Cracking of Concrete - Univ. Illinois, Bulletin

7/29/2019 Cracking of Concrete - Univ. Illinois, Bulletin

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HILL IN 0 SUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

PRODUCTION NOTEUniversity of Illinois at

Urbana-Champaign LibraryLarge-scale Digitization Project, 2007.


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ABSTRACT

THE CRACKING OF CONCRETE IN HIGH-WAY PAVEMENTS AND STRUCTURES IS UNDESIR-ABLE SINCE CRACKING OF THE CONCRETE ISASSOCIATED WITH THE DETERIORATION OFBOTH THE CONCRETE AND REINFORCING STEEL.MANY STUDIES ON THE PHENOMENON OF CRACK-ING IN PLAIN AND REINFORCED CONCRETEHAVE BEEN CONDUCTED; HOWEVER, THESEINVESTIGATIONS HAVE CORRELATED THECRACKING OF CONCRETE WITH VARIOUS PARA-METERS OF THE CONCRETE AND THE ENVIRON-MENT, BUT HAVE NOT CONSIDERED THE MECHA-NISM OF CRACKING.

A THREE-PHASE INVESTIGATION WASUNDERTAKEN TO PROVIDE A BETTER UNDER-STANDING OF THE INITIATION AND GROWTHOF CRACKS IN CONCRETE, WHICH IS ESSEN-TIAL IF CRACKING OF CONCRETE STRUCTURESIS TO BE CONTROLLED. THE EFFECT OFSEVERAL CONCRETE PARAMETERS ON THE FRAC-TURE TOUGHNESS (MATERIAL'S RESISTANCETO PROPAGATION OF AN EXISTING FLAW) ISPRESENTED. A SYSTEMS-TYPE ANALYSIS ISPRESENTED TO DESCRIBE THE COMPLEX CRACK-ING MECHANISM IN CONCRETE STRUCTURES,AND MODELS ARE DEVELOPED FOR STUDYINGCRACKING IN CONCRETE BEAMS AND RIGIDPAVEMENTS. AN APPROXIMATE SOLUTION FORTHE PROBLEM OF SHRINKAGE STRESSES INPLAIN AND REINFORCED CONCRETE MEMBERSWHICH ARE EXTERNALLY LOADED IS DEVELOPED.


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ACKNOWLEDGMENTS

This study was conducted as a part of the researchunder the Illinois Cooperative Highway Research ProgramProject IHR-92, "The Control of Cracking of Concrete."The project has been undertaken by the EngineeringExperiment Station of the University of Illinois incooperation with the Illinois Division of Highways ofthe State of Illinois and the U.S. Department of Trans-portation, Federal Highway Administration, Bureau ofPublic Roads.

On the part of the University, the work coveredby this report was carried out under the general admin-istrative supervision of D. C. Drucker, Dean of theCollege of Engineering, R. J. Martin, Director of theEngineering Experiment Station, T. J. Dolan, Head ofthe Department of Theoretical and Applied Mechanics,and Ellis Danner, Director of the Illinois CooperativeHighway Research Program and Professor of Civil Engi-nee ring.

On the part of the Illinois Division of Highways,the work was under the administrative direction ofR. H. Golterman, Chief Highway Engineer, and J. E. Burke,Engineer of Research and Development.

Technical advice was provided by a Project AdvisoryCommittee consisting of the following personnel:

Representing the Illinois Division of Highways:J. E. Burke, Engineer of Research and

DevelopmentR. L. Duncan, Field EngineerW. Griffin, Structural Design Engineer

Representing the University of Illinois:J. L. Lott, formerly Assistant Professor of

Theoretical and Applied MechanicsG. M. Sinclair, Professor of Theoretical

and Applied MechanicsC. E. Kesler, Professor of Theoretical and

Applied Mechanics and of Civil Engineering and David Raecke,Research Associate in the Department of Theoretical andApplied Mechanics, served as Chairman and Secretary,respectively, of the Project Advisory Committee.


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CONTENTS

I. INTRODUCT ION . . . . . . . . . . . . . . . . . . . . I1.1 Genera l . . . . . . . . . . . . . . . . . . .1.2 Object . . . . . . . . . . . . . . . . . . . . . 11.3 Scope . . . . . . . . . . . . . . . . . . . . 11.4 Notation . . . . . . . . . . . . . . . . . . . . 2

II. EFFECT OF CONCRETE PARAMETERSON FRACTURE TOUGHNESS . . . .. . . . . . . . . . . . . 42.1 Introduction . . . . . . . . . . . . . . . . . . 42.2 Experimental Investigation . . . . . . . . . . . 52.3 Experimental Results . . . . . . . . . . . . . . 62.4 -Discussion of Results. . . . . . . . . . . . . . 8

III. CRACK MECHANISM FOR CONCRETE STRUCTURES . . . . . . . 103.1 Introduction . . . . . . . . . . . . . . . . . . 103.2 Fracture System. . . . . . . . . . . . . . . . . 10

3.3 Fracture of Concrete Structures. . . . . . . .. . 11IV. ANALYTICAL STUDY OF CRACK DEVELOPMENT

ASSOCIATED WITH VOLUME CHANGE . . . . . . . . . . . . 144.1 Introduction . . . . . . . . . . . . . . . . . . 14


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4.2 Development of Stiffness Matrix forFinite Element Analysis. . . . . . . . ..

4.3 Application of the Method andBoundary Conditions. . . . . . . . . . .

V. PRACTICAL APPLICATIONS. . . . . . . . . . . .5.1 Effect of Concrete Parameters

on Fracture Toughness . . . . . . . .5.2 Crack Mechanism for Concrete Structures.5.3 Analytical Study of Crack Development

Associated with Volume Change. . . . . .VI. SUMMARY AND CONCLUSIONS . . . . . . . . . . .

6.1 Object and Scope . . . . . . . . . . . .6.2 Results of Investigation . . . . . . . .6.3 Conclusions . . . . . . . . . . . . ..

VII. SUGGESTIONS FOR FUTURE RESEARCH . . . . . . .VI I I. REFERENCES . . . . . . . . . . . . . . . ..

IX. APPENDIX I, USER'S GUIDE FORCOMPUTER PROGRAM IN FORTRAN IV. . . . . . . .

X. APPENDIX II, COMPUTER PROGRAM IN FORTRAN IVFOR DETERMINATION OF VOLUME CHANGE STRESSESIN PLAIN AND REINFORCED CONCRETE USINGFINITE ELEMENT ANALYSIS . . . . . . . . . . .

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FIGURES

Test Setup.Typical Load - Deformation Curves: Concrete.Maximum Load and Effective Fracture Toughness vs a/w.Effect of w/c Ratio on K .c,Effect of Air Content on K .Effect of Curing Time on K : Mortars and Pastes.c -IEffect of Curing Time and Type of Coarse Aggregate on K : Concretes.Effect of Fine Aggregate on K': Mortars.cEffect of Fine Aggregate on K : Concretes.

cEffect of Fineness Modulus of Coarse Aggregate on K : Concretes.- I cEffect of Coarse Aggregate on K : Concretes.Schematic of Fracture System.Cracked Concrete Element from Reinforced Concrete Body.Reinforced Concrete Tension Member.Cracked Concrete Element from Tension Member.Load, T, vs a /d e for Different Unbonded Lengths, A .Cracked Rigid Pavement.A Typical Element.Reinforced Concrete Model.


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I. INTRODUCTION

1.1 GENERALUndesirable cracking of concrete

in highway pavements and structures isassociated with the deterioration ofboth the concrete and the reinforcingsteel. Corrective maintenance is costlyand inconvenient so that ideal designsshould minimize the size of cracks inhardened concrete. Such a control canbe improved through a basic understand-ing of crack development in concrete.

Many studies of cracking in plainand reinforced concrete have been con-ducted. However, these investigationshave correlated the cracking of concretewith various parameters without consid-ering how cracking occurs.

1.2 OBJECTThe object of this investigation

is to determine the effect of concreteparameters (mix design) on the crackingof concrete, to study the complex crack-ing mechanism in concrete structures,and to develop an analytical solutionfor the problem of volume-change stressesfor plain and reinforced concrete. Theresult, a better understanding of theinitiation and growth of cracks in con-crete, is essential to control crackingconcrete structures.*Fracture toughness is the material'sresistance to propagation of an existingflaw.

1.3 SCOPE1.3.1 Effect of Concrete Parameters

on Fracture ToughnessThe fracture toughnesses* of sever-

al pastes, mortars, and concretes weredetermined by flexural tests of speci-mens containing flaws of various depthscast at the center of the tensile sur-face. Variables in the tests were:water-cement ratio, air content, degreeof hydration, sand-cement ratio, gravel-cement ratio, and gradation and type ofcoarse aggregate.

1.3.2 Crack Mechanism for ConcreteStructures

A systems-type analysis was usedto describe the complex cracking mecha-nism that occurs in concrete structures.The cracking mechanism in a reinforcedbeam subjected to a pure moment, andthe cracking mechanism in a reinforcedconcrete member with the steel loadedin tension was examined by this approach.

1.3.3 Analytical Study of CrackDevelopment Associated withVolume Change

An approximate solution for theproblem of shrinkage stresses in plainand reinforced concrete was developedusing finite element analysis. Themethod can be used to calculate stressesin members which are externally loaded.


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Cracking is incorporated into the analy-sis, and crack width and spacing can becalculated.

1.4 NOTATION= area of steel reinforcement

a

a 1 ,

a'

Bb[C]C1,

c

C , .

