H IL LINO S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Urbana-Champaign Library Large-scale Digitization Project, 2007. ABSTRACT CREEP AND SHRINKAGE OF CONCRETE ARE INTERRELATED TIME-DEPENDENT PHENOMENA WHICH ARE INFLUENCED, TO A LARGE EXTENT, BY THE PROPORTIONS OF THE CONSTITUENTS IN THE CONCRETE MIX, BY THE EXCHANGE OF MOISTURE WITH THE ENVIRONMENT, AND BY THE ENVIRONMENTAL CONDITIONS. CREEP OF CONCRETE AS USED HEREIN IS THE SUM OF A BASIC CREEP COMPONENT AND A DRYING CREEP COMPONENT. BASIC CREEP IS CONSID- ERED TO BE THAT PORTION OF THE LOAD IN- DUCED TIME-DEPENDENT DEFORMATION WHICH OCCURS IN THE ABSENCE OF MOISTURE GAIN OR LOSS. DRYING CREEP IS CONSIDERED TO BE THAT PORTION OF THE LOAD INDUCED TIME-DEPENDENT DEFORMATION, IN ADDITION TO BASIC CREEP, RESULTING UNDER CONDI- TIONS OF A CHANGE IN MOISTURE CONTENT OF THE LOADED SPECIMEN AND WHOSE CHAR- ACTERISTICS ARE SIMILAR TO THOSE OF CONCRETE SHRINKAGE. FREE DRYING SHRINKAGE IS DEFINED AS THE DEFORMATION, IN ADDITION TO CREEP, ELASTIC AND THERMAL DEFORMATIONS, WHICH OCCURS AS A RESULT OF A CHANGE IN THE MOISTURE CON- TENT OF THE CONCRETE. A REVIEW OF LITERATURE CONCERNED WITH THE EFFECT OF VARIOUS PARAMETERS ON THE VOLUME STABILITY OF BOTH LOADED AND UNLOADED SPECIMENS IS PRESENTED. THE PARAMETERS WHICH ARE GENERALLY CON- SIDERED TO BE SIGNIFICANT ARE DISCUSSED IN DETAIL. PROCEDURES ARE PRESENTED FOR ESTI- MATING BASIC CREEP, DRYING CREEP AND FREE DRYING SHRINKAGE OF CONCRETE IN TERMS OF THE VOLUMETRIC COMPOSITION, DEGREE OF HYDRATION, SPECIMEN SIZE, AND AMBIENT RELATIVE HUMIDITY. ESTIMATED VALUES OF BOTH CREEP AND SHRINKAGE COM- PARE REASONABLY WELL WITH EXPERIMENTAL DATA FROM VARIOUS SOURCES IN THE LITER- ATURE. BASED ON AVAILABLE KNOWLEDGE ABOUT THE MICROSTRUCTURE OF HYDRATED CEMENT, HYPOTHESES WHICH APPEAR CAPABLE OF QUALITATIVELY EXPLAINING CREEP AND SHRINKAGE ARE PRESENTED. CONCLUSIONS ARE PRESENTED AS TO HOW AGGREGATE TYPE, CONTINUED HYDRATION, TEMPERATURE, CYCLIC MOISTURE CONDITIONS, AND VARIOUS LOADING CONDITIONS RELATE TO THE CREEP STRAINS AS PREDICTED BY THE PROPOSED METHOD AND IN ADDITION AREAS REQUIRING FURTHER RESEARCH ARE INDICATED. SUGGESTIONS ARE PRESENTED AS TO WHERE THE PROPOSED PREDICTION METHODS CAN BE UTILIZED. ACKNOWLEDGMENTS This report is based on an investigation under the Illinois Cooperative Highway Research Program Project IHR-72, "Prediction of Creep in Structural Concrete from Short-Time Tests." The project was undertaken by the Engineering Experiment Station of the University of Illinois, in cooperation with the Illinois Division of Highways, State of Illinois, and the U.S. Department of Transportation, Bureau of Public Roads. The general administrative supervision was provided by W. L. Everitt, Dean of the College of Engineering; R. J. Martin, Director of the Engineering Experiment Station; T. J. Dolan, Head, Department of Theoretical and Applied Mechanics; and Ellis Danner, Director, Illinois Cooperative Highway Research Program and Professor of Civil Engineering. The administrative supervision by the Division of Highways, State of Illinois, was provided by Virden E. Staff, Chief High- way Engineer and John E. Burke, Engineer of Research and Develop- ment. The Project Advisory Committee consisted of the following personnel: Representing the Illinois Division of Highways: J. E. Burke, Bureau of Research and Development C. E. Thunman, Jr., Bureau of Design R. B. Dellert, Bureau of Materials Representing the U.S. Bureau of Public Roads: J. L. Hirsch, Bridge Engineer Representing the University of Illinois: 0. M. Sidebottom, Professor of Theoretical and Applied Mechanics Mete A. Sozen, Professor of Civil Engineering Clyde E. Kesler, Professor of Theoretical and Applied Mechan- ics and of Civil Engineering, served as Chairman of the Project Advisory Committee. Edward M. Wallo, Research Associate, Depart- ment of Theoretical and Applied Mechanics, served as Secretary of the Project Advisory Committee and as Project Investigator. The authors thank Professor H. K. Hilsdorf, who served as consultant to the project, for his aid in preparing this report. Appreciation is also extended to the reviewers of this Bulletin: John E. Burke, Engineer of Research and Development, Illinois Division of Highways; Omar M. Sidebottom, Professor of Theoretical and Applied Mechanics, University of Illinois; and Thomas W. Kennedy, Assistant Professor of Civil Engineering, University of Texas. CONTENTS I. INTRODUCTION . . . . . . . . . . * * * * * * . . . . 1 1.1 Statement of the Problem . . . . . . . . . . . 1 1.2 Purpose of the Investigation . . . . . . ... 2 1.3 Terminology . . . . . . . ...... * * * * * * . . . . 2 II. HISTORY OF RESEARCH . . . . . . . . . . ........... 5 2.1 General . . . * * * * * * * * * * . 5 2.2 Shrinkage of Concrete . . . . . . . . . . . . 5 2.3 Creep of Concrete . . . . . . . . . . . . . . 10 III. FACTORS AFFECTING CREEP AND SHRINKAGE . . . . . . . . 16 3.1 General . . . . * * * * * * * * . . . . . . . 16 3.2 Mix Constituents and Properties . . . . . . . 16 3.3 Mixing Time and Consolidation . . . . . . . . 28 3.4 Age at Loading, Degree of Hydration, and Curing . . . . . . * * * * * * * * . . . . 29 3.5 Moisture Content . . . . * * * * * * * * . . . 29 3.6 Level, Type, and Distribution of Stress . . . 30 3.7 Ambient Air Velocity . . . . . .... . . . . . 31 3.8 Ambient Humidity . . . . . * * * * . . . . . . 33 3.9 Ambient Temperature . . . . . . . . . . . . . 33 3.10 Size and Shape of Specimen . . . . . . . . . . 37 3.11 Strength . . . . . . . . . . . . . . . . . . . 37 3.12 Reinforcement . . . . . . . . . . . . . . . . 38 IV. PREDICTION METHODS . . . . . . . . . . . . . . . . . 39 4.1 General . . . . . . . . . . . . . . . . . . . 39 4.2 Prediction of Creep . . . . . . . . . . . . . 40 4.3 Prediction of Shrinkage . . . . . . . . . . . 42 4.4 Prediction Methods and Temperature . . . . . . 46 4.5 Creep Prediction - Stresses Other than Uniform Uniaxial Stress . . . . . . . . . 46 V. CONCLUSIONS AND APPLICATIONS . . . . . . . . . . . . 48 5.1 Conclusions . . . . . . . . . . . . . . . . . 48 5.2 Applications . . . . . . . . . . . . . . . . . 51 VI. APPENDIX: ESTIMATION OF CREEP AND SHRINKAGE . . . . 53 6.1 General . . . . . . . . . . . . . . . . . . . 53 6.2 Computations of Volume Concentration . . . . . 53 6.3 Computation of Basic Creep . . . . . . . . . . 54 6.4 Computation of Free Drying Shrinkage . . . . . 54 6.5 Computation of Drying Creep . . . . . . . . . 55 6.6 Total Creep . . . . . . . . . . . . . . . . . 56 VII. REFERENCES . . . . . . . . . . . . . . . . . . . . . 57 FIGURES Figure 1. Effect of Mineralogical Character of Aggregates Upon Creep and Shrinkage (Reference 28). Figure 2. Effect of Composition and Fineness of Cement Upon Creep and Shrinkage (Reference 28). Figure 3. Effect of Air Entrainment on Shrinkage of Concrete (Reference 35). Figure 4. Effect of Continued Mixing and Retempering on Shrinkage of Concrete Containing Niles Valley Aggregates (Reference 35). Figure 5. The Effect of Relative Humidity on Shrinkage for Various Ages After Removal from the Mold at Two Days (Reference 63). Figure 6. The Effect of Relative Humidity on Shrinkage for Various Ages After an Initial 624 Days in Water (Reference 63). Figure 7. Total Drying Shrinkage Versus Maximum Curing Temperature for Different Steaming Periods (Reference 37). Figure 8. Effects of Moisture Conditions of Storage Upon Creep and Shrinkage (Reference 28). Figure 9. Effect of Temperature Conditions of Storage Upon Flow (Reference 129). Figure 10. Figure 11. Figure 12. Figure 13. Figure 14. Effect of Temperature Conditions of Storage Upon Shrinkage (Reference 129). Variation of Shrinkage with Time for Unsealed Specimens Maintained at Various Constant Temper- atures (Reference 135). The Variation of Specific Creep with Temperature and Time After Loading for Sealed Concrete First Loaded at 10 Days (Reference 139). Typical Curves of Total Strain Under Sustained Stress at Various Temperatures -- Sealed Speci- mens (Reference 140). Total Strain at 210 Days Versus Aggregate-Cement Ratios with Stress-Level as Parameter (Reference 160). Figure 15. Concrete as a Composite Material (References 126, 147). Figure 16. Variation of the Degree of Hydration with Curing Time (Reference 126). Figure 17. Schematic Representation of Creep with and Without Moisture Exchange (References 126, 147). Figure 18. Normalized Unit Basic Creep at 100 Per Cent Relative Humidity (References 126, 147). Figure 19. Proposed Rheological Model for Basic Creep of Concrete (References 126, 147). Figure 20. Normalized Unit Drying Creep Under Compression at 50 Per Cent Relative Humidity (References 126, 147). Figure 21. Comparison of Analytical Expressions to Free Shrinkage Data for Mix A (Reference 146). 28 Figure 22. Comparison of Analytical Expression to Free Shrinkage Data for Mix B (Reference 146). 28 Figure 23. Comparison of Analytical Expression to Free Shrinkage Data for Mix S (Reference 133). Figure 24. Comparison of Analytical Expression to Free Shrinkage Data for Mix W (Reference 133). Figure 25. Comparison of Analytical Expressions to Free Shrinkage Data. Figure 26. Comparison of Extrapolated to Estimated Ulti- mate Shrinkage. Figure 27. Normalized Unit Basic Creep in Flexure at 40OF (4.5°C). Figure 28. Normalized .Unit Basic Creep in Flexure at 70OF (21°C). Figure 29. Normalized Unit Basic Creep in Flexure at 110OF (430C). Figure 30. Comparison of Computed and Measured Shrinkage Curves (Reference 29). Figure 31. Comparison of Observed and Calculated Long- Time Creep (References 126, 147). TABLES Table la. Kiln Reactions in Portland Cement Manufacturing (Reference 98). Table lb. Proportions of Major Compounds in the Four Basic Types of Portland Cement. Table 2. Transformations of Compounds by Hydration (Reference 98). Table 3. Crystallographic Data of Calcium Silicate Hydrates (Reference 97). Table 4. Tensile Strengths of Cement Compounds Stored in Water (Reference 100, Page 80). Table 5. Tensile Strengths of Mixes of Cement Compounds with Gypsum and Sand (Reference 100, Page 81). Table 6. Compressive Strength of Mortar Mixes of Cement Compounds (Reference 100, Page 81). Table 7. Comparison of Permeabilities of Rocks and Cement Pastes (Reference 102). Table 8. Creep and Shrinkage Data at Various Temperatures (Reference 136). Table 9. Deformation of Rocks Caused by Moisture Movement. I. INTRODUCTION 1.1 STATEMENT OF THE PROBLEM Concrete possesses many behavioral properties which are, as yet, not thoroughly understood or even completely defined. When used as a structural material the problems associated with this lack of understanding have, to date, been solved by empirical methods. Much has been accomplished toward ad- vancing our knowledge in many areas of concrete technology in recent years. The phenomenon of creep of concrete, conventionally defined as the difference between the time-dependent deformations of a specimen under load and an "identical" unloaded control specimen in the same environment, remains as a classic example of such properties and is the primary concern of this study. Extensive tests aimed at studying the time-dependent deformation of loaded and unloaded concrete and mortar have continued since the first decade of this century. During the ensuing years the definition of, and even the very existence of, creep has been questioned ' and debated by various investigators. As a result of past in- vestigations numerous hypotheses have been presented which attempt to explain the creep phenomena, but most are *Superscript numbers in parentheses refer to entries in References, Chapter VII. deficient in some respect and, as such, do not explain completely this aspect of the physical behavior of concrete. A review, however, of these hypotheses will be of aid in our understanding of the time-dependent behavior. In the conventional definition of creep we find the subtle implication that other behavioral phenomena are interrelated; therefore a study of the one automatically precipitates a need for an investigation of a number of related phenomena. If one were to list all the parameters affecting creep of concrete, one needs list all the items associated with the manufacture and utilization of this material. Many of these factors, however, contribute only imperceptibly. Therefore all major discussion will be restricted only to factors which are considered to have primary influence. In addition to the obvious (at least upon superficial study) interrelated effects of different mix proportions, constituent materials, curing conditions, and stress conditions, there are factors such as moisture exchange and variation, temperature change and variation, specimen size and shape, and admixtures. An under- standing of the interrelation of all the pertinent factors which can alter the creep properties of concrete is an essential prerequisite to the development of a method of predicting the time- dependent behavior of concrete under a sustained load. Many investigators have presented, as a result of analysis of measured behavior, methods and formulas for the prediction of creep of concrete. A study of these prediction methods, even though they are not all completely ade- quate for general use, will provide added insight into the requirements of a thorough prediction technique. In the following text a discussion of the more critical parameters involved in the various hypotheses formulated for the explanation of the creep phenomena will be presented, in addition to a review of much of the pertinent literature available. A review of prediction methods presented by various authors is included. 1.2 PURPOSE OF THE INVESTIGATION The general objective of this in- vestigation was to present a procedure to predict the creep that will occur in a structural concrete under various environmental conditions. It was also proposed that in addition to the primary objective the investigation would yield information concerning the viscous nature of concrete, the mechanism of creep, and a theoretical model repre- senting this behavior. 1.3 TERMINOLOGY 1.3.1 Definitions In the light of the large number of researchers and the numerous volumes of literature available on time-dependent behavior of concrete, a confusion of terminology is possible. The following definitions will be implied when the terms are used in the ensuing discussion. Creep: difference, at any time, be- tween the time-dependent deformations of a specimen under load and an "identical" unloaded control specimen in the same environment. Unit creep: creep strain under unit stress. Basic creep: creep occurring under conditions of no change in moisture content of the loaded specimen. Unit basic creep: basic creep under unit stress. Drying creep: creep, in addition to basic creep, resulting under conditions of a change in moisture content of the loaded specimen. Unit drying creep: drying creep under unit stress. Free drying shrinkage: deformation of plain concrete, in addition to thermal deformation, which occurs in the absence of applied load. 1.3.2 Notation The symbols used in the text are listed for ease of reference. All symbols are defined in the text when first introduced and thereafter only when the possibility of confusion arises. = a constant = a coefficient, may be a con- stant or a variable, also used as subscript or deline- ate aggregate = a constant = a constant = a coefficient, may be constant or variable = a constant = creep coefficient, the ratio of creep strain to initial strain c = E /o = unit creep = cb + cd, also used as a subscript to delineate creep deformation cb = bc/a = unit basic creep cd = Edc/o = unit drying creep D = maximum dimension of the aggregate or the granular material d = effective depth of a beam; also used as a coefficient, may be a constant or a variable; also used as a subscript to delineate drying E = modulus of elasticity E = reduced modulus of concrete e = base of the natural loga- rithm = 2.718 F(R) = a function to account for the effects of ambient relative humidity on shrinkage f' = compressive strength of concrete, psi g = constant representing the volume of gel produced upon full hydration of a unit volume of cement = 2.2 h = degree of hydration of the cement K = a coefficient, may be a con- stant or a variable n = an exponent, may be a con- stant or a variable P = percent of cement paste in c the concrete P = proportion of portland cement replaced by admixture, on a volume basis p = ratio of the area of tension reinforcement to the effec- tive area of concrete, also used as a subscript to delineate cement paste p' = ratio of the area of com- pression reinforcement to the effective area of concrete q = an exponent, may be a con- stant or a variable R = ambient relative humidity, in percent S = surface area of the concrete exposed to drying in in.2 s = a subscript used to delin- eate shrinkage deformation t = time, days V = volume of the concrete specimen exposed to drying in in. 2 V a = (volume of aggregate)/ (volume of concrete) = volume- concentration of aggregate V = volume concentration of C cement V = volume concentration of g granular material V = volume concentration of m inert mineral admixture V = volume concentration of s sol ids V = volume concentration of air (voids) V = volume concentration of w water V = volume concentration of hydrated cement including the gel pores V = volume concentration of uc unhydrated cement w = weight of concrete, lb/ft3 a = exponent representing a function of the surface to volume ratio, also spring compliance in the rheolog- ical model when used with a subscript a = a function of i and E of the constituent materials 8 = gel compliance factor AE-P = the change in shrinkage r occurring when the admixture content is in- creased from 0 to P percent by volume r A(w/c)o-P = the change in the water- r cement ratio accompanying the increase of admixture from 0 to P percent by volume e = total creep strain at any time, t S = elastic strain of concrete e under unit stress e. = total initial elastic strain 6 = free drying shrinkage strain 5 at time, t i . = free drying shrinkage strain in 50 percent relative humidity at time, t E = creep strains on the extreme compression fiber E = creep strain of hardened cement paste Cbc basic creep strain at any time, t edc = drying creep strain at any time, t e = free drying shrinkage of sp hardened cement paste E£ = ultimate free drying shrinkage strain at t = E = ultimate shrinkage of con- crete per 1 percent of cement paste i = Poisson's ratio, also used as a multiple = 10-6 a = stress (normal stress unless more specifically defined) a = actual sustained stress on the concrete 0sh = magnitude of sustained stress which will give a creep strain equal to the shrink- age of the concrete at the given relative humidity T = retardation time of Kelvin dashpot when used with a subscript 4 = fluidity of the dashpot or viscous element in the rheological model 4c = curvature in a beam caused by creep of the concrete 4. = initial curvature in a beam Osh = curvature in a beam caused by shrinkage of the concrete II. HISTORY OF RESEARCH 2.1 GENERAL There are on record investigations of time-dependent behavior of plain concrete which were initiated as far back in the technological history of concrete in the United States as 1905 when Woolson(3) reported on the flow of concrete under pressure. In 1906 Muritz reported the study of the time-dependent behavior of reinforced concrete beams. Since that time there have appeared, in the literature, various reviews of research and annotated bibliographies on creep or shrinkage or both. Extensive work by many notable researchers both in the United States and abroad during the first six decades of this century has been reviewed in a report on the rheology of concrete by Ali and Kesler.(5) This review, published in 1965, presents a critique of 207 publi- cations dealing with creep of plain and reinforced concrete which appeared in the literature up to about 1962. Other bibliographic information is available in a publication by the Cement and Concrete Association which contains a bibliography of more than 800 references on factors influ- encing creep and shrinkage of concrete. In addition the bibliographies by Corley,(7) Meyers, (8) and Slate,(9) and the references in the publications of L'Hermite, (10) Fluck and Washa,(ll) and Wagner, collectively provide a reasonably complete listing of all the creep and shrinkage research publica- tions up to 1964. The present dis- cussion aims at supplementing the review by Ali and Kesler(5) and adding to the history additional information found in the literature to date. The following review is by no means complete but is considered comprehensive. 