ACI James Domingo J. Chairman, Subcommittee II James J. Beaudoin John R. Dan E. Clyde E. Kesler Bruce R. Gamble William R. Frederic H.G. Geymayer Jack A. John Brij B. Michael A. Ward Brian B. Hope Corresponding Members: John W. H.K. Hilsdorf A. Daye Akthem Al-Manaseer Bernard Meyers James J. Beaudoiu K. Ghosh Karim W. Nasser Dan Mikael PJ. Olsen Domingo J. Carreira Baldev R. Seth Chem Stacy K. Hirata Menashi D. Cohen Joe Huterer Liiia Robert L Day Marzouk Chapter pg. almost independently from the rest of the report. l.l-Scope report of the problem 1.3-Definitions of terms Chapter 2-Material response, pg. girders; lightweight-aggregate concretes; modulus of elasticity; moments of inertia; and elastic properties 2.3-Theory for predicting creep and shrinkage of con- shrinkage; strains; stress relaxation; structural design; temperature; thermal expansion; two-way slabs: volume change; warpage. crete creep and shrinkage equations ACI Committee Reports, Guides, Standard Practices, and for standard conditions Commentaries are intended for guidance in designing, plan- ning, executing, or inspecting construction and in preparing specifications. References to these documents shall not be made the Project Documents. If items found in these documents are desired to be a part of the Project Docu- ments, they should be phrased in mandatory language and incorporated into the Project Documents. J 209R-2 ACI COMMITTEE REPORT factors for conditions other than the cases standard concrete composition factors for concrete composition 2.7-Example Acknowledgements, pg. methods for prediction of creep and 2.9-Thermal expansion coefficient of concrete cited in this report Notation, pg. Chapter 3-Factors affeating the structural response Tables, pg. assumptions and methods of analysis, pg. 3.2-Principal facts and assumptions 3.3-Simplified methods of creep analysis l-GENERAL 3.4-Effect of cracking in reinforced and prestressed members l.l-Scope compression steel in members This report presents a unified approach to predicting 3.6-Deflections due to warping the effect of moisture changes, sustained loading, and 3.7-Interdependency between steel relaxation, creep temperature on reinforced and prestressed concrete and shrinkage of concrete structures. Material response, factors affecting the struc- tural response, and the response of structures in which Chapter of structures in which time the time change of stress is either negligible or significant change of stresses due to creep, shrinkage and tem- are discussed. perature is negligible, pg. Simplified methods are used to predict the material 4.1-Introduction response and to analyze the structural response under 4.2-Deflections of reinforced concrete beam and slab service conditions. While these methods yield reasonably 4.3-Deflection of composite precast reinforced beams good results, a close correlation between the predicted in shored and unshored constructions deflections, cambers, prestress losses, etc., and the 4.4-Loss of prestress and camber in noncomposite measurements from field structures should not be ex- prestressed beams pected. The degree of correlation can be improved if the of prestress and camber of composite pre- prediction of the material response is based on test data cast and prestressed-beams unshored and shored for the actual materials used, under environmental and constructions loading conditions similar to those expected in the field 4.6-Example 4.7-Deflection of reinforced concrete flat plates and These direct solution methods predict the response be- havior at an arbitrary time step with a computational ef- 4.8-Time-dependent shear deflection of reinforced fort corresponding to that of an elastic solution. They concrete beams have been reasonably well substantiated for laboratory 4.9-Comparison of measured and computed deflec- conditions and are intended for structures designed using tions, cambers and prestress losses using pro- the 318 Code. They are not intended for the analy- cedures in this chapter sis of creep recovery due to unloading, and they apply primarily to an isothermal and relatively uniform en- Chapter of structures with signigicant time change of stress, pg. Special structures, such as nuclear reactor vessels and 5.l-Scope containments, bridges or shells of record spans, or large aging and the age-adjusted effective ocean structures, may require further considerations modulus method which are not within the scope of this report. For struc- relaxation after a sudden imposed defor- tures in which considerable extrapolation of the mation the-art in design and construction techniques is achieved, relaxation after a slowly-imposed defor- long-term tests on models may be essential to provide a mation sound basis for analyzing serviceability response. Refer- of a change in statical system ence 109 describes