= a column matrix representingthe nodal forces in an ele-ment resulting from volumechange

= critical energy-release rateat the onset of rapid, un-stable crack propagation

= crack length of an edge-cracked specimen or halfthe crack length of acenter-cracked specimen

a 2 = constants relating the vol-ume change strains at anypoint in the element to they-coordinate of the point

= crack length from level ofreinforcement to the cracktip

= flexure specimen width= height of an element= matrix relating stresses

to strainsC 2 = coefficients for differentcrack lengths

= cement content of a partic-ular mix, by weight

S.,c8 = constants relating the dis-placements of a point in anelement to the coordinatesof that point

= a matrix relating strainsto displacements

= length of an element= effective depth of reinforced

concrete tension member= effective modulus of elas-

ticity of the concrete= a column matrix representing

the forces at the nodalpoints of one element

= a column matrix representingforces at all nodes

= bond forces= a force at a point i in the

elemen t

= depth of pavement= stress intensity factor at

the tip of a flaw= change in stress intensity

factor due to a load cycle= effective stress intensity

factor in the matrix of aheterogeneous materialthat is assumed to be homo-geneous

= symmetrical stiffness ma-trix for one element

= symmetrical stiffness ma-trix for the entire model

= critical stress intensityfactor at the onset ofrapid, unstable crackpropagation

= effective fracture tough-ness

= average effective fracturetoughness for a test series

= stress intensity factorfor the concrete subjectedonly to the resultant for-ces at the level of thesteel

= stress intensity factorfor the concrete subjectedonly to moments M

= stress intensity factorfor the concrete subjectedonly to the axial force P

= stress intensity factorresulting from a series offorces and/or moments

= shear span= length of reinforced con-

crete tension member= unbonded length of steel

reinforcement in concretetension member= applied bending moment= moments applied to a rein-

forced concrete beam= total number of nodes in

the model= applied load= polar coordinates= one-half distance over

which concrete is examinedin reinforced concrete beamsubjected to constant moment


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= force transmitted acrosscracked section in tensionmember for equilibrium con-ditions

= force which concrete trans-mits across the crackedsection in reinforced con-crete tension member

= force in steel at crackedsection in concrete tensionmember

= thickness of reinforced con-crete tension member

= column matrix representingthe displacements at thefour nodes of an element

= a column matrix of displace-ments at all nodes

= x-displacement of a pointin the element

= y-displacement of a pointin the element

= flexure specimen depth= water content of a particu-

lar mix, by weight= cartesian coordinate system

of axes with origin at lowerleft-hand node of an element

= shear strain in an element= shear volume change strain

in an element

T s

t

[u]

[u]ux

uyww

x,y

"xyYxys

A

[E]TE. I

x

y

xsEys[1

[o]

x

T xy

= crack opening displacementat level of reinforcement

= a column matrix representingthe strains in an element

= transpose of matrix of com-patible strains due to aunit displacement in thedirection of F.I= normal strain in x-directionin an element

= normal strain in y-directionin an element

= normal volume change strainin the x-direction

= normal volume change strainin the y-direction

= nondimensionalized coordi-nate y/b

= nondimensionalized coordi-nate x/d

= a column matrix representingthe stresses in an element

= a matrix representing volumechange stresses

= stress normal to y-z planein x-direction

= stress normal to x-z planein y-direction

= shear stress on plane perpen-dicular to x-axis in y-direction


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II. EFFECT OF CONCRETE PARAMETERS ON FRACTURE TOUGHNESS

2.1 INTRODUCTION2.1.1 Linear-Elastic Fracture Mechanics

Linear-elastic fracture mechanicsis a study of the stress and displace-ment fields near the tip of a flaw in anideal, homogeneous, elastic material atthe onset of rapid, unstable crack prop-agation, i.e., fracture. Its conceptsare most applicable to brittle materialsin which the inelastic region near thecrack tip is small compared to flaw andspecimen dimensions so that elasticstress field equations provide a goodapproximation ( :*

K e [sn 3Sa = - cos [1-sin 2in 3

S=-- cos [+sin - sin --8 , (1)y C s 2 2

K . 9 6 36r = -- sn - cos - cos -- ,xy 2-- 2 2 2where r and 8 are polar coordinates withorigin at the crack tip.

Equation (1) indicates that thestress and displacement fields can beexpressed in terms of a stress intensityfactor K which is a function of loadingand crack geometry. The evaluation ofK at the onset of rapid, unstable crack

*Superscript numbers in parenthesesrefer to entries in References, Chap-ter VII.

propagation yields the critical stressintensity K which is assumed to be amaterial property called the fracturetoughness, i.e., the material's resis-tance to propagation of an existingflaw. Fracture can thus be predictedfor a structure since crack propagationwill occur when the stress intensityfactor reaches its limiting conditionK .c

As the ratio of plastic zone sizeto specimen dimensions increases, theinelastic region becomes significantand adjustments must be made to correctfor effects of plastic strains adjacentto the crack tip region.(2) An exactsolution to correct for the zone ofyielding is presently unknown; however,an approximate solution can be attainedby assuming a crack tip extension tothe central portion of the inelasticregion and solving the problem withelastic stress field equations for theincreased crack length.

2.1.2 Applications of Fracture Mechanicsto Concrete

Several applications of linear-elastic fracture mechanics have been madeto pastes, mortars, and concretes.Concrete, a polyphase material, has amore complex fracture process than ahomogeneous, ideally brittle material.Fracture of the concrete can occur by


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fracture of the cement paste, fractureof the aggregate, failure of the bondbetween the cement paste and aggregate,or any combination of these mechanisms.

Kaplan was the first to applyfracture mechanics to concrete when heinvestigated one mortar and two con-cretes. An analytical and experimentalapproach, both neglecting slow crackpropagation prior to fracture, wereused to evaluate the critical strainenergy release rate G . The resultsobtained by Kaplan indicated that G cwas influenced by the mix proportions,specimen dimensions, and loading.

Lott and Kesler 6 ) conducted astudy to develop a hypothesis forpropagation of cracks in plain concreteand to compare the hypothesis to resultsof an experimental investigation ofcrack propagation in several mortarsand concretes. It was suggested thatthe critical stress intensity factorK for plain concrete was derived fromthe stress intensity factor of thepaste and a crack arresting mechanismdeveloped by the heterogeneity of theconcrete. Since the critical stressintensity factor for the paste was amaterial constant, variations in thecritical stress intensity factor of theconcrete were reflected through thearresting function. The effects ofseveral concrete parameters (water-cement ratio, sand-cement ratio, andgravel-cement ratio) on the fracturetoughness of the concrete were evalu-ated .

For the range of variables investi-gated, it was found that: the criticalstress intensity factor was independentof water-cement ratio for the threemortars and for various concretes where

the aggregate percentages remained con-stant; the critical stress intensityfactor was independent of fine aggre-gate percentage for three mortars withthe same water-cement ratio; the criticalstress intensity factor varied directlywith coarse aggregate content for con-cretes with the same water-cement ratioand fine aggregate content; and thecritical stress intensity factor forconcrete was found to be approximately20 per cent greater than that for amortar with the same water-cement ratioand fine aggregate content.

2.2 EXPERIMENTAL INVESTIGATION2.2.1 General

The fracture toughnesses of sever-al pastes, mortars, and concretes weredetermined by flexural tests of speci-mens containing flaws of various depthscast at the center of the tensile sur-face. 7 ) Parameters investigated in-cluded: water-cement ratio, air con-tent, degree of hydration, sand-cementratio, gravel-cement ratio, and grada-tion and type of coarse aggregate.

2.2.2 MaterialsType I portland cement was used

in all mixes. The fine aggregate usedwas a Wabash River sand from near Coving-ton, Indiana. Two gravels were used inthe concrete series: a Wabash Rivergravel from near Covington, Indiana,and a crushed limestone which was ob-tained locally.

The air-entraining agent used wasa proprietary compound consisting of anaqueous solution of salts of sulfonatedhydrocarbons containing a catalyst.

2.2.3 Specimen Description


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Nominal dimensions of the pasteand mortar flexural specimens were 2 by2 by 14 in., and nominal dimensions ofthe concrete specimens were 4 by 4 by12 in. A flaw was cast at the centerof the tensile surface of the specimens.The flaw was formed with a 0.003-in.-thick piece of teflon-coated fiberglasscloth. Nominal flaw depths were: 0.25in., 0.5 in., and 1.0 in. for the pasteand mortar specimens, and 0.5 in., 1.0in., and 1.5 in. for the concrete speci-mens. Actual dimensions of the flexurespecimens were measured after testingsince variations in nominal dimensionsoccurred in fabrication.

2.2.4 Fabrication and CuringA two-cubic-foot horizontal pan

mixer was used. The dry ingredientswere blended one minute before water wasadded to the mix. After addition ofthe water, the mixing was continued forthree minutes. When air-entrainingagents were used, they were added to themix water.

The flexure specimens were castwith the plane of the flaw in a verticalposition. The molds were filled in onelift and compacted on a vibrating table.A total of twenty flexural specimenswere cast in steel forms for each seriesof the paste and mortar series, and atotal of eight flexural specimens werecast in plywood forms for each seriesof the concrete series. The exposedsurface of all specimens was troweledsmooth immediately after casting.

Two to four hours after casting thespecimens were covered with wet burlapand plastic sheeting to prevent the lossof moisture. Approximately twenty-fourhours after casting, the specimens were

demolded and stored in a moisture roomfor curing at 100 per cent relativehumidity. The specimens were removedfrom the moisture room at various agesand stored in water until they weretested.

2.2.5 Testing ProcedureA hydraulic testing machine was

used for the flexural tests. Figure Ishows the test setup. The lower loadingplate acted as a dynamometer to measureload applied to the specimen. A defor-meter, supported by needlepoint screws,was used to measure elongation of thetensile surface.

