Cracking of Concrete

H ILL IN 0 S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

PRODUCTION NOTE

University of Illinois at Urbana-Champaign Library Large-scale Digitization Project, 2007.

ABSTRACT

THE CRACKING OF CONCRETE IN HIGH- WAY PAVEMENTS AND STRUCTURES IS UNDESIR- ABLE SINCE CRACKING OF THE CONCRETE IS ASSOCIATED WITH THE DETERIORATION OF BOTH THE CONCRETE AND REINFORCING STEEL. MANY STUDIES ON THE PHENOMENON OF CRACK- ING IN PLAIN AND REINFORCED CONCRETE HAVE BEEN CONDUCTED; HOWEVER, THESE INVESTIGATIONS HAVE CORRELATED THE CRACKING 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 WAS UNDERTAKEN TO PROVIDE A BETTER UNDER- STANDING OF THE INITIATION AND GROWTH OF CRACKS IN CONCRETE, WHICH IS ESSEN- TIAL IF CRACKING OF CONCRETE STRUCTURES IS TO BE CONTROLLED. THE EFFECT OF SEVERAL CONCRETE PARAMETERS ON THE FRAC- TURE TOUGHNESS (MATERIAL'S RESISTANCE TO PROPAGATION OF AN EXISTING FLAW) IS PRESENTED. A SYSTEMS-TYPE ANALYSIS IS PRESENTED TO DESCRIBE THE COMPLEX CRACK- ING MECHANISM IN CONCRETE STRUCTURES, AND MODELS ARE DEVELOPED FOR STUDYING CRACKING IN CONCRETE BEAMS AND RIGID PAVEMENTS. AN APPROXIMATE SOLUTION FOR THE PROBLEM OF SHRINKAGE STRESSES IN PLAIN AND REINFORCED CONCRETE MEMBERS WHICH ARE EXTERNALLY LOADED IS DEVELOPED.

ACKNOWLEDGMENTS

This study was conducted as a part of the research under the Illinois Cooperative Highway Research Program Project IHR-92, "The Control of Cracking of Concrete." The project has been undertaken by the Engineering Experiment Station of the University of Illinois in cooperation with the Illinois Division of Highways of the State of Illinois and the U.S. Department of Trans- portation, Federal Highway Administration, Bureau of Public Roads. On the part of the University, the work covered by this report was carried out under the general administrative supervision of D. C. Drucker, Dean of the College of Engineering, R. J. Martin, Director of the Engineering Experiment Station, T. J. Dolan, Head of the Department of Theoretical and Applied Mechanics, and Ellis Danner, Director of the Illinois Cooperative Highway 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 of R. H. Golterman, Chief Highway Engineer, and J. E. Burke, Engineer of Research and Development. Technical advice was provided by a Project Advisory Committee consisting of the following personnel: Representing the Illinois Division of Highways: J. E. Burke, Engineer of Research and Development R. L. Duncan, Field Engineer W. Griffin, Structural Design Engineer

Representing the University of Illinois: J. L. Lott, formerly Assistant Professor of Theoretical and Applied Mechanics G. M. Sinclair, Professor of Theoretical and Applied Mechanics C. E. Kesler, Professor of Theoretical and Applied Mechanics and of Civil Engineering and David Raecke, Research Associate in the Department of Theoretical and Applied Mechanics, served as Chairman and Secretary, respectively, of the Project Advisory Committee. CONTENTS

I. INTRODUCT ION ...... I

1.1 Genera l ......

1.2 Object ...... 1

1.3 Scope ...... 1

1.4 Notation ...... 2

II. EFFECT OF CONCRETE PARAMETERS ON FRACTURE TOUGHNESS ...... 4

2.1 Introduction ...... 4

2.2 Experimental Investigation ...... 5

2.3 Experimental Results ...... 6

2.4 -Discussion of Results...... 8

III. CRACK MECHANISM FOR CONCRETE STRUCTURES ...... 10

3.1 Introduction ...... 10

3.2 Fracture System...... 10

3.3 Fracture of Concrete Structures...... 11

IV. ANALYTICAL STUDY OF CRACK DEVELOPMENT ASSOCIATED WITH VOLUME CHANGE ...... 14

4.1 Introduction ...... 14

iv 4.2 Development of Stiffness Matrix for Finite Element Analysis...... 14

4.3 Application of the Method and Boundary Conditions...... 19

V. PRACTICAL APPLICATIONS...... 20

5.1 Effect of Concrete Parameters on Fracture Toughness ...... 20

5.2 Crack Mechanism for Concrete Structures. . . . . 21

5.3 Analytical Study of Crack Development Associated with Volume Change...... 2 1

VI. SUMMARY AND CONCLUSIONS ...... 23

6.1 Object and Scope ...... 23

6.2 Results of Investigation ...... 23

6.3 Conclusions ...... 25

VII. SUGGESTIONS FOR FUTURE RESEARCH ...... 27

VI I I. REFERENCES ...... 28

IX. APPENDIX I, USER'S GUIDE FOR COMPUTER PROGRAM IN FORTRAN IV...... 29

X. APPENDIX II, COMPUTER PROGRAM IN FORTRAN IV FOR DETERMINATION OF VOLUME CHANGE STRESSES IN PLAIN AND REINFORCED CONCRETE USING FINITE ELEMENT ANALYSIS ...... 33

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 -I Effect of Curing Time and Type of Coarse Aggregate on K : Concretes. Effect of Fine Aggregate on K': Mortars.c

Effect of Fine Aggregate on K : Concretes. c Effect of Fineness Modulus of Coarse Aggregate on K : Concretes. - I c Effect 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.

I. INTRODUCTION

1.1 GENERAL 1.3 SCOPE Undesirable cracking of concrete 1.3.1 Effect of Concrete Parameters on Fracture Toughness in highway pavements and structures is associated with the deterioration of The fracture toughnesses* of sever- both the concrete and the reinforcing al pastes, mortars, and concretes were steel. Corrective maintenance is costly determined by flexural tests of speci- and inconvenient so that ideal designs mens containing flaws of various depths should minimize the size of cracks in cast at the center of the tensile sur- hardened concrete. Such a control can face. Variables in the tests were: be improved through a basic understand- water-cement ratio, air content, degree ing of crack development in concrete. of hydration, sand-cement ratio, gravel- Many studies of cracking in plain cement ratio, and gradation and type of and reinforced concrete have been con- coarse aggregate. ducted. However, these investigations have correlated the cracking of concrete 1.3.2 Crack Mechanism for Concrete Structures with various parameters without consid- ering how cracking occurs. A systems-type analysis was used to describe the complex cracking mecha- 1.2 OBJECT nism that occurs in concrete structures. The object of this investigation The cracking mechanism in a reinforced is to determine the effect of concrete beam subjected to a pure moment, and parameters (mix design) on the cracking the cracking mechanism in a reinforced of concrete, to study the complex crack- concrete member with the steel loaded ing mechanism in concrete structures, in tension was examined by this approach. and to develop an analytical solution for the problem of volume-change stresses 1.3.3 Analytical Study of Crack Development Associated with for plain and reinforced concrete. The Volume Change result, a better understanding of the An approximate solution for the initiation and growth of cracks in con- problem of shrinkage stresses in plain crete, is essential to control cracking and reinforced concrete was developed concrete structures. using finite element analysis. The *Fracture toughness is the material's method can be used to calculate resistance to propagation of an existing stresses flaw. in members which are externally loaded. Cracking is incorporated into the analy- = depth of pavement sis, and crack width and spacing can be = stress intensity factor at the tip of a flaw calculated. = change in stress intensity factor due to a load cycle 1.4 NOTATION = effective stress intensity A = area of steel reinforcement factor in the matrix of a heterogeneous material a = crack length of an edge- that is assumed to be homo- cracked specimen or half geneous the crack length of a center-cracked specimen = symmetrical stiffness matrix for one element a 1 , a 2 = constants relating the volume change strains at any = symmetrical stiffness ma- point in the element to the trix for the entire model of the point y-coordinate = critical stress intensity a' = crack length from level of factor at the onset of reinforcement to the crack rapid, unstable crack tip propagation B = flexure specimen width = effective fracture toughness b = height of an element = average effective fracture = matrix relating stresses [C] toughness for a test series to strains = stress intensity factor C1, coefficients for different C 2 = for the concrete subjected crack lengths only to the resultant for- c = cement content of a partic- ces at the level of the ular mix, by weight steel

C , . S.,c8 = constants relating the dis- = stress intensity factor placements of a point in an for the concrete subjected element to the coordinates only to moments M of that point = stress intensity factor = a matrix relating strains for the concrete subjected to displacements only to the axial force P = length of an element = stress intensity factor resulting from a series of = effective depth of reinforced forces and/or moments concrete tension member = shear span = effective modulus of elasticity of the concrete = length of reinforced concrete tension member = a column matrix representing the forces at the nodal = unbonded length of steel points of one element reinforcement in concrete tension member = a column matrix representing forces at all nodes = applied bending moment = bond forces = moments applied to a reinforced concrete beam = a force at a point i in the elemen t = total number of nodes in the model = a column matrix representing the nodal forces in an ele- = applied load resulting from volume ment = polar coordinates change = one-half distance over rate = critical energy-release which concrete is examined at the onset of rapid, un- in reinforced concrete beam stable crack propagation subjected to constant moment = force transmitted across A = crack opening displacement cracked section in tension at level of reinforcement member for equilibrium con- = a column matrix representing ditions [E] the strains in an element = force which concrete trans- T E. = transpose of matrix of com- mits across the cracked I patible strains due to a section in reinforced con- unit displacement in the crete tension member direction of F. T = force in steel at cracked I s = normal strain in x-direction section in concrete tension x in an element member = normal strain in y-direction t = thickness of reinforced con- y in an element crete tension member = normal volume change strain [u] = column matrix representing xs in the x-direction the displacements at the four nodes of an element E = normal volume change strain ys in the y-direction [u] = a column matrix of displacements at all nodes [1 = nondimensionalized coordinate y/b u = x-displacement of a point x in the element = nondimensionalized coordinate x/d u = y-displacement of a point y in the element [o] = a column matrix representing the stresses in an element w = flexure specimen depth = a matrix representing volume w = water content of a particu- change stresses lar mix, by weight = stress normal to y-z plane = cartesian x x,y coordinate system in x-direction of axes with origin at lower left-hand node of an element = stress normal to x-z plane in y-direction "xy = shear strain in an element T = shear stress on plane perpen- = shear volume change strain xy Yxys dicular to x-axis in y- in an element direction II. EFFECT OF CONCRETE PARAMETERS ON FRACTURE TOUGHNESS