2.2 SHRINKAGE OF CONCRETE White, (13) in 1911, recorded that Considere, (14 in 1899, studied the behavior of neat cement bars and mortar over a period of 63 days, and reports shrinkage in air and expansion in water ranging from 0.132 to 0.050 mm/100 mm, and 0.079 to 0.028 mm/100 mm, respec- tively. Campbell and White,(15) in 1906, presented a study of neat cement which showed 0.19 percent expansion after five years in water and 0.39 percent contraction after five years in air. White(13) himself contributed to the pioneering study of the varia- tion in volume and length which accompanied changes in moisture content of concrete. In 1913 Goldbeck(16) presented one of the earliest reports of field ob- servations of expansion and contraction of concrete roadways, in order to obtain data to permit more efficient spacing of expansion joints, followed by a report presented in February, 1914 to the National Conference on Concrete Road Building in Chicago by Wig, et al. (17) The latter reported that the condition and character of the sub-base, the moisture transmitted to and from the the concrete, the action of frost, and friction between the slab and sub-base all contribute to or influence expan- sion and contraction. Jesser, in 1913, reported on one of the earliest tests of volume changes using a controlled relative humidity environment. Matsumoto, 19) in 1921, demon- strated the relationship between absorbed moisture and the amount of expansion of mortar and concrete. Pearson, 20) in 1921, observed shrinkage of portland cement mortars and concluded that added aggregate produces lower shrinkage; Chapman,(21) in 1924, added a more detailed study of the influence of aggregate upon shrink- age to the ever growing list of reports dealing with volume changes of concrete. Hatt 22) in a 1925 report listed what may be considered a concise state- ment of what was known, as a result of research during the first quarter of the twentieth century, about volume changes. He listed the following factors which were then thought to influence shrinkage and expansion of concrete: (a) Cement composition (b) Fineness and quality of aggregate (c) Wetness and proportioning of the mix (d) Specimen size and forms (whether absorptive or not) (e) Exposure (relative humidity and temperature) (f) Specimen age and duration of test (g) The base for measurements (i.e., before or after initial or final set) In addition, Hatt recommended the following precautions to minimize volume changes: (a) Minimize the amount of mixing water (b) Provide early protection and curing (c) Use rich concrete to inhibit ingress and egress of water (d) Avoid the use of aggregates containing dust or expansive material A review of the literature on shrinkage following this first 25 years shows that the trend is toward defining the shrinkage mechanism itself and toward developing methods of prediction of concrete shrinkage. (23) In 1935 Lyse reported an in- vestigation which studied the effects of the quality, quantity, and degree of hydration of cement paste on shrinkage of concrete in addition to the effect of type of cement used. In 1943 Ross(24) presented a rheological model to describe the behavior of concrete. The model he proposed consisted of three elements. The first element was a spring which deforms instantaneously upon applica- tion of load. Connected to this spring, in series, was a parallel arrangement of the other two elements, a second spring and a dashpot with a porous piston, which provided the time- dependent response. The second spring was placed within the cylinder of the dashpot so that it supported the porous piston. The cylinder was filled with a viscous liquid of fluidity, 4. To depict the shrinkage response, a fictitious stress was assumed to act on the piston. Thus the resulting equation for evaluating shrinkage of concrete has the form: E = caa 1-e / , (1) where E = free drying shrinkage strain at time, t; a = a fictitious stress causing shrinkage, or force in the model causing shrinkage; a = spring compliance in the rheological model; e = base of the natural logarithm = 2.718; = fluidity of the dashpot or viscous element in the rheological model; and t = time, days. The coefficients for this rela- tionship were derived directly from measured data. In 1944 an extended study by Ross, 25) designed to show the effects of shape and size on shrinkage, re- sulted in the conclusion that the shrinkage varies enormously with the size and shape and is related by a function of the ratio of the surface to the volume. In 1956 Picket(26) proposed a relationship of the form: s = a (1-V )n s sp a where e = free drying shrinkage of hardened cement paste, V = volume concentration of a aggregate, and n = a function of V a On the basis of his test results, he concluded that the first shrinkage cycle is greater than any subsequent shrinkage or expansion caused by moisture movement, and that for a given amount of aggregate the shrinkage is proportional to the water-cement ratio. He also noted that after the first shrinkage cycle, volume changes are for the most part independent of water-cement ratio. Dutron, (27) in 1957, studied shrinkage in conjunction with creep and proposed that for constant tempera- ture and relative humidity and for slow progressive drying the following general formula provided a reasonable approximation of shrinkage: S= e 1-V K) (v + V s sp \ g / \ c m - n + V + V,. 1-e ,' , w v/ \ / where V = volume concentration of g granular material, K = coefficient depending on the maximum dimension, D, of the granular material, expressed in millimeters according to the formula K = 0.4 log (D + 1) for D < 40 mm, V = volume concentration of cement, V = volume concentration of inert m mineral admixture, V = volume concentration of water, V = volume concentration of air v (voids), and q and n vary with the relative humidity, composition of the con- crete, and the specimen dimensions. He also presented a equation for adjusting the shrinkage at 50 percent relative humidity to shrinkage at other relative humidities. This equa- tion will be dealt with in the section on prediction of shrinkage in Chapter IV. Troxel, et al., (28) in 1958, reported a very extensive test series to determine the effects of aggregate- cement ratio, composition and fineness of cement, mineralogical character and size of aggregates, magnitude of stress, curing, size, age at and duration of loading, and reinforcement on shrinkage and creep. These authors present ex- tensive data and analysis to support their conclusions with respect to each of the above mentioned factors. Shideler,(29) in 1958, reported on tests of shrinkage of concretes made with various lightweight aggre- gates . In 1959 Alexander and Wardlaw(30) described experiments which show that when water is lost continuously from cement pastes and mortars the resulting shrinkage takes place in two stages. Curves of drying shrinkage, in percent, versus water loss, in percent, show a definite cusp in the area of from 40 to 60 percent water loss. They con- clude that increasing the duration of moist curing increases both Stage I and Stage II drying shrinkage, and that increasing the percent of mixing water produces a linear increase in drying shrinkage. This increase in shrinkage caused by raising the water-cement ratio is said to be primarily a result of water loss in Stage I, and the drying shrinkage occurring during Stage I I water loss is independent of the water-cement ratio. In 1959 Alexander and Wardlaw(31) also reported results of tests to determine the effect of powdered minerals and fine aggregate on shrink- age of cement paste. Different water- cement ratios, paste contents, and aggregates were used. Among their results the authors cite two simulta- neous changes which occur as a result of replacing some portion of the cement with an "inert" mineral at cement fineness such as powdered basalt. One is the change which may occur in the water-cement ratio. There is also an additional change resulting from a variation in the admixture cement ratio. Thus, if part of the cement is re- placed by powdered mineral and the water-cement ratio held constant, the ultimate shrinkage decreases as a result of the decrease in total paste content and increased restraint from the added solids. If no change is made in the amount of mixing water when the powdered mineral is added, the ultimate shrinkage will increase as a result of the increased water-cement ratio. These effects are represented by the following equation: log A E roP = log A(W) r o-P r + a'P + b' r where A o-p = the change in shrink- r age when the admixture content is increased from 0 to P percent by volume; r A () = the change in water- o-P cement ratio accom- panying the increase of admixture from 0 to P percent by volume; P = proportion of portland cement replaced by powdered admixture, on a volume basis; and a', b' = constants. Keeton,(32) in 1960, as a result of analysis of shrinkage data, con- cluded that the shrinkage versus surface area per unit volume relation- ship can be reasonably extrapolated both to larger and smaller sizes. This conclusion is based on a study of only three specimen sizes, and therefore should be interpreted as being only roughly approximate. Lyse, 33 in 1960, proposes the following relationship for shrinkage: E = E (l-e "t)P , (5) where eo = ultimate shrinkage of con- crete per 1 percent of cement paste, n = an exponent defining the rate of shrinkage and de- pending on the ambient relative humidity, and P = percent of cement paste in the concrete. Based on his test results, Lyse concludes that shrinkage of concrete is approximately proportional to the amount of cement paste regardless of its composition. The above equation is considered as giving reasonably accurate estimates of shrinkage deformation after e and a have been uo determined experimentally. In 1961 Higginson(34) studied the properties of concrete steam-curing at various temperatures. His results indicate that the shrinkage decreases with increasing temperature of steam- curing and with increasing length of time of steam-curing. He also notes that the weight loss during drying showed no significant differences com- pared to moist cured specimens despite differences in the drying shrinkage. Polivka,(35) in 1962, reported a study of the effect of aggregate type and continued mixing and retempering on the shrinkage and cracking character- istics of concrete. His results indicate generally that aggregate type and source have a major influence on shrinkage and that continued mixing and retempering increases shrinkage. His results also indicate that air entrainment reduces shrinkage somewhat. Campbell-Allen,(36) in 1963, presented a discussion of the variables which he believes have a major effect on shrinkage. The variables are listed as mix proportions, cement properties, aggregate properties, compaction, curing conditions, specimens size, ambient conditions, and time. In 1964, Christensen and Skovgaard(37) reported that low-pressure steam-curing at high temperatures, about 80°C (176°F), can reduce the shrinkage of concrete by as much as 50 percent. Rutledge and Neville, (38) in 1966, studied the deformational characteris- tics of lightweight aggregate concrete and found that its shrinkage is con- siderably higher than concrete made with normal-weight aggregate exposed to the same conditions. A comprehensive four-year study on the influence of size and shape of members on drying shrinkage is reported by Hansen and Mattock(39) in 1966. Their results, from specimens which covered a range of values of volume to surface ratio from about I in. to 6 in., showed that though this parameter does not reflect perfectly the variations of both size and shape, it is satisfactory for purposes of practical design. Hansen and Nielsen, in 1966, presented a theory of the influence of aggregate properties on shrinkage of concrete. A relationship is presented from which the shrinkage of a concrete may be computed if the volume concen- tration of aggregate, the shrinkage of the cement paste, the shrinkage of the aggregate, and the modulus of elasticity of both the cement paste and the aggregates are known. The authors agree that the shrinkage of concrete is influenced by the modulus of elasticity of normally acceptable aggregates, but they state that it is their belief, "that the importance of such variations often is exaggerated." Basically, the approach used by the authors is that presented by Pickett.( They have extended this theory to include the effect of the shrinkage of aggregate and the modulus of elasticity of both the cement paste and the aggregate. 2.3 CREEP OF CONCRETE Tests by Woolson,3) in 1905, of concrete filled, steel, thin wall tubes indicated that concrete will "flow" under a sustained load. Though this is not strictly an example of creep of plain concrete, nevertheless it did point out a most interesting behavioral phenomenon which has stimulated exten- sive research since that time. Early studies of the creep phenomenon(4,41) made no attempt at controlling the ambient conditions or other major parameters, and were con- cerned only with gross deformations caused by sustained loads. In addition, all early tests deal ing with this phenomenon were conducted on reinforced concrete. One of the earliest studies reporting the simultaneous observation of time-dependent deformation of both loaded and unloaded specimens was that by McMillan(42) in 1915. Smith, '" in 1916, used deforma- tions measured on four gage lengths 8 in. apart on the top of a reinforced concrete beam under sustained loads, and showed that the flow of concrete was proportional to stress. In addition, Smith also studied deformations of 6- x 24-in. companion cylinders both loaded and unloaded; the purpose of the unloaded cylinders being to obtain "contraction data for correcting the deformation of the loaded cylinders." In this paper Smith also makes mention of the beneficial aspects of the "flow" of concrete, using the example of a concrete beam with confined or restrained ends subjected to a temperature stress. He notes that since the temperature expansion (caused by ordinary air temperature fluctuations) is slow, there is time for flow to occur and thus relieve the stress by an amount which may be from 25 to 80 percent of the computed temperature stress. Goldbeck and Smith, 44) in 1916, found that creep of concrete under water was less than that exposed to air and that a part of the creep was recoverable. Davis, (45) in 1928, reported testing of plain concrete cylinders to study the effect of mix proportions, gradation of aggregate, age, stress, and moisture conditions on flow. He reported that saturated concrete flows less than that which is dry and reaches equilibrium more quickly, and that larger deformations occur with leaner mixes and higher water-cement ratios. His conclusion with respect to creep of wet and dry specimens is not in agreement with the results of other investigators. His erroneous conclu- sion is probably a result of the fact that his so-called dry specimens were actually drying and as a result the drying creep he measured was greater than the creep (basic) of the saturated specimens, as it should have been. The study of shrinkage has also prompted researchers to interpret the time-dependent deformational response to sustained load as nonuniform shrinkage. Maney1l) stated that the confusion lay in incorrect interpreta- tion of the differential time- dependent deformations of loaded and unloaded specimens. In 1960, Ross 46) investigated the effects of using fly ash as a replacement for cement in ordinary gravel concrete. Cylinders 4-1/2 in. in diameter and 12 in. long were used for both creep and shrinkage tests. He concluded that creep was unaffected by the fly ash for mixes using 10, 17.5, and 25 percent of fly ash. Laginha Serafim and Guerreiro,(47) in 1960, investigated the effects of temperature on mass cured concrete prisms. Two ambient temperatures were used: 45°C (113°F) and room temperature. Their results showed an effect of the high temperature only during the age of four to seven days after loading. Keeton,(32) in 1960, conducted creep and shrinkage tests on specimens of three different sizes and at relative humidities of 20, 50, and 100 percent with a constant temperature of 73.40F (23°C) for all tests. He concluded that the creep strains did not appear to be greatly influenced by size in the range of sizes used (3-, 4-, and 6-in.-diameter cylinders) and that creep increased with decreasing humidity. Lyse, 48 in 1960, after a study of the effect of the amount of cement paste in the concrete proposed the following formula which he states is to be used for practical design purposes. (I -nt) (1 - = E 1-e 1 c uo + p , s h\ in which E uo (l-e-nt) where =e total creep strain at any time, t, o = actual sustained stress on a the concrete, a sh = magnitude of sustained stress which will give a creep strain equal to the shrink- age of the concrete at the given relative humidity. The unknown factors in this creep equation are c uo, a, and osh. The value of uo must be obtained from a family of experimental shrinkage curves for a particular time of observation t and where a is evaluated using values at various percentages of relative humid- ity. The value of osh, which gives a creep deformation of magnitude equal to the shrinkage at the same relative humidity, must be evaluated from an experimentally determined plot of stress level versus relative humidity for the particular materials in question. Ishai ', in 1962, reported a study of the influence of sand concen- tration on the deformational character- istics of slender mortar beams. The beams were loaded to one, two, four, and six times the dead load moment and observed for 130 days. Ishai concluded that both deflection magnitude and rate were higher for the mortar with the higher sand concentration. Addi- tional beams were tested in cycles consisting of one week under load followed by about three weeks recovery. The results of these tests showed the irrecoverable creep increased with each cycle but at a reduced rate. Glucklich and Ishai (50) in 1961, reported results of both long- and short-time creep tests of cement paste. They postulated a mechanism to explain permanent set as being one in which increased hydration deposits products on the surfaces of cracks, newly formed as a result of the sustained load, which prevents the closing of such cracks upon removal of the load. To be true, such a mechanism requires time for additional hydration. Their tests showed that for up to eight hours of loading there was almost no irrecov- erable creep. They also concluded that recoverable and irrecoverable creep are a result of two distinct mechanisms. Thorne,(51) in 1961, reported a decrease in the elastic modulus of concrete as the temperature rises and comments that for mass structures the stress relief afforded by creep could be considered, thus reminding us of the beneficial aspect of creep of concrete. Neville, (52) in 1962, presented a paper at a meeting of the Reinforced Concrete Association in London which was a summary of the then current knowledge on creep and shrinkage. He stated that ambient relative humidity, specimen size, strength, age at loading, type of cement, fineness of cement, and water cement ratio are factors which will affect creep. The upper limit at which creep remains proportional to applied stress is given as one-half the ultimate strength. Gluckl ich and Ishai ,(53) in 1962, reported that the results of their tests show a close connection between evaporable water content and the shrinkage, instantaneous deformation, and creep. They report that a linear relationship exists between the gel water and creep rate and note the almost complete absence of creep in specimens with most of the evaporable water removed. Campbell-Allen, (36) in 1963, presented a summary of the factors which affect volume changes and creep which he says shows that these pro- perties both depend on all the variables associated with concrete manufacture and use. Mullen, (54) in 1963, reported the results of creep tests on sealed, paste hollow cylindrical specimens with a wall thickness of 0.1 in. His results of tests on specimens dried at 1 10°C (230°F) to constant weight, loaded in compression, exhibited no load induced time-dependent deformation. Richard,(55) in 1964, reviewed data from 76 different concretes. It was shown that relatively high values of creep and drying shrinkage are not always associated with lightweight aggregates. He concluded that