Prior to each series of flexuraltests, the deformeter and dynamometerwere calibrated. After calibration,the deformeter was placed between theneedlepoint screws of the first specimenand precompressed to a pseudozero point.The recorder was zeroed and load wasapplied at a rate of approximately250 lb per minute for the paste andmortar specimens and 1500 lb per minutefor the concrete specimens until failureoccurred.

2.3 EXPERIMENTAL RESULTS2.3.1 Load-Deformation Curves

During each test, a recorder plot-ted a continuous record of deformationresponse against load response untilfailure of the flexural specimen. Typi-cal load-deformation curves for a con-crete series are presented in Figure 2.

2.3.2 Stress Intensity FactorBrown and Srawley used boundary

value collocation calibrations to devel-op the following expression for thestress intensity factor K for a single-


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edge-cracked specimen subjected to purebending:

K = Y 6Ma (2)BW 2

where

Y = 1.99 - 2.47 (a/W) + 12.97 (a/W)2

- 23.17 (a/W) 3 + 24.80 (a/W) 4 ,P2.M - ,2

and a is the flaw depth, W is the speci-men depth, P is the applied load, Z isthe shear span, and B is the specimenwidth.

In the evaluation of K usingcEquation (2), it was assumed that thematerial was homogeneous and the flawdepth at failure was equal to the castflaw depth. Since concrete is hetero-geneous and the stress intensity factoris a function of the instantaneous crackdepth, the analysis yields an effectivestress intensity factor K rather thanthe actual stress intensity factor.

The effective fracture roughnessK , a measure of concrete's resistanceto propagation of an existing crack, isthe determination of K at M frommax IEquation (2). Figure 3 presents K andP as a function of a/W for a concretemaxseries. The horizontal line in Figure 3represents the mean value of effectivefracture toughness for the particulartest series, K

2.3.3 Effect of Concrete Parameterson Effective Fracture ToughnessWater-Cement Ratio

In the paste series there was adecrease in K of 43.3 per cent when thewater-cement ratio was increased from0.27 to 0.36, while in the mortar series

K' decreased 18.3 per cent when thecwater-cement ratio was increased from0.45 to 0.60 as shown in Figure 4.However, in the concrete series K wascindependent of the water-cement ratiofor the range of water-cement ratiosinvestigated as shown in Figure 4.

Air ContentIn the paste series there was a

23.4 per cent decrease in K when thecair content was increased from 2.0 to8.0 per cent as shown in Figure 5. Inthe mortar series K' decreased by 19.2

cper cent when the air content was in-creased from 3.0 to 9.0 per cent asshown in Figure 5. K decreased byc8.2 per cent when the air content inthe concrete series was increased from2.0 per cent to 12.0 per cent as shownin Figure 5.

Curing TimeFor 28 days moist cure K wasI C6.5 per cent greater than K for six

days moist cure as shown in Figure 6for the paste series. For the mortarseries there was a 47.5 per cent increasein K when the length of moist cure wascincreased from three days to 92 days asshown in Figure 6. When the length ofmoist cure was increased from three daysto 28 days for concrete using a rivergravel coarse aggregate, K increasedC54.2 per cent. However, the increasein Kc was only 7.7 per cent when thelength of moist cure was increased from28 days to 90 days as shown in Figure 7.When a crushed limestone coarse aggre-gate was used, K increased 23.0 percent with an increase in moist curefrom three days to 28 days; however,there was no apparent change in K whenc


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the length of moist cure was increasedfrom 28 days to 90 days as shown inFigure 7. The percentage increases inK when the moist curing period wascincreased from six days to 28 days forthe paste series, the mortar series,the concrete series cast with a crushedlimestone coarse aggregate, and theconcrete series cast with a river gravelcoarse aggregate were 6.5 per cent,12.4 per cent, 21.3 per cent and 24.2per cent, respectively.

Fine Aggregate ContentIn the mortar series there was a

16.2 per cent increase in K when thefine aggregate content was increasedfrom 55.0 per cent to 70.0 per cent asshown in Figure 8. However, for theconcrete series there was a 2.3 per centdecrease in K when the fine aggregateccontent was increased from 35.0 per centto 50.0 per cent as shown in Figure 9.

Gravel Content, Gradation,and Type

For the concretes cast with crushedlimestone coarse aggregate, K increased13.3 per cent when the fineness moduluswas increased from 6.3 to 7.1 as shownin Figure 10. When the percentage ofcoarse aggregate was increased from0.0 per cent to 50.0 per cent there was

a 37.0 per cent increase in K c as shownin Figure 11.

For the concrete series cast witha crushed limestone K was 28.9 perccent, 17.7 per cent, and 1.7 per centhigher than K for the concrete seriescast with a river gravel coarse aggre-gate at ages of three days, six daysand 28 days, respectively. However, atan age of 90 days, K c for the concrete

series cast with a river gravel coarseaggregate was 5.0 per cent higher thanthe concrete series cast with a crushedlimestone coarse aggregate as shown inFigure 7.

2.4 DISCUSSION OF RESULTS2.4.1 Behavior of Fracture Toughness

SpecimensThe load-deformation curves (Fig-

ure 2) illustrate the stages of behaviorof the concrete near the tip of theflaw: linear stage where the cementpaste matrix has no crack extension;slow cracking stage in which stablecracking occurs to result in a decreas-ing slope of the load-deformation curve;and fracture stage where unstable crackpropagation occurs and results in thedeformation increasing without anincrease in applied load.

2.4.2 Effect of ConcreteParameters on RcWater-Cement Ratio

There was a decrease in the effec-tive fracture toughness of the pasteand mortar series with increasing water-cement ratio because the fracturetoughness was dependent on the strengthof the cement paste matrix which was afunction of gel-space ratio. Withincreasing water contents the gel-spaceratio decreased resulting in a reductionof strength and effective fracturetoughness. The fine aggregate of themortar series reduced the effect of thewater-cement ratio because of the crackarresting phenomenon of the fine aggre-gate particles.

The effective fracture toughnessof concrete depended on both the frac-ture toughness of the paste and the


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presence of coarse aggregate. The rangeof water-cement ratios apparently didnot affect the effective fracture tough-ness of the concrete because the effectof the aggregate as a crack arrestingfunction was more significant than theeffect of the water-cement ratio on thepaste matrix strength.

Air ContentIncreasing the air content of the

matrix resulted in a decrease in effec-tive fracture toughness because of areduced matrix strength. With increasingaggregate contents, the decrease wasnot as significant because of the crackarresting phenomenon of the aggregate.

Curing TimeThe increase in effective fracture

toughness with age was the result ofcontinuing hydration of the cement parti-cles to produce a higher strength.

Fine Aggregate ContentThe effective fracture toughness

for the mortar increased with an increas-ing amount of fine aggregate because ofan increased concentration of crackarresting particles in the matrix. Theeffective fracture toughness of the con-crete was not significantly affected byan increasing fine aggregate content

because the coarse aggregate particleswere much better crack arresters andthus concealed the effect of the fineaggregate.

Gravel Content, Gradationand Type

The effective fracture toughnessincreased with an increase in maximumsize particles because the larger aggre-gate particles are more effective ascrack arresters. However, a maximumsize can be reached in conjunction witha poor gradation that will produce alower effective fracture toughnessbecause of the effects of segregationas shown in Figure 10 for a finenessmodulus of 7.45.

An increased gravel content in-creased the effective fracture toughnessbecause the larger gravel content en-larged the concentration of crack arrest-ers in the matrix.

The effective fracture toughnessfor the crushed limestone coarse aggre-gate was greater than the effectivefracture toughness for the river gravelcoarse aggregate until 28 days, indicatingthat the crushed limestone apparentlydeveloped greater bond strength. How-ever, after an age of 28 days the bondstrengths for the two types of coarseaggregate appeared to be equivalent.


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III. CRACK MECHANISM FOR CONCRETE STRUCTURES

3.1 INTRODUCTIONControl of cracking in concrete

structures subjected to varying loadand environment requires a basic under-standing of the crack mechanism to corre-late laboratory data with service con-ditions. The fracture process in con-crete structures is similar to thefracture that occurs in the fracturetoughness specimens of Chapter II, inthat both crack growths are associatedwith a fracture phenomenon that occursin the highly stressed region surroundingthe crack tip. In the fracture tough-ness specimen, the crack propagates whenthe stress intensity factor of Equation(2) reaches the effective fracture tough-ness K . There is no simple stresscintensity factor for a crack in a con-crete structure since the structuralcomponents interact with each other andwith the stress field surrounding thecrack tip. A systems-type analysis ofthe fracture process is used to describethe complex cracking mechanism forconcrete structures.

3.2 FRACTURE SYSTEMHahn and Rosenfield(0O) have pre-

sented a systems-type analysis of thefracture problem and applied it to thefracture of metal plates and incorporatedthe effect of yielding in the region ofthe crack tip. A fracture system of

processes that responds to outsidestimuli and interacts with each otherwas developed. Quantitative analysisof the system requires that the stimuliand responses of the various processesbe expressed in compatible terms(stress), which can be either theoreti-cal or empirical.

A similar system is useful foranalysis of cracking of concretestructures and is shown in Figure 12.The structure ) consists of the struc-tural elements such as concrete, rein-forcement, and supports, includingpavement base materials. The relationbetween load, environment, and thestresses in the various structuralelements is required. Stress-strainmodifiers 1 1 ) include flaws, cracks,inclusions, and other stress concentra-tors. The general level of stress isintensified locally near these modifiers.The linear-elastic fracture mechanicstechniques are used to evaluate theelastic stress field surrounding sharpflaws. Inelastic deformations (I II)may occur in regions that are highlystressed relative to strength. Theseinelastic deformations modify the rela-tive stiffness of the structural elementsand cause a redistribution of stress inthe structure. Cracking mechanisms (IV)are initiated when critical conditionsdevelop in the region of a crack tip,


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and crack growth takes place. Thiscrack growth also modifies the relativestiffness of the structural elementsand results in a stress redistribution.