2.1 INTRODUCTION propagation yields the critical stress 2.1.1 Linear-Elastic Fracture Mechanics intensity K which is assumed to be a Linear-elastic fracture mechanics material property called the fracture is a study of the stress and displace- toughness, i.e., the material's resis- ment fields near the tip of a flaw in an tance to propagation of an existing ideal, homogeneous, elastic material at flaw. Fracture can thus be predicted the onset of rapid, unstable crack prop- for a structure since crack propagation agation, i.e., fracture. Its concepts will occur when the stress intensity are most applicable to brittle materials factor reaches its limiting condition in which the inelastic region near the K . c crack tip is small compared to flaw and As the ratio of plastic zone size specimen dimensions so that elastic to specimen dimensions increases, the stress field equations provide a good inelastic region becomes significant approximation ( :* and adjustments must be made to correct for effects of K e [sn 3S plastic strains adjacent a = - cos [1-sin sin 3 2 to the crack tip region.(2) An exact solution to correct for the zone of --8 , (1) S=--- cos [+sin - sin yielding is presently unknown; however, y C os 2 2 an approximate solution can be attained K . 9 6 36 by assuming a crack tip extension to r xy = --2-- sn 2- cos -2 cos --2 , the central portion of the inelastic region and solving the problem with where r and 8 are polar coordinates with elastic stress field equations for the origin at the crack tip. increased crack length. Equation (1) indicates that the stress and displacement fields can be 2.1.2 Applications of Fracture Mechanics to Concrete expressed in terms of a stress intensity Several applications of linear- factor K which is a function of loading elastic fracture mechanics have been made and crack geometry. The evaluation of to pastes, mortars, and concretes. K at the onset of rapid, unstable crack Concrete, a polyphase material, has a more complex fracture process than a *Superscript numbers in parentheses homogeneous, ideally brittle material. refer to entries in References, Chap- ter VII. Fracture of the concrete can occur by fracture of the cement paste, fracture the aggregate percentages remained con- of the aggregate, failure of the bond stant; the critical stress intensity between the cement paste and aggregate, factor was independent of fine aggre- or any combination of these mechanisms. gate percentage for three mortars with Kaplan was the first to apply the same water-cement ratio; the critical fracture mechanics to concrete when he stress intensity factor varied directly investigated one mortar and two con- with coarse aggregate content for concretes. An analytical and experimental cretes with the same water-cement ratio approach, both neglecting slow crack and fine aggregate content; and the propagation prior to fracture, were critical stress intensity factor for used to evaluate the critical strain concrete was found to be approximately energy release rate G . The results 20 per cent greater than that for a obtained by Kaplan indicated that G mortar with the same water-cement ratio c was influenced by the mix proportions, and fine aggregate content. specimen dimensions, and loading. Lott and Kesler 6 ) conducted a 2.2 EXPERIMENTAL INVESTIGATION study to develop a hypothesis for 2.2.1 General propagation of cracks in plain concrete The fracture toughnesses of sever- and to compare the hypothesis to results al pastes, mortars, and concretes were of an experimental investigation of determined by flexural tests of speci- crack propagation in several mortars mens containing flaws of various depths and concretes. It was suggested that cast at the center of the tensile sur- the critical stress intensity factor face. 7 ) Parameters investigated in- K for plain concrete was derived from cluded: water-cement ratio, air con- the stress intensity factor of the tent, degree of hydration, sand-cement paste and a crack arresting mechanism ratio, gravel-cement ratio, and grada- developed by the heterogeneity of the tion and type of coarse aggregate. concrete. Since the critical stress intensity factor for the paste was a 2.2.2 Materials material constant, variations in the Type I portland cement was used critical stress intensity factor of the in all mixes. The fine aggregate used concrete were reflected through the was a Wabash River sand from near Coving- arresting function. The effects of ton, Indiana. Two gravels were used in several concrete parameters (water- the concrete series: a Wabash River cement ratio, sand-cement ratio, and gravel from near Covington, Indiana, gravel-cement ratio) on the fracture and a crushed limestone which was ob- toughness of the concrete were evalu- tained locally. ated . The air-entraining agent used was For the range of variables investi- a proprietary compound consisting of an gated, it was found that: the critical aqueous solution of salts of sulfonated stress intensity factor was independent hydrocarbons containing a catalyst. of water-cement ratio for the three mortars and for various concretes where 2.2.3 Specimen Description Nominal dimensions of the paste demolded and stored in a moisture room and mortar flexural specimens were 2 by for curing at 100 per cent relative 2 by 14 in., and nominal dimensions of humidity. The specimens were removed the concrete specimens were 4 by 4 by from the moisture room at various ages 12 in. A flaw was cast at the center and stored in water until they were of the tensile surface of the specimens. tested. The flaw was formed with a 0.003-in.- thick piece of teflon-coated fiberglass 2.2.5 Testing Procedure cloth. Nominal flaw depths were: 0.25 A hydraulic testing machine was in., 0.5 in., and 1.0 in. for the paste used for the flexural tests. Figure I and mortar specimens, and 0.5 in., 1.0 shows the test setup. The lower loading in., and 1.5 in. for the concrete speci- plate acted as a dynamometer to measure mens. Actual dimensions of the flexure load applied to the specimen. A defor- specimens were measured after testing meter, supported by needlepoint screws, since variations in nominal dimensions was used to measure elongation of the occurred in fabrication. tensile surface. Prior to each series of flexural 2.2.4 Fabrication and Curing tests, the deformeter and dynamometer A two-cubic-foot horizontal pan were calibrated. After calibration, mixer was used. The dry ingredients the deformeter was placed between the were blended one minute before water was needlepoint screws of the first specimen added to the mix. After addition of and precompressed to a pseudozero point. the water, the mixing was continued for The recorder was zeroed and load was three minutes. When air-entraining applied at a rate of approximately agents were used, they were added to the 250 lb per minute for the paste and mix water. mortar specimens and 1500 lb per minute The flexure specimens were cast for the concrete specimens until failure with the plane of the flaw in a vertical occurred. position. The molds were filled in one lift and compacted on a vibrating table. 2.3 EXPERIMENTAL RESULTS A total of twenty flexural specimens 2.3.1 Load-Deformation Curves were cast in steel forms for each series During each test, a recorder plot- of the paste and mortar series, and a ted a continuous record of deformation total of eight flexural specimens were response against load response until cast in plywood forms for each series failure of the flexural specimen. Typi- of the concrete series. The exposed cal load-deformation curves for a con- surface of all specimens was troweled crete series are presented in Figure 2. smooth immediately after casting. Two to four hours after casting the 2.3.2 Stress Intensity Factor specimens were covered with wet burlap Brown and Srawley used boundary and plastic sheeting to prevent the loss value collocation calibrations to devel- of moisture. Approximately twenty-four op the following expression for the hours after casting, the specimens were stress intensity factor K for a single- edge-cracked specimen subjected to pure K' decreased 18.3 per cent when the c bending: water-cement ratio was increased from 0.45 to 0.60 as shown in Figure 4.

K = Y 6 Ma (2) However, in the concrete series K was BW 2 c independent of the water-cement ratio where for the range of water-cement ratios investigated as shown in Figure 4. Y = 1.99 - 2.47 (a/W) + 12.97 (a/W)2

3 24.80 (a/W) 4 - 23.17 (a/W) + , Air Content P2. In the paste series there was a M - , 2 23.4 per cent decrease in K when the c and a is the flaw depth, W is the speci- air content was increased from 2.0 to men depth, P is the applied load, Z is 8.0 per cent as shown in Figure 5. In the shear span, and B is the specimen the mortar series K' decreased by 19.2 c width. per cent when the air content was in- In the evaluation of K using creased from 3.0 to 9.0 per cent as c Equation (2), it was assumed that the shown in Figure 5. K c decreased by material was homogeneous and the flaw 8.2 per cent when the air content in depth at failure was equal to the cast the concrete series was increased from flaw depth. Since concrete is hetero- 2.0 per cent to 12.0 per cent as shown geneous and the stress intensity factor in Figure 5. is a function of the instantaneous crack depth, the analysis yields an effective Curing Time stress intensity factor K rather than For 28 days moist cure K was I C the actual stress intensity factor. 6.5 per cent greater than K for six The effective fracture roughness days moist cure as shown in Figure 6 K , a measure of concrete's resistance for the paste series. For the mortar to propagation of an existing crack, is series there was a 47.5 per cent increase the determination of K at M from in K when the length of moist cure was max I c Equation (2). Figure 3 presents K and increased from three days to 92 days as P as a function of a/W for a concrete shown in Figure 6. When the length of max series. The horizontal line in Figure 3 moist cure was increased from three days represents the mean value of effective to 28 days for concrete using a river fracture toughness for the particular gravel coarse aggregate, K increased C test series, K 54.2 per cent. However, the increase in Kc was only 7.7 per cent when the 2.3.3 Effect of Concrete Parameters length of moist cure was increased from on Effective Fracture Toughness 28 days to 90 days as shown in Figure 7. Water-Cement Ratio When a crushed limestone coarse aggre- In the paste series there was a gate was used, K increased 23.0 per decrease in K of 43.3 per cent when the cent with an increase in moist cure water-cement ratio was increased from from three days to 28 days; however, 0.27 to 0.36, while in the mortar series there was no apparent change in K when c the length of moist cure was increased series cast with a river gravel coarse from 28 days to 90 days as shown in aggregate was 5.0 per cent higher than Figure 7. The percentage increases in the concrete series cast with a crushed K when the moist curing period was limestone coarse aggregate as shown in c increased from six days to 28 days for Figure 7. the paste series, the mortar series, the concrete series cast with a crushed 2.4 DISCUSSION OF RESULTS limestone coarse aggregate, and the 2.4.1 Behavior of Fracture Toughness Specimens concrete series cast with a river gravel coarse aggregate were 6.5 per cent, The load-deformation curves (Fig- 12.4 per cent, 21.3 per cent and 24.2 ure 2) illustrate the stages of behavior per cent, respectively. of the concrete near the tip of the flaw: linear stage where the cement Fine Aggregate Content paste matrix has no crack extension; In the mortar series there was a slow cracking stage in which stable 16.2 per cent increase in K when the cracking occurs to result in a decreas- fine aggregate content was increased ing slope of the load-deformation curve; from 55.0 per cent to 70.0 per cent as and fracture stage where unstable crack shown in Figure 8. However, for the propagation occurs and results in the concrete series there was a 2.3 per cent deformation increasing without an decrease in K when the fine aggregate increase in applied load. c content was increased from 35.0 per cent to 50.0 per cent as shown in Figure 9. 2.4.2 Effect of Concrete Parameters on R c Water-Cement Ratio Gravel Content, Gradation, and Type There was a decrease in the effec- For the concretes cast with crushed tive fracture toughness of the paste limestone coarse aggregate, K increased and mortar series with increasing water- 13.3 per cent when the fineness modulus cement ratio because the fracture was increased from 6.3 to 7.1 as shown toughness was dependent on the strength in Figure 10. When the percentage of of the cement paste matrix which was a coarse aggregate was increased from function of gel-space ratio. With 0.0 per cent to 50.0 per cent there was increasing water contents the gel-space resulting in a reduction a 37.0 per cent increase in K c as shown ratio decreased in Figure 11. of strength and effective fracture For the concrete series cast with toughness. The fine aggregate of the K was 28.9 per mortar series reduced the effect of the a crushed limestone c cent, 17.7 per cent, and 1.7 per cent water-cement ratio because of the crack higher than K for the concrete series arresting phenomenon of the fine aggre- cast with a river gravel coarse aggregate particles. fracture toughness gate at ages of three days, six days The effective on both the frac- and 28 days, respectively. However, at of concrete depended ture toughness of the paste and the an age of 90 days, K c for the concrete presence of coarse aggregate. The range because the coarse aggregate particles of water-cement ratios apparently did were much better crack arresters and not affect the effective fracture tough- thus concealed the effect of the fine ness of the concrete because the effect aggregate. of the aggregate as a crack arresting function was more significant than the Gravel Content, Gradation and Type effect of the water-cement ratio on the The effective fracture paste matrix strength. toughness increased with an increase in maximum size particles because the Air Content larger aggre- more effective as Increasing the air content of the gate particles are a maximum matrix resulted in a decrease in effec- crack arresters. However, size can be reached in conjunction with tive fracture toughness because of a that will produce a reduced matrix strength. With increasing a poor gradation lower effective fracture toughness aggregate contents, the decrease was because of the effects of segregation not as significant because of the crack shown in Figure 10 for a fineness arresting phenomenon of the aggregate. as modulus of 7.45. gravel content in- Curing Time An increased fracture toughness The increase in effective fracture creased the effective gravel content en- toughness with age was the result of because the larger of crack arrest- continuing hydration of the cement parti- larged the concentration cles to produce a higher strength. ers in the matrix. The effective fracture toughness aggre- Fine Aggregate Content for the crushed limestone coarse than the effective The effective fracture toughness gate was greater for the river gravel for the mortar increased with an increas- fracture toughness 28 days, indicating ing amount of fine aggregate because of coarse aggregate until an increased concentration of crack that the crushed limestone apparently How- arresting particles in the matrix. The developed greater bond strength. of 28 days the bond effective fracture toughness of the con- ever, after an age for the two types of coarse crete was not significantly affected by strengths appeared to be equivalent. an increasing fine aggregate content aggregate III. CRACK MECHANISM FOR CONCRETE STRUCTURES