models and modeling techniques of buckling deflections of an eccentrically concrete structures. For mass-produced concrete mem- compressed member bers, actual size tests and service inspection data will cantilevers of unequal age connected at time result in more accurate predictions. In every case, using by a hinge 5.8 loss of compression in slab and test data to supplement the procedures in this report will deflection of a steel-concrete composite beam result in an improved prediction of service performance. PREDICTION OF CREEP 209R-3 1.2-Nature of the problem with time of concrete volume. The decrease is clue to Simplified methods for analyzing service performance changes in the moisture content of the concrete and are justified because the prediction and control of physico-chemical changes, which occur without stress at- dependent deformations and their effects on concrete tributable to actions external to the concrete. The con- structures are exceedingly complex when compared with verse of shrinkage is which denotes volumetric the methods for analysis and design of strength perfor- increase due to moisture gain in the hardened concrete. mance. Methods for predicting service performance in- Shrinkage is conveniently expressed as a dimensionless volve a relatively large number of significant factors that strain (in./in. or m/m) under steady conditions of relative are difficult to accurately evaluate. Factors such as the humidity and temperature. nonhomogeneous nature of concrete properties caused by The above definition includes drying shrinkage, the stages of construction, the histories of water content, genous shrinkage, and carbonation shrinkage. temperature and loading on the structure and their effect on the material response are difficult to quantify even for Drying shrinkage is due to moisture loss in the structures that have been in service for years. concrete The problem is essentially a statistical one because Autogenous shrinkage is caused by the hydration most of the contributing factors and actual results are in- of cement herently random variables with coefficients of variations Carbonation shrinkage results as the various of the order of 15 to 20 percent at best. However, as in cement hydration products are carbonated in the the case of strength analysis and design, the methods for presence of CO, predicting serviceability are primarily deterministic in nature. In some cases, and in spite of the simplifying Recommended values in Chapter 2 for shrinkage assumptions, lengthy procedures are required to account strain are consistent with the above definitions. for the most pertinent factors. According to a survey by Committee 209, most time-dependent increase of strain in hardened designers would be willing to check the deformations of concrete subjected to sustained stress is defined as creep. their structures if a satisfactory correlation between com- It is obtained by subtracting from the total measured puted results and the behavior of actual structures could strain in a loaded specimen, the sum of the initial in- be shown. Such correlations have been established for stantaneous (usually considered elastic) strain due to the laboratory structures, but not for actual structures. Since sustained stress, the shrinkage, and the eventual thermal concrete characteristics are strongly dependent on en- strain in an identical load-free specimen which is sub- vironmental conditions, load history, etc., a poorer cor- jected to the same history of relative humidity and tem- relation is normally found between laboratory and field perature conditions. Creep is conveniently designated at service performances than between laboratory and field a constant stress under conditions of steady relative strength performances. humidity and temperature, assuming the strain at loading With the above limitations in mind, systematic design (nominal elastic strain) as the instantaneous strain at any procedures are presented which lend themselves to a The above definition treats the initial instantaneous for predicting the initial and time-dependent average strain, the creep strain, and the shrinkage as additive, response (including ultimate values in time) of structural even though they affect each other. An instantaneous members of different weight concretes. change in stress is most likely to produce both elastic and The procedures in this report for predicting inelastic instantaneous changes in strain, as well as dependent material response and structural service per- time creep strains (10 to 100 minutes of duration) which formance represent a simplified approach for design are conventionally included in the so-called instantaneous purposes. They are not definitive or based on statistical strain. Much controversy about the best form of “prac- results by any means. methods