the most significant factors affecting creep of concrete are stress-strength ratio, which overshadows all other variables, and the type of aggregate used . In 1964, Zia and Stevenson(56) reported on a series of creep tests on tee, triangular, and square cross- sectioned members with four different stress distributions, namely, uniform with a magnitude of 2000 psi com- pression, linear varying from 0 to 2000 psi compression, another linear varying from 150 psi tension to 2000 psi compression, and trapezoidal vary- ing from 200 psi to 1200 psi compres- sion. The primary objective of the study was to determine the effects of a nonuniform stress distribution and the shape of the cross section on the creep magnitude. The results of the investigation indicated that the specific creep under a nonuniform stress distribution may be related to that under uniform stress by a factor R to be determined for the particular nonuniform stress distribution. Their results showed that the greatest specific creep response occurred at the most highly stressed fiber in the specimen with the largest stress gradient. The values of Rt suggested for the third stress distribution listed above under long time load were 1.27 for the tee section, 1.44 for the triangular section, and 1.2 for the square section. The mechanical model used for uniform stress distribution consisted of two Kelvin bodies connected in series. Calculation of time effects on deflections of a prestressed con- crete beam, using the method proposed by Scordelis, et al., was modified to include the effects on creep of nonuniform stress. Their computed camber at 100 days is approximately 2.26 times the initial camber. If the effect of nonuniform stress is neglected the computer camber Is approximately 1.96 times the initial camber. The report also contains an annotated bibliography on time-dependent effects in concrete consisting of 64 references. Neville, 58) in 1964, presented a relationship for creep of concrete as a function of its paste content. He proposed the following creep relation: d E a c m dV = 1-V a ' c a in which a = p m 1 + p + 2 (1-2pa) E/E p awhere where , (8) a = a function of p and E of the constituent materials, p = Poisson's ratio of cement paste, a = Poisson's ratio of aggregate, E = modulus of elasticity of the cement paste, E = modulus of elasticity of the aggregate, which is patterned after Pickett's shrinkage equation [Equation (2). Con- sidering a to be a constant is not a serious error, according to Neville, and he gives values of E/E = 1/2, a Ia = 0.12, and p = 0.10, which yields a value of am = 1.37. Integration of Equation (7) yields: S= Sp (-Vma) m (9) c cp a (9) where c is the creep strain of hardened cement paste. Neville then uses the hyperbolic time function first proposed by Ross(59) and relates the parameters of that equation for two different mixes using the value of a m corresponding to the time used, thus: (A + B t)(1-V )a m 1 1 al = (A + B t) (-V ) m. (10) 2 2 a2 With a known for two values of t, m both A and B can be determined from 2 2 A and B ; or alternately using only 1 1 one value of a and the assumption m (59) proposed by Ross that A = 100B, B 2 can be determined directly from B . 1 A prediction of creep for any mix at any time t is thus possible by using the creep data of one basic mix and a knowledge of the value of a for the m aggregate used. At the International Conference on the Structure of Concrete held in London in 1965, many papers dealing with both creep and shrinkage of con- crete were presented by many notable men whose research in the area of volume changes in concrete is well known. (See References 60 through 67.) These papers, in general, present a summary of their work to date. Illston, 68) in 1965, reported on creep tests of concrete in tension. The effects of stress level, age at loading, time under load, and ambient humidity was studied and compared to behavior under compressive stress. In general the influence of the factors studied was the same for both stress conditions; however, the rates of deformation were slightly different. His tests show linearity of stress and creep strain for stresses less than 50 percent of the ultimate strength. The tension tests show slow delayed elastic strain and also that the strain due to the second cycle of load is unaffected by a previous cycle under stress of the opposite sense. In 1965 a series of three technical reports by Keeton from the U.S. Naval Civil Engineering Laboratory, Port Hueneme, California covered five phases of a study of shrinkage and creep of specimens varying from small cylinders to full size cross section of I, box, and tee beams. The report of phase 1 69) studied the effect of specimen size, storage humidity, and stress magnitude on both shrinkage and creep for one concrete mix. The con- clusions reported are that creep is a result of viscous flow and gel water seepage and that the time-dependent properties of different size and shape specimens can be estimated using exposed surface area to volume as a parameter. Techniques and equations are developed to predict time-dependent properties of any specimen with the same concrete mix as was used in the study or using short-time creep tests from the actual mix. The phase 2 re- port(70) is basically that of the study of the same variable as in phase 1, except the specimen used is a hollow-box beam. The phase 3( report covered the study of prestressing losses on small rectangular beams of sufficient size to provide surface area to volume values for extrapolation curves. The effect of this parameter on shrinkage of cylinders, prisms, tee- sections, and I-sections was studied in phase 4. (71) Phase 5,(71) in addition to the creep and shrinkage study, contains a report of the use of photoelastic coatings on concrete as a method of measuring strains. Some examples of maximum shear strain contours for specimens which were allowed to shrink in 20-percent relative humidity are shown. Neville and Staunton, (72) in 1965, presented analytical and numerical methods for computing creep of concrete when the stress-strength ratio varies with time which are based on a "stand- ard" creep curve for a concrete of constant strength. The methods pro- posed consider either a change in strength or a change in stress. England, 73) in 1965, also pre- sented a method of estimating creep and shrinkage of concrete from the properties of its aggregate and matrix. The model proposed by England is a two- phase material consisting of unit cubes of aggregate surrounded symmetri- cally by a matrix which can be either cement paste or mortar. Factors in- cluded are volume concentration of aggregate, elastic modulus of both matrix and aggregate, creep of both matrix and aggregate, water-cement ratio, degree of cement hydration, con- ditions of storage, and absorption of the aggregate. An experimentally determined creep time curve for the matrix is used to predict the time- dependent strains for concretes of many mix proportions but containing the same matrix. The model in the proposed form has not been used for aggregates that creep, or which are naturally absorbent, and the authors conclude that further experimental data is needed for such cases. The comparison of test data to values predicted by the model are in very good agreement for those mixes investigated by the author. Test results on concretes con- taining cast iron, glass, or flint gravel as aggregate showed that creep strains were less than those of the parent matrix; whereas for concrete using polythene (polyethylene) as an aggregate the creep strain was 2.8 times greater than .the parent matrix for aggregate concentrations of 50 percent. Creep in concrete and damping in concrete when subjected to cyclic stress are thought to be aspects of the same basic phenomenon of the inelastic behavior of concrete. On this premise investigations into the possibility of correlating the creep and dynamic properties and of predicting the former from the latter were conducted.(164-166) Normalized unit creep values at a given age of loading were found to be corre- lated with the saturated-state loga- rithmic decrement, but were independent of the dynamic modulus of elasticity. The correlations obtained were for a particular mix at a given age of loading and no reliable means of extending a prediction method which could cover a wider range of possible mix proportions for any age if loading could be found. III. FACTORS AFFECTING CREEP AND SHRINKAGE 3.1 GENERAL When one considers the complexity of concrete and its time-dependent structural and chemical nonhomogeneity, the conclusion that any change in its composition or its environment will, as a consequence, affect its volume stability is inescapable. The role of research on creep and shrinkage, generally, has been to determine the degree of influence of the various parameters on the time-dependent behavior of concrete. The parameters which are generally considered signifi- cant, though inseparable in actuality, will be dealt with separately in the following discussion under eleven general categories. The discussion in each of these categories will deal with the effect of the particular parameter on both shrinkage and creep. The order of these categories is not intended as a measure of their degree of influence, but was chosen only for convenience of study as follows: (1) Mix Constituents and Proportions (a) Aggregate: Content Gradation Particle size and shape Permeability Absorption Mineral composition Modulus of elasticity Unit weight Time-dependent properties (b) Cement and water: Paste content Type of cement Chemical content or composition of cement Fineness of cement Water-cement ratio (c) Admixtures (2) Mixing Time and Consolidation (3) Age at Loading, Degree of Hydration, and Curing (4) Moisture Content (5) Level, Type, and Distribution of Stress (6) Ambient Air Velocity (7) Ambient Humidity (8) Ambient Temperature (9) Size and Shape of Specimen (10) Strength (11) Reinforcement 3.2 MIX CONSTITUENTS AND PROPORTIONS In a composite material such as concrete the main variables are the constituent elements which, when taken together, comprise the whole. In order to properly understand how the com- posite material will respond to partic- ular stimuli, the effect of each com- ponent upon that response is of primary importance. 3.2.1 Aggregate In 1924 Chapman,(21) after studying the influence of aggregate upon shrinkage, concluded that the relation between cracking (as a result of shrinkage) of concrete and the kind and proportions of aggregate is a matter deserving of much more extended and careful study. Research since that time reveals that aggregate will have a primary, dual effect on both creep and shrinkage of conrete. First, the volumetric content, gradation, and particle size of aggregate in a given mix and second, the composition and mechanical properties of the aggregate are all known to have an influence to a greater or lesser degree. Work by Neville(58) and Pickett(26) demonstrated that the volumetric con- tent of aggregate in concrete is re- lated to creep and shrinkage, respec- tively. The volumetric concentration of the aggregate in a given mix is one of the principal variables used in the prediction formulas for creep and shrinkage presented in this report. Gradation and particle size may be considered under the same category as volumetric content, since all three are interrelated in a given mix. Research by Davis and Davis 74) indi- cates that, "for the same richness of mix, the flow is greater in a concrete for which the aggregate contains a large amount of fines (resulting in a higher water-cement ratio) than in a similar concrete for which the aggre- gate has less fines (resulting in a lower water-cement ratio)." Davis(75) reported creep increases the leaner the mix and smaller the fineness modulus. Other research using 1-1/2-in, and 6-in. maximum size aggregate indicated lower creep values for the larger aggregate size; however, the unit paste content for the large aggregate was lower. It is evident that the results of these tests(74-76) are probably a consequence of changes in parameters other than and including aggregate gradation and particle size. More recent opinion 2) indicates that the primary influence of variations in gradation and particle size and shape is in the effect on the aggregate con- tent and as a result on the paste con- tent. The more finely graded aggregate also tends to have less thickness in the paste layer between individual aggregate particles, which is considered as a part of the reason for the smaller dimensional changes. In 1930, Davis 77) reported that variations in the type of mineral aggre- gate have a marked effect on the volume changes of concrete due to variations in moisture condition and Davis and Davis74, 78) reported that the charac- ter of the mineral aggregate has a marked effect on flow and listed in order of increasing flow sandstone, basalt, granite, quartz, and limestone. In 1934, Troxell 79) stated that the relative shrinkage of concrete made of gravel (with a large percentage of sandstone particles) sandstone, trap rock, slag, granite, quartz, or lime- stone decreases in that order. As a result of analysis of data from a number of sources, Shank(80) derived creep coefficients of concretes contain- ing different aggregates to be, for silica 0.085, for gravels and limestone 0.13, for sandstone 0.16, and for granite 0.3. Davis and Troxell (81) in 1954, reported that the mineral character of the aggregate has a larger influence on drying shrinkage and creep than is generally realized. Con- cretes containing hard, dense aggregates of low absorption and high modulus of elasticity exhibit less creep and shrinkage than do concretes having weaker, less rigid more absorbent rock types. Troxell , et al. , reported creep was greatest for concrete using crushed sandstone aggregate followed by basalt, gravel, granite, quartz, and limestone in descending order of magnitude; shrinkage was greatest for concrete using crushed sandstone also followed by gravel, basalt, granite, limestone, and quartz in descending order (Figure 1). Powers(82) states that not all aggregates have the same restraining effect, and the less com- pressible the particles the greater the restraint on shrinkage. This re- straining effect was also pointed out by Carlson. (83) Shideler(29) showed that a light-weight coarse aggregate such as expanded shale may give as much as 38 percent more shrinkage than a concrete of the same strength made with ordinary weight aggregate. RUsch, Kordina, and Hilsdorf 84) also demon- strated the influence of the petrolog- ical type of aggregate on creep. Their tests indicate that this influence acts primarily through the modulus of elasticity. They list the following aggregates in increasing order of creep magnitude as: basalt, quartz, gravel, marble, granite, and sandstone. Their results also show that the absorption of aggregate and its modulus of elasticity are related. Lyse concluded that the shrinkage of concrete is approximately proportional to the amount of cement paste which implies dependence upon the volume concentration of the aggregate. Ishai(85) reports creep magnitude and rate of mortar beams loaded in flexure increased with increase in sand concentration. In a summary of critical parameters in volume changes and creep, Campbell- Allen 36) indicates that the restrain- ing effect of aggregate is of great importance in conjunction with type, mineral composition, and surface con- dition of aggregate. Aggregate size and concentration affect drying shrinkage as they in- fluence the total amount of water and cement in the mix and further, concretes made with aggregates of loosely cemented granular structure such as some sandstones creep more than those made with dense aggregates such as quartz or limestone. Washa lists mix proportions and the mineral composition of the aggregate among the more important factors that influence drying shrinkage and the maximum size, grading, and mineral character of aggre- gate as all having appreciable effects on creep of portland cement concretes. Philleo 88) states, "Work at the Water- ways Experiment Station of the Corps (76) of Engineers and at the University of California 89) has demonstrated that the creep of sealed specimens, having identical water-cement ratios and air contents in the mortar phase but different amounts and sizes of a given coarse aggregate is proportional to the paste content." Detailed discussions of the effect of aggregate on shrinkage and creep can be found in References 90 and 91. Clay present in poorly washed aggregate will increase the shrinkage of the resulting concrete considerably. Hveem and Tremper(92) report that con- crete made with dirty sand may shrink twice as much as concrete made of the same sand when clean, and unwashed coarse aggregate increased shrinkage as much as 70 percent over the thoroughly washed aggregate. According to Hansen the different permeability, and its in- fluence on the rate of moisture move- ment, of various aggregates is respon- sible for the differences observed in the creep and shrinkage behavior of concretes made with different aggre- gates. He suggests that the effect of the rheological properties of the aggregates on creep of concrete is negligible when these aggregates have little or no creep themselves. In summary, it can be stated that the primary influence of aggregate on the shrinkage and creep characteristics of concrete is in its restraining effect on the potential volume insta- bility of the products of hydration of the cement paste. Variations in particle size and grading as such permit the use of leaner mixes, thereby reducing the magnitude of these de- formations. Different petrological types of aggregate will present varying degrees of restraint which depend on their modulus of elasticity. The pore character of aggregate indicates the amount of water they can absorb, the rate of absorption and drying, and its strength and elasticity, and thus porosity of aggregates becomes a significant factor with the more porous aggregates enhancing shrinkage and drying creep of concrete. The inference should not be made that the values associated with a particular concrete aggregate, classi- fied as granite, limestone, etc., in the previous paragraphs, may be ex- pected to occur in the same order of magnitude in another concrete contain- ing aggregate with the same petrolog- ical name. No rock name implies any- thing specific insofar as characteris- tics such as permeability and/or creep properties are concerned. 3.2.2. Cement and Water Cement and Paste Structure The role of hydrated cement as the prime agent responsible for the volume instability of concrete and mortar has long been accepted. The preceeding discussion on the effect of aggregate on volume changes emphasizes the role of hydrated paste in concrete. The reason for volume changes of the hydrated cement can be better under- stood if one first looks at the general physical and chemical nature of the gel formed when cement hydrates. Portland cement is composed mainly of four major constituents (Table 1); dicalcium silicate (C S),* tricalcium 2 silicate (C S),* tricalcium aluminate 3 (C A)* and tetracalcium aluminoferrite 3 (C AF).