3.1 INTRODUCTION processes that responds to outside Control of cracking in concrete stimuli and interacts with each other structures subjected to varying load was developed. Quantitative analysis and environment requires a basic under- of the system requires that the stimuli standing of the crack mechanism to corre- and responses of the various processes late laboratory data with service con- be expressed in compatible terms ditions. The fracture process in con- (stress), which can be either theoreti- crete structures is similar to the cal or empirical. fracture that occurs in the fracture A similar system is useful for toughness specimens of Chapter II, in analysis of cracking of concrete that both crack growths are associated structures and is shown in Figure 12. with a fracture phenomenon that occurs The structure ) consists of the struc- in the highly stressed region surrounding tural elements such as concrete, rein- the crack tip. In the fracture tough- forcement, and supports, including ness specimen, the crack propagates when pavement base materials. The relation the stress intensity factor of Equation between load, environment, and the (2) reaches the effective fracture tough- stresses in the various structural ness K . There is no simple stress elements is required. Stress-strain c intensity factor for a crack in a con- modifiers 1 1 ) include flaws, cracks, crete structure since the structural inclusions, and other stress concentra- components interact with each other and tors. The general level of stress is with the stress field surrounding the intensified locally near these modifiers. crack tip. A systems-type analysis of The linear-elastic fracture mechanics the fracture process is used to describe techniques are used to evaluate the the complex cracking mechanism for elastic stress field surrounding sharp concrete structures. flaws. Inelastic deformations (I I I) may occur in regions that are highly 3.2 FRACTURE SYSTEM stressed relative to strength. These Hahn and Rosenfield(0O) have pre- inelastic deformations modify the rela- sented a systems-type analysis of the tive stiffness of the structural elements fracture problem and applied it to the and cause a redistribution of stress in fracture of metal plates and incorporated the structure. Cracking mechanisms (IV) the effect of yielding in the region of are initiated when critical conditions the crack tip. A fracture system of develop in the region of a crack tip, and crack growth takes place. This separately for each action on the free crack growth also modifies the relative body. The resultant stress field is stiffness of the structural elements obtained by summing the individual and results in a stress redistribution. stress intensity factors K., and the The general fracture process of a condition of crack instability occurs concrete structure is as follows: when the resultant K equals the effec- The input (A) of load and environ- tive fracture toughness of the concrete ment to the structure (1) causes stresses K . c to develop within the structural ele- i=m ments. The general stress levels are 7 K. = K - K / I r transmitted (B) to any modifiers (I I) i= in the system. The stresses are increased and transmitted (D) to the 3.3 FRACTURE OF CONCRETE STRUCTURES inelastic deformations (Ill) and trans- The analysis of concrete cracking mitted (F) to the fracture mechanisms in structures is based on a resultant (IV). At a critical stress level the stress intensity factor for a crack in inelastic deformations occur and are the concrete, which is the stress modi- transmitted back (E) to the modifiers, fier associated with the cracking and at some critical condition existing mechanism that interacts with the loads cracks propagate, and the effect of and the other elements of the structure. increased crack lengths are fed back (G) A concrete body containing the crack to the modifiers. These effects on the is isolated, and the stress intensity modifiers are reflected back (C) to the factors for the various actions are structure as changes in relative stiff- determined using available expressions ness and result in stress redistribution. and summed to obtain the resultant

Inelastic deformations tend to stress intensity factor K r . Equilibrium increase the relative stiffness of con- crack conditions, which relate crack crete and promote cracking, while crack geometry and load, correspond to the growth tends to reduce the relative limiting condition of Equation (3), stiffness of the concrete and arrests K = K. r c (3-1) crack growth. The systems-type analysis of 3.3.1 Crack in Constant Moment Region of Reinforced Concrete Beam cracking concrete structures is based on a free body diagram of the concrete The cracking mechanism in a rein- portion of the structure, which is the forced concrete beam subjected to a structural element that contains the constant moment M is analyzed by consid- crack that will propagate. The effects ering the concrete within a distance s of load, environment, reinforcement, of the crack as shown in Figure 13. and other structural elements on concrete The concrete is subjected to moments M c fracture are obtained by superposition. and axial compressive forces P which The stress intensity factor describing are the actions of the adjacent concrete, the stress field surrounding the tip and of resultant bond forces Fb at the of the crack in the concrete is evaluated level of the reinforcement, which are the net forces transferred to the con- quantitative analysis of crack equili- crete over the interval s. The resul- brium is based on the concrete element tant stress intensity factor K which of Figure 15. The only forces acting describes the stress field surrounding on the concrete are the bond forces Fb b the crack tip in the beam is that develop as the reinforcement elongates. The unbonding at K = KMc + Kp + K (4) the free ends and at the cracked section affect the where KMc is the stress intensity factor magnitude of the bond forces. The for the concrete subjected only to the bond forces cause an opening A of the moments Mc, Kp is the stress intensity crack at the level of the reinforcement. factor for the concrete subjected only The stress intensity factor K may to the axial forces P, and KF is the be expressed in terms of the bond stress intensity factor for the concrete forces Fb or the opening A. ( 1 2) subjected only to the resultant forces CIFb at the level of the steel. Expressions C2 Ec K = - 1 = A for these stress intensity factors are t d' d 2 e e available. (1 ' 1 2) However, the magnitudes of Mc, P, and Fb are functions of where t is the thickness, d is the e the forces in the reinforcement. The effective depth, E is the modulus of c steel forces are dependent upon the elasticity of the concrete, and C 1 and

inelastic deformations associated with C 2 are coefficients that are evaluated unbonding and cannot be defined with for various crack lengths a , effective (12) sufficient accuracy for a quantitative depths d , and specimen lengths .( e s analysis of cracking. The maximum equilibrium crack A qualitative analysis of cracking length corresponds to a stress intensity indicates that KMc is the parameter that factor K that is equal to the effective

tends to cause crack extension; Kp is fracture toughness of the concrete K C', negative for compressive forces and K = K tends to arrest crack growth; K F is c negative when the bond forces Fb act The force which the concrete transmits toward the crack and tends to arrest across the cracked section, T , is cracking and is positive and tends to equal to the bond force Fb' cause cracking when the load forces act K t d away from the cracks. T = F = c e c b

3.3.2 Crack in Reinforced Concrete and is usually small relative to the Tension Member force in the steel T . The wedge open- s A reinforced concrete member with ing L which corresponds to an equili- the steel loaded in tension, Figure 14, brium crack is has been suggested as a useful model of 1 the cracking mechanism in beams, 3) K d 2 c e and crack development under increasing 4 C 2 E c load has been investigated.(1 ) A and is assumed to be equal to the elon- Crack data from Reference (14) is gation of the steel reinforcement over shown in Figure 16 for increasing loads. an equivalent unbonded length £ , and The first cracks initiated in two different specimens T P at the 20 kip load S EU (9) level. They corresponded to equivalent S S unbonded lengths of 0.5 and 1.2 in. where T is the force in the steel at At s the higher load levels of 25 and the cracked section, A is the steel 35 kips, the crack lengths corresponded area, and E is the modulus of elastic- to unbonded lengths of 1.2 to 1.5 in. ity for steel. The total force T trans- At crack initiation there was a large mitted across the cracked section for range in the unbonded length. As the equilibrium conditions is load increased, the unbonded lengths increased and the range was reduced. K td K A E d This is an example T = T + T = c e + c s s e (10) of the interaction c s Ci C 2 u E of an inelastic deformation, the unbonding, with the concrete cracking T is unique for a given equilibrium mechanism. crack length, and T s varies inversely with the unbonded length A . The total 3.3.3 Crack in Rigid Pavement load T may be calculated for a given Cracking in rigid pavements may crack length a by substituting various be analyzed by using the cracked beam unbonded lengths £u into Equation (10) . on an elastic foundation of Figure 17. This has been done for the specimen The stress intensity factor should vary geometry and material properties of with the inverse of crack length crack specimens of Reference (14), and the relationships between T and the K = f ( ) , (ll)

ratio of crack length to effective a /d since an increase in the crack length are given in Figure 16 for different a transfers more load to the elastic unbonded lengths. The effective frac- foundation in the region of the crack ture toughness has been assumed to be and reduces the stresses on the cracked approximately 0.6 ksiv'-n . section. The limiting condition is a The relationships of Figure 16 in- crack through the pavement depth (a = h), dicate the following: and the load is still transferred to (a) The equilibrium crack lengths the foundation. This is a stable crack a increase with total force T if the growth condition. Crack growth arrests unbonded length 2u is constant; additional cracking until the applied (b) The total force T transmitted load is increased. Crack growth may across a given cracked section decreases also be caused by repeated loadings, as the unbonded length increases; and this model should find applications (c) An increased unbonded length in the fatigue of rigid concrete pave- is associated with an increased crack ments. length corresponding to a virtual load increase. IV. ANALYTICAL STUDY OF CRACK DEVELOPMENT ASSOCIATED WITH VOLUME CHANGE

4.1 INTRODUCTION 4.2 DEVELOPMENT OF STIFFNESS MATRIX FOR FINITE ELEMENT ANALYSIS The volume of concrete changes with 4.2.1 Assumptions age through shrinkage or swelling associated with moisture movement. Nonuni- The following assumptions are made form volume change of concrete takes in developing the finite element model: place because of nonuniform moisture (a) Concrete and steel have linear exchange. Changes occur in the shape stress-strain diagrams; of concrete members, and stresses are (b) Loads and deformations are induced. The nonuniform shrinkage of applied to nodal points; concrete has not been studied to the (c) Shrinkage strains are applied same extent as uniform shrinkage, and to elements; more information is available in the (d) A perfect bond exists between literature on uniform shrinkage of steel and concrete; concrete than on nonuniform or relative (e) Concrete and steel are homo- shrinkage of concrete. geneous and isotropic materials and each Theoretical analysis of shrinkage has an identical stress-strain relation stresses in concrete involves a tedious in tension and compression (except con- solution of partial differential equa- crete is assumed to fail at a limiting tions of diffusion and compatibility. stress in tension but not in compression); If the analysis of reinforced concrete (f) The steel element has no is desired, these differential equations physical dimensions and is assumed to become more complex. be present on a horizontal line of nodes In recent years the solutions to only; the problems that have been extremely (g) Loading is one directional. difficult to solve by means of analytical approaches have been obtained by 4.2.2 Concrete Element numerical computations through the use Elements of various shapes may be of digital computers. In particular, used in the finite element analysis. the analysis of shrinkage stresses in However, the shape of the element should plain and reinforced concrete can be be selected so it fits the needs of the performed by use of the finite element analysis. In this analysis a rectan- method. gular element was selected since it fits the structural shapes (beams and slabs) on the position of the horizontal rein- in which shrinkage is to be studied. forcement other than being restricted A rectangular element also allows appli- to a single line. cation of linearly varied shrinkage strains to any particular element. Fig- 4.2.4 Analytical Model ure 18 depicts a typical rectangular The complete reinforced concrete element. The four corners of the rec- model used in this study is shown in tangle are called the nodes. The forces Figure 19. The model is assumed to be and displacements that are to be applied of unit thickness, although this is not to the structural element being analyzed a necessary requirement for the analysis. by the finite element method must be Boundary conditions specified at applied through these nodal points. the nodal points on the model can be The stresses and strains in any varied to fit a specific problem, i.e., one element are not constants, but a fixed condition at the left end of depend on the coordinates of the point the beam is realized when the displace- at which they are evaluated. The coor- ments in the x- and y-directions are dinates of a point in any one element set equal to zero for all the nodes at are measured from a set of axes with the left-most side of the model. Loads the origin at the lower left node of are applied at the nodes, and distri- the element as shown in Figure 18. buted loads are represented by a series