The general fracture process of aconcrete structure is as follows:

The input (A) of load and environ-ment to the structure (1) causes stressesto develop within the structural ele-ments. The general stress levels aretransmitted (B) to any modifiers (I I)in the system. The stresses are in-creased and transmitted (D) to theinelastic deformations (Ill) and trans-mitted (F) to the fracture mechanisms(IV). At a critical stress level theinelastic deformations occur and aretransmitted back (E) to the modifiers,and at some critical condition existingcracks propagate, and the effect ofincreased crack lengths are fed back (G)to the modifiers. These effects on themodifiers are reflected back (C) to thestructure as changes in relative stiff-ness and result in stress redistribution.

Inelastic deformations tend toincrease the relative stiffness of con-crete and promote cracking, while crackgrowth tends to reduce the relativestiffness of the concrete and arrestscrack growth.

The systems-type analysis ofcracking concrete structures is based ona free body diagram of the concreteportion of the structure, which is thestructural element that contains thecrack that will propagate. The effectsof load, environment, reinforcement,and other structural elements on concretefracture are obtained by superposition.The stress intensity factor describingthe stress field surrounding the tipof the crack in the concrete is evaluated

separately for each action on the freebody. The resultant stress field isobtained by summing the individualstress intensity factors K., and thecondition of crack instability occurswhen the resultant K equals the effec-tive fracture toughness of the concreteK .c

i=m7 K. = K - Ki=

3.3 FRACTURE OF CONCRETE STRUCTURESThe analysis of concrete cracking

in structures is based on a resultantstress intensity factor for a crack inthe concrete, which is the stress modi-fier associated with the crackingmechanism that interacts with the loadsand the other elements of the structure.A concrete body containing the crackis isolated, and the stress intensityfactors for the various actions aredetermined using available expressionsand summed to obtain the resultantstress intensity factor Kr . Equilibriumcrack conditions, which relate crackgeometry and load, correspond to thelimiting condition of Equation (3),

K = K.r c (3-1)3.3.1 Crack in Constant Moment Region

of Reinforced Concrete BeamThe cracking mechanism in a rein-

forced concrete beam subjected to aconstant moment M is analyzed by consid-ering the concrete within a distance sof the crack as shown in Figure 13.The concrete is subjected to moments M cand axial compressive forces P whichare the actions of the adjacent concrete,and of resultant bond forces Fb at thelevel of the reinforcement, which are

/ I r


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the net forces transferred to the con-crete over the interval s. The resul-tant stress intensity factor K whichdescribes the stress field surroundingthe crack tip in the beam is

K = KMc + Kp + K (4)

where KMc is the stress intensity factorfor the concrete subjected only to themoments Mc, Kp is the stress intensityfactor for the concrete subjected onlyto the axial forces P, and KF is thestress intensity factor for the concretesubjected only to the resultant forcesat the level of the steel. Expressionsfor these stress intensity factors areavailable. (1 '12) However, the magni-tudes of Mc, P, and Fb are functions ofthe forces in the reinforcement. Thesteel forces are dependent upon theinelastic deformations associated withunbonding and cannot be defined withsufficient accuracy for a quantitativeanalysis of cracking.

A qualitative analysis of crackingindicates that KMc is the parameter thattends to cause crack extension; Kp isnegative for compressive forces andtends to arrest crack growth; K F isnegative when the bond forces Fb acttoward the crack and tends to arrestcracking and is positive and tends tocause cracking when the load forces actaway from the cracks.

3.3.2 Crack in Reinforced ConcreteTension Member

A reinforced concrete member withthe steel loaded in tension, Figure 14,has been suggested as a useful model ofthe cracking mechanism in beams, 13)and crack development under increasingload has been investigated.(1 4 ) A

quantitative analysis of crack equili-brium is based on the concrete elementof Figure 15. The only forces actingon the concrete are the bond forces Fbbthat develop as the reinforcement elon-gates. The unbonding at the free endsand at the cracked section affect themagnitude of the bond forces. Thebond forces cause an opening A of thecrack at the level of the reinforcement.

The stress intensity factor K maybe expressed in terms of the bondforces Fb or the opening A. ( 12)

CIFb C2 EcK = - 1 = At d' d 2

e e

where t is the thickness, d is theeeffective depth, E is the modulus ofcelasticity of the concrete, and C1 andC 2 are coefficients that are evaluatedfor various crack lengths a , effective

(12)depths d , and specimen lengths .(e sThe maximum equilibrium crack

length corresponds to a stress intensityfactor K that is equal to the effectivefracture toughness of the concrete K ,'

K = KcThe force which the concrete transmitsacross the cracked section, T , isequal to the bond force Fb'

K t dT = F = c ec band is usually small relative to theforce in the steel T . The wedge open-sing L which corresponds to an equili-brium crack is

K d 2c e

C 2 E c


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and is assumed to be equal to the elon-gation of the steel reinforcement overan equivalent unbonded length , and

T PS EU (9 )S S

where T is the force in the steel atsthe cracked section, A is the steelarea, and E is the modulus of elastic-ity for steel. The total force T trans-mitted across the cracked section forequilibrium conditions is

K td K A E dT = T + T = c e + c s s e (10)c s Ci C2 u ET is unique for a given equilibriumcrack length, and T s varies inverselywith the unbonded length A . The totalload T may be calculated for a givencrack length a by substituting variousunbonded lengths u into Equation (10) .This has been done for the specimengeometry and material properties ofcrack specimens of Reference (14), andthe relationships between T and theratio of crack length to effective a /dare given in Figure 16 for differentunbonded lengths. The effective frac-ture toughness has been assumed to beapproximately 0.6 ksiv'-n .

The relationships of Figure 16 in-dicate the following:

(a) The equilibrium crack lengthsa increase with total force T if theunbonded length 2u is constant;

(b) The total force T transmittedacross a given cracked section decreasesas the unbonded length increases;

(c) An increased unbonded lengthis associated with an increased cracklength corresponding to a virtual loadincrease.

Crack data from Reference (14) isshown in Figure 16 for increasing loads.The first cracks initiated in two dif-ferent specimens at the 20 kip loadlevel. They corresponded to equivalentunbonded lengths of 0.5 and 1.2 in.At the higher load levels of 25 and35 kips, the crack lengths correspondedto unbonded lengths of 1.2 to 1.5 in.At crack initiation there was a largerange in the unbonded length. As theload increased, the unbonded lengthsincreased and the range was reduced.This is an example of the interactionof an inelastic deformation, the un-bonding, with the concrete crackingmechanism.

3.3.3 Crack in Rigid PavementCracking in rigid pavements may

be analyzed by using the cracked beamon an elastic foundation of Figure 17.The stress intensity factor should varywith the inverse of crack length

K = f ( ) , (ll)since an increase in the crack lengtha transfers more load to the elasticfoundation in the region of the crackand reduces the stresses on the crackedsection. The limiting condition is acrack through the pavement depth (a = h),and the load is still transferred tothe foundation. This is a stable crackgrowth condition. Crack growth arrestsadditional cracking until the appliedload is increased. Crack growth mayalso be caused by repeated loadings,and this model should find applicationsin the fatigue of rigid concrete pave-ments.


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IV. ANALYTICAL STUDY OF CRACK DEVELOPMENT ASSOCIATED WITH VOLUME CHANGE

4.1 INTRODUCTIONThe volume of concrete changes with

age through shrinkage or swelling asso-ciated with moisture movement. Nonuni-form volume change of concrete takesplace because of nonuniform moistureexchange. Changes occur in the shapeof concrete members, and stresses areinduced. The nonuniform shrinkage ofconcrete has not been studied to thesame extent as uniform shrinkage, andmore information is available in theliterature on uniform shrinkage ofconcrete than on nonuniform or relativeshrinkage of concrete.

Theoretical analysis of shrinkagestresses in concrete involves a tedioussolution of partial differential equa-tions of diffusion and compatibility.If the analysis of reinforced concreteis desired, these differential equationsbecome more complex.

In recent years the solutions tothe problems that have been extremelydifficult to solve by means of analyti-cal approaches have been obtained bynumerical computations through the useof digital computers. In particular,the analysis of shrinkage stresses inplain and reinforced concrete can beperformed by use of the finite elementmethod.

4.2 DEVELOPMENT OF STIFFNESS MATRIXFOR FINITE ELEMENT ANALYSIS

4.2.1 AssumptionsThe following assumptions are made

in developing the finite element model:(a) Concrete and steel have linear

stress-strain diagrams;(b) Loads and deformations are

applied to nodal points;(c) Shrinkage strains are applied

to elements;(d) A perfect bond exists between

steel and concrete;(e) Concrete and steel are homo-

geneous and isotropic materials and eachhas an identical stress-strain relationin tension and compression (except con-crete is assumed to fail at a limitingstress in tension but not in compression);

(f) The steel element has nophysical dimensions and is assumed tobe present on a horizontal line of nodesonly;

(g) Loading is one directional.

4.2.2 Concrete ElementElements of various shapes may be

used in the finite element analysis.However, the shape of the element shouldbe selected so it fits the needs of theanalysis. In this analysis a rectan-gular element was selected since it fits


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the structural shapes (beams and slabs)in which shrinkage is to be studied.A rectangular element also allows appli-cation of linearly varied shrinkagestrains to any particular element. Fig-ure 18 depicts a typical rectangularelement. The four corners of the rec-tangle are called the nodes. The forcesand displacements that are to be appliedto the structural element being analyzedby the finite element method must beapplied through these nodal points.