are needed tical creep equations” stems from the fact that no clear to accurately estimate the variability of all factors in- and inelastic strains) and the creep strain. Also, the creep definition lumps together the basic creep and the drying 1.3-Definitions of terms creep. Thefollowing terms are defined for general use in this report. It should be noted that separability of creep and Basic creep occurs under conditions of no shrinkage is considered to be strictly a matter of defin- moisture movement to or from the environment ition and convenience. The time-dependent deformations Drying creep is the additional creep caused by of concrete, either under load or in an unloaded speci- men, should be considered as two aspects of a single complex physical phenomenon. considering the effects of creep, the use of either a Shrinkage unit strain, (creep per unit stress), or creep coefficient, Shrinkage, after hardening of concrete, is the decrease (ratio of creep strain to initial strain), yields the same conditions = accurate of the material response in the Creep strain = and it is to use material will provide a range of variations the The choice of either of or is a matter of tent, temperature. and loading histories in predicting 1.3.3 Relaxation stress increasing with time at a 2.2-Strength 1.3.4Modulus of elasticity (1 minutes) stress-strain (2-l) for 1.3.5 Contraction and expansion = (2-1) where in days and are constants, iations caused by heat of of cement and by in 2.1-Introduction = procedures used predict the of 5 on the prediction of the material response representing parameters are not available for local materials and Experimental of the response of normal weight, sand lighweight, egate does appear affect these accurate prediction of structural service response is 7 are given in Table Values for the time-ratio, than testing actual materials under and PREDICTION OF CREEP cured refer in The modulus of rupture depends on the shapeof the 132 and C 511. Temperatures other 3 F (23 loading conditions 1.7 C) and relative humidities 35 in. (150 x in ASTM 78, Where much the tension is remote the constant on Table 2.2.1 for curing. in of girders of temperature on the compressive and or I-beams, the modulus of rupture approaches of etes direct tensile strength. with different types of cement with and by and is used in admixtures at various temperatures between of Reference 27. modulus of 25 F (-3.9 C) and 120 F (48.9 ( C) were studied in Ref- is determined with erence Constants in Table 2.2.1 are not applicable to con- such as mass concrete, containing or understood is not the truly instantaneous modulus, but V cementsor containing blends of and a modulus which corresponds to loads of one to five slower and may continue over periods well beyond one year age. “Steam cured” curing with at atmosphericpressure at temperatures 212 The that affect are discussed in detail in References 3,6,13-16, and are Experimental References 1-6arecompared in 7 and all fall within 20 for predicting creep and shrinkage: refers to percent of the by (2-l) and for constants and in and cycle employed in steam curing may sub- in References 3, 7, 17, and 83. stantially affect the strength-time ratio early days 15,18-21, the following general procedure is suggested 2.2.2 for predicting creep and of time. Eqs. (2-3), (2-4),and (2-5) are considered satisfactory in most cases for computing average values for modulus of rupture, direct tensile strength, and secant mod- (2-6) respectively of different weight = = where (cid:72) and (in days), and are considered con- stants for a given member shape and size which define = the time-ratio part, is the ultimate creep coefficient defined as ratio of creep strain to initial strain, is For the unit weight of concrete, w in pcf and the com- the ultimate shrinkage strain, and is the time after pressive strength, in psi loading in Eq. (2-6) and time from the end of the initial curing in Eq. (2-7). = 0.60 to 1.00 (a conservative value of =0.60 When and may be used, although a value = 0.60 to the familiar hyperbolic equations of and 0.70 is more realistic in most cases) in slightly different form. The form of these equations is thought to be conven- = 33 ient for design purposes, in which the concept of the ultimate (in time) value is modified by the time-ratio to For w in and in yield the desired result. The increase in creep after, say, 100 to 200 days is usually more pronounced than shrink- = 0.012 to 0.021 conservative value of = age. In percent of the ultimate value, shrinkage usually 0.012 may be used, although a value of increases more rapidly during the first few months. Ap- 0.013 to 0.014 is more realistic in most cases) propriate powers of in Eqs. (2-6) and (2-7) were found in References 6 and 7 to be 1.0 for shrinkage (flatter hyperbolic form) and 0.60 