A set of stresses, x' , T xy, is of concentrated loads acting at the calculated for each node and also for nodes. If the model is loaded by in- the center of the element. Principal ducing deformations on it, the deforma- stresses are calculated at the center tions must also be applied at the nodes. of the element. In order to detect the Only strains are applied to the ele- occurrence of cracking in an element, ments. the maximum principal stress is compared with the limiting stress at which con- 4.2.5 Derivation of Stiffness Matrix for One Element crete is assumed to crack. In this study the finite element 4.2.3 Steel Element problem is restricted to two dimensions The physical size of steel is resulting in either a plane stress or usually small compared to concrete. In plane strain condition. The stiffness particular, when one considers shrinkage matrix is derived for the general case reinforcement, this difference in size partly from the work of Przemieniecki(15) becomes very noticeable. In this study, and will be valid for both plane stress the steel element is assumed to have no and plane strain. physical dimension but only mechanical Consider the element in Figure 18. properties. The steel is assumed to be To simplify the analysis, the nondimen-

present along a line of horizontal nodal sionalized coordinates e = x/d and rn = points as shown in Figure 19. No steel y/b are used.(15) The displacements at is assumed to be present inside the con- any point in an element are functions crete elements. There is no restriction of the coordinates of the point and will be assumed to be of the following to displacements and [u] is a column nature: matrix representing the displacements at the four u = C nodes of an element. It is 1 s + C2Cn + C 3 n + C 4 x (12) important to note that the strain in u = Cs5 + C 6 Vn + C7r1 + C8 each element is not a constant, but is where: dependent on the coordinates of the ux, u are displacements in the x- point at which it is to be evaluated.

and y-directions and C 1 , C 2 ,..., The stress-strain relationship for

C 8 are constants which can be eval- the element can be represented in matrix uated from the following boundary notation as conditions: [a] = [C] [e] (14) at (0, 0) u = ui, u = u2 x y where [C] is a matrix relating stresses (0, b) u = u3, u = U 4 x y to strains and [e] is a column matrix (d, b) u = us, u = u6 x y representing the strains in an element. (d, 0) u = uy, u = uO x y Substitution of Equation (13a) into The displacements will then be repre- Equation (14) yields sented by [c] = [C] [D] [u] , (15) u = (l-c)(l-n)u + (l-O)nu3 + Tnus 1 which relates the stresses to the nodal + C(1-n)u7 , (12a) displacements for one element.

u = (I-0)(l-nl)u2 + (I-O)nu4 + ýTu6 A typical force F. at a point i + C(l-n)u8 . in the element can be calculated by the unit displacement theorem ( 1 The total strains can be determined by 5) differentiating the displacement equa- F. = f T a dV (16) tions

au au x d x where: Sx = x = d ' T C. = transpose of matrix of compat- Du I au ible strains due to a unit E y = y bb= •rn- ' (13) displacement in the direction of F.,

au Du x I u uIu a = stress matrix resulting from Yxy ay ax b 3n d ' all forces acting on the element, dV = element of volume in the where E and e are the normal strains x y element, in the x- and y-directions, respective- = integration over total volume ly, and xy is the shear strain. Sof the element. The strain-displacement relation- v ship for the rectangular element becomes The forces at the nodal points of in matrix notation the elements can be found by substituting the matrices into Equation (16) and then [E] = [D] [u] (13a) integrating the product over the total where [D] is a matrix relating strains volume of the element. to displacements and [u] is a column

FIGURE 1. TEST SETUP 6.6

6.0

5.4

4.8

4.2

3.6 3.0

2.4

1.8 - a/w=0.125

1.2

0.6

0 V 0 0.001 0.002 0.003 0 0.001 0002 0.003 Deformation, A., in. Deformation, AA, in.

3.'

2. 2.'

1.1

O.c

O.(

0.: I 0 0.001 0.002 0.003 0 0.001 0.002 0003 Deformation, A&, in. Deformation, tA, in.

FIGURE 2. TYPICAL LOAD-DEFORMATION CURVES: CONCRETE 0K

0 E nCL

'Uo, E 0 E

0 0.1 0.2 0.3 0.4 o/w

FIGURE 3. MAXIMUM LOAD AND EFFECTIVE

FRACTURE TOUGHNESS VS A/W.

0.8 o Concrete - a Pastes ----- * Mortors - 0.7

0.6

5-.--.

-S.-- -S. 04 -5

,' S -S.--.- 5- 0.3

0.2

. I

0 0.25 0.30 035 0.40 045 050 0.55 0.60 065 0,70 w/c

FIGURE 4. EFFECT OF W/C RATIO ON R c 0 2.0 4.0 6.0 80 10.0 12.0 14.0 Air Content, Percent

FIGURE 5. EFFECT OF AIR CONTENT ON R c

Log Days

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

Log Days

FIGURE 7. EFFECT OF CURING TIME AND TYPE

OF COARSE AGGREGATE ON K c: CONCRETES

0.8

0.7

0.6

0.3

0.2-

0.1

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

FIGURE 8. EFFECT OF FINE AGGREGATE ON K c: MORTARS 0.8

0.7

0.6 -

02------

d . 71 50 69 ., ,. 40 67 30 3 65 63 Fine Aggregate by Weight, per cent Fineness Modulus

FIGURE 9. EFFECT OF FINE AGGREGATE FIGURE 10. EFFECT OF FINENESS MODULUS

ON R c: CONCRETES OF COARSE AGGREGATE ON R c: CONCRETES

08 -

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

0.6

0.4 ------

0,3------tOAD I ENVIRONMENT 03

0.1

0o o 10 20 30 40 50 Coarse Aggregate by Weight, per cert

FIGURE 11. EFFECT OF COARSE AGGREGATE FIGURE 12. SCHEMATIC OF

ON K c: CONCRETES FRACTURE SYSTEM Ic (

FIGURE 13. CRACKED CONCRETE ELEMENT FROM REINFORCED CONCRETE BODY

FIGURE 14. REINFORCED CONCRETE TENSION MEMBER

FIGURE 15. CRACKED CONCRETE ELEMENT FROM TENSION MEMBER 0 01 02 03 04 05 0.6 0.7 0e 09 1.0 FIGURE16.LOAD,/dT,VSFOR

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

FIGURE 17. CRACKED RIGID PAVEMENT Id 4 d

FIGURE 18. A TYPICAL ELEMENT

I-JL-^ -Steel

m FIGURE 19. REINFORCED CONCRETE MODEL

r•- a -'-i

Yxys = shearing shrinkage strain, a, and a2 = constants, [F] = ( ; [B]T [C] [B] dV [u] n = y/b. v (17) Equation (18) can be written in matrix notation as: where [F] is a column matrix representing the forces at the four nodal points al + a2n of the element, [D] is the transpose xs of [D], and the remaining expressions al + a2nT (18a) are as defined above. Yxys 0 The final equation can be expressed in matrix notation as follows: The reason for assuming e and e xs ys being equal at any particular point in [F] = [K) [u] (17a) the element is that these shrinkage where: strains are very similar in nature to T [K] = f [D] [C] [D] dV. thermal strains. Other assumptions v concerning the distributions of shrink- [K] is called the stiffness matrix age stresses can be incorporated into for one element and it relates the nodal the analysis. point forces to the nodal point displace- The forces that are induced at ments. The stiffness matrix of the en- each node as a result of the shrinkage tire system can be obtained by directly strains must be found in order to incor- adding the contribution of each individ- porate the effect of shrinkage strains ual element stiffness in the proper in the model. Consider Figure 18 and location. assume that the element is acted upon

4.2.6 Incorporation of Shrinkage by a state of shrinkage strain of the Strains in the Model type described above. The forces pro- The free shrinkage strain at any duced by this state of strain can be point in an element is a function of found by using the fact that the work the relative humidity at that point. done by the external forces must equal For purposes of consistency of the dis- the change of the internal energy. placements u , u , the shrinkage strains for a specimen drying from one side only [F ]T[u] = f [a ]T[e] dV are assumed to be defined by the follow- (19) ing formulas: where: n xs = al + a2 , [u] = matrix of unit displacements at the nodes of the element; e = ai + a21 , (18) ys [F i = transpose of the matrix of forces that are produced at y = 0 , xys the nodes as a result of where : shrinkage strains; T E = normal shrinkage strain in [ao] = transpose of the matrix of xs x-direction stresses produced by shrinkage strains; e = normal shrinkage strain in ys y-direction [e] = matrix of strains produced [u] = a 2 n x 1 column matrix of by unit displacements at the displacements at all nodes; nodes; [E] = [D] since the magnitude of the displacement [F ] = a 2n x 1 column matrix of is unity; -- forces induced at nodal points by shrinkage strains; dV = element of volume in the element; n = total number of nodes. Eouation (23) represents a system of = integration over total volume 2n of the element. simultaneous equations which can be v solved for the nodal displacements. From the stress-strain relationship These equations are derived from the force equilibrium equations [os] = [C][ei] in the x- and y-directions, i.e., the sum of all and the forces in the x- and y-directions ]T[C]T , [ T = [ at any node must equal zero unless a boundary condition however, since [C] is a symmetric matrix is defined at the [C] T = [C]. Therefore, node. When a boundary condition is defined at a node the sum of the forces [0 T = [ i] T C] (20) at the node will equal the external The forces resulting from shrinkage load applied at that node. strains can be found by substituting Equation (20) into Equation (19), thus 4.2.7 Development of Cracks Stresses are [Fs]T = f [ ]T[c][D] dV. calculated at five v (21) points for every element -- the four corners and the center of the element. The final equation that relates nodal The principal stresses and the direction forces to nodal displacements and shear- of the maximum principal stress are ing strains is calculated at the center of every ele- [F] = [K][u] + [Fs]. (22) ment. The maximum principal stress at the As a result of shrinkage strains, center is compared to a limiting nodal forces are produced which may be stress for cracking and if it exceeds the limiting stress, obtained for the entire model by adding the element is assumed to have the contributions of individual elements cracked and thus does not in the proper locations. The equation carry any tensile stresses. When cracking does relating nodal forces to nodal displace- occur, the element can be completely ignored since ments and shrinkage strains for the en- the loading is assumed tire model is to be one directional. If the stresses in two or more horizontally [F] = [K] [u] + [F ] (23) adjacent elements exceed the limiting where: stress at the same time, the element [F] = a 2n x I column matrix of with the largest stress is assumed to forces at all nodes; be cracked. Cracks are found through [K] = a 2n x 2n symmetrical stiff- an iteration process and every time a ness matrix for the entire body; new crack appears the analysis is repeated in order to find other cracks that might have appeared as a result of assumed that all of the elements in a the new crack. Thus the crack pattern row are under the influence of the same in the model is developed and the direc- shrinkage strain. The volume change tion of each crack is also calculated. strain is assumed to be positive if it It should be noted, however, that since produces expansion and negative if it a cracked element is assumed to carry produces contraction. no stress, no two horizontally adjacent The steel stresses are limited by elements can be cracked. This requires the assumption of perfect bond between that the length of each element not the steel and concrete. exceed one-half of the expected crack The method is not limited to con- spacing. crete but can be used to evaluate the stresses for any material. 4.3 APPLICATION OF THE METHOD AND BOUNDARY CONDITIONS 4.3.2 Boundary Conditions 4.3.1 Application of Method Boundary conditions are limited The method developed can be applied only in the sense that the conditions to any member with a shape that can be are applied at the nodes. To represent approximated by rectangular elements. a roller, the vertical displacement at The assumptions that were made in devel- the node on the roller is set equal to oping the method must also be reasonably zero. To represent a pin connection, valid, i.e., since it is assumed that both the horizontal and vertical dis- the stress-strain diagram for concrete placements at the node are set equal to is linear, the maximum compressive stress zero. To represent a fixed end, the in the concrete must remain below a horizontal and vertical displacements reasonable limit. of all the nodes at that end are set The loads are applied to the nodes equal to zero. Other boundary conditions and the directions of the loads are may be applied similarly. Boundary governed by the set of axes assumed for conditions that partially limit the the model, i.e., loads acting in the movement of a node, such as a spring, positive x- and y-directions are assumed can be applied if modifications are positive. If there are any applied dis- made in the digital computer program. placements, they follow the same sign convention. Volume change strain is 4.3.3 Computer Program for Deterimination applied to the elements and two values of Volume Change Stresses in Plain and Reinforced Concrete of volume change strain are specified, Using Finite Element Analysis one at the top side and one at the bot- The computer program VCSC (Volume tom side of the element. It is assumed Change Stresses in Concrete), prepared that there is a linear strain variation in FORTRAN IV, and a User's Manual are between the two values of volume change contained in the Appendix. strain for the element. It is further V. PRACTICAL APPLICATIONS

