Inverse Analysis Method for Concrete Shrinkage Prediction from Short-Term Tests

ACI Materials Journal/May-June 2012 293

Title no. 109-M27

ACI MATERIALS JOURNAL TECHNICAL PAPER

ACI Materials Journal, V. 109, No. 3, May-June 2012.MS No. M-2011-019.R1 received October 13, 2011, and reviewed under Institute

publication policies. Copyright © 2012, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the March-April 2013 ACI Materials Journal if the discussion is received by December 1, 2012.

Inverse Analysis Method for Concrete Shrinkage Prediction from Short-Term Testsby Mohammad Shekarchi, Farnam Ghasemzadeh, and Siavash Sajedi

this model in shrinkage prediction. They commented that the errors in the ACI 209R-8215 model are due to the influence of size and shape of the specimen on shrinkage. Ojdrovic and Zarghamee16 used short-term test results to adjust both the Bažant-Panula (BP-KX)17,18 and ACI 209R-8215 models to predict long-term concrete shrinkage and creep. The results showed that the coefficient of variation (COV) of long-term measured shrinkage and creep relative to the predic-tions obtained by modified forms of such models using the short-term tests can be significantly reduced. Ojdrovic and Zarghamee16 mentioned that using the early-age shrinkage and creep data within the first few days after exposure for long-term prediction will increase the number of errors. Also, this method is appropriate for low-water concretes without additives.

One method to predict the long-term shrinkage of concrete by using the short-term measurements can be obtained by modifying existing models using inverse analysis (IA). IA is concerned with the estimation of unknown parameters from available information. This method provides some algorithms that lead to the best fit between the experimental measurements and the corresponding computed data.19,20 IA has been used in various fields of structural engineering. Almeida et al.21 used this method to fit the long-term results of shrinkage and creep prediction models to long-term experimental measurements. These modified models can only be used for concretes with mixture proportions similar to their specimens.

In this paper, a new method is proposed in which the long-term shrinkage can be estimated by using the results of the short-term shrinkage data and the existing shrinkage predic-tion models. The accuracy of the method is examined through the comparison of the obtained results to the test results.

RESEARCH SIGNIFICANCEShrinkage plays an important role in the design of

prestressed concretes and the cracking tendency of massive concretes and repaired members. Due to the complicated process of experimental programs in measuring the shrinkage of concretes in the laboratory and the lack of long-term test data for concrete structures at the design stage, numerical methods can be used as a suitable tool for the prediction of long-term shrinkage. This investigation studies the IA method for the prediction of long-term shrinkage of various mixture designs from short-term measurements.

The accurate long-term prediction of shrinkage in concrete struc-tures has been an important issue for a long time. Although several models for the prediction of shrinkage strains already exist, a reli-able model suitable for different concretes is still needed. In this study, the total shrinkage of four different mixture designs was measured in the laboratory; subsequently, the long-term shrinkage of these concretes was predicted from the short-term measure-ments using the inverse analysis (IA) method applied to six existing shrinkage models. Two different mixtures from two previous studies were chosen to verify the proposed method. The results show a good agreement between the experimental measurements and the estimated values.

Keywords: inverse analysis; long-term prediction; short-term measure-ments; shrinkage.

INTRODUCTIONDrying shrinkage is the decrease of concrete volume

with time after the hardening of the concrete, which has a direct influence on the behavior of concrete members, such as prestress losses of prestressed concrete members, long-term deflection of girders, and cracking tendency of repair systems.1,2 It is well known that cracking due to shrinkage can reduce the performance of a concrete structure, espe-cially in corrosive environments, because it can create a direct path for penetration of the corrosive ions into the concrete and reduce concrete durability. Thus, to perform a proper design of concrete structures, predicting the long-term deformations of the concrete is a major issue. Several methods have been proposed to predict the long-term shrinkage of concrete.1,3-10 Because each of these models is based on limited tests and due to the variety of the charac-teristics of the concrete mixture design, there is no globally reliable method available for the shrinkage evaluation of various concretes.

For a close prediction of time-dependent deformations, short-term measurements of the given concrete should be provided.11 In this case, the estimation is done using the following three steps12:

1. Choosing a representative model;2. Calibrating the parameters of the model using the short-

term measurements; and3. Estimating the long-term behavior of concrete from the

calibrated model.In some studies, the shrinkage prediction models have

been modified by using short-term measured data to mini-mize the errors between the experimental values and the esti-mated curves. Brooks and Neville13 conducted a short-term test using specimens made from the actual concrete. These authors extrapolated the measured time-dependent deforma-tion values to obtain the long-term values. Almudaiheem and Hansen14 used a modified ACI 209R-8215 equation using the short-term measurement of shrinkage to reduce the errors of

294 ACI Materials Journal/May-June 2012

0.52

0

0

1 / ( 1) ( )COV

(1 / )

n

et

n

n R

n J

− ∑ = ∑

(2)

where n is the number of variations; Ret is the residual values at time t; and J is the experimental values.

