 mm). To obtain similar crack widths fortensile members, the z-factors of 145 and 175 for flexuralmembers should be multiplied by the ratio of coefficientsin Eq. (3.7) and (3.6) (= 0.91). Using the same definitionof z for both tensile and flexural members, this producesz-values of 132 and 160, respectively, for tensile members.

Rizkalla and Hwang18” reported recently on tests indirect tension and presented an alternative procedure forcomputing crack widths and crack spacing based on ex-pressions given by Beeby19 and Leonhardt.20

CHAPTER 4-EFFECT OF CRACKINGON AXIAL STIFFNESS

4.1-Axial stiffness of one-dimensional membersWhen a symmetrical uncracked reinforced concrete

member is loaded in tension. the tensile force is dis-

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tributed between the reinforcing steel and concrete inproportion to their respective stiffnesses. Total load atstrain E is given by

P = PC + P, = (Ed, + nE& e= EP, (1 - p + n p) E = (EA)uCe (4.1)

in which EC = modulus of elasticity of concrete. Loadscarried by the concrete and reinforcing steel are, re-spectively

PC’ 2-- Pi I1 + np

and

p,” np p( 11 + np (4.3)

Cracking occurs when the strain E corresponds to the ten-sile strength of concrete. If the ascending branch of thetensile stress-strain curve is assumed to be linear E = Et= &‘/EC, in which frl’ is the tensile stress causing the firstcrack. The total load at cracking PCr is carried across thecrack entirely by the reinforcement. If the applied forceremains unchanged, steel stress after the crack occurs&is given by

f,, = % = fr; (f - 1 + n] (4.4)

The load carried across the crack by the reinforcementis gradually transferred by bond to the concrete on eachside of the crack. As the applied load increases, addi-tional cracks form at discrete intervals along the memberas discussed in Chapter 3. The contribution of concretebetween cracks to the net stiffness of a member is knownas tension stiffening. The gradual reduction in stiffness

Axial Strain

Fig. 4.1-Tensile load versus strain diagram

Axial Load

steel plus concrete

shaded area representscontribution of concreteto overall stiffness

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due to progressive cracking is referred to as strain softening.

The stiffening effect of the concrete between crackscan be illustrated by considering the relationship betweenthe load and the average strain in both the uncrackedand cracked states. A tensile load versus strain curve isshown in Fig. 4.1. In the range P = 0 to P = Pcr , the
member is uncracked, and the response follows the lineOA. The load-strain relationship [Eq. (4.1)] is given by
P = EP, (I - p + np) E = (EA),,e (4.5)

If the contribution to stiffness provided by the con-crete is ignored, the response follows the line OB, andthe load-strain relationship is given by

P = EfiSe = n E, pA,cl = (EA),E (4.6)

For loads greater than Pcr,, the actual response is inter-mediate between the uncracked and fully cracked limits,and response follows the line AD. At Point C on AD,where P is greater than P,,, a relationship can be de-veloped between the load P and average strain in themember E,,,

The term (m), can be referred to as the effectiveaxial cross-sectional stiffness of the member. This termcan be written in terms of the actual area of steel As, andan effective modulus of elasticity Esln of the steel bars

p = EsmAs% (4.8)

or

f,E =;sm (4.9)

m

Several methods can be used to determine E,. Forexample, the CEB Model Code gives

(4.10)

in which f,, is given by Eq. (4.4), f, = PIA,, E; = LIE,,and k = 1.0 for first loading and 0.5 for repeated or sus-tained loading.

Combining Eq. (4.9) and (4.10)

Esm= l_kf”’

i 01 (4.11)

f,

The CEB expression is based on tests of direct tensionmembers conducted at the University of Stuttgart.20

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Other methods for determining E, are reviewed byMoosecker and Grasser.21

An alternative approach is to write the effective stiff-ness (EA)c in terms of the modulus of elasticity of theconcrete and an effective (reduced) area of concrete, i.e.

P=E/I,c, (4.12)

This approach is analogous to the effective moment ofinertia concept for the evaluation of deflections de-veloped by Branson and incorporated in AC1 318.