* About 70 to 80 percent of portland cement is a mixture of alite *Shortened notation, used by cement chemists, describes each oxide by one letter. and belite, the impure modifications of C S and 63-C S;* therefore it is 3 2 reasonable to expect that the predom- inant gel phases in hydrated cement paste would be very similar to the gel phases observed in the pure silicate pastes, and according to Grudemo,(95) this has been shown to be true. isobars** from the principal com- pounds of portland cement, hydrated separately, show that the hydrates of pure C S and C S resemble the hydrates 3 2 of portland cement. According to Bernal, two forms of hydrated calcium silicates are stable at low temperatures having compositions of C SH and C SH . The 1-1.5 2.5-0.5 2 4-2 first of these forms, which is commonly referred to as tobermorite, has been identified with a group of somewhat ill-defined, naturally occurring minerals from Crestmore, California; more recently from Ballycraighy in Northern Ireland and from Tobermory in Mull, Scotland. Because of the structural similarity of the silicate hydrates to this natural mineral, they are sometimes referred to as tobermorite gel. From the study of the chemistry of portland cement and the chemistry of the products of its reaction with water 98) it has also been determined that the reaction of the calcium silicates produces a substance that has a composition (Table 2) and a crystal structure resembling that of the natural mineral Tobermorite; hence, *a-C2S is used to distinguish this low temperature form from that produced at high temperatures. **Isobar: a plot of water content versus temperature at constant vapor pressure. ***l A - 10-10 m. this cement gel is also referred to as tobermite gel. The gel is considered to be the main cementing compound of concrete. Various properties of con- crete such as setting, hardening, strength, dimensional stability, and elastic modulus are dependent upon the properties of its tobermorite gel phase. The literature on low temperature paste hydration of C S and a-C S shows 3 2 that the composition and particle habits are essentially the same for both materials with the exception of the amount of crystalline calcium hydroxide which is higher in the C S (97) 3 pastes. Bernal reports that the cell dimentions for the calcium sili- cate hydrates have been determined for many of the phases. The two forms which are stable at low temperatures, C SH and C SH , are said 1-1.5 2.5-0.5 2 4-2 to occur in the form of extremely thin fibrous crystals similar to those found in gels in general. The struc- tures of these compounds show the presence of a fiber repeat unit of about 7.3 A*** with marked pseudo-halving (Table 3). This is thought to imply the existence of silicate tetrahedra joined by hydrogen bonds. They also show a layered structure, the spacing of which varies from 14 to 9 A, upon loss of water, in a way similar to clay minerals. This characteristic may explain the shrinking property of concrete. These compounds, which seem to represent the main products formed in portland cements hydrated under mild hydrothermal conditions, show both properties of needle crystals, with the property of rapid formation of networks simulating gels, and those of a platelike mineral with its capacity for swelling and shrinking. Another respect in which calcium silicate hydrate crystals resemble swelling clays is the evidence that the lath- shaped needles are extremely thin, probably less than 5 molecules in width or 50 A. This may account for their colloidal properties, their tendency to oriented aggregation, and their ability to hold water in excess of their own bulk. According to Neville,(91) other studies have also indicated the existence of fibrous particles with sheaf-like ends similar to the swelling clay halloysite. Many transitional forms are believed to exist, including some spherical parti- cles, but they finally become fibrous and appear as sheets or foils aggre- gated as fluffy masses. It is interesting to observe that the hy- drates of C S and C S show strength 2 3 developments which are similar to portland cement. The products of hydration in C S (95) 3 pastes cured at normal temperatures all show the same structure. Generally, the structure consists of irregular masses possibly composed of small plates. These irregular masses are identified as being almost certainly very ill-crystallized tobermoritic (99) material. Bogue presents an accurate history of the controversy which existed for about 50 years over the alite of portland cement. It is now generally accepted that the alite is tricalcium silicate. According to Lea,(100) tricalcium silicate has all the essential properties of portland cement. It undergoes initial and final set; mixes of C S and water are 3 less plastic than portland cement and water, and the addition of gypsum renders it more plastic and has some effect on the setting time. If it is kept in moist air the mass sets and hardens producing fair strength. Tables 4, 5, and 6 list some represent- ative strength values for tricalcium silicate pastes and mortars. The tricalcium silicate attains the greater part of its strength in seven days and little increase occurs there- after. The products of hydration of ý-C S(95) paste at 70° and 100°C (158° and 212°F) and a water to solid ratio of 0.5 consists largely of fibrous C SH type tobermorite. The 2 2 pastes prepared at 500C (1220F) were composed of dense aggregates of thin fibers, whereas in those prepared at 100°C (212*F) longer and thicker needles or rods were observed. Lea(100) states that dicalcium silicate exhibits no definite setting time and the mass sets slowly in a few days. The addition of gypsum produces little change. The strengths attained by pure pastes and mortars are listed In Tables 4, 5, and 6. Dicalcium silicate produces little strength until after 28 days but gains strength steadily at later ages and approaches the strength of tricalcium s i 1 icate. Powers(101) describes the hydra- tion process as starting at the cement grain-water interface and the gel growing outward and inward simultane- ously, with the cement grain residue encased in the surrounding gel as long as such residue exists. Additional water needed for further hydration then must diffuse inward through the gel pores while the components of hydrated cement in solution diffuse to the outer border of the gel layer. At the outer border the new gel either adds to existing crystals or starts new ones. It is presumed that about 55 percent of the newly formed gel is transported outwardly, and 45 percent stays inside. It is known that the hydration products of a paste contain- ing an excess of cement, for example, a water-cement ratio of 0.38, can eventually produce a gel with a poros- ity equal to about 28 percent. This same degree of density is possible even though the cement is not in excess. Since the inward growth of gel is pro- duced under the greatest concentration of material, at least 45 percent of the gel in any specimen has the mini- mum porosity. Furthermore, the cement and water mixture is initially in a flocculent state as hydration begins; therefore all of the cement grains in the fresh paste are in contact with neighboring cement grains at many points. The outward growing gel should, therefore, reach minimum poros- ity in the vicinity of these points of contact; however, whether the minimum porosity attained by the in- ward growing and the outward growing gels will be the same is questionable. This growth will result in a range of pore sizes, the smallest being possibly monomolecular in dimension and the largest just smaller than the smallest in which initiation of nucleation would be possible. Any region where the porosity exceeds this character- istic minimum porosity contains pores large enough to allow nucleation of new crystals or growth of existing crystals as the components of hydrated cement diffuse outward. It seems, therefore, that as a given region approaches minimum porosity, the capillary pores remaining will gradually be closed up with additional hydration products if hydration continues. Most investigations show that the products of hydration of one cubic centimeter of cement require about 2.2 cm3 of space upon complete hydration. The volume of the paste when set is approximately equal to the volume of the cement plus the volume of the water; therefore the paste volume is a function of the ratio of the mixing water to the cement. The final volume is composed of products of hydration, water, and voids. Cement gel is de- scribed by Powers(102) as being com- posed of gel particles with small voids among these gel particles which are called gel pores. The paste structure is not identical to the gel structure in that a specimen of paste contains gel, crystals of calcium hydroxide, some minor components, and some of the original water-filled spaces in the fresh paste. What remains of these original water-filled spaces existing in the hardened paste appears as inter- connected channels or, in dense structures, as cavities which are interconnected by the gel pores. These residual channels or cavities are called capillary pores or capillary cavities. All of these void spaces are submicroscopic in size. High capillary porosity is indicative of continuous interconnected networks of capillary pores through the paste. At normal paste porosity the capillary cavities are interconnected only by gel pores. A value of porosity of about 28 percent is the value generally considered as an average. Many impor- tant characteristics, in particular volume stability, are affected by the porosity and the effects of water in the capillary and gel pores. Table 7 contains a comparison of the permeabil- ities of some common rocks and cement pastes. Grudemo(95) reports that the largest part of the cement pastes examined consisted of exceedingly ill- formed colloidal products in which it is difficult to see any definite form or structure. The paste materials having been lightly ground were com- posed of rounded irregular aggregates of diameters ranging from 0.5 to 2 mpi. These aggregates possibly correspond to the initial cement particles the grinding and milling operation having caused the material to split along zones of contact between hydrated particles. Similar to pure C S and f-C S pastes, the structure in the 2 interior of the aggregate particles is a mixture of fibrous or acicular elements and elements of the shape of small and thin rounded flakes on the order of a few hundred angstroms in diameter. The thin flakes being the predominant phase and the fibrous phase having, in general, a finer more dis- torted texture than the fibrous phase of pure silicate pastes. The distance between surfaces of solid particles approaches zero, and evidence shows that the gel particles are in contact with each other at many points, some of which are chemically bonded. The specific surface, as reported by Powers,(102) of the solid part of th gel is about 700 m2/cm3 of solids. The specific surface, as reported by Brunauer and Copeland, of one gram of tobermorite gel is some 3 million cm2, implying that the average diameter of a gel particle is about one-thousandth that of a typical cement grain. Brunauer, et al.,(103) state that tobermorite crystallite being thin sheets with a thickness of only two or three unit cells (20 to 30 A) will have a specific surface area of 377 m2/g if they are two unit cells thick, and 251 m2/g if they are three unit cells thick. Powers(101) estimated the width of pores in cement paste from the ratio of pore volume to surface area; that is, the hydraulic radius as being 10 A. This means that the average width of gel pore lay between 20 and 40 A, probably closer to 20 A. The corres- ponding average distance between solid surfaces in the gel is between 14 and 28 A, 18 A, or 14 atomic diameters being considered a reasonable estimate of the average width. This distance is about five times the diameter of an unbonded oxygen atom or about thirteen and one-half times the diameter of a single bonded oxygen atom. The un- bonded diameter of 3.6 A is approxi- mately the same as that of a water molecule. Only recently a report on the measurement of the size of capillary pores has indicated that the capillary pores are orders of magnitude larger than the gel pores. Results of absorptivity tests also indicate that the capillary pores are much larger than gel pores, except in pastes in e which the gel nearly fills the available space. 6) In a hardened paste there are three possible classes of water 96) which are categorized generally as nonevaporable water, gel water, and capillary water. Nonevaporable water is defined as that part of the total water which does not evaporate at a vapor pressure of not over about 6 x 10 4 mm of mercury at 23°C (73.5*F). It is a constituent of the solid material of the paste existing in part as OH groups in calcium hydroxide and some as water of crystallization in calcium sulfoaluminate. There is an additional portion combined in the solid phase whose exact form of combina- tion is not yet known. It is thought that of this additional portion some may be chemically combined with the hydrous silicates and alumina-bearing compounds and some held by physical forces such as van der Waal forces. In volume-analysis the volume of the nonevaporable water is added to the volume of the original cement to give the absolute volume of the solid phase in the paste. The specific volume of the nonevaporable water is taken to be about 0.75. The gel water is defined as the water contained in the gel pores. The gel pores being very small, the gel water is within the range of the surface forces of the solid phase. This is indicated by the value of 0.99 for the mean specific volume of the gel water. The vapor pressure of gel water is a function of the degree of saturation of the gel at any fixed temperature. The lowest pressure just exceeds the upper limit defined for the nonevaporable water, and the highest pressure is nearly equal to that of pure water in bulk at the same temperature. The weight of gel water in a saturated paste is equal to four times the weight required to form a complete monomolecular absorbed layer on the solid phase. Capillary water like gel water is a solution of alkalies and other salts. It is defined as the water which occupies the space in the paste other than that occupied by the solid phase and the gel pores. Practically all the capillary water lies beyond the range of the surface forces of the solids; hence, in saturated paste, the capil- lary water is under no stress. Its specific volume is the same as the specific volume of a solution of the same composition. Brunauer, et al.,(103). , reported that the surface energy of tobermorite obtained by room temperature hydration of C S or 1-C S having a composition of 3 2 Ca Si 0 .2H 0* is 386 t 20 ergs/cm2 3 2 4 2 at 23.50C (740F). This value is inter- mediate between the surface energies of calcium hydroxide, 1180 ergs/cm2, and hydrous amorphous silica, 129 ergs/ cm2. Since the calcium silicate hydrate is structurally and chemically intermediate between calcium hydroxide and hydrous silica, this value is expected. Calcium hydroxide consists of nearly perfect crystals and hydrous silica is completely amorphous while the tobermorite was very poorly crystallized. The surface energy of tobermorite having the composition Ca Si 0 .3H 0 is difficult to obtain 3 2 7 2 experimentally, but a rough estimate *Equivalent in cement chemist shortened notation is Ca Si 0 .2H 0 = C S H . 3 2 4 2 3 2 2 is possible. Tobermorite has, like clay minerals, a layered structure and at a vapor pressure of 5 x 10-4 mm of mercury it contains two moles of combined water per mole with a distance of 9.3 A between layers. At a vapor pressure of 8 x 103 mm of mercury the distance between layers increases to 10.4 A. Therefore all or part of the third mole of combined water is between the layers. The surface energy of Ca3Si207.3H2 0 would be about 386 ergs/ cm2 if all of the third mole of combined water were between the layers; if not, the surface energy would be about 130 ergs/cm2 lower which, because of previous work, Brunauer, et al.,(103) believe is the case. The surface energy of Ca Si 0 .3H 0 tobermorite 3 2 7 2 is then in the range 320 t 70 ergs/cm2. In summary, then, from a study of the chemistry of hydration and the hydration products, the following general conclusions are valid for portland cement. The great surface area of tobermorite gel gives rise to the forces which are responsible for the cementing action of cement paste. Adhesion is one of the forces of attrac- tion between closely spaced surfaces which is a function of surface area. The gel particles adhere to each other and to the surface of any other particles mixed with the paste such as aggregate, thereby cementing the whole mixture together. The cementing action depends on the high surface energy of the tobermorite gel. Besides surface area and forces, porosity plays an important part in determining the strength of hardened portland cement paste. The volume of the total pore space depends on the water-cement ratio. When cement paste sets, it acquires an initial stable volume approximately equal to the sum of the volumes of the cement and the water. Since, after hydration, a given volume of cement will produce a certain volume of hydration products, the higher the water-cement ratio, the greater the final volume and the more porous the final paste. Higher paste porosity produces weaker concrete. Porosity is also an important factor in determining the dimensional stability of a concrete. The problem of volume changes associated with the movement of moisture through the paste during shrinkage and swelling is thus dependent on the structure of the tobermorite gel which plays a dominant part. Effect of Cement Type on Shrinkage and Creep Woods and Steinour(104) and Bogue, et al.,(105) have studied the relative effects of the principal com- pounds in cement on drying shrinkage, and their general findings are: (a) C S has relatively little, 2 if any, volume change during moist curing and it shrinks at about twice the rate of portland cement paste drying in air. (b) C S undergoes a slight expansion during curing and its shrink- age in air is about 30 percent less than the shrinkage of ordinary portland cement paste under the same conditions. (c) C A undergoes considerable expansion during moist curing and rapid shrinkage in air. Tests by Yoshida(106) correlate volume changes, resulting from variations in moisture conditions, with the corresponding changes in water content of the cement paste. His results indicate that the nonevaporable water in a hardened paste increases with age under moist conditions at a decreasing rate and remains substan- tially constant (except for loss of water released by carbonation) under dry conditions. He concludes that at 100 percent relative humidity the hardened paste expands somewhat, at 95 percent it remains unchanged, and in air less than 95 percent relative humidity it shrinks at a decreasing rate. Arnstein and Reiner(107) investi- gated creep of hardened cement paste and mortar beams and concluded that the creep viscosity of mortar and con- crete can be predicted if the creep viscosity of the neat cement in the mixes is known. In 1959 Alexander and Wardlaw(31) studied shrinkage of hardened portland cement paste and concluded that the ultimate drying shrinkage increases, for increased water-cement ratio, linearly with the percentage of water present by volume. (48) Lyse concluded that the shrinkage of concrete is approximately proportional to the amount of cement paste in the concrete, regardless of its composition. The rheological behavior of hardened cement paste was studied extensively by Glucklich(108) and Glucklich and Ishai, 50) resulting in a rheological model representing the linear elastic instantaneous response and the time-dependent deformations of the paste. Campbell-Allen(36) states that there is no clear evidence as to the effect of cement types on shrinkage. Contrary to this, Troxel, et (28) al . , show results of tests indi- cating that concrete made with low heat cements, Type IV, shrinks more than that made with Type I cement (Figure 2). From the figure it is evident that this is, in part, a result of differences in cement grain size which is in agreement with the work of Graf.(109) Graf reported shrinkage increased from 1170 to 1690 p in./in. at three months when the specific surface varied from 1355 to 2280 cm2/g for the same portlant cement. Shrinkage of cement increases in the following order: ordinary portland cement, aluminous cement, slag, and high early strength cement. L'Hermite (110) reports shrinkage of 2200 p in./in. for ordinary portland cement paste at about three years, 2500 p in./in. for aluminous cement, and 3500 p in./in. for high early strength cement. Davis, et al. , reported that low heat cement exhibited about 80 per- cent greater creep than normal portland cement. As a result of his study of normal, rapid hardening, and an aluminous cement, Glanville(ll2) concluded that the cement which hardened most rapidly had the least amount of creep at a given age for air-stored specimens. Troxel, et al.,(28) show results of tests which indicate that concrete made with low heat cements, Type IV, creeps more at any age than that made with Type I cement (Figure 2). From the figure it is also evident that the Oyr Time After Loading (Log Scale) FIGURE 1. EFFECT OF MINERALOGICAL CHARACTER OF AGGREGATES UPON CREEP AND SHRINKAGE (REFERENCE 28). 2800 2400 2000 0 E 1600 0 1200 a- n 800 C 400 0t 160 S120 0 E 80 0 40 1600 0 1200 E g 800 0 400 0 _ A V I 1600 Specific Surfoce 2200 _ -. 1200 Type I" (Low Hoot)-- . -- - -" O 1300 oo -- I 300 --- 0Type I (Normal) 0 - I Iuodays 28daoys 90uudays lyr. 2yr. yr. 10uyr. : Time After Loading (LogScole) FIGURE 2. EFFECT OF COMPOSITION AND FINENESS OF CEMENT UPON CREEP AND SHRINKAGE (REFERENCE 28). 