The stresses and strains in anyone element are not constants, butdepend on the coordinates of the pointat which they are evaluated. The coor-dinates of a point in any one elementare measured from a set of axes withthe origin at the lower left node ofthe element as shown in Figure 18.

A set of stresses, x' , T y, iscalculated for each node and also forthe center of the element. Principalstresses are calculated at the centerof the element. In order to detect theoccurrence of cracking in an element,the maximum principal stress is comparedwith the limiting stress at which con-crete is assumed to crack.

4.2.3 Steel ElementThe physical size of steel is

usually small compared to concrete. Inparticular, when one considers shrinkagereinforcement, this difference in sizebecomes very noticeable. In this study,the steel element is assumed to have nophysical dimension but only mechanicalproperties. The steel is assumed to bepresent along a line of horizontal nodalpoints as shown in Figure 19. No steelis assumed to be present inside the con-

There is no restriction

on the position of the horizontal rein-forcement other than being restrictedto a single line.

4.2.4 Analytical ModelThe complete reinforced concrete

model used in this study is shown inFigure 19. The model is assumed to beof unit thickness, although this is nota necessary requirement for the analysis.

Boundary conditions specified atthe nodal points on the model can bevaried to fit a specific problem, i.e.,a fixed condition at the left end ofthe beam is realized when the displace-ments in the x- and y-directions areset equal to zero for all the nodes atthe left-most side of the model. Loadsare applied at the nodes, and distri-buted loads are represented by a seriesof concentrated loads acting at thenodes. If the model is loaded by in-ducing deformations on it, the deforma-tions must also be applied at the nodes.Only strains are applied to the ele-ments.

4.2.5 Derivation of Stiffness Matrixfor One ElementIn this study the finite element

problem is restricted to two dimensionsresulting in either a plane stress orplane strain condition. The stiffnessmatrix is derived for the general casepartly from the work of Przemieniecki(15)and will be valid for both plane stressand plane strain.

Consider the element in Figure 18.To simplify the analysis, the nondimen-sionalized coordinates e = x/d and rny/b are used.(15) The displacements atany point in an element are functionsof the coordinates of the point and willrete elements.


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be assumed to be of the followingnature:

u = C 1 s + C2Cn + C 3 n + C 4x (12)u = Cs5 + C 6 Vn + C7r1 + C8

where:ux, u are displacements in the x-and y-directions and C 1 , C 2 ,...,C8 are constants which can be eval-uated from the following boundaryconditions:at (0, 0) u = ui, u = u2x y(0, b) u = u3, u = U 4x y(d, b) u = us, u = u6

x y(d, 0) u = uy, u = uOx yThe displacements will then be repre-sented by

u = (l-c)(l-n)u1 + (l-O)nu3 + Tnus+ C(1-n)u7 , (12a)

u = (I-0)(l-nl)u2 + (I-O)nu4 + Tu6+ C(l-n)u8 .

The total strains can be determined bydifferentiating the displacement equa-tions

au aux d xSx = x = d 'Du I au

E = b= - y b rn ' (13)au Du x I u uIYxy ay ax b 3n d '

where E and e are the normal strainsx yin the x- and y-directions, respective-ly, and xy is the shear strain.

The strain-displacement relation-ship for the rectangular element becomesin matrix notation

[E] = [D] [u] (13a)where [D] is a matrix relating strainsto displacements and [u] is a column

to displacements and [u] is a columnmatrix representing the displacementsat the four nodes of an element. It isimportant to note that the strain ineach element is not a constant, but isdependent on the coordinates of thepoint at which it is to be evaluated.

The stress-strain relationship forthe element can be represented in matrixnotation as

[a] = [C] [e] (14)where [C] is a matrix relating stressesto strains and [e] is a column matrixrepresenting the strains in an element.Substitution of Equation (13a) intoEquation (14) yields

[c ] = [C] [D] [u] , (15)which relates the stresses to the nodaldisplacements for one element.

A typical force F. at a point iin the element can be calculated bythe unit displacement theorem ( 15)

F. = f T a dV (16)

where:TC. = transpose of matrix of compat-

ible strains due to a unitdisplacement in the directionof F.,

a = stress matrix resulting fromall forces acting on the ele-ment,

dV = element of volume in theelement,

= integration over total volumeSof the element.

vThe forces at the nodal points of

the elements can be found by substitutingthe matrices into Equation (16) and thenintegrating the product over the totalvolume of the element.


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FIGURE 1. TEST SETUP


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0 0.001 0.002 0.003Deformation, A., in.

0 0.001 0.002 0.003Deformation, A&, n.

0 0.001 0002 0.003Deformation, AA, in.

0 0.001 0.002 0003Deformation, tA, in.

FIGURE 2. TYPICAL LOAD-DEFORMATION CURVES: CONCRETE

6.6

6.05.44.84.23.63.02.41.81.20.6

0

- a/w=0.125

3.3.'2.2.'2.

1.1

O.c

O.(0.:I

V


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0K

0Eno,CL

EE

'U

0

0 0.1 0.2 0.3 0.4o/w

FIGURE 3. MAXIMUM LOAD AND EFFECTIVEFRACTURE TOUGHNESS VS A/W.

0.25 0.30 03 5 0.40 045 050w/c

0.55 0.60 065 0,70

FIGURE 4. EFFECT OF W/C RATIO ON R c

0.8

0.7

0.6

05

04

0.3

0.2

o Concrete -a Pastes -----* Mortors -

,' S

5-.--.-S.-- -S. -5

-S.--.- 5-

0

. I


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0 2.0 4.0 6.0 80 10.0 12.0 14.0Air Content, Percent

FIGURE 5. EFFECT OF AIR CONTENT ON R c

Log Days

FIGURE 6. EFFECT OF CURING TIME ON K c: MORTARS & PASTES


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A-)

Log Days

FIGURE 7. EFFECT OF CURING TIME AND TYPEOF COARSE AGGREGATE ON K c: CONCRETES

50 55 60 65 70 75Fine Aggregate by Weight, per cent

FIGURE 8. EFFECT OF FINE AGGREGATE ON K c: MORTARS

0.8

0.7

0.6

05

0.3

0.2-

0.1

0 ___


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0.8

0.7

0.6 -

05

02

02-- --- -- --- --

30 40 50Fine Aggregate by Weight, per cent

FIGURE 9. EFFECT OF FINE AGGREGATE

ON R c: CONCRETES

08 -

0.7 ---- --- _--- ---

0.6

0.4 ------ -------

0,3----------- -03

02

0.1

0o

FIGURE 11.

ON

10 20 30 40 50Coarse Aggregate by Weight, per cert

EFFECT OF COARSE AGGREGATE

K c: CONCRETES

Fineness Modulus

FIGURE 10. EFFECT OF FINENESS MODULUSOF COARSE AGGREGATE ON R c: CONCRETES

tOAD I ENVIRONMENT

FIGURE 12. SCHEMATIC OF

FRACTURE SYSTEM

o

63 65 6769 71 d 3. ., ,.


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( IcFIGURE 13. CRACKED CONCRETE ELEMENT FROM REINFORCED CONCRETE BODY

FIGURE 14. REINFORCED CONCRETE TENSION MEMBER

FIGURE 15. CRACKED CONCRETE ELEMENT FROM TENSION MEMBER


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0 01 02 03 04 05 0.6 0.7 0e 09 1.0FIGURE6.OAD,,SORd

FIGURE 16. LOAD, T, VS 6/d e FOR DIFFERENT UNBONDED LENGTHS, k

FIGURE 17. CRACKED RIGID PAVEMENT Id 4d

FIGURE 18. A TYPICAL ELEMENT

r- a -'-i

-Steel

FIGURE 19. REINFORCED CONCRETE MODEL

LI-JL-^

m


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42/70


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[F] = ( ; [B]T [C] [B] dV [u]v (17)

where [F] is a column matrix represent-ing the forces at the four nodal pointsof the element, [D] is the transposeof [D], and the remaining expressionsare as defined above.

The final equation can be expressedin matrix notation as follows:

[F] = [K) [u] (17a)where:

[K] = f [D] T [C] [D] dV.v

[K] is called the stiffness matrixfor one element and it relates the nodalpoint forces to the nodal point displace-ments. The stiffness matrix of the en-tire system can be obtained by directlyadding the contribution of each individ-ual element stiffness in the properlocation.

4.2.6 Incorporation of ShrinkageStrains in the ModelThe free shrinkage strain at any

point in an element is a function ofthe relative humidity at that point.For purposes of consistency of the dis-placements u , u , the shrinkage strainsfor a specimen drying from one side onlyare assumed to be defined by the follow-ing formulas:

xs = al + a2 n ,e = ai + a21 , (18)ys

y = 0 ,xyswhere :

E = normal shrinkage strain inxs x-direction

e = normal shrinkage strain inys y-direction

Yxys = shearing shrinkage strain,a, and a2 = constants,

n = y/b.Equation (18) can be written in matrixnotation as:

xs

Yxys

al + a2nal + a2nT (18a)

0

The reason for assuming e and exs ysbeing equal at any particular point inthe element is that these shrinkagestrains are very similar in nature tothermal strains. Other assumptionsconcerning the distributions of shrink-age stresses can be incorporated intothe analysis.

The forces that are induced ateach node as a result of the shrinkagestrains must be found in order to incor-porate the effect of shrinkage strainsin the model. Consider Figure 18 andassume that the element is acted uponby a state of shrinkage strain of thetype described above. The forces pro-duced by this state of strain can befound by using the fact that the workdone by the external forces must equalthe change of the internal energy.