Using the same form of the equation as used for theeffective moment of inertia, the effective cross-sectionalarea for a member can be written as

Ae =Ag($r + A$ -(+'I (4.13)

where A9 = gross cross-sectional area and Acr = nA,.The term Ag could be replaced by the transformed

area Ae to include the contribution of reinforcing steel tothe uncracked system [At = AC + n& = Ag + (n - 1)AsI*

The load-strain relationships obtained using the CEBexpression and the effective cross-sectional area [Eq.(4.13)] compare quite favorably, as shown in Fig. 4.2.

A third approach that has been used in finite elementanalysis of concrete structures involves a progressivereduction of the effective modulus of elasticity of con-crete with increased cracking.

4.2-Finite element applicationsExtensive research has been done in recent years on

the application of finite elements to modeling the be-havior of reinforced concrete and is summarized in areport of the ASCE Task Committee on Finite ElementAnalysis of Reinforced Concrete.23 Two basic ap-proaches, the discrete crack approach and the smearedcrack approach, have been used to model cracking andtension stiffening.

In the discrete crack approach, originally used by Ngoand Scordelis,244 individual cracks are modeled by usingseparate nodal points for concrete elements located atcracks, as shown for a flexural member in Fig. 4.3. This
allows separation of elements at cracks. Effects of bonddegradation on tension stiffening can be modeled bylinear24 or nonlinear 255bond-slip linkage elements con-necting concrete and steel elements.
The finite element method combined with nonlinearfracture mechanics was used by Gerstle, Ingraffea, andGergely26 to study the tension stiffening effect in tensionmembers. The sequence of formation of primary and se-condary cracks was studied using discrete crack modeling.A comparison of analyses with test results is shown inFig. 4.4.

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240

180

Lood P(kips)

120

--- CEB- - - - - - Effective Area, A ,- - - Steel Alone

l200

Axial Strain (millionths)

Fig. 4.2-Tensile load versus strain diagrams based on CEB and effective cross-sectional area expressions

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STEEL ELEMENT

Fig. 4.3-Finite element modeling by the discrete crackapproach24

opyrovio reproduction or networking permitted without license from IHS

300 Elastic analysis;secondary cracking

‘;;g 2 5 0

ignored

-Analysis including ’ ’

Elongation, (mm)

Fig. 4.4-Steel stress versus elongation curves for tensionspecimen based on nonlinear fracture mechanics ap-proach26

In the smeared crack approach, tension stiffening ismodeled either by retaining a decreasing concretemodulus of elasticity and leaving the steel modulusunchanged, or by first increasing and then graduallydecreasing the steel modulus of elasticity and setting theconcrete modulus to zero as cracking progresses. Scanlonand Murray’ introduced the concept of degrading con-crete stiffness to model tension stiffening in two-wayslabs. Variations of this approach have been used infinite element models by a number of researchers.28-31

Gilbert and Warner31 used the smeared crack conceptand a layered plate model to compare results using thedegrading concrete stiffness approach and the increasedsteel stiffness approach. Various models compared byGilbert and Warner are shown in Fig. 4.5. Satisfactory
results were obtained using all of the models considered.However, the approach using a modified steel stiffnesswas found to be numerically the most efficient. Morerecent works32 has shown that the energy consumed infracture must be correctly modeled to obtain objectivefinite element results in general cases. While most ofthese models have been applied to flexural members, theright American Concrete Institute
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same general approach can be used for members indirect tension.

4.3-SummarySeveral methods have been proposed in the literature

to estimate the axial stiffness of cracked reinforcedconcrete members. The CEB Model Code approach in-volves the modification of the effective steel modulus ofelasticity and appears to be well-established in Europe.An alternative approach, suggested in this report,involves an expression for the effective cross-sectionalarea that is analogous to the well-known effectivemoment of inertia concept. Both of these approachesappear to be acceptable for the analysis of one-dimensional members.

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- Layer Containing the Tensile Steel-.-.- Layer Once Removed from the Steel

1 - - - Layer Twice Removed from the Steel

b) Gradually Unloading ResponseAfter Cracking

Alternative Stress-Strain Diagram for Concrete in Tension

a) I f Ey>E, b) I f Ey’C,

Material Modelling Law :E,I I I I Icz ca &b El E( 1 c,kr 1%5.&j 3.&J S.&j 8.a ll.CJU.C,

El E2 E2 EC Es Es ,

~0.~27Es~20.E&6.E&15E&Y+

Fig. 4.5-Tension stiffening models proposed for smearedcrack finite element approach31 (a, = tensile strength ofconcrete, fv = yield strength of reinforcment, u = stress,and E = strain)

ige

--

For more complex systems, finite element analysis pro-cedures have been used successfully to model the be-havior of cracked reinforced concrete, using a variety ofstiffness models.