0yr. Z2 1000 800 600 400 200 0 Days at 70 F and 50% R.H. FIGURE 3. EFFECT OF AIR ENTRAINMENT ON SHRINKAGE OF CONCRETE (REFERENCE 35). 1200 1000 800 600 400 200 0 0 10 20 30 40 50 60 70 80 90 Days at 70 F and 50% R.H. FIGURE 4. EFFECT OF CONTINUED MIXING AND RETEMPERING ON SHRINKAGE OF CONCRETE CONTAINING NILES VALLEY AGGREGATES (REFERENCE 35). 0 r Cr U,> I5 o 6 0" c 0 u1 Ip FOR VARIOUS 600 500 o 400 300 S200 a 2 Cu Cr E UI C TUE E F FECT O F Relative Humidity, per cent RELATIVE HUMIDITY ON SHRINKAGE AGES AFTER REMOVAL FROM THE MOLD AT TWO DAYS (REFERENCE 63). Relative Humidity, per cent FIGURE 6. THE EFFECT OF RELATIVE HUMIDITY ON SHRINKAGE FOR VARIOUS AGES AFTER AN INITIAL 624 DAYS IN WATER (REFERENCE 63). F= 32 68 104 140 176 212 Temperature FIGURE 7. TOTAL DRYING SHRINKAGE VERSUS MAXIMUM CURING TEMPERATURE FOR DIFFERENT STEAMING PERIODS (REFERENCE 37). O yr Time After Loading (Log Scale) FIGURE 8. EFFECTS OF MOISTURE CONDITIONS OF STORAGE UPON CREEP AND SHRINKAGE (REFERENCE 28). 2400 -c -2000 1600 (. :o 1200 a- O S800 400 n 1200 c 800 0 E a 400 0 1200 800 0 400 0 c() 400 Relative Humidity 50%0/ 70% Wag_  .r- - "Water 100 F (38G),25 percent Relative Humidity !elotive Humidity 70 F (2 1 C),U 50 OF (-18 A-92 F (33C),Wet ----F (1 CWet 70F 1 21 Ci Wit r 0 20 40 60 80 /1~ 140 160 Time Since Application of Stress to Loaded Specimen, days EFFECT OF TEMPERATURE CONDITIONS OF STORAGE UPON FLOW (REFERENCE 129). >ercent C),Dry 1000 800 600 400 200 to 200 0 20 FIGURE 10. 40 60 80 100 120 140 Time Since Application of Stress to Loaded Specimen, days EFFECT OF TEMPERATURE CONDITIONS OF STORAGE UPON SHRINKAGE (REFERENCE 129). 1000 800 J 600 400 FIGURE 9. 100 F (38 C), Dry, 25 per cent Relative Humidity ..--* o 50 per cent Relative Humidity 0 F(-18 C),Dry ' ° L_ 70 F (21 G), Wet S0 92 F (33 C), Wet _ _ _ _ _ _ _ _ _ _ _ II _ f //00, /00 r- r I t - 70 F 2. e c FIGURE 11. VARIATION OF SHRINKAGE WITH TIME FOR UNSEALED SPECIMENS MAINTAINED AT .a C 'S ZL C (0 0) u 0 Un 400 200 U 0 10 20 30 40 50 60 Time t, days VARIOUS CONSTANT TEMPERATURES (REFERENCE 135). F - 32 68 104 140 176 212 248 284 Temperoture FIGURE 12. LOADING 0 z 0 z 0 (n THE VARIATION OF SPECIFIC CREEP WITH TEMPERATURE AND TIME AFTER FOR SEALED CONCRETE FIRST LOADED AT 10 DAYS (REFERENCE 139). DAYS AFTER LOADING FIGURE 13. TYPICAL CURVES OF TOTAL STRAIN UNDER SUSTAINED STRESS AT VARIOUS TEMPERATURES -- SEALED SPECIMENS (REFERENCE 140). 100 C * (212 F) S80 C (176 F) 50 C (122 F) -20 C (68 F) z ( Unit Stress 1 I I 1 1 c C C 0 V 0 u 0 a U 5, 'i VhC VhC v - c V-I -Voa -Vuc Unit Area 0 0.4 0.8 1.2 1.6 20 Cement Paste / Total Agg. (by Absolute Volume) FIGURE 15. CONCRETE AS A COMPOSITE MATERIAL (REFERENCES 126, 147). FIGURE 14. TOTAL STRAIN AT 210 DAYS VERSUS AGGREGATE-CEMENT RATIOS WITH STRESS-LEVEL AS PARAMETER (REFERENCE 160). JC 0 0 I 0 C7 ,<: 8 Curing Time ,(log scale) FIGURE 16. VARIATION OF THE DEGREE OF HYDRATION WITH CURING TIME (REFERENCE 126). Aggregate yUnhydroted Cement /////////////////////////// Hydrated Cement (Gel Gel Pores) Voids I -V uC T t0 in .± Basic Creep TS l Swelling E Shrinkage FIGURE 17. SCHEMATIC REPRESENTATION OF CREEP WITH AND WITHOUT MOISTURE EXCHANGE (REFERENCES 126, 147). 0 10 FIGURE 18. NORMALIZED UNIT 400- - - - - - - - - - 350- - - - ---- - 300- 1 A--I-I 250 .250----- ------------------------ 2A °o -5-- [225(l-e-+11-5(l-e-i)+ .3t] x io-9 200- a 150 s, _ * Mix A, Solid Cyl. 100 a Mix A, Hollow Cyl. o Mix 9, Solid Cyl. Compression 04 6 Mix B, Hollow Cyl. A Mix C, Solid Cyl. 50 a Mix D, Solid Cyl. a Mix A, U-Specimen " Flexure a Mix B, U-Specimen f 0 .. . . . . . . . . 20 30 40 50 60 Time t, days 70 80 90 100 BASIC CREEP AT 100 PERCENT RELATIVE HUMIDITY (REFERENCES 126, 147). AC a (a) Compression T Stable (b) Tension 0 0 Moisture Gaio [ii Moisture Loss C C 1' 0 b N 'S IE m m echanism for stantoneous eformation R3a, FIGURE 19. PROPOSED RHEOLOGICAL MODEL FOR BASIC CREEP OF CONCRETE (REFERENCES 126, 147). )9a2 350 300- (EcVa.hc/9,)t= (2.99t* 7.73)xi10-4 200 Ol 150 -- - - - 100 ----_ ----_ --_ __--- - - -- - - -- - - 100 * Mix A, Solid Cyl. S- n Uiv a nSolid r l 0 10 20 30 40 50 60 Time t, Doys 70 80 90 100 FIGURE 20. NORMALIZED UNIT DRYING CREEP UNDER COMPRESSION AT 50 PERCENT RELATIVE HUMIDITY (REFERENCES 126, 147). S&3U, o- - Mix , Solid Cyl. I I A Mix D, Solid Cyl. o______ x , Solid Cyl. 'U C 'U :1. 'U Time t, days FIGURE 21. COMPARISON OF ANALYTICAL EXPRESSIONS TO FREE SHRINKAGE DATA FOR MIX A (REFERENCE 146). 28 600 - 500 --- ^^ I I * 300 /, o4-0- 03'"- e C GE.. Reft 148 200 _ _ _ ---- ---- ---- ---_--- ---_-- / 0 dia cylinder ( J" se prism 100 Environment 5I 0% R.H 70 F 10 -- -- -- -- -- -- --;- 00 - 4-- S42(l-e-0-1t' el) " ~ ,00 ----0 -z-,-- */ C- 1 RE fIE4 200 ____---_-- ---- --- --- ---- --- 0 o0 4dia. cylinder 0 4" sq prism 100 0 Environment 50% R.H. 70 F 0 Time t, days FIGURE 22. COMPARISON OF ANALYTICAL EXPRESSION TO FREE SHRINKAGE DATA FOR MIX B (REFERENCE 146). 28 1200 - o !" sq prism cured for 3 days 1000 prior to drying Environment 75% R H. 70 F 80C 00s =I 540 (1- e-0.40to") 60 0 --- ---- --a---- | --- --- | --- --- -- 400 SC.E B. C  -o ,- Ret. 148 200(- --- --- 0 10 20 30 40 50 60 70 80 90 Time t , days FIGURE 23. COMPARISON OF ANALYTICAL EXPRESSION TO FREE SHRINKAGE DATA FOR MIX S (REFERENCE 133). 1200 --- --- --- --- --- --- --- --- --- 1000 Baa or E/so 836 (1-e-0.40to.es 00 40 a ... o sq prism 4- ...- CEB j Ret 148 cured for 3 days 200-- prior to drying / Environment / 25% RH 0 70 F E =L u c :=L 'U c C :4- In '4, Time t, days FIGURE 24. COMPARISON OF ANALYTICAL EXPRESSION TO FREE SHRINKAGE DATA FOR MIX W (REFERENCE 133). Time t , days FIGURE 25. COMPARISON OF ANALYTICAL EXPRESSIONS TO FREE SHRINKAGE DATA. c *s c =t. (A \U FIGURE 26. COMPARISON OF EXTRAPOLATED TO ESTIMATED ULTIMATE SHRINKAGE. Estimated Esoo , /L in./in. (Eq.21) Time t, days FIGURE 27. NORMALIZED UNIT BASIC CREEP IN FLEXURE AT 40°F (4.5°C). 0. cl -. c- ZL b U 'JO =U VI 0. C c Qz C b N. n Time t, days FIGURE 28. -* o 0° f0 NORMALIZED UNIT BASIC CREEP IN FLEXURE AT 70°F (210C). 0 0 0 Ebc t t ] ) = 480(I-e 0o)+|5(I-e 2)+0.3t - S U I I- 0 0 o Mix A o Mix B a Mix C * Mix D * Mix A Time t days 80 90 NORMALIZED UNIT BASIC CREEP IN FLEXURE AT 1100F (43°C). 0 a * S U U * 0 -a---- * * * U 0 3- --- VI ii • FIGURE 29. .4, Time t, days FIGURE 30. COMPARISON OF COMPUTED AND MEASURED SHRINKAGE CURVES (REFERENCE 29). FIGURE 31. COMPARISON OF OBSERVED AND CALCULATED LONG-TIME CREEP (REFERENCES 126, 147). w - CM CN N Un - 1o %D OD co LA\ - 1% N tn C4 -T" m* - U. N in A) 4 < u 0 B * U LJ ( U I O Em U) *-) - U E - U L *- E- 4- 0 * - a- Uo I- E e - 0 N U. + 0 0 N N + + 0 0 u u 1 o - - 0 L- M" N L. - 4) Q x UL U.0o a" I- Iz 0 x LA0 L) U- 0 zu- U) I- 0 Z 0r < O a .Z 0.. U. LL. 00 Uv) U Oa- a- l N 4-j 5- 3 c V --i 0 eOE O( U -0 + 4) 0 E e C u c 0 In -1J eU C 0 E N 2 + e N u 0- V/I EU t >. TABLE 2 TRANSFORMATIONS OF COMPOUNDS BY HYDRATION (REFERENCE 98) 2(3CaO-SiO ) 2 2(2CaO-Si02) + 6H 0 2 = 3CaO-2SiO -3H 0 + 2 2 (Tobermorite Gel) = 3CaO-2SiO23H 20 + (Tobermorite Gel) + 4H 0 3Ca (OH) 2 (Calcium Hydroxide) Ca(OH)2 (Calcium Hydroxide) 4CaO-Al 20 Fe 0 + 1OH 0 + 2Ca(OH)2 2 3 2 3 2 2 3CaOAl 203 3CaO-Al 0 6CaO-Al 0 -Fe 0 *12H 0 2 3 2 3 2 (Calcium Aluminoferrite Hydrate) + 12H20 + Ca(OH) = 3CaO-Al 0 3Ca(OH) -12H 0 (Tetracalcium Aluminate Hydrate) + 10H 0 + CaSO -2H 0 = 3CaO-Al 0 3CaSO *12H2 0 (Calcium Monosulfoaluminate)* *Reaction occurring in the presence of gypsum. 0 0 0 0 o 0 0 0 1 0 0 . IN •0 N N I 0O 0 0 0 0 0 0 0 0 0 0 • r1 1 r^1 r^ r1 U) U0 U0 LL LL L. 0 0 0 1- *1 *- L. LL I. X X X 0. 0. 0. L. L. L. a. CL a. 0. a. QL Eu EU Eu N 0 I= = En En Ed) LO U) X) X) X 4)4 41 41 *- *- 0 0 E E L. 1- 4)4 .0 .0 0 0 I- I- '4 En N En o ' o UU = En N En '4 *. U me X I V) .-a N ^--s Z tr t/ U(M uJ ' (- *- 0 ) in LC EO ID L= 4-1 0 L r Ln I- Ul D. C 4- L 4)n z C I- 3. >» -le U) N 40 3 co >- -0 A U) S) * ~- E Lil I I r- I I NC I I M I I N I I I I -- - I 0 a 0 0 0~ _ I '0 LA1 N - I CN i4 M m r- I o Ln 0 N a I L 0m N 4-1 4- 0 I 0 0 01 O ta I - r4 - U)l N I M4~ - 00 N I --T e co - -a I - o r'. L,- 00I 0 %0 N %0 r m I o - '0 - uil I o Lt1 r- '. M0 I E1 l in CO ) I' 1O N O - I co r-. %o r-. o L1 %D I n 'r% %0 %o Lt i 1 mW1O0 1 N LA N CO n 01 i- N 0) 0 N S% - - 0% m 4-1 4- o0 L W1 LA N 0 OP c1 0 .0 '0 U) '0 - - '0 E1 4-, oE- 0 o oo r-o 'o 0 - 0 01 0 .00 U) -.r - N - < m < n 4 0u) 01 0O 0 - - Ln + + + + + tn n En En Eo U U '0 0 u M Ln LA n VIn < Ma e 4 m m- - 0 c0 0 U U Uo co LA -T -T TABLE 5 TENSILE STRENGTH OF MIXES OF CEMENT COMPOUNDS WITH GYPSUM AND SAND (REFERENCE 100, PAGE 81) Cementing Mix 1:3 Sand Mortars -- Water Stored Tensile Strength, psi 3 days 7 days 28 days C S 3 C S + 5% Gypsum C A 3 C AF 75% C S + 25% C A 3 3 75% C S + 25% C A + 2.5% Gypsum 3 3 43 86 284 455 270 TABLE 6 COMPRESSIVE STRENGTH OF MORTAR MIXES OF CEMENT COMPOUNDS (REFERENCE 100, PAGE 81) Cementing Mix 85% C S + 15% 6-C S 3 2 85% C S + 15% C A 3 3 85% C S + 15% C AF 3 4 1 :3 Sand Mortars -- Water Stored Compressive Strength, psi 3 days 7 days 28 days 6 months 2040 2620 4100 4960 1720 2650 3850 4600 1600 2450 2900 4300 TABLE 7 COMPARISON OF PERMEABILITIES OF ROCKS AND CEMENT PASTES (REFERENCES 101 AND 154) Water-Cement Ratio of Mature Paste Permeability of Permeability of Having the Same Kind of Rock Rock, Darcys Rock, cm/sec Permeability Dense Trap Quartz Diorite Marble Marble Granite Sandstone Granite 2.57 8.56 2.49 6.00 5.57 1 .28 1 .62 10-9 10-9 10-8 10-7 1 1 -6 10-s 10-s 3.45 1. 15 3.34 8.05 7.48 1.72 2.18 10-13 10-1 3 10-12 10-11 10-10 10-9 10-9 0.38 0.42 0.48 0.66 0.70 0.71 0.71 TABLE 8 CREEP AND SHRINKAGE DATA AT VARIOUS TEMPERATURES (REFERENCE 136) Shrinkage in fi in./in. at Creep in f in./in. at 300C 1050C 110°C 300°C 105C 110oC Time (860F) (221-F) (230°F) (860F) (2210F) (230*F) TABLE 9 DEFORMATION OF ROCKS CAUSED BY MOISTURE MOVEMENT Environment Relative Type Shrinkage Swelling Temperature Humidity Source of Rock in Percent in Percent in Percent Ref. 155 Ref. 155 Ref. 156 Ref. 156 Ref. 157 Basal t Flint Basalt Chert Doleri te Ref. 158 Dolerite*' 0.015 to 0.020 0.006 0 to 0.065 (Average of 0.027) 0.06 200°C (680F) 20°C (680F) 0.020 0.005 *Altered dolerite containing approximately 30 percent of the clay mineral montronite. difference in creep magnitudes is not a result of variation in grain size, since for Type IV cement the concrete with the cement having the larger grain size creeps most, and for Type I cement concrete the reverse is true. It is interesting to note that Sheikin and Baskakov(ll3) reported that for the same stress-strength ratio the creep appeared to be independent of cement composition. They reported that the lowest creep occurred when cement consisting mostly of C S was used and the highest creep with cement consisting largely of C S. The effect 2 of mineralogical composition ceased after about seven months. 3.2.3 Admixtures The effect of admixtures on the volume stability of concrete has not been investigated to the extent that other factors have. In 1911 White (13) reported that integral waterproofing compounds did not help to reduce volume changes of specimens. The influence of lime as an ad- mixture was investigated in 1917 by Scofield and Stinchfield (114 and by Davis and Troxell (115) in 1928, re- sulting in the conclusion that it had no appreciable effect on shrinkage. In 1930 Davis(ll6) presented a summary of research on volume changes and con- cluded that high-magnesium hydrated lime used as an admixture appears to have little effect on expansion or shrinkage. With respect to the use of integral waterproofing compounds, he concludes they are not effective in preventing or reducing volume changes. Finally, the substitution of a dolo- mitic or high magnesium lime for cement in small quantities, up to 10 percent by weight of the cement, reduces shrinkage, and substitutions of clay indicates an increase in the volume changes as the clay content increases. In 1935 Menzel (117) reported that tests of steam-cured concretes and mortars containing finely divided silica bearing materials, (100 percent passing the No. 200 sieve) such as Haydite, flue* ash, lava, and fine silica, showed volume changes ranging from about 1/8 to 1/2 that of the correspond- ing mixtures moist-cured but without the admixtures. We should note here that the effect of steam-curing is the dominant factor. Davis and Troxell 81) support the use of admixtures which may improve early strength, reduce the mixing-water requirement, improve workability and plasticity, and reduce bleeding and segregation, provided that they do not increase creep and shrinkage. At the same time they caution that accelerators and water reducing admixtures do not behave the same for all concrete mixes and all cements, and in addition they may substantially increase drying shrinkage when used in normal amounts. They also report that the use of well- recognized air entraining agents has resulted in no evidence of either shrinkage or creep being appreciably affected and that calcium chloride, when used as an accelerator in normal amounts, increases drying shrinkage. In general they caution against the use of any admixture unless its effects on *Commonly known as fly ash. strength, shrinkage, and creep, for the particular concrete materials used, is investigated in advance to determine whether its use may or may not be advantageous. In 1956 Washa 87) reported that very little work has been done to determine how creep of concrete might be affected by admixtures. He states that approved air-entraining agents have no effect on creep, and that con- crete made with pozzolonic material generally exhibit greater creep. He notes that, "where creep is an impor- tant factor, proprietary compounds should not be used unless their effects have been previously determined." With regard to shrinkage he notes that dis- persing and wetting agents, in nominal amounts, have little effect on con- crete shrinkage, and replacement of cement with fly ash of low carbon con- tent has little effect on shrinkage. He also reports that the use of pozzolanic material as a replacement for part of the cement in mortar may produce as much as 50 percent more shrinkage. With regard to accelerators, Washa states that the use of up to 2 percent calcium chloride by weight of cement may also increase shrinkage from 10 to 50 percent over that of the corresponding mortar without calcium chloride. As a result of recent tests on creep of concrete, Ross concluded that creep is unaffected by the use of fly ash as a replacement of up to 25 percent of the cement in the concrete mix. Abdun-Nur (118) in 1961 reported that replacement of cement in concrete with up to 30 percent low-carbon fly ash results in about the same or slightly lower shrinkage than would occur with normal concrete. Tremper and Spellman(119) in 1963 report that the use of triethanolamine, to increase rate of strength development, results in increased drying shrinkage. Washa 120) in a recent publication (1966) summarizes the general knowledge about the effect of admixtures on drying shrinkage. In general the statement can be made that admixtures that increase the unit water content of concrete may be expected to increase the drying shrinkage; however, the contrary is not strictly true. Also, concretes having some part of the cement replaced by pozzolons such as pumicite or raw diatomaceous earth shrink more than concrete without such substitutions. In addition, the substitution of low- carbon, high-fineness fly ash in quantities of up to 30 percent of the cement produces concrete having the same or lower shrinkage potential than the corresponding concrete without fly ash. Finally, of the more common admixtures, calcium chloride in normal amounts increases and entrained air (Figure 3) in normal amount decreases drying shrinkage. 3.3 MIXING TIME AND CONSOLIDATION Investigations concerning mixing and consolidation with regard to effect on volume stability are rather scarce. Polivka 35) in 1962 reported that the effect of continued mixing and re- tempering was to increase the shrinkage. He states that the longer the mixing time the larger the increase in shrink- age; however, his results are complicated by the fact that added water for re- tempering increased the water-cement ratio. He states that the increase in shrinkage he observed was primarily a result of the increased water-cement ratio but also allows the fact that the observed increase in shrinkage is in part caused by the longer mixing times (Figure 4) . With respect to compaction, Campbell-Allen(36) stated that incom- plete compaction may increase shrinkage values. Mixing time and consolidation are factors which may affect the strength of concrete and as such become indi- rectly related to creep and possibly even shrinkage. 3.4 AGE AT LOADING, DEGREE OF HYDRATION, AND CURING Age at loading is a factor in con- sidering the volume stability of con- crete in that it influences the degree of hydration and strength as indicated by the correlation of creep with matu- rity by Ross. Since the principal part of creep and shrinkage of concrete is considered by most investigators to be occurring in the hydrated cement gel, age at loading, degree of hydra- tion, and curing are relevant only insofar as they affect the amount and quality of that gel. Of the three factors named above curing is, no doubt, of primary impor- tance, since different curing methods can produce different degrees of hydration at the same ages or the same degree of hydration at different ages. We can safely conclude that for a given concrete cured under given conditions, the amount of creep varies inversely with the age at loading, i.e., the same concrete loaded at an early age yields larger creep strains for a given time under load compared to the same concrete loaded at a later age. This effect on creep strains, also true for shrinkage, can be emphasized by comparing Figures 5 and 6 as presented by L'Hermite and Mamillan. (63) Higginson(34) indicates that steam-cured concrete will decrease the drying shrinkage of the hardened con- crete and that the decrease is depend- ent on both the temperature of the steam-curing and the length of time of the steam-curing. A more recent re- port indicates that low temperature, 40*C (104*F), short-term (nine hours) steam-curing has little effect on the total drying shrinkage, but long-term (48 hours) high temperature, 80°C (176°F), steam-curing can reduce drying shrinkage by as much as 50 percent (Figure 7). 3.5 MOISTURE CONTENT Moisture content is a relevant parameter indirectly insofar as the rate of moisture loss from a specimen of concrete is itself a function of its moisture content and, therefore, the rate and magnitude of drying shrinkage and drying creep will be dependent on the specimen moisture content at the time the specimen is loaded. In addi- tion, since the amount of continued hydration is a function of moisture content, the final values of both drying shrinkage and drying creep will be affected. It was noted by Powers(122) that hydration virtually ceases when the vapor pressure falls below 0.3 of the saturation vapor pressure and proceeds very slowly below 0.8 of the saturation vapor pressure. Neville(52) concludes that the content of mixing water increases shrinkage as it reduces the volume of the aggregate. However, water content per se is not believed to be a primary factor. At least part of the total creep is believed to be due to migration of water within the voids of the specimen caused by the action of the loads and it is known(53) that creep diminishes as the free water content diminishes. Tests by Mullen 54) on cement paste specimens loaded in compression after being dried to contant weight exhibited no time-dependent deformation. 3.6 LEVEL, TYPE, AND DISTRIBUTION OF STRESS The greater majority of research on creep behavior of concrete deals with specimens under compressive stress because of the relative ease of in- vestigating this condition. As a con- sequence, only limited information is available concerning creep of speci- mens using other stress conditions. Generally, the results indicate that the qualitative creep behavior is essentially the same under all basic states of stress. 