5.1 EFFECT OF CONCRETE PARAMETERS ON The effective fracture toughness FRACTURE TOUGHNESS of concrete was found to be directly The effective fracture toughness proportional to both coarse aggregate was not significantly affected by the content and gradation of coarse aggre- fine aggregate content (30.0 per cent gate. Thus, by increasing or decreasing to 50.0 per cent, by weight), air con- the percentage of coarse aggregate, or tent (4.0 per cent to 10.0 per cent), increasing or decreasing the maximum and water-cement ratio (5.7 gal/sack to aggregate size, or a combination of 7.3 gal/sack). Since the range of the both, the fracture toughness can be ad- parameters investigated was inclusive justed. However, if the fracture tough- of most mix designs, they can be neglect- ness is to be increased by using a larged in designing for a mix of high or low er maximum size coarse aggregate, the fracture toughness (material's resis- gradation of coarse aggregate must be tance to propagation of an existing uniform to minimize segregation and its flaw). Although only two types of detrimental effects. Limitations will coarse aggregate were used in the inves- be placed on maximum aggregate size by tigation, the results suggest that the design considerations, i.e., size and effect of type of coarse aggregate on shape of the concrete members, amount the fracture toughness was similar to and distribution of reinforcing steel, the effect of type of coarse aggregate etc. on the bond strength between coarse Since the major aim of crack con- mortar.(16) aggregate and cement paste or trol is to minimize crack width by Thus, high quality aggregates (homoge- increasing the number of cracks in neous, low absorption, high modulus of hardened concrete, the design of a elasticity relative to cement paste, concrete mix to maximize the high frac- etc.) should be used to develop mixes ture toughness is not necessarily the with high fracture toughness values answer. It also may be advisable to (gradation requirements previously make a sacrifice in the desired fracture stated would apply to all types of ag- toughness value so that small flaws can gregate). The variables significantly form to act as stress relievers. These affecting the fracture toughness of con- small flaws would prevent the buildup crete can be limited to the coarse aggre- of stress values in the concrete that gate content and gradation of coarse can lead to formation of large cracks aggregate. which could allow the ingress of water to cause corrosion of the reinforcement forced concrete member with the steel which could then result in rapid deter- loaded in tension. Since the only ioration of the concrete. However, in forces acting on the concrete are the the design of all mixes the objectives bond forces which develop as the rein- of required qualities of hardened con- forcement elongates, the stress inten- crete, workability of fresh concrete, sity factor can be expressed in terms and economy should be maintained even of either the bond forces or the crack if it means a sacrifice in fracture opening displacement at the level of toughness. the reinforcement. The total force transmitted across the cracked section 5.2 CRACK MECHANISM FOR CONCRETE for equilibrium conditions for a given STRUCTURES crack length and different lengths of In order to control the cracking unbonding can be calculated from the of concrete structures due to varying specimen geometry and material proper- load and environment, a systems-type ties for the cracked specimen. analysis can be used to describe the Cracking in rigid pavements may complex cracking mechanism for a rein- be analyzed by using a cracked beam on forced concrete beam subjected to a con- an elastic foundation. The stress in- stant moment, for a reinforced concrete tensity factor varies inversely with member with the steel loaded in tension, crack length since an increase in the and for a cracked beam on an elastic crack length transfers more load to the foundation. structure in the region of the crack A reinforced beam subjected to a and thus reduces the stresses on the constant moment may be analyzed using cracked section. This crack growth the approach developed. The resultant arrests additional cracking until the stress intensity factor is the sum of load is increased. The limiting condi- the individual stress intensity factors tion is reached when the crack has for the concrete being subjected only propagated through the pavement depth to a moment, for the concrete being sub- while the load is still transferred to jected only to axial forces, and for the foundation (stable crack growth the concrete being only subjected to condition). Also, this model finds the resultant forces at the level of application to crack growth under re- the reinforcement. Since expressions peated random loads in rigid concrete for the individual stress intensity pavements, but quantitative results factors are available in the litera- cannot be derived until data becomes ture l l '1 2 ) the resultant stress inten- available on the change in crack length sity factor can be determined. When as a function of stress intensity factor the resultant stress intensity factor related to repeated loads. becomes equal to or exceeds the critical stress intensity factor the crack will 5.3 ANALYTICAL STUDY OF CRACK DEVELOP- MENT ASSOCIATED WITH VOLUME CHANGE propagate until it is arrested. The cracking mechanism in beams The computer program VCSC (Volume can be anproximated by using a rein- Change Stresses in Concrete) is prepared in FORTRAN IV for the solution of stres- and may lead to cracks if the rate of ses caused by volume change in plain and drying is rapid enough. Under such reinforced concrete. The program is conditions this method can be used to capable of solving cases in which the predict cracking patterns and stress model is subjected to external and/or distributions for various strengths of internal displacements as well as volume concretes, amount of reinforcement, and change strains. The program uses the thickness of concrete cover, when the finite element method of analysis. The shrinkage stresses are dominant. elements are rectangles of equal size. The boundary conditions used to Nodal displacements, normal stresses, simulate a bridge deck or a highway and shearing stress at each node of each pavement could be rollers on the bottom element, normal stresses, shearing stress of the model with a corner node fixed maximum principal stress, minimum prin- in both the x- and y- directions. The cipal stress, and direction of maximum model should be long enough so the com- principal stress at the center of the putation of stresses and displacements same element, and steel stresses for would converge to their actual values each element of steel result from the over a range of length which is suffi- solution of each problem in the order ciently long for a crack pattern to given. develop. In general it is suggested In their early life, highway pave- that the stresses and deformations not ments and bridge decks are subjected to be taken at points which are closer than shrinkage stresses that may be the two columns of elements to a side bound- primary stresses acting on the members ary. VI. SUMMARY AND CONCLUSIONS

6.1 OBJECT AND SCOPE In the paste and mortar series The objective of this study is to there was a decrease in effective frac- gain an increased understanding of crack ture toughness with increasing water- initiation and growth in concrete, which cement ratio, while in the concrete is essential to improved control of series there was no apparent effect of cracking of concrete structures, i.e., varying the water-cement ratio on effec- to acquire a better understanding of tive fracture toughness for the range the effect of concrete parameters on of water-cement ratios investigated. crack development in concrete and to Increasing the air content de- correlate crack development in concrete creased the effective fracture toughness with various types of distress. for the paste, mortar, and concrete The investigation was divided into series. three major divisions: (1) experimental In the paste, mortar, and concrete investigation of fracture toughness, series there was an increase in effec- effect of concrete parameters on the tive fracture toughness with age. This fracture toughness of pastes, mortars, increase was significant up to an age and concretes; (2) crack mechanism for of 29 days, but for curing times of concrete structures, systems-type analy- greater than 29 days the change in sis description of complex cracking effective fracture toughness from the mechanism that occurs in concrete struc- 29-day value was not significant. tures; and (3) analytical study of crack There was an increase in effective development associated with volume fracture toughness in the mortar series change, approximate solution for problem with increasing sand-cement ratio, how- of shrinkage stresses in plain and rein- ever, the change in effective fracture forced concrete was developed. toughness with increasing sand-cement ratio in the concrete series was not 6.2 RESULTS OF INVESTIGATION significant for the range of sand- 6.2.1 Effect of Concrete Parameters cement ratios investigated. on Fracture Toughness The effective fracture toughness The effective fracture toughness of the concrete series increased with K was based on the assumption that the an increase in the maximum size of concrete was homogeneous and the flaw coarse aggregate and also with an depth at failure was equal to the cast increased gravel-cement ratio. However, flaw depth. there was a decrease in the effective fracture toughness when a large amount act toward the crack thus causing crack of maximum size aggregate was used in arrest, and positive when load forces the mix. This was probably attributable act away from the crack to cause crack to segregation. propagation). Crack propagation occurs The effective fracture toughness when the resultant stress intensity of the concrete series cast with a factor reaches the critical value. river gravel coarse aggregate was lower A model was presented for inves- than the effective fracture toughness tigating the cracking mechanism in of the concrete series cast with a beams. The model was used for a quanti- crushed limestone coarse aggregate for tative analysis of crack equilibrium. ages of three days and six days; how- The only forces acting on the element ever, at ages of 29 days and 92 days the are the bond forces, which can be difference in their effective fracture examined by the opening displacement toughness was not significant. at the level of the reinforcement and are affected by unbonding. The stress 6.2.2 Crack Mechanism for Concrete intensity factor can be expressed in Structures terms of the displacement, or the bond A fracture system of processes forces. Applying the results of this that respond to outside stimuli and approach to the specimen geometry and to describe the interact was developed material properties of Reference (14) cracking mechanism in concrete complex yielded the following: was applied to structures. The system (a) The equilibrium crack length of the concrete the free body diagram increases with the total force trans- portion of the structure, which is the mitted across the cracked section by contains the structural element that the concrete and steel if the unbonded propagate. crack that will length is constant; in a rein- The cracking mechanism (b) The total force transmitted subjected to a forced concrete beam across the cracked section by the con- A quali- constant moment was analyzed. crete and steel decreases as the un- the cracking indi- tative analysis of bonded length increases; cated that the resultant stress intensi- (c) An increased unbonded length the stress ty factor which describes is associated with an increased crack in the beam field surrounding the crack length during a virtual load increase. was a function of the stress intensity The cracking in rigid pavements being sub- factor due to the concrete can be analyzed using a cracked beam (causes crack jected only to a moment on an elastic foundation. The stress intensity factor extension), the stress intensity factor would vary inversely load due to the effect of an axial with the crack length since an increase (negative for compressive forces and in the crack length transfers more load thus tends to arrest crack propagation), to the elastic foundation in the region due to and the stress intensity factor of the crack and reduces the stresses the level of the the resultant forces at on the cracked section. steel (negative when the bond forces 6.2.3 Analytical Study of Crack concrete structures due to various types Development Associated with of distress, i.e., distresses due to Volume Change environment or loading. The resistance An analytical method for the eval- to propagation of the flaws inherent uation of volume change stresses was in concrete can be adjusted by modifying presented which can be used for both the mix design, i.e., varying the coarse plain and reinforced concrete. The aggregate content or gradation of coarse finite element method was utilized with aggregate, or type of coarse aggregate. the aid of a digital computer program Since the experimental investigation to construct an approximate solution to covered a range of concrete parameters the problem. The method predicts crack- that would be inclusive of most mix de- ing, crack patterns, and magnitude and signs, an approximate effective fracture distribution of stresses in members in toughness value can be determined from which shrinkage stresses are the primary this investigation for most designs used. stresses. From the effective fracture tough- The effects of various parameters ness value for the mix design chosen on volume change stresses can be pre- and the stress intensity factors due dicted. For example, the effect of to the type of load distress, the resul- concrete strength, concrete cover, tant stress intensity factor can be amount of reinforcement, etc., can be determined for a reinforced concrete studied by using various parameters in beam subjected to a constant moment. the computer program, thus saving consid- The resultant stress intensity factor erable time over that required in an can then be used to describe the crack- extensive experimental program. ing mechanism; i.e., crack propagation, The method can also consider the which will occur when the resultant effects of loads and displacements on stress intensity factor reaches the the model. critical value. The method will be more effective The model developed for investi- as additional input data on shrinkage, gating the cracking mechanism in con- particularly nonuniform shrinkage, crete beams under load can be used for become available. If reliable infor- obtaining quantitative results. As an mation on shrinkage strains at various example of its application, this approach times becomes available, then time to was used in conjunction with the results cracking can be predicted. of Reference (14) for studying the The bond between steel and concrete equilibrium crack mechanism in reinforced was assumed to be perfect. As a first concrete members. aoproximation this is justifiable because Cracking in rigid pavements due to volume change stresses are relatively external load can be studied using a sma ll. . cracked beam on an elastic foundation. This model can also be used for studying 6.3 CONCLUSIONS the crack mechanism of pavements due to The results of this investigation random loading, but further data is can be used to describe the cracking of required on the change in stress intensity factor with loading cycles. stresses are the primary stresses. The The computer program developed can effects of various parameters on volume be used for evaluation of volume change change stresses, as well as the effects stresses resulting from either environ- of concrete cover, concrete strength, mental distress or load distress. The amount of reinforcement, etc., can be method predicts cracking, crack patterns, studied by using various parameters in and magnitude and distribution of the program. stresses in members in which shrinkage VII. SUGGESTIONS FOR FUTURE RESEARCH