CHAPTER 5-CONTROL OF CRACKINGCAUSED BY DIRECT TENSION

5.1-IntroductionThe three previous chapters emphasized predicting be-

havior when cracking due to direct tension occurs in re-inforced concrete members. However, a major objectiveof design and construction of concrete structures shouldbe to minimize and/or control cracking and its adverseht American Concrete Institute

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effects. This chapter is intended to provide guidance toassist in achieving that objective.

The recommendations contained in ACI 224R applywhere applicable. This chapter deals more specificallywith members loaded in direct tension.

5.2-Control of cracking caused by applied loadsThe main concern of crack control is to minimize

maximum crack widths. In the past, tolerable crackwidths have been related to exposure conditions (ACI224R). However, at least in terms of protecting rein-forcement from corrosion, the effect of surface crackwidth appears to be relatively less important than be-lieved previously (ACI 224.1R). For severe exposures, itis preferable to provide a greater thickness of concretecover even though this will lead to wider surface cracks.Tolerable crack widths may also be related to aestheticor functional requirements. Based on experience usingthe z-factor for flexural cracking (ACI 318), a crack widthof 0.016 in. (0.4 mm) may be acceptable for appearancein most cases. Functional requirements such as liquidstorage (ACI 350R) may require narrower crack widthssuch as 0.008 in. (0.2 mm) for liquid-retaining structures.

Eq. (3.6) and (3.8) for members in direct tension maybe used to select and arrange reinforcement to limitcrack widths.

Since crack width is related to tensile stress in rein-forcement, cracks attributed to live loads applied forshort periods may not be as serious as cracks due tosustained load, since the cracks due to live load may beexpected to close or at least decrease in width uponremoval of the load. If acceptable crack control cannotbe achieved by the use of mild reinforcement alone, pre-stressing can be used to reduce tensile stresses in astructure. Shrinkage-compensating concrete placed inaccordance with ACI 223 can also be effective.

While measures can be taken at the design stage tocontrol cracking, it is equally important to apply properconstruction procedures to insure the intended perfor-mance of the structure. This requires avoiding over-loading the structure during construction. Carefulplacement of reinforcement is also essential, includingprovision for properly designed lap splices (ACI 318).

5.3-Control of cracking caused by restraint of volumechange

Cracking due to the restraint of volume change duringthe early life of a structure can be minimized by pro-tecting new concrete for as long as practical from dryingor temperature drop of the surface of the member thatwould cause tensile stress in the member greater than itsearly age strength.

Drying shrinkage at early ages can be controlled byusing proper curing procedures. Shrinkage-compensatingconcrete can be very effective in limiting cracks due todrying shrinkage (ACI 223).

Temperature changes related to the heat of hydrationcan be minimized by placing concrete at lower than nor-

ot for Resale


CopProvNo r

--``,`,

ma1 temperatures (precooling). For example, placing con-crete at approximately 50 F (10 C) has significantlyreduced cracking in concrete tunnel linings.33 It shouldbe noted that concrete placed at 50 F (10 C) tends todevelop higher strength at later ages than concreteplaced at higher temperatures.

Circumferential cracks in tunnel linings (as well ascast-in-place conduits and pipelines) can be greatlyreduced in number and width if the tunnel is kept bulk-headed against air movement and shallow ponds of waterare kept in the invert from the time concrete is placeduntil the tunnel goes into service (see Fig. 35 ofReference 34).

To minimize crack widths caused by restraint stresses,bonded “temperature” reinforcement should be provided.As a general rule, reinforcement controls the width andspacing of cracks most effectively when bar diameters areas small as possible, with correspondingly closer spacingfor a given total area of steel. Fiber reinforced concretemay also have application in minimizing the width ofcracks induced by restraint stresses (ACI 544.1R).