3.6.1 Creep Under Sustained Tensile and Compressive Loading Various investigators have re- ported on tests of creep in tension and compression. (See Reference 28, 60, 111, 112, 123, and 124.) Their work suggested that the limiting value of tensile creep is probably equal to that for compressive creep, both at the same stress, with the only difference being that for tensile stresses the creep rate is higher in the first few weeks after loading. Considerable problems arise in measuring tensile creep strains, since the stress applied is smaller than that which can be applied to the same con- crete in compression by an order of magnitude. The practice of subtracting the shrinkage strain from the total strain of the loaded specimen can be very inaccurate since the shrinkage strain is generally large in magnitude. In spite of this, it is considered that tensile creep strain increases when simultaneous shrinkage occurs.60, 124) 3.6.2 Creep Under Sustained Torsional Loading Creep of specimens loaded in torsion was studied by Anderson,(125) Ruetz, 60, 124) Ali and Kesler.(126) Their results indicate that the quali- tative behavior was essentially the same as creep in tension and compression. In tests by Glucklich and Ishai(53) and Ishai,(127) conducted primarily toward a study of the creep mechanism, specimens loaded in torsion were used. No comparison is made with creep under any other state of stress. Duke and Davis(128) report that creep under shearing stress is about double the value of creep in compression. Ruetz 60) states that the average viscosity values calculated from compressive and torsional tests were in agreement. From this agreement of the values of viscosity he concludes that the creep process is a pure shear process as in the plastic flow found in plastic materials. 3.6.3 Creep Under Sustained Multiaxial Loading The effect of multiaxial stress conditions on creep is a subject which is still in doubt. The problem, though it has been investigated, is compli- cated by disagreement about the value of Poisson's ratio for creep strains. Various investigators (84,91 128,129) report that Poisson's ratio for creep strains does exist, and it is con- cluded that it has about the same value as for elastic deformations. Others(130) have reported that the value of Poisson's ratio for creep is zero. Also, work by Glanville and Thomas (123) shows in one case that the lateral creep was about 5 percent of the longitudinal creep, and in another case there was no appreciable lateral creep. It is evident that before any conclusions can be made regarding multiaxial stress effects on creep much additional research is needed. 3.6.4 Creep Under Sustained Flexural Loading The results of investigations into creep of plain concrete beams loaded in pure flexure by Davis, et al., indicate that the fibers on the tension side of the beam creep somewhat more than the fibers on the compression side, and also that the resulting movement on the neutral surface is negligibly small and may be considered as remaining substantially in a fixed position. They observed that at an age of 40 days for beams stored in air the total deformation of the extreme compression fiber was three times as great, and that of the extreme tension fiber was ten times as great as the corresponding deformation observed on the beam stored in moist air. The differences in the creep in tension and compres- sion here are probably a result of the effects of shrinkage-creep inter- action and also the presence of a definite strain gradient under this type of load. As a result of tests of mortar beams in flexure, Ishai 49) states that the deformability decreased with increased sand concentration. More recent work 56) on the effect of a nonuniform stress distribution on creep of concrete indicates that the highest creep occurs in the fiber of maximum stress on the specimen with the highest stress gradient. 3.7 AMBIENT AIR VELOCITY Basic creep is defined as the time-dependent deformation under a sustained load of a concrete specimen while it is in hygrometric equilib- rium with its environment, therefore its value, by definition, will be independent of air velocity. Drying creep, i.e., time- dependent deformation of a drying specimen under a sustained load, and free drying shrinkage both are a function of the rate of moisture loss and as such could possibly be dependent on the ambient air velocity over the specimen. Ruetz 60,124) comments that his experience indicated that during the initial drying stages the rate of drying of cement specimens depends on the amount of air movement. He states further that a specimen in an environ- ment of 70 percent relative humidity with air moving at 0.5 m/sec will dry more rapidly than one in a dessicator containing stagnant air at 10 percent relative humidity. This statement is misleading since the results of other investigators on drying of concrete seem to imply the opposite. Hansen reports test results of specimens drying in calm air and specimens drying in a wind tunnel with air flow at 5 m/sec. In both cases the temperature was 20°C (68°F) and the relative humidity of the air was 50 percent. His shrinkage data for 100 days drying show that the shrinkage curve from specimens drying in calm air is slightly higher, about 0.01 thousandths, throughout the test period. Drying creep data also reported by Hansen for two different series of specimens show in one case that after about 20 days under load the creep of specimens in calm air is above that of specimens in air moving at 5 m/sec; whereas prior to that, it is below, with the largest difference being about 20 p in./in. in 300 P in./in. In the other case the creep of the specimen in air moving at 5 m/sec was greater in magnitude than in calm air with the difference being about 30 1 in./in. in 400 p in./in. Hansen's tests were made on small mortar specimens having a large surface to volume ratio and a low aggregate cement ratio of about 1.4 by weight, in order that any differences in shrinkage and creep which might occur would be magnified. He con- cluded that the effect of air velocity on creep and drying shrinkage is insignificant. The problem of drying of concrete in air moving at different velocities has also been studied by Pihlajavaara.(132) His tests were conducted on small mortar discs, 13.5 cm in diameter and 1.2 cm thick. His results for tests of 0.1, 0.5, 1.0, and 5.5 m/sec air velocity show no difference in the specimen moisture content at any air velocity during 32 days drying. The fact that the velocity of the ambient air does not influence shrink- age and creep is surprising. However, studies by Pihlajavaara(132) explain this phenomenon. In evaporation drying, generally a distinction is made between three stages. (a) The constant rate period: the surface is completely wet and drying is isothermal. (b) The first falling-rate period: the surface is only partially wet and drying is no longer isothermal. (c) The second falling-rate period: the surface of the drying member appears to be dry. In the first two stages the drying rate depends on the air velocity. In the third stage, the second falling-rate period water can, even in calm air, evaporate at a faster rate than it can be transported to the surface. Drying during the second falling-rate period, therefore, is independent of the air velocity. Because of the low moisture conductivity of concrete, the first two stages are completed after a very short period of drying, usually less than 24 hours. From there on drying and consequently shrinkage and drying creep are independent of the air velocity. 3.8 AMBIENT HUMIDITY Ambient humidity has been shown to influence the rate and the magnitude of both free drying shrinkage and creep Dutron(27) derived a function to adjust the shrinkage at 50 percent relative humidity to a different relative humidity state all at about 700F. For specimens originally having a water content at the time of loading, which is such that the specimen loses its internal moisture to the environ- ment, creep has been found to increase in magnitude with the increase being greater the greater the difference in relative humidity between the environ- ment and the specimen. Research has shown(28) that in atmospheres of 70 and 50 percent relative humidity the creep strains are about double and triple, respectively, those at 100 percent relative humidity (Figures 6, 7, and 8) . Increased shrinkage and creep strains at decreased relative humidity of storage have been shown by numerous investigators. (See References 28, 32, 60, 129, and 133.) 3.9 AMBIENT TEMPERATURE A review of the available liter- ature dealin.g with the effects of temperature on creep and shrinkage reveals only a limited number of reports. Though few in number, useful information regarding temperature effects can be obtained from these reports. As with many other aspects of research on the behavior of concrete, there appear to be differing views on certain aspects of the behavior of concretes at high temperatures. Davis, et al., (129) in 1934 reported the results of tests (Figures 9 and 10) of creep and shrinkage of concrete using 4-by 10-in. cylindrical specimens at 0°, 70°, and 100I F (-18°, 210, and 38*C) stored in air, and at 700 and 920F (210 and 33°C) under water. The relative humidity for the 70°F (21*C) storage was 50 percent, and for IOO1 F (38°C) storage was 25 percent with the specimens allowed to come to constant temperature in each of the environments before loading. The results indicate that for the specimens in water the creep at 92*F (33°C) was approximately equal to that at 70°F (21°C). The authors concluded that this indicated that the effect of temperature variation, within the range studied, on the flow is slight or negligible if the moisture conditions are the same. They also note that the flow of the specimens in air at O°F (-180C) is of the same order of magni- tude as the water stored specimens, which is possibly a result of the low rate of evaporation of moisture at the freezing temperature. The creep and shrinkage of specimens at the highest temperature is greatest but this is, in the main, a result of the very low relative humidity. Theuer(134) in 1937 reported a series of tests of two different con- cretes using saturated, semi-dry, and dry cylindrical sealed specimens at temperatures ranging from 26° to 123°F (-3.5* to 50.5°C) and a stress-strength ratio of 0.2. He reports that instan- taneous deformation at 0.2 hours were less for wet than for dry and semi-dry specimens, and they increased with increasing temperatures. Time-dependent deformation of the oven dry concrete was small and differed only by a small amount at the various temperatures. Theuer concluded on the basis of his test results that for both strong and weak concretes and for all three moisture conditions, deformations were greater the higher the temperature. He also notes that with the exception of the dry specimens the residual deformations upon unloading were greater the higher the temperature. Laginha Serafin and Guerreiro in 1960 reported a study of the in- fluence of temperature on creep of mass-cured concrete prisms, 20 by 20 by 60 cm, sealed in copper jackets. Specimens were tested at room temper- ature and at 45°C (113°F) and at ages of three and eight days. Companion unloaded prisms were also kept in the same environments and deformations of the loaded prisms were corrected for the length changes of the unloaded prism. The authors concluded that the influence of a high constant tempera- ture on the creep strains is slight, being felt only during the first four to seven days after loading. England and Ross(135) in 1962 investigated both shrinkage and creep at various temperatures. Figure 11 shows the results of their study of the effects of temperature on shrinkage. The authors concluded that shrinkage increased with increase in temperature. No mention is made, however, of the relative humidity of the environment at the various temperatures. If we assume that the relative humidity was the same at all temperatures, then the increase in shrinkage is caused by the increase in temperature. If both temperature and relative humidity varied, then the increase in shrinkage noted is a result of the combined effects of both parameters. As a result of their creep studies the authors conclude that elevated temper- atures up to 140°C (2840F) may cause the creep of both sealed and unsealed concrete to be several times greater than at normal temperatures. They also indicate that for sealed specimens the influence of temperature is more pronounced in the temperature range of 200 to 60°C (680 to 140°F) than in the range of 100* to 140°C (2120 to 284°F). We should note here that the conclusions of England and Ross(135) agree with those of Theuer, 134) even extended up to a temperature of 1400°C (284°F), i.e., that the higher the temperature, the greater the magnitude of the creep, other factors being equal. Hollaway and Wajda(136) in 1965 reported results of creep and shrinkage tests at 85*F (29.5*C) and approximately 44 percent relative humidity for about 60 days, and at 105*C (221°F) and 110°C (230°F) for periods of 32 to 44 days. The authors give no information about the ambient relative humidity at the higher temperatures. Their results are plotted as bands representing the scatter of the test results at a particular temperature and not as indi- vidual deformation-time curves. The high temperature creep bands cross each other at about 10 days with the 1100C (2300F) higher before and lower after that time. The slopes of these bands at the termination of the tests indicates another probable crossing is eminent. Table 8 contains some representative average values from their data. shai (64) states that temperature changes have a considerable influence on shrinkage and creep even in the range in which the gel composition remains unaffected which he cites as 10° to 60°C (50° to 1400F). He lists the following effects of an increase in temperature: "(1) A decrease in the viscosity of the liquid phase. This results in accelerated creep and shrinkage and is reflected in lower viscosity param- eters and retardation times .... "(2) A decrease in the binding energy within the crystals and between the gel particles. This is reflected mainly in an increase in elastic defor- mation and recoverable creep, giving lower instantaneous elastic and inelastic rheological parameters. "(3) A decrease in the surface tension of the liquid phase, resulting in a partial relief of these forces. This is reflected in the swelling of heated specimens under prolonged exposure to a dry atmosphere (50 to 70 percent R. H.), which is in addition to normal thermal expansion. This would appear to be one of the reasons for the varying coefficient of thermal expansion of hardened cement paste under different moisture conditions. "(4) An accelerated drying and emptying of the gel and capillary pores. "(5) A decrease in the adhesive forces between the solid particles and the confined liquid layers. "(6) An increase in the rate of hydration." In the light of these effects, even though some tend to increase and others to decrease volume change, one would expect that the dominant result is that an increase in temperature would produce increases in both creep and shrinkage, which agrees with previous investigations, and increased elastic deformation, also in agreement with previous investigations. Ishai(64) further comments, "the fifth (influence) contributes decisively to the compact- ing or squeezing mechanism and its rate, which is reflected in the con- siderable increase in the irrecoverable creep compared with the recoverable component.'(137) He also notes that some of the irrecoverable creep and shrinkage is due to the increased hydration. Results of creep tests performed at different temperatures ranging from 700 to 2050F (210 to 96°C) by Nasser and Neville(138) seem to indicate two different types of behavior. Specimens were tested at three different stress levels, 35, 60, and 70 percent, at each temperature. Those tested at the 35 percent stress level indicate that creep increases with increasing temper- ature for the range of temperatures stated above. Those tested at the 60 and 70 percent stress level indicate that creep increases with increasing temperature only up to about 160°F (71°C) and thereafter decreases. Their results indicate a creep recovery inde- pendent of temperature and of the magni- tude of stress. The results of creep tests at 35 percent stress level agree with the results of creep tests by Theuer(134) and England and Ross,(135) while the results of the creep tests at 60 and 70 percent stress level do not. In addition, the conclusion by Nasser and Neville 138) with respect to creep recovery does not agree with that of Theuer(134) and Ishai(64) who both indicate creep recovery as being influenced by temperature. Figure 12 represents specific creep data collected by Ross, England, and Suan(139) at Kings College, London. The trend of these results indicates an increase in specific creep with an increase in temperature for a range of temperatures from 20° to 1400°C (680 to 284°F), for time intervals up to 80 days. Ross, England, and Suan con- clude that considering only the range of temperatures between 200 and 100°C (68° to 2120F) creep can be represented approximately as directly proportional to temperature in degrees centigrade. Tests reported by Ross(140) at the 1960 Colloquium on the Nature of Inelastic Behavior of Concrete and Its Structural Effects at Cornell University also indicate the dependency of creep on temperature, Figure 13. In addition, Ross 140) reported on a series of 62-in.-long specimens which were sealed on all but one end. The sealed end was maintained at 80°C (176*F) and the unsealed end at 20°C (680F). The concrete dried and contracted at the heated and sealed end and also at the cool open face. The shrinkage penetrated only about 12 in. in 400 days indicating that for deep penetration of shrinkage much longer drying times are required. Ruetz(60, 124) reports that the creep rate of specimens will increase with increasing temperature up to about 40°C (104°F) and any further temperature increase will accelerate the hydration process which tends to compensate for the increased creep rate. He further states that dried specimens or speci- mens subjected to high temperature prior to testing will exhibit a steady increase in creep with increasing temperatures. This again is not com- pletely in agreement with other investigators. Hansen and Erikson (141) note that for tests of beams stored in water, at 40°C (104°F), and extended over 100 days there is a large effect of temperature change on the deflections which take place while the temperature is in- creasing. They note no effect of previous high temperature history on subsequent deflections at constant temperatures. A critical consideration in the behavior of concrete at other than normal temperatures is the effect of temperature on strength. Malhotra(142) states that a decrease in strength results from an increase in temperature. The same decrease is noted if the specimen is tested at the increased temperature or allowed to cool to room temperature and then tested. The per- cent of unheated strength is about 5 to 20 percent less for specimens tested after cooling. Thorne 51) stated that the elastic modulus decreases as the temperature rises. Campbell-Allen and Low report a loss of 15 percent of compressive strength when the con- crete is exposed to one cycle at 300°C (572*F); however, for exposure of one thermal cycle at 2000°C (3920F) and 16 percent at 300°C (5720F) after one thermal cycle were reported. Campbell- Allen and Low (143) also report a 55 percent reduction in the secant modulus after treatment to 250°C (4820F) and 300°C (5720F) for one cycle, and note that this reduction of elastic modulus will be accompanied by an increase in the rate of creep. We should note here that the temperatures at which significant strength reductions may occur are well above the range of temperatures for which creep data is available; strength reduction as such may not be a consideration in the inter pretation of these data. The reduction in the elastic modulus of concrete may, however, be a significant parameter since various investigators have reported increased elastic deformations with an increase in temperature occurring at all temperature ranges. 3.10 SIZE AND SHAPE OF SPECIMEN Investigations into effects of size and shape on creep and shrinkage indicate that if the shapes are relatively similar and the size roughly the same, the magnitude of the time- dependent deformations will be about equal, all other factors being the same. Work by Ross,(25) Weil,(144) and (39) Hansen and Mattock present typical results showing the effect of specimen size on shrinkage. All agree that the ratio of the volume to surface of the drying specimen is a reasonable param- eter for use in estimating shrinkage. Since the rate at which a specimen will dry in a given environment is a function of size, then drying creep of specimens will also be dependent on specimen size to a large degree. The effect of specimen size on contin- ued hydration mentioned previously is also a factor which will be reflected in different magnitudes of deformation under load for specimens having dif- ferent sizes. With the exception of the extremes such as a relatively thin slab compared to a stocky square or round column section the effect of shape is.secondary in nature(39'145'146) so long as surface to volume ratio is held constant. 