for creep (steeper curve for ACI REPORT larger values of t). This can be seen in Fig. (2-3) and form of the creep coefficient, (ratio of creep strain to (2-4) of Reference 7. initial strain), as compared to creep strain per unit stress, Values of v ,, and can be determined This is because the effect of concrete stiffness is in- u by fitting the data obtained from tests performed in cluded by means of the initial strain. accordance to ASTM C 512. Eqs. (2-6) and (2-7) 2.4-Recommended creep and shrinkage equations for were found to standard conditions Equations and (2-10) are recommended = 0.40 to 0.80, for predicting a creep coefficient and an unrestrained (cid:33)6 to 30 days, shrinkage strain at any time, including ultimate values.6-7 = 1.30 to 4.15, They apply to normal weight, sand lightweight, and all = 0.90 to 1.10, lightweight concrete (using both moist and steam curing, = 20 to 130 days, and Types I and III cement) under the standard condi- = 415 x 10” to 1070 x (m/m) tions summarized in Table 2.2.2. Values of and need to be modified by the These constants are based on the standard conditions correction factors in Sections 2.5 and 2.6 for conditions in Table 2.2.2 for the normal weight, sand lightweight, other than the standard conditions. and all lightweight concretes, using both moist and steam Creep coefficient, for a loading age of 7 days, for curing, and Types I and III cement as in References 3-6, moist cured concrete and for 1-3 days steam cured con- crete, is given by Eq. (2-8). Eqs. and (2-10) represent the average values for these data. These equations were compared with the data (120 creep and 95 shrinkage specimens) in (2-8) Reference 7. The constants in the equations were deter- mined on the basis of the best fit for all data individually. Shrinkage after age 7 days for moist cured concrete: The average-value curves were then determined by first obtaining the average of the normal weight, sand light- weight, and all lightweight concrete data separately, and then averaging these three curves. The constants and recommended in References 7 and96 were approx- Shrinkage after age 1-3 days for steam cured concrete: imately the same as the overall numerical averages, that is v 2.35 was recommended versus 2.36; = 800 for moist cured con- crete, and 730 x versus 788 x for steam cured concrete. In Eq. is time in days after loading. In Eqs. The and shrinkage data, based on 20-year mea- surements7,18 for normal weight concrete with an initial (2-9) and (2-l0), is the time after shrinkage is con- sidered, that is, after the end of the initial wet curing. time of 28 days, are roughly comparable with Eqs. (2-8) In the absence of specific creep and shrinkage data for to (2-10). Some differences are to be found because of local aggregates and conditions, the average values sug- the different initial times, stress levels, curing conditions, gested for and are: and other variables. However, subsequent with 479 creep data = and points and 356 shrinkage data points resulted in the same average for = 2.35, but a new average for = 780 x 10-6 (m/m), for both moist and steam cured x concrete. It was found that no consistent distinction in the ultimate shrinkage strain was apparent for moist and where and represent the product of the applicable steam cured concrete, even though different time-ratio correction factors as defined in Sections 2.5 and 2.6 by terms and starting times were used. Equations (2-12) through (2-30). These values correspond to reasonably well shaped independently evaluated and recommended in Reference aggregates graded within limits of ASTM C 33. Aggre- 60, in which a comprehensive experimental study was gates affect creep and shrinkage principally because they made of the various parameters and correction factors influence the total amount of cement-water paste in the for different weight concrete. concrete. No consistent variation was found between the dif- The time-ratio part, [right-hand side except for and ferent weight concretes for either creep or shrinkage. It of Eqs. and (2-l0), appears to be was noted in the development of Eq. (2-8) that more applicable quite generally for design purposes. Values consistent results were found for the creep variable in the from the standard Eqs. (2-8) to (2-10) of and PREDICTION OF CREEP are shown in Table 2.4.1. Note that v is used where is the loading age in days. Representative val- in Eqs. and hence, = ues are shown in Table 2.51. Note that in Eqs. for the age of the precast beam concrete at the slab and the Creep correction factor must be used when computing the ultimate creep coefficient of It has also been that the time-ratio part of the present beam corresponding to the age when slab is Eqs. (2-8) and (2-10) can be used to extrapolate cast, v That is: us creep and shrinkage data determined experimentally