If tensile forces in a restrained concrete member willresult in unacceptably wide cracks, the degree of restraintcan be reduced by using joints where feasible or leavingempty pour strips that are subsequently filled with con-crete after the adjacent members have gained strengthand been allowed to dry. Flatwork will be restrained bythe anchorage of the slab reinforcement to perimeterslabs or footings. When each slab is free to shrink fromall sides towards its center, cracking is minimized. Forslabs on ground, contraction joints and perimetersupports should be designed accordingly (ACI 302.1R-80). Frequent contraction joints or deep grooves must beprovided if it is desired to prevent or hide restraintcracking in walls, slabs, and tunnel linings [ACI 224R-80(Revised 1984), ACI 302.1R-80].

A

A,A,As

dc

ECEsEml

f,fsfSW

fc’fi’n

NOTATION

area of concrete symmetric with reinforcingsteel divided by number of bars, in.’effective cross-sectional area of concrete, in.’gross area of section, in.2

area of nonprestressed tension reinforcement,in.2

distance from center of bar to extreme tensionfiber, in.modulus of elasticity of concrete, ksimodulus of elasticity of reinforcement, ksieffective modulus of elasticity of steel to thatof concretemodulus of rupture of concrete, psistress in reinforcement, ksisteel stress after crack occurs, ksicompressive strength of concrete, psitensile strength of concrete, psithe ratio of modulus of elasticity of steel toyright American Concrete Institute

ided by IHS under license with ACI

-`-`,,`,,`,`,,`---

eproduction or networking permitted without license from IHS

P =PC =Pu =PS =s =

t, =

wc =w =mar

=; =

Em=

ES =

P =

that of concreteaxial loadaxial load carried by concreteaxial load at which cracking occursaxial load carried by reinforcementbar spacing, in.effective concrete cover, in.unit weight of concrete, lb/ft3

most probable maximum crack width, in.factor limiting distribution of reinforcementratio of distance between neutral axis andtension face to distance between neutral axisand centroid of reinforcing steel = 1.20 in.beamsaverage strain in member (unit elongation)tensile strain in reinforcing bar assuming notension in concretereinforcing ratio = A#$

CONVERSION FACTORS-SI EQUIVALENTS

1 in. = 25.4 mm1 lb (mass) = 0.4536 kg1 lb (force) = 4.488 N1 lb/in.2 = 6.895 kPa

1 kip = 444.8 N1 kip/in.2 = 6.895 MPa

Eq. (3.5)

2

Wmax = 0.02 fSdc x 10-3

Eq. (3.6)

Wmax = 0*0145f, ‘&X x 10-3

Eq. (3.7)

Wmax = O.OlIf, ‘g-x x 10-3

W,nar in mm, f, in MPa, dc in mm, A in mm2, and s inmm.

CHAPTER 6-REFERENCES

6.1-Recommended referencesThe documents of the various standards-producing or-

ganizations referred to in this report are listed below withtheir serial designation.

American Concrete Institute

209R

223

Prediction of Creep, Shrinkage, and Temper-ature Effects in Concrete StructuresStandard Practice for the Use of Shrinkage-

Not for Resale


CoPrNo

224R224.1R

302.1R

318

350R

544.1R

Compensating ConcreteControl of Cracking in Concrete StructuresCauses, Evaluation, and Repair of Cracks inConcrete StructuresGuide for Concrete Floor and Slab Construc-tionBuilding Code Requirements for ReinforcedConcreteEnvironmental Engineering Concrete Struc-turesState-of-the-Art Report on Fiber ReinforcedConcrete

ASTM

C 78

C 293

C 496

Standard Test Method for Flexural Strength ofConcrete (Using Simple Beam with Third-Point Loading)Standard Test Method for Flexural Strength ofConcrete (Using Simple Beam with Center-Point Loading)Standard Test Method for Splitting TensileStrength of Cylindrical Concrete Specimens

Co&e’ Euro-International du Be’tonlFbd&ation Inter-nationale de Ia Pre’contrainte

CEB-FIP Model Code for Concrete Structures

These publications may be obtained from:

American Concrete InstitutePO Box 19150Detroit, MI 48219

ASTM1916 Race St.Philadelphia, PA 19103

Comite Euro-International du B&onEPFL Case Postale 88CH 1015 Lausanne, Switzerland

6.2-Cited references1. Scanlon, Andrew, and Murray, David W., “Practical

Calculation of Two-Way Slab Deflections,” Concrete In-ternational: Design & Construction, V. 4, No. 11, Nov.1982, pp. 43-50.