3.11 STRENGTH Creep of concrete is a function of the strength of the concrete indirectly as a result of the factors previously discussed under Sections 3.2, 3.3, 3.4, 3.5, and 3.9; that is, any factors which affect the strength will indirectly or directly, depending on the point of view, affect both creep and shrinkage rates and magnitudes. More reasonably, the stress level can be considered as the parameter rather than strength. If one were to consider the creep of concrete under a given sustained stress, then the rate and magnitude of the expected creep deformations are definitely a function of the strength (Figure 14). Experimental results by many researchers indicate there exists a region of linear proportionality between creep and the applied stress. The upper limit on the region of linear propor- tionality has been variously reported as being between 30 and 85 percent of the strength of the concrete. Since severe internal cracking is known to take place in concrete compression specimens at about 40 to 60 percent of the ultimate strength, it is evident that other behavioral phenomena will be affected also. The following dis- cussion will consider the upper limit for the linearity of creep and applied stress to be 40 percent of the ultimate strength for both tension and compres- sion. 3.12 REINFORCEMENT The effect of steel reinforcing bars on both creep and drying shrinkage is obviously that of restraint. The magnitude of the restraint is dependent upon the percentage of reinforcement present in the cross section and also its distribution. As a result of the time-dependent deformations of concrete, there will be stress redistributions which in general will be an increase in the magnitude of the steel stress and a corresponding decrease in magnitude and redistribution of the stress in the concrete. IV. PREDICTION METHODS 4.1 GENERAL The ideal prediction method would be one which would include all the possible variables which bear on the final value of creep and shrinkage or both; however, this approach is, as yet, not practical. A more reasonable approach at this time is one which will reflect the effect of any changes in the major parameters insofar as the magnitude and time variation of creep and shrinkage are concerned. Effects of variables such as mixing time and consolidation, for example, cannot reasonably be included since in practice the range of these factors is generally not known. The prediction methods presented in the following paragraphs are based on the assumption that the total time- dependent, load induced, deformation of the aggregate and the total drying shrinkage or swelling of the aggregate combined are negligibly small. The proposed prediction methods therefore may be used to predict the behavior of any concrete made with such aggre- gates. Whether concrete may be considered as an ideal composite hard material or an ideal composite soft material has been questioned by many investigators. Hansen (6193) shows that for concretes using normal weight aggregates the behavior is that of a composite soft material. The modulus of elasticity for a composite soft material may be estimated on the assumption that the stress is the same in the more rigid particle phase and the less rigid matrix and the constituents carry. the stress in series; that is, the compliances are added. The cement paste, on the other hand, with the gel and capillary pores can be considered as a composite hard material with the paste matrix having a higher modulus of elasticity than the voids and under load the two phases of a composite hard material share the load in parallel. A schematic repre- sentation, Figure 15,(126,147) shows the aggregate and unhydrated cement grains as the more rigid phase of a composite soft material in series with the hydrated cement paste matrix, which is itself considered as a composite hard material with the hydrated cement paste as the more rigid matrix enclosing the less rigid voids. For a stress of unity the strain in the composite body can be written as: V V (1-V -V )2 S-a +uc a uc (ll) e E E V E e E a uc Vhc "hc in which Vhc = ghVc, Vuc = (1-h)Vc, where E = elastic strain of concrete e under unit stress, V = volume concentration of aggregate, E = elastic modulus of aggregate, a V = volume concentration of u c unhydrated cement, E = elastic modulus of unhydrated cement particles, Vhc = volume concentration of hydrated cement including the gel pores, E hc= elastic modulus of hydrated cement, g = a constant representing the volume of gel produced upon full hydration of a unit volume of cement = 2.2, h = degree of hydration of the cement, V = volume concentration of c cement. The last term in this equation represents the contribution of the hydrated gel to the total strain of the composite material and can be written in terms of the degree of hydration as follows: 1(-V -(1-h)V 2 2 S) a c 8 e hc ghV cEh E h The gel compliance factor 8 here representing the ratio of the deforma- tion of the gel component of the con- crete to the deformation of a hypo- thetical specimen of pure gel subjected to the same stress as the concrete. Figure 16 shows the variation of the degree of hydration with moist curing time for Type I portland cement.(126) This. curve is applicable for mixes having a water-cement ratio of 0.5 to 0.8 by weight. 4.2 PREDICTION OF CREEP 4.2.1 General Of all the proposed hypotheses, few, if any, are in complete accord with all the observed phenomena related to the creep behavior of concrete. The hypothesis proposed by Ali and Kesler(126,147) appears to explain con- vincingly the creep behavior of con- crete under moderate stress levels. This hypothesis, like others, considers creep of concrete to be divisible into two components which, for purposes of investigation and prediction, are con- sidered to be independent and are referred to as basic creep and drying creep. Basic creep is considered to be that part of creep which can occur independently of the loaded specimen's moisture loss or gain. Drying or wetting creep is considered to be the creep, in addition to basic creep, which re- sults from the simultaneous moisture loss or gain of the loaded specimen. Since by this definition drying or wetting creep is linked to moisture loss, the problem of prediction of drying or wetting creep will be com- plicated by the shrinkage or swelling of a specimen not in hygrometric equi- librium with the environment. A sche- matic representation of creep with and without moisture exchange is given in Figure 17. 4.2.2 Basic Creep Under Normal Stress Unit basic creep, that is creep strain per unit stress, varies con- siderably for different concrete mixes under the same environmental and loading conditions. In order to reduce the basic creep of the different mixes to some common value, unit basic creep was divided by the gel compliance factor, 8. Figure 18 shows that the data normalized in this manner is grouped very closely together with little difference between the mixes. The data in this figure were taken on both concrete and mortar specimens of varying compositions and for both axial compressive and flexural loads with all specimens loaded in an environ- ment of 100 percent relative humid- ity. (126) The rheological model for basic creep, Figure 19 of the form chosen by Ali and Kesler, (26,147) which con- forms to the viscoelastic nature of basic creep defined in their hypothesis, will be used since it was found to fit the creep curves with a minimum number of basic rheological elements. The basic creep equation for such a model takes the form: Ebc = a I (l-e-t/T1) + a2 (I e t/'2 )+ t , (13) where Ebc = basic creep strain at any time, t , a = normal stress, a ,a = compliance of the Kelvin springs, T ,T = retardation times of the Kelvin dashpots,, = effective fluidity of the free dash pot, t = time after loading in days. For the normalized data in Figure 18 the rheological constants were found by graphical curve fitting and the corresponding function is plotted through the data. The following values provided the best fit to the normalized data for a temperature of 70*F and a relative humidity of 100 percent. a, = 225 x 10-9, a2 = 115 x 10-9, Ti = 35, T2 = 2, 0 = 0.30 x 10-9. This analysis is based on the assumption that all the basic creep arises in the hydrated cement gel; that is, the time- dependent deformations of the aggregate (if any) are neglected. It is further assumed that the stress levels are low enough to insure linearity of creep strain with sustained stress; that is, on the order of 10 to 40 percent of the ultimate strength value. Finally, it is assumed that no significant additional hydration occurs after loading. An example of the application of Equation (13) is shown in the Appendix. 4.2.3 Drying Creep Under Normal Stress Drying creep and drying shrinkage are two phenomena which are observed to be functions of the moisture migration from the specimen. Ali and Kesler(147) considered shrinkage as being a measure of the volume of the gel moisture withdrawn from a drying specimen, therefore a single relation expressing drying creep as a product of the shrinkage per unit volume of gel, the gel compliance factor 8, and some function of time was proposed. The classical hyperbolic function of time first proposed by Ross(59) provided a simple relation which yields a reasonable fit to many creep time functions. Drying creep then may be represented by the following equation: Edc = oe s a + , (14) where dc = drying creep strain at any time, t, s = free drying shrinkage strain at time, t, a = a constant, assumed indepen- dent of mix proportions, b = a constant, assumed inde- pendent of mix proportions, t = time after loading in days. Values of the constants in the function of time were evaluated for data measured at 70°F and 50 percent relative humidity and reasonable agree- ment was found using a = 2.99 x lO" and b = 7.73 x 10 . Figure 20 shows a plot of the normalized unit drying creep, i.e., (Edc Vhc) t/(oB E ) versus time. The reasonable fit obtained appears to justify the assumption that drying creep is proportional to the corresponding free shrinkage. Although the constants were evaluated at only one temperature and relative humidity, it seems reasonable to assume that any adjustment for different environmental conditions would be in the form of a product of edc and a function of tem- perature or relative humidity or both. An example of the application of Equation (14) is shown in the Appendix, Chapter VI. The application of Equation (14) to the general case, where the time at which drying was initiated and the time at which the specimen was loaded do not coincide, requires special atten- tion. The value of free drying shrink- age, e , to be used is the change in shrinkage which takes place from the day the specimen is loaded. 4.2.4 Total Creep Under Normal Stress The total creep strain, c , due to a sustained normal stress is defined as the sum of the two components basic creep and drying creep; that is: e = 6bc + Fdc c bc dc (15) A complete example of all calcula- tions required to develop a creep-time curve is presented in the Appendix. 4.3 PREDICTION OF SHRINKAGE 4.3.1 General A knowledge of the free drying shrinkage of concrete in advance of actual construction would be of inter- est both in the analysis and design of concrete structures. This information is necessary, first for the prediction of drying creep, as was described in Section 4.2.3, and second for the estimation of long time deflections caused by shrinkage of the concrete. One method of predicting shrinkage of a particular concrete is to deter- mine its deformation experimentally over a short period of time and extrap- olate the shrinkage at some later date from these data. Conversely, there are a few methods available to analyti- cally predict shrinkage of concrete from given mix proportions and environmental conditions. The reliability of these methods, however, remains doubtful. The investigations reported by Wallo, et al., (145,146) were undertaken with the following objectives: (a) To ascertain experimentally the time variation of free drying shrinkage with particular reference to specimen size and shape, curing time, and relative effect of volumetric com- position of the mix. (b) To investigate possible correlations between these variables and the observed free drying shrinkage on the basis of experimental data. (c) To develop empirically an analytical relation which would allow the prediction of the free drying shrinkage at any time as a function of volumetric composition, extent of curing, a size-shape parameter, and the ambient conditions, specifically relative humidity. (d) To compare this analytical relation with experimental results from various investigations. An empirical equation is presented which can be used to predict free drying shrinkage as a function of time in terms of volumetric composition of the mix, curing time, specimen size and shape, and ambient relative humidity. In addition, the results as predicted by this equation are compared to measured shrinkage data taken from various sources in the literature. 4.3.2 Free Drying Shrinkage It is generally agree that the time-dependent shrinkage deformation of commonly used normal weight aggregate is much smaller than that of the cement paste in any concrete mix (Table 9). Therefore, in the present discussion, all time-dependent shrinkage deformations will be thought of as being caused by the cement paste matrix. It is known(100) that shrinkage of concrete increases with a decrease in the diameter of the pores from which water evaporates. The distribution of pore sizes in the cement paste varies with the degree of hydration, and as small pores causing large shrinkage deformation are concentrated in the gel, a shrinkage analysis must consider not only the total amount of cement but also the amount of hydrated cement gel as well. The degree of hydration increases with the life of the specimen, and the rate of hydration at any time during that life is a function of the internal relative humidity. Hydration generally ceases if the internal relative humidity of the concrete is less than about 80 percent.(122) Because of the low moisture diffusivity of concrete, a considerable time elapses until the internal relative humidity of a drying specimen is equal to or less than 80 percent. Therefore a concrete speci- men continues to hydrate for some time after it has been exposed to drying. This continued hydration period is longer for large specimens than for small ones. However, as an approxima- tion it is assumed in the following analysis that, for all specimens, hydration ceases at the end of the moist curing period. Restraint against shrinkage will be considered as being dependent on all aggregate and unhydrated cement grains present; that is, the volume of solids, V s, which is defined by the following equation. Vs = V + Vc = V + (1-h) V c. (16) It is now assumed that the ulti- mate shrinkage, es., is proportional to the concentration of hydrated cement paste and decreases with increasing concentration of restraining particles at a particular relative humidity. Therefore Esm = (A - BVhc)(1-Vs) F(R) , (17) where e = ultimate free drying ~s shrinkage strain at t = , A = a constant, assumed inde- pendent of mix proportions, B = a constant, assumed inde- pendent of mix proportions, F(R) = a function to account for the effects of ambient relative humidity on shrinkage. The ultimate shrinkage as defined by Equation (17), which is similar in form to that used by Dutron,(27) re- flects the interaction of the restraint provided by the solids, which is re- presented by the term (1-V s) and the shrinkage potential of the products of hydration in terms of the parameter V hc. The evaluation of Equation (17) for e resulted in a negative value for the coefficient B. This is probably because of the dual effect of the degree of hydration, h. All other parameters being constant, an increase in the degree of hydration, h, results In an Increased volume of gel and a reduction of the amount of restraining particles of unhydrated cement. Conse- quently, the shrinkage deformations of the concrete increase. This tendency has been kept in mind when setting up Equation (17). On the other hand, an increase of h results in an increase of strength and modulus of the cement paste. Thus the resistance of the system to deformations caused by the evaporation of water has been increased and the shrinkage of the concrete becomes smaller. It should be noted that Vs and Vhc are expressed as volume concentrations, therefore, any change in Vs produces a corresponding change in V hc. If a change of aggregate con- centration is made, then the volume concentration of cement will change and as a result both Vs and Vhc change. If for a given mix there is a change in the curing history the amounts of both the hydrated and unhydrated cement change and again Vs and Vhc both change. In the present analysis the effect of size and shape of a drying specimen upon its ultimate shrinkage, esm, has not been taken into consideration. Ambient conditions, most predoml- nently relative humidity, have been shown to influence the rate and magni- tude of free drying shrinkage. Dutron (27) derived an equation relating the shrinkage at 50 percent relative humidity and the shrinkage at any other relative humidity state, all at about 70*F (21*C), in the following form: ES . sl (0.96 log 05- ) , (18) where es1 = free drying shrinkage strain In 50 percent relative humidity at time, t, R = ambient relative humidity, in percent. In the present analysis the re- lationship presented by Dutron(27) will be used to predict shrinkage at environ- mental relative humidities other than at 50 percent. The ultimate shrinkage equation for a constant temperature of 70°F (21°C) then has the following form: Es» = (A + B Vhc)(1-V ) fit to the measured data is: s = s (l-e-O'l (S/V)to° 65) and e o = (2400-2100 Vhc )(1-V ) (0.96 log 105-R) x 10-6 0.96 log 105-R1 . (19) The direct application of these formulas is limited to concretes made with Type I portland cement. The appli- Time related phenomena frequently cation of Equations (21) and (22) is can be expressed by exponential func- further limited to the range of water- tions as in the equation for basic cement ratios from about 0.4 to 0.8 by creep, Equation (13). A similar weight, to aggregate-cement ratios of approach was taken in this analysis from about 4 to 9 by weight, and to and the following equation is proposed mixes with a cement content of from 376 to depict the shrinkage time relation- to 752 lbs/yd3. An example of the ship. application of Equations (21) and (22) is shown in the Appendix. S= Iat-e ) tn(20) Figures 21 to 25 show the compari- son of the analytical equation for free in which drying shrinkage, Equation (21), to the (s\ measured data for those data used in d ' evaluating the various constants and for data from various sources in the t = time in days from the day literature, as noted. In addition drying was initiated, these figures also contain curves showing n = a constant, assumed independent the shrinkage as predicted by the proce- of mix proportions, d = a constant, assumed independent dure in the Comite Europeen du Beton of mix proportions, (C.E.B.) Recommendations for an Inter- S = surface area of the concrete national Code of Practice.(148) exposed to drying in in.2 Figure 26 shows the comparison of V = volume of the concrete speci- men exposed to drying i.n in.3 the ultimate free shrinkage, esM, as S. obtained by extrapolation of measured For the free shrinkage data collected on three different sized data versus the ultimate free shrinkage collected on three different sized as computed from Equation (22). The specimens and two different mixes, the constants of Equations (16) and (19) dashed lines on Figure 26 enclose those were determined(146) and the form of estimated values which fall within . ± 20 percent of the extrapolated value. these equations which yielded the best The measured data was extrapolated (21 ) (22) using the hyperbolic function, proposed by Ross.