in accordance with ASTM C 512, to complete time curves up to ultimate quite well for creep, and reasonably well for shrinkage for a wide variety of data. It should be noticed that the time-ratio in Eqs. (2-8) to (2-10) does For shrinkage considered for other than 7 days for not differentiate between basic and drying creep nor moist cured concrete and other than l-3 days for steam between drying autogenous and carbonation shrinkage. cured concrete, determine the difference in Eqs. (2-9) Also, it is independent of member shape and size, and (2-10) for any period starting after this time. because and are considered as constant in Eqs. That is, the shrinkage strain between 28 days and 1 and (2-10). year, would be equal to the 7 days to 1 year shrinkage The shape and size effect can be totally considered on minus the 7 days to 28 days shrinkage. In this example the time-ratio, without the need for correction factors. for moist cured concrete, the concrete is assumed to have That is, in terms of the shrinkage-half-time as given been cured for 7 days. Shrinkage factor as in by Eq. (2-35) by replacing by in Eq. (2-9) and by below, is applicable to Eq. (2-9) for concrete moist cured in Eq. (2-8) as shown in 2.8.1. Also by taking during a period other than 7 days. = 1.0 and d = = 26.0 [exp in Eqs. (2-6) and (2-7) as in Reference 23, where is the volume to For shrinkage of concrete moist cured during a period surface ratio, in inches. For in mm use d = = 26.0 of time other than 7 days, use the Shrinkage factor inTable 2.5.3. This factor can be used to estimate differ- References 61, 89, 92, 98 and 101 consider the effect ential shrinkage in composite beams, for example. of the shape and size on both the time-ratio Linear interpolation may be used between the values dependent development) and on the coefficients affecting the ultimate (in time) value of creep and Committee 209, Subcommittee I Report’ is For ambient relative humidity greater than 40 percent, commended for a detailed review of the effects of use Eqs. (2-14) through concrete constituents, environment and stress on age correction factors.7, dependent concrete deformations. = 1.27 for 40 factors for conditions other than the standard concrete composition7 Shrinkage = 1.40 0.0102, for 40 80 All correction factors, y, are applied to ultimate values. However, since creep and shrinkage for any = 3.00 for 80 100 period in Eqs. (2-8) through (2-10) are linear functions of the ultimate values, the correction factors in this procedure may be applied to short-term creep and where is relative humidity in percent. Representative shrinkage as well. values are shown in Table 2.5.4. Correction factors other than those for concrete com- The average value suggested for = 40 percent is position in Eqs. (2-11) through (2-22) may be used in = 780 x (m/m) in both Eqs. (2-9) and conjunction with the specific creep and shrinkage data (2-10). From Eq. (2-15) of Table 2.5.4, for = 70 per- from a concrete tested in accordance with ASTM C 512. cent, = x = 546 x in/in. (m/m), for example. For lower than 40 percent ambient relative For loading ages later than 7 days for moist cured humidity, values higher than 1.0 shall be used for Creep concrete and later than l-3 days for steam cured con- and Shrinkage crete, use Eqs. (2-11) and (2-12) for the creep correction mm) or ratio other than 1.5 in. (38 mm) Themember size effects on concrete creep and shrink- Creep = for moist cured concrete (2-11) (see Equations 2-6,2-7,2-8,2-9,2-10 and2-35). ly, it also affects the ultimate creep coefficient, and Creep = 1.13 for steam cured the ultimate shrinkage strain, concrete Two methods are offered for estimating the effect of ACI COMMITTEE REPORT member size on and The average-thickness Shrinkage = method tends to compute correction factor values that where is the volume-surface ratio of the member in since Creep = Creep = 1.00 for h = 6 in. (150 mm) and = is, when h = Creep (cid:33) 2.5.5.a Average-thickness method The method of treating the effect of member size in Shrinkage (cid:33) 1.2 exp(-0.00472 v/s) terms of the average thickness is based on information from References 3, 6, 7, 23 and 61. where in mm. For average thickness of member less than 6 in. (150 Representative values are shown in Table 2.5.5.2. Table 2.5.5.1. These cor- However, for either method should not be taken respond to the values for small members. For less than 0.2. Also, use 100 x 10” average thickness of members greater than 6 in. (150 (m/m) if concrete is under seasonal wetting and drying mm) and up to about 12 to 15 in. (300 to 380 mm), use cycles and 150 x (m/m) if concrete Eqs. (2-17) to (2-18) through (2-20). is under sustained drying conditions. During the first year after loading: Temperature other than 70 F (21 C) Temperature is the second major environmental