[F ]T[u] = f [a ]T[e] dV (19)where:

[u] = matrix of unit displacementsat the nodes of the element;

[F i = transpose of the matrix offorces that are produced atthe nodes as a result ofshrinkage strains;T[ao] = transpose of the matrix ofstresses produced by shrink-age strains;


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[e] = matrix of strains producedby unit displacements at thenodes; [E] = [D] since themagnitude of the displacementis unity;

dV = element of volume in theelement;= integration over total volume

of the element.v

From the stress-strain relationship[os] = [C][ei]

and

[ T = [ ]T[C]T ,

however, since [C] is a symmetric matrix[C] T = [C]. Therefore,[0 T = [ i] T C] (20)

The forces resulting from shrinkagestrains can be found by substitutingEquation (20) into Equation (19), thus

[Fs]T = f [ ]T[c][D] dV.v (21)

The final equation that relates nodalforces to nodal displacements and shear-ing strains is

[F] = [K][u] + [Fs]. (22)As a result of shrinkage strains,

nodal forces are produced which may beobtained for the entire model by addingthe contributions of individual elementsin the proper locations. The equationrelating nodal forces to nodal displace-ments and shrinkage strains for the en-tire model is

[F] = [K] [u] + [F ] (23)where:

[F] = a 2n x I column matrix offorces at all nodes;

[K] = a 2n x 2n symmetrical stiff-ness matrix for the entirebody;

[u] = a 2 n x 1 column matrix ofdisplacements at all nodes;

[F ] = a 2n x 1 column matrix of-- forces induced at nodal

points by shrinkage strains;n = total number of nodes.

Eouation (23) represents a system of2n simultaneous equations which can besolved for the nodal displacements.These equations are derived from theforce equilibrium equations in the x-and y-directions, i.e., the sum of allthe forces in the x- and y-directionsat any node must equal zero unless aboundary condition is defined at thenode. When a boundary condition isdefined at a node the sum of the forcesat the node will equal the externalload applied at that node.

4.2.7 Development of CracksStresses are calculated at five

points for every element -- the fourcorners and the center of the element.The principal stresses and the directionof the maximum principal stress arecalculated at the center of every ele-ment. The maximum principal stress atthe center is compared to a limitingstress for cracking and if it exceedsthe limiting stress, the element isassumed to have cracked and thus doesnot carry any tensile stresses. Whencracking does occur, the element canbe completely ignored since the loadingis assumed to be one directional. Ifthe stresses in two or more horizontallyadjacent elements exceed the limitingstress at the same time, the elementwith the largest stress is assumed tobe cracked. Cracks are found throughan iteration process and every time anew crack appears the analysis is re-peated in order to find other cracks


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that might have appeared as a result ofthe new crack. Thus the crack patternin the model is developed and the direc-tion of each crack is also calculated.It should be noted, however, that sincea cracked element is assumed to carryno stress, no two horizontally adjacentelements can be cracked. This requiresthat the length of each element notexceed one-half of the expected crackspacing.

4.3 APPLICATION OF THE METHOD ANDBOUNDARY CONDITIONS

4.3.1 Application of MethodThe method developed can be applied

to any member with a shape that can beapproximated by rectangular elements.The assumptions that were made in devel-oping the method must also be reasonablyvalid, i.e., since it is assumed thatthe stress-strain diagram for concreteis linear, the maximum compressive stressin the concrete must remain below areasonable limit.

The loads are applied to the nodesand the directions of the loads aregoverned by the set of axes assumed forthe model, i.e., loads acting in thepositive x- and y-directions are assumedpositive. If there are any applied dis-placements, they follow the same signconvention. Volume change strain isapplied to the elements and two valuesof volume change strain are specified,one at the top side and one at the bot-tom side of the element. It is assumedthat there is a linear strain variationbetween the two values of volume changestrain for the element. It is further

assumed that all of the elements in arow are under the influence of the sameshrinkage strain. The volume changestrain is assumed to be positive if itproduces expansion and negative if itproduces contraction.

The steel stresses are limited bythe assumption of perfect bond betweenthe steel and concrete.

The method is not limited to con-crete but can be used to evaluate thestresses for any material.

4.3.2 Boundary ConditionsBoundary conditions are limited

only in the sense that the conditionsare applied at the nodes. To representa roller, the vertical displacement atthe node on the roller is set equal tozero. To represent a pin connection,both the horizontal and vertical dis-placements at the node are set equal tozero. To represent a fixed end, thehorizontal and vertical displacementsof all the nodes at that end are setequal to zero. Other boundary conditionmay be applied similarly. Boundaryconditions that partially limit themovement of a node, such as a spring,can be applied if modifications aremade in the digital computer program.

4.3.3 Computer Program for Deteriminatiof Volume Change Stresses inPlain and Reinforced ConcreteUsing Finite Element Analysis

The computer program VCSC (VolumeChange Stresses in Concrete), preparedin FORTRAN IV, and a User's Manual arecontained in the Appendix.


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V. PRACTICAL APPLICATIONS

5.1 EFFECT OF CONCRETE PARAMETERS ONFRACTURE TOUGHNESSThe effective fracture toughness

was not significantly affected by thefine aggregate content (30.0 per centto 50.0 per cent, by weight), air con-tent (4.0 per cent to 10.0 per cent),and water-cement ratio (5.7 gal/sack to7.3 gal/sack). Since the range of theparameters investigated was inclusiveof most mix designs, they can be neglect-ed in designing for a mix of high or lowfracture toughness (material's resis-tance to propagation of an existingflaw). Although only two types ofcoarse aggregate were used in the inves-tigation, the results suggest that theeffect of type of coarse aggregate onthe fracture toughness was similar tothe effect of type of coarse aggregateon the bond strength between coarseaggregate and cement paste or mortar.(16)Thus, high quality aggregates (homoge-neous, low absorption, high modulus ofelasticity relative to cement paste,etc.) should be used to develop mixeswith high fracture toughness values(gradation requirements previouslystated would apply to all types of ag-gregate). The variables significantlyaffecting the fracture toughness of con-crete can be limited to the coarse aggre-gate content and gradation of coarseaggregate.

The effective fracture toughnessof concrete was found to be directlyproportional to both coarse aggregatecontent and gradation of coarse aggre-gate. Thus, by increasing or decreasingthe percentage of coarse aggregate, orincreasing or decreasing the maximumaggregate size, or a combination ofboth, the fracture toughness can be ad-justed. However, if the fracture tough-ness is to be increased by using a larg-er maximum size coarse aggregate, thegradation of coarse aggregate must beuniform to minimize segregation and itsdetrimental effects. Limitations willbe placed on maximum aggregate size bydesign considerations, i.e., size andshape of the concrete members, amountand distribution of reinforcing steel,etc.

Since the major aim of crack con-trol is to minimize crack width byincreasing the number of cracks inhardened concrete, the design of aconcrete mix to maximize the high frac-ture toughness is not necessarily theanswer. It also may be advisable tomake a sacrifice in the desired fracturetoughness value so that small flaws canform to act as stress relievers. Thesesmall flaws would prevent the buildupof stress values in the concrete thatcan lead to formation of large crackswhich could allow the ingress of water


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to cause corrosion of the reinforcementwhich could then result in rapid deter-ioration of the concrete. However, inthe design of all mixes the objectivesof required qualities of hardened con-crete, workability of fresh concrete,and economy should be maintained evenif it means a sacrifice in fracturetoughness.

5.2 CRACK MECHANISM FOR CONCRETESTRUCTURESIn order to control the cracking

of concrete structures due to varyingload and environment, a systems-typeanalysis can be used to describe thecomplex cracking mechanism for a rein-forced concrete beam subjected to a con-stant moment, for a reinforced concretemember with the steel loaded in tension,and for a cracked beam on an elasticfoundation.

A reinforced beam subjected to aconstant moment may be analyzed usingthe approach developed. The resultantstress intensity factor is the sum ofthe individual stress intensity factorsfor the concrete being subjected onlyto a moment, for the concrete being sub-jected only to axial forces, and forthe concrete being only subjected tothe resultant forces at the level ofthe reinforcement. Since expressionsfor the individual stress intensityfactors are available in the litera-ture l l '1 2 ) the resultant stress inten-sity factor can be determined. Whenthe resultant stress intensity factorbecomes equal to or exceeds the criticalstress intensity factor the crack willpropagate until it is arrested.

The cracking mechanism in beamscan be anproximated by using a rein-

forced concrete member with the steelloaded in tension. Since the onlyforces acting on the concrete are thebond forces which develop as the rein-forcement elongates, the stress inten-sity factor can be expressed in termsof either the bond forces or the crackopening displacement at the level ofthe reinforcement. The total forcetransmitted across the cracked sectionfor equilibrium conditions for a givencrack length and different lengths ofunbonding can be calculated from thespecimen geometry and material proper-ties for the cracked specimen.

Cracking in rigid pavements maybe analyzed by using a cracked beam onan elastic foundation. The stress in-tensity factor varies inversely withcrack length since an increase in thecrack length transfers more load to thestructure in the region of the crackand thus reduces the stresses on thecracked section. This crack growtharrests additional cracking until theload is increased. The limiting condi-tion is reached when the crack haspropagated through the pavement depthwhile the load is still transferred tothe foundation (stable crack growthcondition). Also, this model findsapplication to crack growth under re-peated random loads in rigid concretepavements, but quantitative resultscannot be derived until data becomesavailable on the change in crack lengthas a function of stress intensity factorrelated to repeated loads.