The prediction models were modified by the coeffi-cients C1 and C2 and were used to predict the long-term shrinkage of concrete. Afterward, the results were assessed by Chauvenet’s criterion22 to recognize and omit the outlier model(s); then, the acceptable models were averaged. The calculated averaged results may be considered as the long-term shrinkage values for the given concrete. Figure 1 pres-ents the flowchart of the proposed method.

Modified form of shrinkage prediction models by IA method

Among the various methods for the shrinkage prediction of concretes, six models were chosen and are presented in this section, together with the modifying coefficients C1 and C2. It should be noted that pozzolanic materials such as silica fume (SF) and ground-granulated blast-furnace slag (GGBS) are assumed as a part of the cement in the shrinkage calculation through these models. These modified models are presented in detail as follows:

ACI 209.2R-08 model3—ACI proposed an empirical model for predicting the shrinkage strain as a function of time. Using IA, the modified model of this method is

−e = e + −

2

1( , )C

csh c shu

c

t tt t C

f t t (3)

where esh(t,tc) is the shrinkage strain (mm/mm); t is the age of the concrete (days); tc is the age of the concrete when drying starts at the end of moist curing (days); eshu is the ultimate shrinkage strain (mm/mm); f is a constant that depends on

ACI member Mohammad Shekarchi is an Associate Professor in the Department of Civil Engineering at the University of Tehran, Tehran, Iran. He received his PhD from the Institut National des Sciences Appliquées de Lyon (INSA de Lyon), Lyon, France. His research interests include the durability of concrete in severe conditions, such as the marine environment and petrochemical plants; fiber-reinforced concrete; shrinkage and creep; and mass concrete.

Farnam Ghasemzadeh is a Research Assistant at the Construction Materials Institute (CMI) at the University of Tehran, where he also received his MSc in civil engineering. His research interests include the durability design of concrete structures in marine environments, repair, and shrinkage and creep in concrete.

Siavash Sajedi is a Research Assistant at CMI at the University of Tehran, where he also received his MSc in civil engineering. His research interests include restrained shrinkage in repaired concrete members, repair and rehabilitation of existing structures, and nonlinear finite element analysis of reinforced concrete structures.

METHODOLOGYIA method for prediction of shrinkage

For predicting the long-term shrinkage of concrete using the IA method, the short-term values for a given concrete should be measured in the laboratory. Applicable shrinkage models are used to predict the shrinkage of concrete for the same period. Afterward, the IA method is applied to modify the prediction models and define the modifying coefficients so that the difference between the experimental results and the predicted values is minimized. The coefficient C1 is applied to modify the entire equation, whereas the coeffi-cient C2 is used to modify the time-function term. The modi-fied form of each model may be represented as

21( ) ( ( ))C

sh t C A f te = (1)

where A is the variable coefficient that depends on the prop-erties of the specimen; and f(t) is the time-function term.

To determine the optimized values of C1 and C2, the shrinkage modifier code (SMC) was developed in Visual Basic. This method is based on the minimization of the COV by comparing the experimental results and the predicted values. The applied COV in the SMC may be defined as

Fig. 1—Process of long-term shrinkage prediction by short-term measurements.


the member shape and size of the specimen (days); and t – tc is the duration of drying (days).

CEB MC90-99 model4—This model subdivides the total shrinkage into two categories: autogenous shrinkage and drying shrinkage. The modified model is as follows

( ) ( )( ) 2

1( , ) ,C

sh c cas cds ct t C t t te = e + e (4)

28( ) ( ) ( )cas caso cm ast f te = e b (5)

28( , ) ( ) ( ) ( )cds c cdso cm RH ds ct t f h t te = e b b − (6)

where ecas(t) is the autogenous shrinkage strain (mm/mm); ecds(t) is the drying shrinkage strain (mm/mm); ecaso(fcm28) is the notional autogenous shrinkage coefficient; bas(t) is a function describing the time development of autogenous shrinkage; ecdso(fcm28) is the notional drying shrinkage coef-ficient; bRH(h) is the correction term for the effect of rela-tive humidity (RH) on shrinkage; and bds(t – tc) is a function describing the time development of drying shrinkage.

B3 model 5—Equation (7) presents the modified form of the B3 model5

21( , ) ( ( ))C

sh c sh h ct t C K S t t∞e = − e − (7)

where esh∞ is the ultimate shrinkage strain (mm/mm); Kh is the correction term for the effect of RH on shrinkage; and S(t – tc) is the time-function term.