2. Cole, Peter P.; Abel, John F.; and Billington, DavidP., “Buckling of Cooling-Tower Shells: State-of-the-Art,”Proceedings, ASCE, V. 101, ST6, June 1975, pp. 1185-1203.

3. Tam, K.S.S., and Scanlon, A., “The Effects ofRestrained Shrinkage on Concrete Slabs,” Structuralpyright American Concrete Institute

ovided by IHS under license with ACI reproduction or networking permitted without license from IHS

Engineering Report No. 122, Department of Civil En-gineering, University of Alberta, Edmonton, Dec. 1984,126 pp.

4. Neville, Adam M., Hardened Concrete: Physical andMechanical Aspects, ACI Monograph No. 6, AmericanConcrete Institute/Iowa State University Press, Detroit,1971, 260 pp.

5. Wright, P.J.F., “Comments on Indirect Tensile Teston Concrete Cylinders,” Magazine of Concrete Research(London), V. 7, No. 20, July 1955, pp. 87-96.

6. Price, Walter H., “Factors Influencing ConcreteStrength,” ACI JOURNAL, Proceedings V. 47, No. 6, Feb.1951, pp. 417-432.

7. Evans, R.H., and Marathe, M.S., “Microcrackingand Stress-Strain Curves for Concrete in Tension,”Materials and Structures, Research and Testing (RILEM,Paris), V. 1, No. 1, Jan.-Feb. 1968, pp. 61-64.

8. Petersson, Per-Erik, “Crack Growth and De-velopment of Fracture Zones in Plain Concrete andSimilar Materials,” Report No. TVBM-1006, Division ofBuilding Materials, Lund Institute of Technology, 1981,174 pp.

9. Broms, Bengt B., “Crack Width and Crack Spacingin Reinforced Concrete Members,” ACI JOURNAL, Pro-ceedings V. 62, No. 10, Oct. 1965, pp. 1237-1256.

10. Broms, Bengt B., “Stress Distribution in Rein-forced Concrete Members With Tension Cracks,” ACIJOURNAL, Proceedings V. 62, No. 9, Sept. 1965, pp. 1095-1108.

11. Broms, Bengt B., and Lutz, Leroy A., “Effects ofArrangement of Reinforcement on Crack Width andSpacing of Reinforced Concrete Members,” ACIJOURNAL, Proceedings V. 62, No. 11, Nov. 1965, pp.1395-1410.

12. Goto, Yukimasa, “Cracks Formed in ConcreteAround Deformed Tension Bars,” ACI JOURNAL,Proceedings V. 68, No. 4, Apr. 1971, pp. 244-251.

13. Goto, Y., and Otsuka, K., “Experimental Studieson Cracks Formed in Concrete Around Deformed Ten-sion Bars,” Technology Reports of the Tohoku University,V. 44, No. 1, June 1979, pp. 49-83.

14. Clark, L.A., and Spiers, D.M., “Tension Stiffeningin Reinforced Concrete Beams and Slabs Under Short-Term Load,” Technical Report No. 42.521, Cement andConcrete Association, Wexham Springs, 1978, 19 pp.

15. Somayaji, S., and Shah, S.P., “Bond Stress VersusSlip Relationship and Cracking Response of TensionMembers,” ACI JOURNAL, Proceedings V. 78, No. 3, May-June 1981, pp. 217-225.

16. Cusick, R.W., and Kesler, C.E., “Interim Report-Phase 3: Behavior of Shrinkage-Compensating ConcretesSuitable for Use in Bridge Decks,” T. & A.M. Report No.409, Department of Theoretical and Applied Mechanics,University of Illinois, Urbana, July 1976, 41 pp.

17. Gergely, Peter, and Lutz, Leroy A., “MaximumCrack Width in Reinforced Concrete Flexural Members,”Causes, Mechanism, and Control of Cracking in Concrete,SP-20, American Concrete Institute, Detroit, 1968, pp.

--``,`,-`-`,,`,,`,`,,`---

Not for Resale

224.2R-1 2 ACI COMMlTTE E REPORT

CoPrNo

87-117.18. Rizkalla, S.H., and Hwang, L.S., “Crack Prediction

for Members in Uniaxial Tension,” ACI JOURNAL, Pro-ceedings V. 81, No. 6, Nov.-Dec. 1984, pp. 572-579.