5.3 ANALYTICAL STUDY OF CRACK DEVELOP-MENT ASSOCIATED WITH VOLUME CHANGEThe computer program VCSC (Volume

Change Stresses in Concrete) is prepared


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in FORTRAN IV for the solution of stres-ses caused by volume change in plain andreinforced concrete. The program iscapable of solving cases in which themodel is subjected to external and/orinternal displacements as well as volumechange strains. The program uses thefinite element method of analysis. Theelements are rectangles of equal size.Nodal displacements, normal stresses,and shearing stress at each node of eachelement, normal stresses, shearing stressmaximum principal stress, minimum prin-cipal stress, and direction of maximumprincipal stress at the center of thesame element, and steel stresses foreach element of steel result from thesolution of each problem in the ordergiven.

In their early life, highway pave-ments and bridge decks are subjected toshrinkage stresses that may be theprimary stresses acting on the members

and may lead to cracks if the rate ofdrying is rapid enough. Under suchconditions this method can be used topredict cracking patterns and stressdistributions for various strengths ofconcretes, amount of reinforcement, andthickness of concrete cover, when theshrinkage stresses are dominant.

The boundary conditions used tosimulate a bridge deck or a highwaypavement could be rollers on the bottomof the model with a corner node fixedin both the x- and y- directions. Themodel should be long enough so the com-putation of stresses and displacementswould converge to their actual valuesover a range of length which is suffi-ciently long for a crack pattern todevelop. In general it is suggestedthat the stresses and deformations notbe taken at points which are closer thantwo columns of elements to a side bound-ary.


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VI. SUMMARY AND CONCLUSIONS

6.1 OBJECT AND SCOPEThe objective of this study is to

gain an increased understanding of crackinitiation and growth in concrete, whichis essential to improved control ofcracking of concrete structures, i.e.,to acquire a better understanding ofthe effect of concrete parameters oncrack development in concrete and tocorrelate crack development in concretewith various types of distress.

The investigation was divided intothree major divisions: (1) experimentalinvestigation of fracture toughness,effect of concrete parameters on thefracture toughness of pastes, mortars,and concretes; (2) crack mechanism forconcrete structures, systems-type analy-sis description of complex crackingmechanism that occurs in concrete struc-tures; and (3) analytical study of crackdevelopment associated with volumechange, approximate solution for problemof shrinkage stresses in plain and rein-forced concrete was developed.

6.2 RESULTS OF INVESTIGATION6.2.1 Effect of Concrete Parameters

on Fracture ToughnessThe effective fracture toughness

K was based on the assumption that theconcrete was homogeneous and the flawdepth at failure was equal to the castflaw depth.

In the paste and mortar seriesthere was a decrease in effective fracture toughness with increasing water-cement ratio, while in the concreteseries there was no apparent effect ofvarying the water-cement ratio on efftive fracture toughness for the rangeof water-cement ratios investigated.

Increasing the air content de-creased the effective fracture toughnefor the paste, mortar, and concreteseries.

In the paste, mortar, and concretseries there was an increase in effec-tive fracture toughness with age. Thiincrease was significant up to an ageof 29 days, but for curing times ofgreater than 29 days the change ineffective fracture toughness from the29-day value was not significant.

There was an increase in effectivfracture toughness in the mortar seriewith increasing sand-cement ratio, however, the change in effective fracturetoughness with increasing sand-cementratio in the concrete series was notsignificant for the range of sand-cement ratios investigated.

The effective fracture toughnessof the concrete series increased withan increase in the maximum size ofcoarse aggregate and also with anincreased gravel-cement ratio. Howevethere was a decrease in the effective


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fracture toughness when a large amountof maximum size aggregate was used inthe mix. This was probably attributableto segregation.

The effective fracture toughnessof the concrete series cast with ariver gravel coarse aggregate was lowerthan the effective fracture toughnessof the concrete series cast with acrushed limestone coarse aggregate forages of three days and six days; how-ever, at ages of 29 days and 92 days thedifference in their effective fracturetoughness was not significant.

6.2.2 Crack Mechanism for ConcreteStructures

A fracture system of processesthat respond to outside stimuli andinteract was developed to describe thecomplex cracking mechanism in concretestructures. The system was applied tothe free body diagram of the concreteportion of the structure, which is thestructural element that contains thecrack that will propagate.

The cracking mechanism in a rein-forced concrete beam subjected to aconstant moment was analyzed. A quali-tative analysis of the cracking indi-cated that the resultant stress intensi-ty factor which describes the stressfield surrounding the crack in the beamwas a function of the stress intensityfactor due to the concrete being sub-jected only to a moment (causes crackextension), the stress intensity factordue to the effect of an axial load(negative for compressive forces andthus tends to arrest crack propagation),and the stress intensity factor due tothe resultant forces at the level of thesteel (negative when the bond forces

act toward the crack thus causing crackarrest, and positive when load forcesact away from the crack to cause crackpropagation). Crack propagation occurswhen the resultant stress intensityfactor reaches the critical value.

A model was presented for inves-tigating the cracking mechanism inbeams. The model was used for a quanti-tative analysis of crack equilibrium.The only forces acting on the elementare the bond forces, which can beexamined by the opening displacementat the level of the reinforcement andare affected by unbonding. The stressintensity factor can be expressed interms of the displacement, or the bondforces. Applying the results of thisapproach to the specimen geometry andmaterial properties of Reference (14)yielded the following:

(a) The equilibrium crack lengthincreases with the total force trans-mitted across the cracked section bythe concrete and steel if the unbondedlength is constant;

(b) The total force transmittedacross the cracked section by the con-crete and steel decreases as the un-bonded length increases;

(c) An increased unbonded lengthis associated with an increased cracklength during a virtual load increase.

The cracking in rigid pavementscan be analyzed using a cracked beamon an elastic foundation. The stressintensity factor would vary inverselywith the crack length since an increasein the crack length transfers more loadto the elastic foundation in the regionof the crack and reduces the stresseson the cracked section.


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6.2.3 Analytical Study of CrackDevelopment Associated withVolume Change

An analytical method for the eval-uation of volume change stresses waspresented which can be used for bothplain and reinforced concrete. Thefinite element method was utilized withthe aid of a digital computer programto construct an approximate solution tothe problem. The method predicts crack-ing, crack patterns, and magnitude anddistribution of stresses in members inwhich shrinkage stresses are the primarystresses.

The effects of various parameterson volume change stresses can be pre-dicted. For example, the effect ofconcrete strength, concrete cover,amount of reinforcement, etc., can bestudied by using various parameters inthe computer program, thus saving consid-erable time over that required in anextensive experimental program.

The method can also consider theeffects of loads and displacements onthe model.

The method will be more effectiveas additional input data on shrinkage,particularly nonuniform shrinkage,become available. If reliable infor-mation on shrinkage strains at varioustimes becomes available, then time tocracking can be predicted.

The bond between steel and concretewas assumed to be perfect. As a firstaoproximation this is justifiable becausevolume change stresses are relativelysma ll. .

6.3 CONCLUSIONSThe results of this investigation

can be used to describe the cracking of

concrete structures due to various typeof distress, i.e., distresses due toenvironment or loading. The resistanceto propagation of the flaws inherentin concrete can be adjusted by modifyinthe mix design, i.e., varying the coarsaggregate content or gradation of coarsaggregate, or type of coarse aggregate.Since the experimental investigationcovered a range of concrete parametersthat would be inclusive of most mix de-signs, an approximate effective fracturtoughness value can be determined fromthis investigation for most designs use

From the effective fracture tough-ness value for the mix design chosenand the stress intensity factors dueto the type of load distress, the resul-tant stress intensity factor can bedetermined for a reinforced concretebeam subjected to a constant moment.The resultant stress intensity factorcan then be used to describe the crack-ing mechanism; i.e., crack propagation,which will occur when the resultantstress intensity factor reaches thecritical value.

The model developed for investi-gating the cracking mechanism in con-crete beams under load can be used forobtaining quantitative results. As anexample of its application, this approawas used in conjunction with the resultof Reference (14) for studying theequilibrium crack mechanism in reinforcconcrete members.

Cracking in rigid pavements due toexternal load can be studied using acracked beam on an elastic foundation.This model can also be used for studyinthe crack mechanism of pavements due torandom loading, but further data isrequired on the change in stress


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intensity factor with loading cycles.The computer program developed can

be used for evaluation of volume changestresses resulting from either environ-mental distress or load distress. Themethod predicts cracking, crack patterns,and magnitude and distribution ofstresses in members in which shrinkage

stresses are the primary stresses. Theeffects of various parameters on volumechange stresses, as well as the effectsof concrete cover, concrete strength,amount of reinforcement, etc., can bestudied by using various parameters inthe program.


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VII. SUGGESTIONS FOR FUTURE RESEARCH

The effects of different environ-mental conditions and different loadingrates on the effective fracture tough-ness of concrete should be investigated.

Application of the systems-typefracture analysis requires: understand-ing the inelastic phenomenon of unbond-ing of cracked sections; theoretical orempirical knowledge of the actual con-crete stresses near cracks; developmentof stress intensity factor expressionsfor various models such as the rigidpavement; and a further look at fatiguecrack growth in terms of the stressintensity factor with loading cycles.

Utilization of the analytic methodfor solution of volume change stressesrequires data which is compatible withthe computer program. One area inwhich there is insufficient data isshrinkage strains in members of largecross section so that nonuniform shrink-age data can be obtained with all but

one or two surfaces sealed Data shouldbe provided for free shrinkage versustime for various drying, environmental,and surface conditions. A load-sliprelationship could be incorporated intothe program to eliminate the assumptionof perfect bond between the steel andconcrete, and thus bond stress can bepart of the program output. The com-puter program could be modified toaccept springs as part of the boundaryconditions so that slip over the sub-grade of the highway pavement can beincorporated. Also, in order to inves-tigate the effects of rate of drying onthe stresses produced by shrinkage, theshape of the stress-strain curve couldbe varied, i.e., study the effect ofthe stress-strain diagram on the effectof the maximum value of strain whichoccurs at the outermost fiber of themember.