factor Creep = in creep and shrinkage. This effect is usually considered to be less important than relative humidity since in most For ultimate values: structures the range of operating temperatures is and high temperatures seldom affect the structures Creep (cid:33) during long periods of time. The effect of temperature changes on concrete During the first year of drying: and shrinkage is basically two-fold. First, they directly influence the time ratio rate. Second, they also affect the Shrinkage (cid:4)(cid:33)(cid:4)1.23-0.038 h, rate of aging of the concrete, i.e. the change of material properties due to progress of cement hydration. At 122 For ultimate values: F (50 C), creep strain is approximately two to three times as great as at 68-75 F (19-24 C). From 122 to 212 F (50 Shrinkage (cid:33)1.17-0.029 h, to 100 C) creep strain continues to increase with tem- perature, reaching four to six times that experienced at where his the average thickness in inches of the part of room temperatures. Some studies have indicated an ap- the member under consideration. parent creep rate maximum occurs between 122 and 176 During the first year after loading: F (50 and 80 There is little data establishing creep rates above 212 F (100 C). Additional information on Creep = temperature effect on creep may be found in References For ultimate values: 2.6-Correction factors for concrete composition Creep Equations (2-23) through (2-30) are recommended for use in obtaining correction factors for the effect of During the first year after loading: slump, percent of fine aggregate, cement and air content. It should be noted that for slump less than 5 in. (130 Shrinkage =1.23-0.00015 h, mm), fine aggregate percent between percent, cement content of 470 to 750 lbs. per (279 to 445 For ultimate values: and air content less than 8 percent, these factors are approximately equal to 1.0. Shrinkage =1.17-0.00114 h, These correction factors shall be used only in con- = 2.35 where h is in mm. and = 780 x (m/m). As recommended in Representative values are shown in Table 2.5.5.1. 2.4, these average values for and should be used 2.5.5.b Volume-surface ratio method only in the absence of specific creep and shrinkage data Thevolume-surface ratio equations (2-21) and (2-22) for local aggregates and conditions determined in accord- were adapted from Reference 23. with ASTM C 512. If shrinkage is known for local aggregates and con- Creep (cid:4)(cid:33) exp(-0.54 v/s)] (2-21) PREDICTION OF CREEP 209R-9 The principal disadvantage of the concrete compo- 2.6.5 Shrinkage ratio of concretes with equivalent paste sition correction factors is that concrete mix charac- teristics are unknown at the design stage and have to be Shrinkage strain is primarily a function of the shrink- estimated. Since these correction factors are normally not age characteristics of the cement paste and of the ag- excessive and tend to offset each other, in most cases, gregate volume concentration. If the shrinkage strain of they may be neglected for design purposes. a given mix has been determined, the ratio of shrinkage strain of two mixes with different content of paste but with equivalent paste quality is given in Eq. Creep =0.82 + 0.067s 1 Shrinkage = 0.89 + (cid:21) where s is the observed For where and are the total aggregate solid volumes per mm use: unit volume of concrete for each one of the mixes. Creep (cid:33)0.82 + 0.00264s 2.7-Example Find the creep coefficient and shrinkage strains at 28, Shrinkage = 0.89 + 90, 180, and 365 days after the application of the load, assuming that the following information is known: 7 days moist cured concrete, age of loading = 28 days, 70 percent ambient relative humidity, shrinkage considered Creep (cid:33) 0.88 + from 7 days, average thickness of member 8 in. (200 For 50 percent 752 lbs. of cement per (446 and 7 percent air Also, find the differential shrinkage strain, Shrinkage = 0.30 + for the period starting at 28 days after the appli- cation of the load, = 56 days. For > 50 percent The applicable correction factors are summarized in Table 2.7.1. Therefore: Shrinkage = 0.90 + = = 1.67 where is the ratio of the fine aggregate to total aggre- gate by weight expressed as percentage. = (780 x = 530 x 2.6.3 Cement content Cement content has a negligible effect on creep co- The results from the use of Eqs. (2-8) and (2-9) or efficient. An increase in cement content causes a reduced Table 2.4.1 are shown in Table 2.7.2. creep strain if water content is kept constant; however, Notice that if correction factors for the concrete data indicate that a proportional increase in modulus of composition are ignored for and they will be 10 elasticity accompanies an increase in cement content. and 4 percent smaller, respectively. If cement content is increased and water-cement ratio is kept constant, slump