The effects of different environ- one or two surfaces sealed Data should mental conditions and different loading be provided for free shrinkage versus rates on the effective fracture tough- time for various drying, environmental, ness of concrete should be investigated. and surface conditions. A load-slip Application of the systems-type relationship could be incorporated into fracture analysis requires: understand- the program to eliminate the assumption ing the inelastic phenomenon of unbond- of perfect bond between the steel and ing of cracked sections; theoretical or concrete, and thus bond stress can be empirical knowledge of the actual con- part of the program output. The com- crete stresses near cracks; development puter program could be modified to of stress intensity factor expressions accept springs as part of the boundary for various models such as the rigid conditions so that slip over the sub- pavement; and a further look at fatigue grade of the highway pavement can be crack growth in terms of the stress incorporated. Also, in order to inves- intensity factor with loading cycles. tigate the effects of rate of drying on Utilization of the analytic method the stresses produced by shrinkage, the for solution of volume change stresses shape of the stress-strain curve could requires data which is compatible with be varied, i.e., study the effect of the computer program. One area in the stress-strain diagram on the effect which there is insufficient data is of the maximum value of strain which shrinkage strains in members of large occurs at the outermost fiber of the cross section so that nonuniform shrink- member. age data can be obtained with all but VIII. REFERENCES

(1) "Fracture Toughness Testing and Its Proceedings, American Concrete Applications," ASTM STP No. 381, Institute, 41, (1946-7), pp. 101- American Society for Testing and 132, 249-503, 549-602, 669-712, Materials (April, 1965). 845-880, 992-993. (2) Irwin, G. R. Proceedings, 1960 (10) Hahn, G. T. and Rosenfield, A. R., Sagamore Research Conference on "A Systems-Type Approach to Ordnance Materials. Washington, Problems on Fracture" in Funda- D.C.: U.S. Office of Technical mental Phenomena in the Material Services. Sciences, Vol. 4. New York: Plenum Press (1967), pp. 33-43. (3) Kaplan, M. F., "Crack Propagation and the Fracture of Concrete," (11) Srawley, J. E. and Gross, B., Proceedings, American Concrete "Stress Intensity Factors for Institute, 58 (1961), pp. 591-611. Crack-line-Loaded Edge-Crack Specimens," NASA TND - 3820 (4) Glucklich, J., "Fracture of Plain (1967), pp. 1-19. Concrete," Proceedings of the ASCE, 89:EM 6 (1963), pp. 127-138. (12) Gross, B., Roberts, E., Jr., and Srawley, J. E. , "Elastic Displace- "Static and Fatigue _ ments for Various Edge-Cracked Fractures of Portland Cement Plate Specimens," NASA TND-4232 Mortars in Flexure," Proceedings, (1967), pp. 1-12. First International Conference on Fracture, Vol. 2, Sendai, Japan (13) Reis, E. E., Jr., Mozer, J., (1965), pp. 1343-1382. Bianchini, A. C., and Kesler, C. E., "Causes and Control of (6) Lott, J. L. and Kesler, C. E., Cracking in Concrete Reinforced "Crack Propagation in Plain Con- with High Strength Steel Bars-- crete," Symposium on Structure of A Review of Research," University Portland Cement Paste and Concrete, of Illinois Engineering Experiment Highway Research Board Special Station Bulletin No. 479, Urbana, Report 90, Washington, D.C., (1966), Illinois (1965), pp. 1-61. pp. 204-218. (14) Broms, B. B., "Stress Distribution, and Lott, J. L., "Frac- (7) Naus, D. J. Crack Patterns and Failure ture Toughness of Portland Cement Mechanisms of Reinforced Concrete Concretes," Theoretical and Applied Members," Proceedings, American Mechanics Report No. 314, Univer- Concrete Institute, 61 (October, sity of Illinois at Urbana-Champaign 1964), pp. 1535-1557. (1968), pp. 1-87. (15) Pzemieniecki, J. S. Theory of (8) Brown, W. F. and Srawley, J. E., Matrix Structural Analysis. "Plane Strain Crack Toughness New York: McGraw-Hill Book Co., Testing of High Strength Metallic (1968). Materials," ASTM STP No. 410, American Society for Testing and (16) Hsu, T. T. C. and Slate, F. 0., Materials (1966), pp. 13-14. "Tensile Bond Strength Between Aggregate and Cement Paste or (9) Powers, T. C. and Brownyard, T. L. , Mortar," Proceedings, ACI , Vol. 60 "Studies of the Physical Properties (April, 1963), pp. 465-486. of Hardened Portland Cement Paste," IX. APPENDIX I: USER'S GUIDE FOR COMPUTER PROGRAM IN FORTRAN IV

User's Guide and volume change strains. The computer program VCSC (Volume The input cards for the program Change Stresses in Concrete) is prepared are discussed below in the order that in FORTRAN IV language for the solution they appear in the program: of "Stresses Caused by Volume Change in READ 242, NPBLM; Plain and Reinforced Concrete." The 242 FORMAT (15). program is capable of solving cases in The value of NPBLM indicates the number which the model is subjected to external of problems that are to be solved. loads and/or external displacements as This will be discussed in more detail well as volume change strains. The later. program uses the finite element method 1000 READ 1, MX, NX, DX, DY, ANU, of analysis. The elements are rectan- E, ES, AS, LFSN, LSN, KIND, gles of equal size. The result of the NSTR solution of each problem is the nodal 1 FORMAT (215, 3F5.2, 2F15.1, displacements, the normal stresses, and F5.3, 415) the shearing stress at each node of each where: element; the normal stresses, the shear- MX = number of elements in each ing stress, the maximum principal stress, row. the minimum principal stress, and the NX = number of elements in each column. direction of maximum principal stress DX = the length of an element. at the center of the same element; and DY = the height of an element. steel stresses for each element of ANU = Poisson's ratio for concrete. steel, in that order. E = modulus of elasticity of The input for the program consists concrete. of the following parameters: ES = modulus of elasticity of steel. (a) The controlling information AS = the area of steel. for the geometry and input LFSN = the first steel node. and output of the results; LSN = the last steel node. (b) description of material KIND = 0 is the problem is plane properties; strain, = any non-zero number up to five digits if the (c) numerical values for external problem is plane stress. loads, external displacements, NSTR = number of points in an element at which the stresses are to KZ2 = 1 if there is a constraint in be evaluated. This is either the y-direction at node JK; I or 5 for evaluating the 0 if there is no constraint stresses at the center of the in the y-direction at node JK. element or at the center and PLOAD the four corners, respectively. = the magnitude of the displace- (2*JK) ment constraint in the y- The value of NPBLM is equal to the direction, zero for supports. number of cards that correspond to the READ 310, N 1000 READ 1 statement. 310 FORMAT (15) READ 166, NANLYS, SIGCR, AINC Here, the number N represents the 166 FORMAT (15, 2FI0.5) total number of nodes at which loads where: are applied. NANLYS = the number of times a problem READ 312, JK, KZI , PPLOD(2*JK-1), is to be repeated with the KZ2, PPLOD(2*JK) load or displacement incre- mented each time, or with 312 FORMAT (215, F10.5, 15, F10.5) shrinkage strains increased. where: SIGCR = limiting tensile stress for JK = the number of the node at concrete. which a load is to be applied. AINC = the increment by which the KZI = 1 if there is a load component loads or displacements are to in the x-direction at node JK; be increased, represented as 0 if there is no load component a fraction of the original in the x-direction at node JK. loads or displacements. PPLOD = magnitude of READ 310, N the load in the (2*JK-1) x-direction at node JK. 310 FORMAT (15) KZ2 = I if there is a load component The number N represents the total in the y-direction at node JK; 0 if there is no load in the number of nodes at which a particular y-direction at node JK. displacement is to be defined such as PPLOD = magnitude of the load in the supports. For example, for a simply (2*JK) y-direction at node JK. supported beam, the value of N would be READ 400, M 2 for the two nodes that are restrained 400 FORMAT (15) in one or both directions, unless dis- The number M is the total of all placements are also applied at other the rows of nodes at which volume change nodes. strains are to be applied. READ 303, JK, KZI, PLOAD (2*JK-1), READ 403, (SHRKG(I), I = 1, M) KZ2, PLOAD(2*JK) 403 FORMAT (5Fl5.10) 303 FORMAT (215, F10.5, 15, Fl0.5) where: where: SHRKG(l) = the value of volume change strain at the rows of nodes I. JK = the number of node at which there is to be a displacement I = the number of the row for constraint. which the strain is being read; the top row of nodes KZI = 1 if there exists a constraint is number 1, the second row in the x-direction at node JK; number 2, and so on. 0 if there is no constraint in the x-direction at node JK. This concludes the definition of PLOAD = the magnitude of the displace- the READ statements. ment constraint in the x- (2*JK-1) The following is an explanation of direction, zero for supports the PRINT statements ir the computer this PRINT statement, the first con- program except for those print state- straint is in the x-direction and the ments that comprise the titles for the second constraint is in the y-direction. printed output: This, for example, would happen for a PRINT 69, MX, NX, DX, DY, ANU, node fixed in both directions for which E, ES, AS, LFSN, LSN, KIND, the magnitudes would be zero in both NSTR directions. The magnitude of the dis- 69 FORMAT (215, 3F5.2, 2F15.1 , placement will include the additional F5.3, 415//) increments, if any. PRINT 660, NANLYS, SIGCR, AINC PRINT 83, I, LCNT(I), NCR(I), 660 FORMAT (119, F17.5, F16.5//) ELD(121), ELD(122) PRINT 661, JK, KZI, PLOAD 83 FORMAT (16, 114, 126, E24.4, (2*JK-1 ) , KZ2, PLOAD(2*JK) E13.4) 661 F0RMAT (16, 113, F17.5, 113, where: F16.5//) I = the number of node PRINT 662, JK, KZI, PPLOD LCNT(I) = the total number of elements connected to node I (2*JK-1 ) , KZ2, PPLOD(2*JK) NCR(I) = the total number of cracked 662 FORMAT (16, 16, F13.2, 16, elements connected to node F12.2//) I. Note that NCR(I) must always be less than LCNT(1) PRINT 663, (SHRKG (I), I = 1, ELD(121), = the magnitude of displace- M) ELD(122) ments at node I in the x- 663 FORMAT (5F15.10//) and y-directions, respectively. The parameters are already defined PRINT 707, J, ((SIGMA (Kl ,I), in the explanation of the READ state- Kl = 1, 3), 1=2, NSTR) ments. These PRINT statements merely 707 FORMAT (15, 6X, 12E10.3) print out the input parameters for where: checking purposes. J = the number of elements. PRINT 314, K, EPOAD SIGMA = the stresses in the elements. FORMAT (16, E15.6//) 314 Here a set of stresses is where : printed out for each node, 1, 2, 3, and 4, of the element. = number of node at which an K Each set of stresses consists external load is applied. of the stresses in the x- and EPOAD = the load acting at node K. y-directions and the shear This includes the increments stress. to the loads if there are any. PRINT 18, J, ICR(J), (SIG(I, J), PRINT 314, K, ELOAD(2*K-2+IJ) 1 = 1, 3), SIGPI, SIGP2, THETA - 314 FORMAT (16, E15.6) 18 FORMAT (215, 2X, 3E13.4, 2X, whe re : 2E13.4, FlO.4/) at which K = the number of node This is a PRINT statement for the there exists a displacement constraint. stresses at the center of the element ELOAD = the magnitude of the dis- in which: at (2*K-2+1J) placement constraint J = element number node K. Zero for supports. ICR(J) = 0 if the element is not cracked, If a node number appears twice in >0 if the element is cracked PRINT 19, KCR, NCRT SIG = a set of stresses at the cen- 19 FORMAT (114, 125) ter of the element. The set where: consists of stresses in the KCR = the number of elements cracked in the immediately preceding x- and y-directions and the round. shear stress NCRT = the total number of elements SIGP1 = maximum principal stress at cracked up to this point. the center of the element PRINT 602, I, SIGST(1) SIGP2 = minimum principal stress at 602 FORMAT (115, E13.4) the center of the element where: THETA = the angle that the maximum I = the number of steel element principal stress makes with SIGST(1) = the steel stress for the steel element I . the positive direction of the x-axis, in degrees. X. APPENDIX II: COMPUTER PROGRAM IN FORTRAN IV FOR DETERMINATION OF VOLUME CHANGE STRESSES IN PLAIN AND REINFORCED CONCRETE USING FINITE ELEMENT ANALYSIS