(59) This procedure has long been accepted as a method which provides reasonable estimates of the ultimate free shrinkage, but it generally is a poor approximation for the shrinkage- time relationship at the beginning of drying. The hyperbolic function can be expressed in the following form: t 1 (t + t ) , s s (23) where t is the time in days for one- I half the ultimate shrinkage to occur. A plot of t/c versus t would yield a straight line with intercept equal to t /I sw and a slope equal to 1/Esm if shrinkage were to follow this law. With the exception of the early time periods the measured data can be represented by a straight line and the slope or the ultimate free shrinkage can be obtained. 4.4 PREDICTION METHODS AND TEMPERATURE The information available on temperature effects on creep and shrink- age is sparse, confusing, and even contradictory. It appears that in general the only agreement among the various reports is that creep and shrinkage both increase with increasing temperature, at least, within a certain range which is, as yet, not defined. Test results of basic creep tests obtained from a pilot study conducted at the University of Illinois(149) using three different ambient water temperatures of 40*, 700, and 110°F (4.5°, 210, and 43*C) are shown in Figures 27, 28, and 29, respectively. These plots of normalized unit basic creep though insufficient in number for quantitative study indicate that at 1100F (43°C) the gel compliance factor B is not an effective normalization factor. Curves representing approxi- mately the average values of the data are shown on these figures. The values obtained for the spring compliances, retardation times, and dashpot fluidity in Equation (13) at the three tempera- tures indicate changes in the directions which one would reasonably anticipate. An increase in temperature will result in an increase in spring compliance with increasing temperature, a decrease in retardation times and an increase in dashpot fluidity. For the purposes of prediction it is evident that additional research is needed before the effect of temperature on the rate and magnitude of creep and shrinkage can be defined analytically. It does not seem feasible, at this time, to attempt to predict the rate or magnitude of creep or shrinkage for temperatures which depart from ordinary room temperature by more than t 20°F (7°C) . 4.5 CREEP PREDICTION -- STRESSES OTHER THAN UNIFORM UNIAXIAL STRESS As a result of the review of the information available concerning creep of specimens under stress conditions other than uniaxial normal stress, it is evident that much additional research is required in order that the related problems be resolved. In particular the question of the value or the existence of the value for the counter- part of Poisson's ratio for creep strains must first be answered. Then, and only then, can any prediction method be developed for the multiaxial stress condi tion. The phenomenon of increased creep strains occurring in specimens having a large stress gradient as compared to those with a uniform stress is still lacking a thorough explanation. The use of the prediction methods developed for uniform uniaxial, normal stress is recommended as a means of obtaining qualitative information under conditions involving large stress gradients. With respect to the prediction of creep under shearing stress a qualita- tive estimate would be obtained by using twice the value predicted for creep under a compressive stress which would be in agreement with the work of Duke and Davis.(128) V. CONCLUSIONS AND APPLICATIONS 5.1 CONCLUSIONS The following general conclusions appear to be in agreement with the information available to date: (a) Basic creep may be considered as a combination of shear deformations in both the cement gel and in the absorbed gel water layers and a possible migration of moisture from the gel pores to the larger capillary pores. Both the shear deformation and the moisture migration may occur as described above but with no consequent loss or gain of total water content in the specimen. This results in a partly viscous and partly delayed elastic behavior. Test results indicate that the basic creep mechanism is tempera- ture sensitive. Basic creep under normal stress computed using Equation (13) provides reasonably accurate pre- dictions of creep strains for a given concrete at a temperature of about 70°F (21°C) using the following values for the coefficients: bc [225(-et/35)+ 15 (I -e-t/2)+ 0.3t x 10-9. (24) This equation is applicable for stress levels up to about 40 percent of the ultimate strength of the concrete. Basic creep is considered to be only partly recoverable. (b) Drying creep may be considered as being caused by a mechanism similar to that involved in free drying shrink- age or in swelling and occurring principally in the hydrated cement gel. The interaction of drying creep and shrinkage can be visualized as being composed of 'two parts. First, the moisture gradient may be affected by the applied stress; that is, under com- pressive load moisture may be "squeezed" outward during drying or have its entry inhibited during wetting and in this manner affect the rate of shrinkage while the reverse effect will occur in each case if the load is tensile. Second, the applied compressive stress in a drying specimen forces the surfaces of the gel closer together thus modify- ing the shrinkage magnitude or reducing the swelling with the effect of a tensile load being a reversal; that is, a separation of the gel surfaces. Drying creep under normal stress, when expressed as a product of the free drying shrinkage as in Equation (14), provides reasonably accurate predictions of creep strains for a given concrete at a temperature of about 700F (21°C) using the following values for the coefficients: dc = B s 2.99 + L.73 x 10-4. (25) d V thc This equation is applicable for stress levels up to about 40 percent of the ultimate strength of the concrete. Drying creep is temperature sensitive, at least to the same degree as the free drying shrinkage. Drying creep is con- sidered irrecoverable except that restoration of the original water con- tent would be expected to produce partial recovery on unloading. (c) Free drying shrinkage may be considered as occurring primarily in the hydrated cement gel as a result of the removal of water by evaporation. Since the gel structure is primarily colloidal in nature, the removal of water brings the gel particles closer together thus mobilizing the strong surface forces resulting in an overall change in length. Both the time rate and magnitude of free drying shrinkage are predicted with reasonable accuracy for a given concrete at a temperature of about 70*F (21°C) and for values of ambient relative humidity ranging from about 25 to 95 percent using the following equation: £ = (2400-2100Vhc)(I-V )(0.96 log 105-R) (le-0.1 (S/V)t- ) 10- . (26) Shrinkage of concrete is temperature sensitive insofar as the temperature will affect the evaporation of water from the gel, decrease the viscosity of the gel water, and decrease the adhesive forces between the gel water and the solid particles. For the range of concrete mixes commonly used in practice, about 70 percent of the total free drying shrinkage is reversible upon restoration of the original water content. The nonrecoverable deformation which accompanies shrinkage is probably a result of the formation of additional bonds within the gel when closer contact between the gel particles is made possible by loss of the gel water. (d) The total creep at any time after loading is obtained by adding the basic creep and the drying creep at that time. The total time-dependent deformation of a loaded specimen at any time is the sum of the basic creep, the drying creep, and the free drying shrinkage. The computed values of long- time creep and shrinkage from the equations discussed above are in reasonable agreement with the measured data as shown in Figures 21 through 26, 30 and 31. (e) It is noteworthy that the agreement of the predicted deformation and the measured values mentioned above are in reasonable agreement, particularly since the equations do not contain a parameter to account for the difference in the aggregate in a direct manner. The equations were derived assuming that the cause of the total time- dependent deformation was to be found mainly within the hydrated cement gel. The constants in these equations were evaluated from data obtained using con- crete made with two aggregates, gravel and limestone, common to central Illinois. Therefore the application of these prediction methods to concretes using aggregate from the same or nearby sources can be expected to yield reliable results. Application of these predic- tion methods to concrete using aggregates from other sources has been found to be as much as ± 50 percent in error. Data from short-time creep and shrinkage tests can be used to check or revaluate the constants in Equations (24), (25), and (26). (f) Aggregates containing large amounts of clay and aggregates such as sandstone when used in concrete will contribute to the time-dependent dimensional changes caused by sustained loads or drying of the concrete or both. Before the above prediction method can be applied to all aggregate types, additional research using different aggregates is desirable. (g) Strictly speaking, the effects of continued hydration of cement on the deformational response of loaded and unloaded specimens should be con- siderable; however, the problems associated with the prediction or estimation of the rate and amount of such additional hydration remain at this time unresolved. Since it is known that the hydration, for practical purposes, ceases when the specimen's internal relative humidity drops below 80 percent, the hydration at different points in a drying member ceases at different times. This additional hydration is also related to a speci- men size and shape as a result of its dependence on drying. It is concluded that any attempt to include the effects of additional hydration in the proposed prediction methods is impractical at the present time. (h) The results of pilot tests and information available in the literature on temperature effects on creep and shrinkage indicate a need for additional research before an attempt is made to extrapolate the proposed prediction methods to tempera- tures which depart from room temperature by more than 20*F (7°C). Generally, it can be stated that both creep and shrinkage increase with an increase in temperature. This increase is approximately linear up to about 80°C (176*F) and nonlinear thereafter, and the magnitude of the creep and shrink- age deformation can be two to three times greater at 800C (1760F) than at 20°C (680F). Pilot tests (145) indicate that the basic creep of specimens under- going some temperature change during the loaded life is greater than the basic creep of specimens under constant temperature conditions, all other factors being equal. (i) The effect of cyclic variations in moisture conditions of the concrete will be to increase the magnitude of the creep deformation. The last drying, after cyclic wetting and drying, extends the shrinkage curve to the same value it had at the end of the first drying. In this light the shrinkage as proposed herein may provide a limiting envelope toward which cyclic drying will eventually extend. (j) Extension of the proposed prediction methods to include multiaxial stress conditions is not believed possible at the present time. (k) Cyclic variations of loading can be expected to increase the creep above the values predicted using the equations presented herein. (1) Direct application of the creep and shrinkage prediction methods proposed herein can be made by the structural designer by incorporating this information with methods of analysis dealing with long-term deflec- tions of structural elements. A few suggestions of such practical applica- tions are contained in Section 5.2. 5.2 APPLICATIONS In all cases of analysis of long- time behavior of reinforced concrete members under sustained loading, the effects of creep and shrinkage deforma- tions should be considered. The use of unit creep strain, c (creep per unit stress), or a creep coefficient, C (ratio of creep strain to initial strain), yield the same results and in either case the modulus of elasticity of concrete, E c, must be included. These three parameters are related according to the following equation: Ct = cEc . (27) Creep effects may also be included by using the reduced modulus of conr crete, E ct, which is defined by the following equation: E c (28) E ct = I + C (28) Using the total creep strain which can be computed using Equations (13), (14), and (15), the unit creep strain can be found. cc c bc dc c - + a a a Computation of the creep coeffi- cient is also facilitated by using the sum of the values obtained from Equations (24) and (25) and the (29) following equation: c C = C t . I (30) where e. is the initial elastic strain. The initial elastic strain can be computed by any method, in particular the value suggested by the American Concrete Institute Standard 318-63(150) may be used; that is, .i E- = w (31) where w is the weight of concrete, lb/ft3, and f' is the compressive strength of concrete, psi. Any analytical method which re- quires a knowledge, in advance, of the time-dependent deformational behavior of concrete can, it is believed, use the equations for creep and shrinkage presented in Chapter IV. One possible application would be the use of the values for creep directly in the method proposed for calculations of long-time beam deflections proposed by Corley and Sozen.(15l These authors propose the following simple expression for the curvature caused by creep of the concrete. c = d C (32) where 0c = curvature in a beam caused by creep of the concrete, c = creep strains on the extreme compression fiber, d = the effective depth of the beam. In this formulation it is assumed that the calculated initial steel strain does not change with time and the strain distribution remains linear The creep strain on the extreme compression fiber is defined by the authors(151) as: C et = Ec , t a cc have a different value the equation may be modified accordingly. In addi- tion it is suggested that the shrinkage time equation presented in Section 4.3 could be used to compute curvature and deflection of a beam resulting from shrinkage of the concrete by modifying Equation (35) as follows: (33) where E. is the instantaneous strain in the beam. For transformed cracked section the creep curvature in terms of the initial curvature is: Oc = Kp. C . (34) Using the proposed methods for predicting creep strain on the extreme fiber, the value for Ct can be found for any time, t, desired and a deflec- tion time curve could be developed. A similar approach is used(151) for curvature caused by shrinkage and the following expression is suggested to estimate shrinkage curvature: sh 035 (p-p ), (35) where Osh = curvature in a beam caused by shrinkage of the concrete p = ratio of area of tension reinforcement to effective area of concrete, P' = ratio of area of compression reinforcement to effective area of concrete. The authors(151) state that this ex- pression was developed for use with concrete which develops a free shrink- age strain on the order of 500 x 10-6 and add that for a concrete known to 0.035 Es (p-p'). sh d 500 x 10-6 (36) The value of such an application is not so much in its estimation of the ultimate shrinkage curvature but in that shrinkage curvature at any time, t, after drying begins can be estimated. In 1966 the ACI Committee 435(152) presented a report which discusses short-time and long-time deflections of reinforced concrete flexural members. In the report several empirical ex- pressions are presented for computing beam deflections caused by shrinkage and creep. Five relationships are presented for computing shrinkage curvature, all of which contain the free shrinkage strain as one of the variables for both singly and doubly reinforced con- crete beams. The committee suggests that an approach such as that proposed by Branson be used in those cases in which the effects of shrinkage and creep are considered separately and where the choice of appropriate shrinkage and creep coefficients is made by the designer. The creep and shrinkage prediction equations suggested in Chapter IV would be usable in computing deflections by such a method. VI. APPENDIX: ESTIMATION OF CREEP AND SHRINKAGE 6.1 GENERAL The following is an example of the computations necessary to estimate creep and shrinkage curves for a concrete using the equations in Chapter IV. The particular mix used in this example is that used by Shideler.(29) The 28-day compressive strength of the mix was reported as 7560 psi. The measured creep curve and the estimated creep curve are shown in Figure 30 labeled 8-7000. The following is a list of data required for estimating creep and shrinkage. Mix proportions by weight (Series 7000 by Shideler) Aggregate = 3351 lbs Cement = 557 lbs Water = 295 lbs Specific gravity of aggregate = 2.65 (Aggregate No. 8 by Shideler) Specific gravity of cement = 3.15 Specific gravity of water = 1 Air content of mix = 0 (by Shideler) Curing = 7 days in fog room Age at loading = 7 days Stress = 2000 psi compression Storage after loading = 73°F, 50 percent relative humidity Specimen size: 6-in.-by 12-in.-long right circular cylinder 6.2 COMPUTATIONS OF VOLUME CONCENTRATION Absolute Unit Volume Volume ft3 ft3 Aggregate 2.36562.4) = 20.26 = 0.728 = V Cement 3.15 2.) = 2.83 = 0.102 = V Water 295 = 4.73 = 0.170 = V Total = 27.72 1.000 Degree of hydration at 7 days moist cure as obtained from Figure 16 is h = 0.7. Therefore from Equation (11): V = ghVc = 2.2 (0.7) 0.102 = 0.157, V = (1-h) V = 0.3 (0.102) = 0.031. The volume of solids found is then: V = V + V = 0.728 + 0.031 = 0.759. s a uc Therefore 1-V = 0.241. S 6.3 COMPUTATION OF BASIC CREEP Using the volume concentrations the gel compliance factor, 1, can now be found using Equation (12). -V -(1-h)V2 = (0.241 ac (0 24l)25 ghV 0.157 = 37 Using Equation (24) and the values of a follows: and 6, the basic creep is computed as E bc = 0.74 [225 (-e-t/35) + 115 (l-e-t/2) + 0.3t days 0.3t 225 -e-t/35) 115 (-e-t/2) Ebc Sv in./in. 10 3 56 114 128 20 6 98 115 162 40 12 153 115 207 80 24 200 115 251 150 45 222 115 282 200 60 225 115 296 300 90 225 115 318 360 108 225 115 332 6.4 COMPUTATION OF FREE DRYING SHRINKAGE For R = 50 percent relative humidity using Equation (22), the ultimate free drying shrinkage is: x 10-6 55 esS = [2400-(2100)0.157](0.241) 0.96 log 105-50 x 10"6 = 500 x 10-6 Compute the surface to volume ratio (neglecting bases as the load plates seal them) S = irdL, V= Td2L V = "--4-- , S - = . = 0.67. Using Equation (21) the shrinkage at various times is: S= 500 (l-e0067t"65) p in./in. days 0.067t06s (l-e-0*067to. 6) in/in. 10 0.299 0.258 129 20 0.469 0.373 187 40 0.737 0.529 264 80 1.16 0.687 344 150 1.74 0.825 412 200 2.16 0.885 442 300 2.81 0.936 468 360 3.08 0.954 476 6.5 COMPUTATION OF DRYING CREEP Using Equation (25) the values of drying creep follows: for various times are as Edc 2000 0.37 (2.99 + 73 10 dc 0.157 e s 2.99 + t73 = 0.471 es (2.99 + 7.73) t Edc u in./in. days 0.471 E 2.99 + 7- 73) 10 61 3.76 229 20 88 3.38 298 40 124 3. 18 394 80 162 3.09 500 150 194 3.04 590 200 208 3.03 630 300 225 3.02 680 360 229 3.01 690 6.6 TOTAL CREEP t Ebc Edc c days i in./in. + in./in. p in./in. 10 128 229 357 20 162 298 460 40 207 394 601 80 251 500 751 150 282 590 872 200 296 630 926 300 318 680 998 360 332 690 1022 VII. REFERENCES 1. Maney, G. 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G., Bibliography on Time-Dependent Effects in Plain and Reinforced Concrete. Urbana, Ill.: Department of Civil Engineer- ing, University of Illinois, 1959. 8. Meyers, B. L., A Review of Litera- ture Pertaining to Creep and Shrink- age of Concrete (Engineering Experiment Station Bulletin). Columbia, Mo.: College of Engi- neering, University of Missouri, February, 1963. 9. Slate, F. 0. Comprehensive Bibli- ography of Cement and Concrete (Engineering Experiment Station Bulletin). Lafayette, Ind.: College of Engineering, Purdue University, 1947. 10. L'Hermite, R., "What Do We Know About the Plastic Deformation and Flow of Concrete?" Annales de L'Institute Technique du Batiment et des Travany Publics, Paris, No. 117, (September, 1957), pp. 778-809. (In French.) 11. Fluck, P. G., and Washa, G. W., "Creep of Plain and Reinforced Con- crete," Proceedings, American Concrete Institute, 54 (1958), pp. 879-895. 12. Wagner, 0., Creep of Plain Concrete. 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