and creep will increase and Eq. 2.8-Other methods for predictions of creep and shrink- Other methods for prediction of creep and shrinkage Shrinkage = 0.75 + are discussed in Reference 61,68, 86, 87,89,93,94,95, 97, and 98. Methods in References 97 and 98 subdivide where c is the cement content in pounds per creep strain into delayed elastic strain and plastic flow For cement content in (two-component creep model). References 88,89,92,99, 100, 102, and 104 discuss the conceptual differences be- Shrinkage = tween the current approaches to the formulation of the creep laws. However, in dealing with any method, it is 2.6.4 Air content important to recall what is discussed in Sections 1.2 and 2.1 of this report. Creep = 0.46 + 2.8.1 but not less than 1.0 special The preceding formulation represents a compromise Shrinkage = 0.95 + between accuracy and generality of application. More ac- curate formulas are possible but they are inevitably not where is the air content in percent. as general. ACICOMMlTTEE REPORT The time curve of creep given by Eq. (2-8) exhibits a characteristic thickness of the cross section, or twice the volume-surface ratio decline of slope in log-t scale for long times. This prop- 2 in mm) erty is correct for structures which are allowed to lose their moisture and have cross sections which are not too Drying diffusivity of the concrete (approx. massive (6 to 12 in., 150 to 300 mm).Structures which 10 mm/day if measurements are unavail- are insulated, or submerged in water, or are so massive they cannot lose much of their moisture during their age dependence coefficient not decreasing at end, but steadily increasing. For example, if Eq. (2-8) were used for extrapolating time creep data for a nuclear reactor containment into long times, the long-term creep values would be seriously 12, if 7, set = 7 underestimated, possibly by as much as 50 percent as if 21, set = shown in Fig. 3 of Ref. 81. It has been found that creep without moisture ex- coefficient depending on the shape of cross age is better section, that is: described by Equation This is called the double power law. 1.00 for an infinite long slab In Eq. (2-33) is a constant, and strain is the sum 1.15 for an infinite long cylinder of the instantaneous strain and creep strain caused by 1.25 for an infinite long square prism unit stress. 1.30 for a sphere 1.55 for a cube temperature coefficient where is a constant which indicates the lefthand asymptote of the creep curve when plotted in log t-scale (time t = 0 is at in this plot). The asymptotic value is beyond the range of validity of Eq. (2-33) and should not be confused with elastic modulus. Suitable T concrete temperature in kelvin values of constants are = and = being the modulus of concrete which does not under- reference temperature in kelvin go drying. With these values, Eq. (2-33) and Eq. (2-8) give the same creep for = 28 days, t= 10,000 days water content in and 100 percent relative humidity = all other correction factors being taken as one. By replacing in Eq. (2-9) shrinkage is expressed Eq. (2-33) has further the advantage that it describes without the need for the correction factor for size in Sec- not only the creep curves with their age dependence, but also the age dependence of the elastic modulus in The effect of drying on creep may then be expressed absence of drying. is given by = for 0.001 by adding two shrinkage-like functions and to the day, that is: double power law for unit Function expresses 1 1 the additional creep during drying and function being = + (0.001)1/8 negative, expresses the decrease of creep by after an initial drying. The increase of creep during drying arises about ten times slower than does shrinkage and so lus, which is given by = when t = days is function is similar to shrinkage curve in Eq. (2-9) with substituted. Since three constants are necessary to de- replaced by 0.1 in Eq. (2-8). scribe the age dependence of elastic modulus (E,, and This automatically accounts also for the size effect, only one additional constant (i.e., is needed to without the need for any size correction factor. The de- describe creep. crease of creep rate due to drying manifests itself only In case of drying, more accurate, but also more com- very late, after the end of moisture loss. This is apparent from the fact that function is similar to shrinkage section size is expressed in terms of the shrinkage half- curve in Eq. (2-9) with replaced by 0.01 Both time, as given in Eq. (2-35) for the age at which con- and include multiplicative correction factors for rela- crete drying begins. tive humidity, which are zero at 100 percent, and func- tion further includes a factor depending on the time lag from the beginning of drying exposure to the begin- = 150 ning of loading. where: 2.9-Thermal expansion coefficient of concrete