/*ID HASSAN M. REJALI // FXFC WATFOO //SYSTN nrn * $nP KP926.TTME'300,PAGFS9100 nDITMENSTON X(200),Y(200) NNF(400,4) ,LCNT(200),NEN(200,4),AK(898) 1,STTF(P?,00 ),K7(700),KKH(1400),COFF(400? S),ELD(400),ELOAD(400) ?STG(3,400).f)ISP(R),CR(S,3,8),SHRKG(400),EPS(40092),SIGMA(3,5), 3SIGST(100),P[OA0)(400),ICR(400),PPL-O0 (400),KA(700),NCR(400) T=1 .0 PT=3. 141~q?6535AR79 RFAD ?42.NPRLM 24P FORMAT(T9) 1000 RFAn 1,MXNX,DX,DY,ANUE,ES,ASLFSNLSNKINDNSTR I FORMAT (?ITr3F5.?qFl5.lF5.3,415) PPTNT 240 240 FnPMAT(1H1) PPINT ?3? P3? FnPMAT(4 INPUT DATA*) PPINT69,MXNX.D)X*Y.ANllFEESASLFSNgLSNqKINDNSTR 69 FORMAT (?IS,3FS.?,2F15.1.F5.3,415//) RFAD 166.NANLYSSIGCR*AINC 166 FORMAT (1.?F1O.5) PRINT 69? 69? FORMAT(f NUMBRFP OF PRORLEMS, CRITICAL STRESS, LOAD INCREMENT*) PRINT 660,NANLYS•SIGCRAINC 660 FORMAT(11 9F17.5,F16.5//) C FOP THE PRIPOSE OF FVALUIATION OF STIFFNESSES REPLACE E BY 1. MT=MX+1 NT=NX+ FF=1.0 STFR=( 1 ?.*FScAS*FE)/(F)X*F) C STFP TS THE STEEL FACTOR COMPARARLE TO AK*S FOR THE CONCRETE C CnFFFICIEMTS PFLATING OISPLACEMENTS TO FORCFS FOR A SINGLE ELEMENT ANhIII= (1 .- ANI I)/ ((1 .+ANll)* (1.-2.*AN I ) ) ANII2=AMtI/( ((1 .+ANIU) * (I .- . *ANU)) IF(KIN).FO.0) GO TO 701 ANII =1 ./(] .- ANIJ*ANU) ANII?=ANII/(1 .- ANIJ*AN!1) 70] ANI3=1 ./?2.*(1 .+ANU) HFTA=FY/OX AK(1 1)=4.*AJNUl]*RFTA+4.*ANU3/RETA AK (1 .2) =3.* (ANUP+ANIJ3) AK(?. 1)=3.*(fNI2?+ ANIi3) AK( 1 3)=?. *ANIJI*LIFTTA-4.*ANll3/RETA AK (3.1 ) =?.*ANIJ *RFTA-4.*ANI13/RETA AK(1,4)=3.*(AN I3-ANI2?) AK (4.1 ) =3.* ( ANII1-AN'I2) AK ( 1 . ) =-2?.* (ANUtJ*HFTA+ ANIJ3/RFTA) AK ( . 1) =-?.. *(AI'IIAI'.FTA+ANi1)/HFTA) AK ( 1 .,) =--3. (ANIU?+ A'IIJ3) AK (6, 1 )=-3.*(ANtJ2+ANU3) AK(1,7)=-4.*ANUI*RETA+2.*ANUJ3/BETA AK(7,1)=-4.*ANUI*BETA+2.*AN tl3/BFTA AK(1 9)=-3.*(ANU3-ANU2) AK(8,1)=-3.*(ANU3-ANU2) AK(2.2)=4.*ANUI/BETA+4.*ANU3*BETA AK(2,3)=-3.*(ANU3-ANU2) AK(32?)=-3.*(ANU3-ANU2) AK(2.4)=-4.*ANUL/RETA+2.*ANU3*BETA AK(4,2)=-4.*ANUil/BETA+2.*ANt)3*BETA AK(2,5)=-3.*(ANU2+ANU3) AK(5,2)=-3.*(AN()2+ANUJ3) AK(2,6)=-?.*(ANUL/BETA+ANU3*PETA) AK(6?2)=-?.*(ANU[/BETA+ANU3*BETA) AK (27)=3.* (ANU3-ANL12) AK(7?2)=3.*(ANU3-ANU2) AK(2,8)=?.*ANU1/BFTA-4.*ANU3*RETA AK(8,2)=2.*ANUI/BETA-4.*ANU3*BETA AK(393)=4.*ANU*BEFTA+4.*ANU3/RETA AK(3,4)=-3.*(ANU2+ANU3) AK(4,3)=-3.*(ANIJ2+ANU3) AK(3,5)=-4.*ANU1*BETA+2.*ANU3/BETA AK(593)=-4.*ANU1*BETA+2.*ANU3/BETA AK(396)=3.*(ANU3-ANI)2) AK(6,3)=3.*(ANU3-ANU2) AK(3,7)=-2.*ANUI*BETA-2.*ANU3/BETA AK(7,3)=-2.*ANU1*BETA-2.*ANUI3/BETA AK(3,A)=3.*(ANU2+ANIJ3) AK (R,3) =3.* (ANU2+AN1l3) AK(4,4)=4.*ANUl/BETA+4.*ANU3*BETA AK(4,5)=-3.*(ANU3-ANIJ2) AK (594)=-3.*(ANU3-ANU2) AK(4,6)=2.*ANUl/BETA-4.*ANU3*RETA AK(6,4)=2.*ANIJ1/RETA-4.*ANU3*RETA AK(4,7)=3.*(ANIJ)+ANU3) AK(794)=3.*(ANUJ2+ANUI3) AK(4,8)=-2.*ANUlI/RETA-?.*ANI)3*BETA AK(8,4)=-2.*ANU1/RETA-2.*ANtU3*BETA AK(595)=4.*ANUI*RETA+4.*ANU3/BETA AK(5.6)=3.*(ANU2+ANU3) AK(6,5)=3.*(ANU2+ANU3) AK(5,7)=2.*ANU1*BETA-4.*ANU3/BETA AK(7,5)=2.*ANU1*BETA-4.*ANU3/BETA AK(5,8)=3.*(ANU3-ANIJ2) AK(8,5)=3.* (ANU3-ANU2) AK(6,6)=4.*ANU1/BETA+4.*ANU3*RETA AK (6,7) =-3.*(ANI3-ANU2) AK(7,6)=-3.*(ANU3-ANU2) AK(6,8)=-4.*ANilJ/RETA+2.*ANtU3*BETA AK(8,6)=-4.*ANIJ/RETA+2.*ANIJ3*RETA AK(7,7)=4.*ANU1 * B FTA+4.*ANU3/BETA AK(79R)=-3.*(ANIJ2+ANU3) AK (A87)=-3.*(ANU2+ANU3) AK(8.8)=4.*ANUl/RFTA+4.*ANU3*BETA C DFTERMINATION OF THE COORDINATES OF THF NODAL POINTS MTI=MT-1 NTI =NT-1 DO 20 M=1,MT MI=M-1 AM = MI DO 20 N=1,NT NI=N-1 AN =N1 L=M] *NT+N X(I ) = AM*OX Y(L) = ANtJDY C DFTFPMINATION OF THE Nn. OF NODES FOR FACH ELEMENT AND NODE NUMBERS 115 IF(M-1) ?o.?20.0 10 IF(N-1) ?20.?20o.151 151 K 1 =(M-2? ) *NT 1 + (N-1) NNF(KI .1)=L -NT-1 NNF(Kl.?)=L-INT NNF (K] *,3) =1_ NKF(Kl.4)=L-1 ?n CONT NIIF L T=I KT=K1 no?211 =1 LT NCR(L )=0 21 ILCNT(L)=0 Do ?5 K=1.KT n002 1=1.4 NT=NNF(K, I) LtCrT (NT )=ICNT(NI) +1 LC1=LCNT(NT) 25 NFN(NTI.C1)=K OrO L=1 LT KA(L)=1 A K7(L)=1 READ 110,N 3]0 FORMAT(IS) PRINT ?13 233 FORMAT(* NODE. CONSTRAINT, X-DISPLACEMFNT, CONSTRAINT, Y-DISP ILACFMF NT*) 00 30? K=1.N RFAr 303.IK,9K71*PLOAD(2*JK-1),K7Z2PLOAD(?*JK) 303 FORMAT(?T5.Fl0.5 IS5.F10.5) PPINT661*.JK K71 PLOAD(?PJK-1) ,KZ2?PLOAD (2*JK) 661 FORMAT ( I6TI 3F17.5 113,F6.5//) 302 K7(JK)=1IK71+2*K7? RFAD 310.N IF(N.FO.0) GO TO 455 PPINT 234 234 FORMAT(* NOOE. X-LOAD, AMOUNT, Y-LOAD, AMOUNT#) 00 315 K=1,N RFAn 3129,IKKZ71,PPLOD(2*JK-1),KZ2,PPLOD(?2*JK) 31? FORMAT (2T-Fl0O.5,5I.F10.5) PRINT662?JKK71,PPLOD(2*JK-1),KZ2,PPLOD(2*JK) 66? FORMAT(T61I6.F1?.?, 6,Fl2.?//) 315 KA(JK)=1+K71l+2K72 455 NSFLT=(LSN-LFSN)/NT = 1 00 401 L ]KT ICR(L)=n EPS(L 1 )=0.0 401 EPS(tL )=0.0 NCRT=0 00 4 NS=1,NAtLYS ANS=NS READ 4O«0.M 400 FORMAT (IS) IF (M.FO.O) GO TO PP PFAD403 (SHPKG(I1),I=1 M) 403 FORMAT (SFl5.10) PRINT 235 231 FORMAT(i SHPINKAGE STRAIN FROM TOP TO ROTTOM ON ROWS OF NODES#) PTINT i3, (SHPKG( I ) I= M) ((3 FORMAT (SFI5.n1//) On). 4Q7 1=1 .', I.L=NT-I+* IF(I.FQ.1) GO TO 405 00 404 J~J=1,MTI J=LL+(J.-1])*NTI 404 EPS(J,2)=SHRKG(I) 405 IF(I.FO.NT) GO TO 407 LL=LL-1 DO 406 JJ=1,MTI J=LL+(J.J-1)*NTI 406 EPS(J.l)=SHRKG(I) 407 CONTINUE 22 KCR=O NSN=LFSN 00 300 K=1ILT ELOAD(2*K-1)=0. ELOAD(2*K)=0. IF (KA(K).EO.l) GO TO 300 II=KA(K)/2 PRINT 236 236 FORMAT(# NODE FORCE*) D0313 I=1II IJ=KA(K)-II-(II-1)*(2-I) ELOAD(2*K-2+IJ)=PPLOD(2*K-2+IJ)*(1.+(ANS-1.)*AINC)*12./E EPOAD=ELOAD(2*K-2+IJ)*E/12. 313 PRINT 314,K,FPOAD 314 FORMAT(16,E15.6//) 300 CONTINUE 00 11 K=1,LT KH=K KL=K LC1=LCNT(K) DO 2 I=1,LC1 NE=NEN(K,I) DO 2 L=1,4 IF(KL-NNE(NF,L)) 6,6,7 7 KL=NNE(NF,L) 6 IF(KH-NNE(NE,L)) 9,2,2 9 KH=NNE(NE,L) 2 CONTINUE KH=2*KH IF(K.EQ.I) GOT05 KKP=2*K-2 IF(KH.LT.