19. Beeby, A.W., “The Prediction of Crack Widths inHardened Concrete,” Structural Engineer (London), V.57A, Jan. 1979, pp. 9-17.

20. Leonhardt, Fritz, “Crack Control in ConcreteStructures,” IABSE Surveys No. S-4/77, InternationalAssociation for Bridge and Structural Engineering,Zurich, Aug. 1977, 26 pp.

21. Moosecker, W., and Grasser, E., “Evaluation ofTension Stiffening in Reinforced Concrete Linear Mem-bers,” Final Report, IABSE Colloquium on AdvancedMechanics of Reinforced Concrete (Delft, 1981), Inter-national Association for Bridge and StructuralEngineering, Zurich.

2 2 . Branson, D . E . , “Instantaneous and Time-Dependent Deflections of Simple and ContinuousReinforced Concrete Beams,” Research Report No. 7,Alabama Highway Department, Montgomery, Aug. 1963,94 pp.

23. Finite Element Analysis of Reinforced Concrete,American Society of Civil Engineers, New York, 1982,545 pp.

24. Ngo, D., and Scordelis, A.C., “Finite ElementAnalysis of Reinforced Concrete Beams,” ACI JOURNAL.,Proceedings V. 64, No. 3, Mar. 1967, pp. 152-163.

25. Nilson, Arthur H., “Nonlinear Analysis ofReinforced Concrete by the Finite Element Method,”ACI JOURNAL, Proceedings V. 65, No. 9, Sept. 1968, pp.757-766.

26. Gerstle, Walter; Ingraffea, Anthony R.; andGergely, Peter, ‘‘Tension Stiffening: A FractureMechanics Approach,” Proceedings, International Con-

--``,`,-`-`,,`

pyright American Concrete Institute ovided by IHS under license with ACI reproduction or networking permitted without license from IHS

,,`,`,,`---

ference on Bond in Concrete (Paisely, June 1982), Ap-plied Science Publishers, London, 1982, pp. 97-106.

27. Scanlon, A., and Murray, D.W., “An Analysis toDetermine the Effects of Cracking in Reinforced Con-crete Slabs,” Proceedings, Specialty Conference on theFinite Element Method in Civil Engineering, EIC/McGillUniversity, Montreal, 1972, pp. 841-867.

28. Lin, Cheng-Shung, and Scordelis, Alexander C.,“Nonlinear Analysis of RC Shells of General Form,”Proceedings, ASCE, V. 101, ST3, Mar. 1975, pp. 523-538.

29. Chitnuyanondh, L.; Rizkalla, S.; Murray, D.W.;and MacGregor, J.G., “An Effective Uniaxial TensileStress-Strain Relationship for Prestressed Concrete,”Structural Engineering Report No. 74, University ofAlberta, Edmonton, Feb. 1979, 91 pp.

30. Argyris, J.H.; Faust, G.; Szimmat, J.; Warnke, P.,and William, K.J., “Recent Developments in the FiniteElement Analysis of Prestressed Concrete ReactorVessels,” Preprints, 2nd International Conference onStructural Mechanics in Reactor Technology (Berlin,Sept. 1973), Commission of the European Communities,Luxembourg, V. 3, Paper H l/l, 20 pp. Also, NuclearEngineering and Design (Amsterdam), V. 28, 1974.

31. Gilbert, R. Ian, and Warner, Robert F., ‘‘TensionStiffening in Reinforced Concrete Slabs,” Proceedings,ASCE, V. 104, ST12, Dec. 1978, pp. 1885-1900.

32. Bazant, Zdenek, and Cedolin, Luigi, “Blunt CrackBand Propagation in Finite Element Analysis,”Proceedings, ASCE, V. 105, EM2, Apr. 1979, pp. 297-315.

33. Tuthill, Lewis H., “Tunnel Lining With PumpedConcrete,” ACI JOURNAL., Proceedings V. 68, No. 4, Apr.1971, pp. 252-262.

34. Concrete Manual, 8th Edition, U.S. Bureau ofReclamation, Denver, 1975, 627 pp.

Not for Resale