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VIII. REFERENCES

(1) "Fracture Toughness Testing and ItsApplications," ASTM STP No. 381,American Society for Testing andMaterials (April, 1965).

(2) Irwin, G. R. Proceedings, 1960Sagamore Research Conference onOrdnance Materials. Washington,D.C.: U.S. Office of TechnicalServices.

(3) Kaplan, M. F., "Crack Propagationand the Fracture of Concrete,"Proceedings, American ConcreteInstitute, 58 (1961), pp. 591-611.

(4) Glucklich, J., "Fracture of PlainConcrete," Proceedings of the ASCE,89:EM 6 (1963), pp. 127-138.

_ "Static and FatigueFractures of Portland CementMortars in Flexure," Proceedings,First International Conference onFracture, Vol. 2, Sendai, Japan(1965), pp. 1343-1382.

(6) Lott, J. L. and Kesler, C. E.,"Crack Propagation in Plain Con-crete," Symposium on Structure ofPortland Cement Paste and Concrete,Highway Research Board SpecialReport 90, Washington, D.C., (1966),pp. 204-218.

(7) Naus, D. J. and Lott, J. L., "Frac-ture Toughness of Portland CementConcretes," Theoretical and AppliedMechanics Report No. 314, Univer-sity of Illinois at Urbana-Champaign(1968), pp. 1-87.

(8) Brown, W. F. and Srawley, J. E.,"Plane Strain Crack ToughnessTesting of High Strength MetallicMaterials," ASTM STP No. 410,American Society for Testing andMaterials (1966), pp. 13-14.

(9) Powers, T. C. and Brownyard, T. L. ,"Studies of the Physical Propertiesof Hardened Portland Cement Paste,"

Proceedings, American ConcreteInstitute, 41, (1946-7), pp. 101-132, 249-503, 549-602, 669-712,845-880, 992-993.

(10) Hahn, G. T. and Rosenfield, A. R.,"A Systems-Type Approach toProblems on Fracture" in Funda-mental Phenomena in the MaterialSciences, Vol. 4. New York:Plenum Press (1967), pp. 33-43.

(11) Srawley, J. E. and Gross, B.,"Stress Intensity Factors forCrack-line-Loaded Edge-CrackSpecimens," NASA TND - 3820 (1967),pp. 1-19.

(12) Gross, B., Roberts, E., Jr., andSrawley, J. E. , "Elastic Displace-ments for Various Edge-CrackedPlate Specimens," NASA TND-4232(1967), pp. 1-12.

(13) Reis, E. E., Jr., Mozer, J.,Bianchini, A. C., and Kesler,C. E., "Causes and Control ofCracking in Concrete Reinforcedwith High Strength Steel Bars--A Review of Research," Universityof Illinois Engineering ExperimentStation Bulletin No. 479, Urbana,Illinois (1965), pp. 1-61.

(14) Broms, B. B., "Stress Distribution,Crack Patterns and FailureMechanisms of Reinforced ConcreteMembers," Proceedings, AmericanConcrete Institute, 61 (October,1964), pp. 1535-1557.

(15) Pzemieniecki, J. S. Theory ofMatrix Structural Analysis.New York: McGraw-Hill Book Co.,(1968).

(16) Hsu, T. T. C. and Slate, F. 0.,"Tensile Bond Strength BetweenAggregate and Cement Paste orMortar," Proceedings, ACI , Vol. 60(April, 1963), pp. 465-486.


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IX. APPENDIX I: USER'S GUIDE FO R COMPUTER PROGRAM IN FORTRAN IV

User's GuideThe computer program VCSC (Volume

Change Stresses in Concrete) is preparedin FORTRAN IV language for the solutionof "Stresses Caused by Volume Change inPlain and Reinforced Concrete." Theprogram is capable of solving cases inwhich the model is subjected to externalloads and/or external displacements aswell as volume change strains. Theprogram uses the finite element methodof analysis. The elements are rectan-gles of equal size. The result of thesolution of each problem is the nodaldisplacements, the normal stresses, andthe shearing stress at each node of eachelement; the normal stresses, the shear-ing stress, the maximum principal stress,the minimum principal stress, and thedirection of maximum principal stressat the center of the same element; andsteel stresses for each element ofsteel, in that order.

The input for the program consistsof the following parameters:

(a) The controlling informationfor the geometry and inputand output of the results;

(b) description of materialproperties;

(c) numerical values for externalloads, external displacements,

and volume change strains.The input cards for the program

are discussed below in the order thatthey appear in the program:READ 242, NPBLM;

242 FORMAT (15).The value of NPBLM indicates the numberof problems that are to be solved.This will be discussed in more detaillater.

1000 READ 1, MX, NX, DX, DY, ANU,E, ES, AS, LFSN, LSN, KIND,NSTR

1 FORMAT (215, 3F5.2, 2F15.1,F5.3, 415)

where:MX = number of elements in each

row.NX = number of elements in each

column.DX = the length of an element.DY = the height of an element.

ANU = Poisson's ratio for concrete.E = modulus of elasticity ofconcrete.

ES = modulus of elasticity of steel.AS = the area of steel.

LFSN = the first steel node.LSN = the last steel node.KIND = 0 is the problem is plane

strain, = any non-zero numberup to five digits if theproblem is plane stress.

NSTR = number of points in an element


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at which the stresses are tobe evaluated. This is eitherI or 5 for evaluating thestresses at the center of theelement or at the center andthe four corners, respectively.

The value of NPBLM is equal to thenumber of cards that correspond to the1000 READ 1 statement.

READ 166, NANLYS, SIGCR, AINC166 FORMAT (15, 2FI0.5)

where:NANLYS = the number of times a problem

is to be repeated with theload or displacement incre-mented each time, or withshrinkage strains increased.

SIGCR = limiting tensile stress forconcrete.

AINC = the increment by which theloads or displacements are tobe increased, represented asa fraction of the originalloads or displacements.

READ 310, N310 FORMAT (15)The number N represents the total

number of nodes at which a particulardisplacement is to be defined such assupports. For example, for a simplysupported beam, the value of N would be2 for the two nodes that are restrainedin one or both directions, unless dis-placements are also applied at othernodes.

READ 303, JK, KZI, PLOAD (2*JK-1),KZ2, PLOAD(2*JK)

303 FORMAT (215, F10.5, 15, Fl0.5)where:

JK = the number of node at whichthere is to be a displacementconstraint.

KZI = 1 if there exists a constraintin the x-direction at node JK;0 if there is no constraintin the x-direction at node JK.

PLOAD = the magnitude of the displace-(2*JK-1) ment constraint in the x-

direction, zero for supports

KZ2 = 1 if there is a constraint inthe y-direction at node JK;0 if there is no constraintin the y-direction at node JK.

PLOAD = the magnitude of the displace-(2*JK) ment constraint in the y-direction, zero for supports.

READ 310, N310 FORMAT (15)

Here, the number N represents thetotal number of nodes at which loadsare applied.

READ 312, JK, KZI , PPLOD(2*JK-1),KZ2, PPLOD(2*JK)

312 FORMAT (215, F10.5, 15, F10.5)where:

JK = the number of the node atwhich a load is to be applied.

KZI = 1 if there is a load componentin the x-direction at node JK;0 if there is no load componentin the x-direction at node JK.

PPLOD = magnitude of the load in the(2*JK-1) x-direction at node JK.

KZ2 = I if there is a load componentin the y-direction at node JK;0 if there is no load in they-direction at node JK.

PPLOD = magnitude of the load in the(2*JK) y-direction at node JK.

READ 400, M400 FORMAT (15)

The number M is the total of allthe rows of nodes at which volume changestrains are to be applied.

READ 403, (SHRKG(I), I = 1, M)403 FORMAT (5Fl5.10)

where:SHRKG(l) = the value of volume change

strain at the rows of nodes I.I = the number of the row for

which the strain is beingread; the top row of nodesis number 1, the second rownumber 2, and so on.

This concludes the definition ofthe READ statements.

The following is an explanation of


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the PRINT statements ir the computerprogram except for those print state-ments that comprise the titles for theprinted output:

PRINT 69, MX, NX, DX, DY, ANU,E, ES, AS, LFSN, LSN, KIND,NSTR

69 FORMAT (215, 3F5.2, 2F15.1 ,F5.3, 415//)PRINT 660, NANLYS, SIGCR, AINC

660 FORMAT (119, F17.5, F16.5//)PRINT 661, JK, KZI, PLOAD(2*JK-1 ) KZ2, PLOAD(2*JK)

661 F0RMAT (16, 113, F17.5, 113,F16.5//)PRINT 662, JK, KZI, PPLOD(2*JK-1 ) KZ2, PPLOD(2*JK)

662 FORMAT (16, 16, F13.2, 16,F12.2//)PRINT 663, (SHRKG (I), I = 1,M)

663 FORMAT (5F15.10//)The parameters are already defined

in the explanation of the READ state-ments. These PRINT statements merelyprint out the input parameters forchecking purposes.

PRINT 314, K, EPOAD314 FORMAT (16, E15.6//)

where :K = number of node at which an

external load is applied.EPOAD = the load acting at node K.

This includes the incrementsto the loads if there are any.

PRINT 314, K, ELOAD(2*K-2+IJ)314 FORMAT (16, E15.6)

whe re :K = the number of node at which

there exists a displace-ment constraint.

ELOAD = the magnitude of the dis-(2*K-2+1J) placement constraint at

node K. Zero for supports.If a node number appears twice in

this PRINT statement, the first con-straint is in the x-direction and thesecond constraint is in the y-directionThis, for example, would happen for anode fixed in both directions for whichthe magnitudes would b

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Cracking of Concrete - Univ. Illinois, Bulletin