(2*K-2+KKH(KK2))) KH=2*K-2+KKH(KK2) 5 KL=2*KL-1 K21=2*K-1 K22=2*K KKH(K21 )=KH-2*K+1 KKH(K22)=KH-2*K KHL=KH-KL*1 C THE FOLLOWING IS THE PROCEDURE RY WHICH THE STIF. MATRIX IS EVALUATED C TWO EQUATIONS AT ATIME 0099 1=1,2 0099 J=1.KHL 99 STIF(I,J)=O. IF(NCP(K).GE.LC1) 60 TO 555 NCR(K)=O 00 29 I=1.LCl NE=NEN(K,I) IF(ICP(NF).GT.O) GO TO 2A 00 3 L=1,4 IF(K-NNE(NE,IL)) 3,?3,3 CONTITIlE JJ=I- 00 9 L= 1,4 NN=INF (NF .L) KM=2*NIN-KL DO 92 N=],? D[ 92 M=l,2 KMM=KM+M-1 JJ?=? *JJ+N-? LM?=2?L+M-2 9? STIF(N,KMM)=STIF(N,KMM)+AK(JJ2,LM2) Cl =T*DY*(ANIJA+ANiI2)/6. C2?=T*DX*(ANI11+ANU2)/4. AJI=1-?*(JJ/3) AJ2=(JJ/2?)-2?(JJ/4)*+l AJ3=1-?*((JJ/2)-2*(JJ/4)) ELOAr)D(?K-I)=ELOAD(2?*K-I)-Cll*AJle(EPS(NE,2)*(3.-AJ2)+EPS(NEl)* IAJ?)*1?. ELOAD(?*K)=ELOAD(2*K)-C?2*AJ3*(EPS(NE,1)+EPS(NE,2))*12. GO TO 29 ?A NCR(K)=NCR(K)+1 ?9 CONTINUE 555 IF (K.NF.NSN.OR.NSN.GT.ILSN) GO TO 598 IF(NCR(K).GE.LCI) NCR(K)=NCR(K)+1 NSN=NSN+NT STIF(192*K-KI_)=STIF(1,2*K-KL)+2*STFR KSTI=2*K-Ki_-2*NT KST2=2*K-KL+2*NT IF (K.EQ.LFSN) KST1=2*K-KL IF (K.FO.LSN) KST2=?*K-KL STIF(1IKSTI)=STIF(1,KSTI)-STFR STIF(1,KST2)=STIF(1,KST?)-STFR C PONDARY CONDTTIONS IF(NCR(K).EO.LC1) GO TO 11 598 IF(KZ(K).FQ.1) GO TO 76 IT=K7(K)/2 DO 87 I=1.11 IJ=K7(K)-II-(II-1)*(2-I) IF(NCR(K).GT.LC1.AND.IJ.EQ.?) GO TO 556 DO AS JK=1,KHL A8 STIF(IJJK) = 0. ELOAD(?K-?+T,.J)=PLOAD(2*K-2+IJ)*(1.+(ANS-1.)*AINC) PRINT 241 241 FORMAT(# NODE, DISPLACEMENTS) PRINT 314,KELOAD(2*K-2+IJ) 87 STTF(IJ,2*K-KLI+J-1)=1.0 C ElIMINATION OF THE OFF DIAGONAL TERMS 556 IF(NCR(K).EO.LC1) GO TOll 76 00 100 I=1,2 IF(NCR(K).GT.LC1.ANO.I.EQ.2) GO TO 11 KI?=2?*K-2+I KI?1=KI?-1 IF(K.FQ.l.AND.I.FQ.l) GO TO 64 DO 61 KK=KL*KI21 KHK=KKH(KK) DO 62 JK=IKHK KKLI=KK-KL+1 JKKL=KKL1JK 6? STTF(TJKKL) = STIF(I,JKKL)+STIF(IKKLI)*COEF(KKJK) 61 EI.OAD(KT?) = FLOAD(KI2)+STIF(T,KKL1)*ELD(KK) 64 K?L]=Kr?-KI *1 FID( KIP) = -ELOAO( KI2)/STIF(IK2?L1 KKK?=KKH(KT?) IF(K.FO.LT.AND.I.FO.?) GOTO 100 DO 63 J J=1,KKK2 JPKt.=J J+KI?-KL+1 63 COFF(KI2T,j,J) = -STIF(IqJ2KL)/STIF(I 9 K?LI) 100 CONTINUE 11 CONTINUF C BACK SURSTITUTION AND EVALUATION OF THF DISPLACFMENTS CALLED -ELD- LT2 = 2*LT 00 81 K2=2,LT2 NK=2*LT-K2+1 KKK?=KKH(NK) DO 81 JJ=I,KKK2 NKJ=NK+JJ 81 ELD(NK) = FLD(NK)+CnEF(NK,JJ)*ELDO(NKJ) PRINT 999 999 FORMAT (IHI) PRINT 34 34 FORMAT(# NODE, NO. OF ELEMENTS CONNECTED. NO. OF ELEMENTS CRACK lED, DISPLACEMENTS UIV•) DO 900 1=19LT2 900 ELD(I)=-ELD(I) DO 85 I=1,LT I21 = 2*I-1 I22=2*I PRINT 83,I,LCNT(I),NCR(I),ELD(I21),ELD(122) 83 FORMAT(16,114,126,E24.4,E13.4) 85 CONTINUE C EVALUATION OF THE STRESSES FROM THE DISPLACEMENTS PRINT 504 504 FORMAT (1HI) DO 94 I=1,NSTR PSI=I/4 ATA=I/3-I/5 IF(I.GT.1) GO TO 95 PSI=0.5 ATA=0.S 95 PSI1=1.-PSI ATAI=1.-ATA PSI=PSI*E ATA=ATA*E PSII=PSII*E ATA1=ATA1*E C COEFFICIENTS RELATING DISPLACEMENTS TO STRESSES FOR A SINGLE ELEMENT CR(II,11)=-ANU1*ATAI/DX CR (I1,2)=-ANU2*PSII/DY CR (I,13)=-ANU1*ATA/DX CB(I,1,4)=ANU2*PSI1/DY CR(I.1,5)=ANU1*ATA/DX CR (I1,6)=ANU2*PSI/DY CR (I,17)=ANUI*ATAl/DX CR(I1,18)=-ANU2*PSI/DY CR(I2,?,1)=-ANU2*ATA1/DX CR(I,2,2)=-ANUI*PSII/DY CR (I2,3)=-ANU2*ATA/DX CR(I,2,4)=ANUI*PSII/DY CB(I,?,5)=ANU2*ATA/nX CR(1,2,6)=ANIJ]*PSI/nY CR(l,2,7)=ANIJ2*ATA1/DX CR(I»2,8)=-ANUI*PSI/DY CR(I,3.1)=-AN1I3*PSII/DY CR(T,3.2)=-ANI3*ATA1/DX CR(I,3,3)=ANI)3*PSIl/DY CR(13,4)=-ANU3*ATA/DX CR(I,3.5)=ANII3*PSI/nY CR(I,3,6)=AKNlJ3*ATA/Ox CR( 1,3,7)=-ANU3*PSI/DY CR(I,3,R)=ANU3*ATAI/OX 94 CONTINUE PRTNT 237 237 FORMAT(# FLFMENT NO.. X,Y, AND SHEAR STERESSES AT FOUR CORNERS#) PRINT 23A 238 FORMAT(# ELEMENT NO., CRACKING NO.* XY, SHEAR STRESSES, MAX., I MIN. PPIN. STRESS, DIRFCTION MAX. PRIN. STRESS AT CENTERS) DO 101 J=1KT 00 98 JH = 1,3 SIG(JH,J)=0.0 DO 98 T=1,5 98 STGMA(JH,I)=0.0 DO 97 I=1,4 DISP(2*I-1)=FLD(2*NNE(JI)-I) 97 DISP(?*I)=ELD(2*NNE(J,I)) DO 103 I=1,NSTR 00 103 KI=1,3 DO 102 K2=1,8 102 SIGMA(Kl,I)=SIGMA(KlI)+CB(I,K1,K2)*DISP(K2) II=-(I/3)*(I/5)+2 IF(K1.EQ.3) GO TO 104 IF(I.FQ.I) GO TO 972 SIGMA(KI)=SIGMA(KII)-EPS(JII)*(ANU1+ANU2)*E GO TO 103 972 SIGMA(KlI)=SIGMA(Kl,I)-(EPS(J,1)/2.*EPS(J,2)/2.)*(ANU1+ANU2)*E 104 IF(I.NE.I) GO TO 103 SIG(KlJ)=SIGMA(Kl1I) 103 CONTINUE IF (NSTR.EO.I) GO TO 905 PRINT 707,J,(( SIGMA(KlI),KI=1,3),I=2,NSTR) 707 FORMAT(I5,6X,12E10.3) 905 IF(ICR(J).EQ.O) GO TO 906 ICR(J)=ICR(J)+1 GO TO 101 906 PPl=(SIG(1»J)+SIG(2,J))/2. PP2=(SIG(1,J)-SIG(2.J))/2. PP3=SQRT(PP2*PP2+SIG(3,J)*SIG(3,J)) SIGPI=PPI+PP3 SIGP2=PPI-PP3 THETA=(ATAN?(SIG(3,J),PP2))*90./PI IF(SIGPI-SIGCR) 15.16,16 16 ICR(J)=1 KCR=KCR+1 15 PRINT IA,J,ICR(J),(SIG(I,J),1=1,3),SIGP1,SIGP2,THETA 18 FORMAT(215.2X,3E13.4.2X,2E13.4,F10.4/) 101 CONTINUE DO 201 T=1,NTI J=l 211 NC=0 IJ=I+(J-1)*NTI IF (ICR(IJ).NE.1) GO TO 212 IF(ICR(IJ-NTI).GT.1) GO TO 219 215 IN=IJ+NTI IF(IN.GT.KT) GO TO 204 IF(ICR(IN).NE.1) GO TO 204 NC=NC+1 IJ=IN GO TO 215 204 IF(NC.FQ.O) GO TO 216 NC1=NC IF(ICP(TN).I.F.I) GO TO 231 ICP(IJ)=0 IJ=TJ-NTI NCI=NC-1 231 IF(NC.EO.I) GO TO 217 PMAX=(SIG(1,IT )+÷ IG(2,IJ))/2.+SQRT ((SIG(19IJ)-SIG(2,IJ))*(SIG(I1 1 TJ)-SIG(?. 1. 1)) /4.+SIG( I. I J)*STG(3, TJ)) IJr=TJ ICR(IJ)=0 DO 210 NNl=.NCI 1J=TJ-NTI ICR(IJ)=0 PPS =(STG(19IJ)+cTG(2,IJ))/?.*SORT ((SIG( 1, IJ)-SIG(2. IJ))*(SIG(1 , IIJ)-SIG(?,IJ)) /4.+SIG(3.IJ)*SICG(3,IJ)) IF(PMAX.GT.PRS) GO TO ?10 PMAX=PPS IJC=IJ 210 CONTINUE KCR=KCR-NC ICR(IJC)=] GO TO ?21 216 IF(ICR(IN).LE.1) GO TO 217 ICR(IJ)=0 KCR=KCR-1 217 KCR=KCR-NC 218 J=J+1+NC GO TO 212 219 ICR(I J)=n KCR=KCR-1 212 J=J+1 IF(J.LT.MTI) GO TO ?11 201 CONTINUE NCRT=NCRT+KCR PRINT 239 239 FORMAT(# ELEMENTS CRACKED THIS ROUND, TOTAL CRACKED*) PRINTI9,KCR,NCRT 19 FORMAT(I14,I?5) IF (NSELT.EO.O) GO TO 13 PRINT 603 603 FORMAT(lH1) PRINT 601 601 FORMAT (* STEEL ELEMENT*g* STRESS*) LST2=2*LFSN-1 00 600 I=1,NSFLT LSTI=LST2 LST2=LST1I2*NT SIGST(I)=(ELD(LST2)-ELD(LSTI))*ES/DX PRINT 602?ISIGST(I) 602 FORMAT (I15,F13.4) 600 CONTINUE 13 IF(KCR.NE. 0) GO TO 22 4 CONTINUF PRINT 777 777 FORMAT(* END OF PRORLEM#) NPPLM=NPRlM-1 IF(NPRLM.NE.0) GO TO 1000 STOP END SENTRY DATA DECK

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