Model proposed by Huo et al.1—Huo et al.1 modified the ACI 209.2R-083 equation by incorporating the strength correction factor

e = − e′

−× − + −′

2

1( , ) (1.2 0.0073 )

(45 0.3626 )

sh c ci shu

C

c

ci c

t t C f

t tf t t

(8)

where fci′ is the 28-day compressive strength of concrete (MPa).Model proposed by Tadros et al.8—Tadros et al.8 proposed a

simple and compact model for predicting the shrinkage strain

−e = × × × −

− − × × × + − + −′

2

61

'

( , ) 480 10 (2 0.0143 )

1064 3.7( ) 5 735 1 0.145 61 0.58

sh c

C

c

ci ci c

t t C HV

t tSf f t t

(9)

where H is the ambient RH (decimals); and V/S is the volume-to-surface area ratio (mm).

Model proposed by Mazloom9—Mazloom9 proposed an equation for estimating the shrinkage of sealed and drying specimens made of cement with SF for high-strength concretes. The following equation presents the modified form of the Mazloom9 equation after applying the IA method.

−e = × × ×

−× − + × +

2

61( , ) 516 10

(0.3 12.6)

sh c

C

c

c

t t C Y

t tt t SF

(10)

where for sealed specimens, Y = 0.014SF + 0.39; and for drying specimens, Y = 1.14 – 0.007(V/S) ≥ 0.014SF + 0.39, where SF is the percentage of cement replaced by SF.

EXPERIMENTAL PROGRAMMaterials

Table 1 shows the chemical composition of the used cement, SF, and GGBS. The coarse aggregate was gravel with a maximum particle size of 16 mm (0.63 in.) and the fine aggregate was silica sand with a fineness modulus of 3.2.

Mixture proportionsFour high-performance concrete (HPC) mixtures with

the same workability and same water-binder ratios (w/b = 0.38) were designed. The mixture proportions are listed in Table 2. Concrete Mixture Design C-C is a control concrete. In Concrete Mixture Design C-SF, 7.5% of cement weight

Table 1—Chemical compositions and specific gravity of binding materials

Binder SiO2, % Al2O3, % CaO, % MgO, % Fe2O3, % SO3, % Na2O, % K2O, % Specific gravity

Cement 21.00 5.00 63.00 1.800 3.500 1.60 0.50 0.60 3.15

GGBS 35.50 10.00 36.50 9.500 0.700 1.86 0.50 0.53 2.86

SF 93.16 1.13 — 1.60 0.72 0.05 — — 2.11

Table 2—Mixture proportions

Concrete

Mixture proportioning Workability

Cement, kg/m3

(lb/ft3)GGBS, kg/m3

(lb/ft3)SF, kg/m3

(lb/ft3)SP, % by

cement contentCoarse aggregate, kg/m3

(lb/ft3)Fine aggregate, kg/m3

(lb/ft3) w/bSlump, mm

(in.)

C-C 420 (26.2) — — 0.5 793 (49.5) 1000 (62.4) 0.38 145 (5.7)

C-SF 390.5 (24.4) — 29.5 (1.84) 0.6 793 (49.5) 988 (61.7) 0.38 155 (6.1)

C-S 315 (19.7) 105 (6.55) — 0.5 793 (49.5) 992 (61.9) 0.38 160 (6.3)

C-SFS 285.5 (17.8) 105 (6.55) 29.5 (1.84) 0.5 793 (49.5) 980 (61.2) 0.38 150 (5.9)


Free shrinkage testing was conducted according to ASTM C157. The test method involved measuring the changes of the length of 100 x 100 x 500 mm (3.9 x 3.9 x 19.7 in.) concrete prisms. After fabrication, the specimens were covered with wet burlap for 24 hours and then removed from the steel molds. The initial length change measurement of the speci-mens was taken by a comparator according to ASTM C490 (Fig. 2). Then, the specimens were placed in the curing water at 23°C ± 1°C (73.4°F ± 33.8°F) for 6 days and eventually removed and placed in a controlled environment of 25°C ± 1°C (77°F ± 33.8°F) and 50 ± 2% RH. The length changes were measured for a period of 500 days. Strains were calculated by dividing the change in length by the initial length. Figure 3 shows a view of the creep and shrinkage testing room of the Construction Materials Institute (CMI).

Verifying proposed methodTo verify the proposed method, two different concrete

mixture designs from two different studies (HSC8-3A23 and LTHSC 3B24) were chosen; the capability of this method is assessed in this section. Concrete Mixture Design HSC8-3A is a high-strength concrete with a w/b of 0.3 and Concrete Mixture Design LTHSC 3B is a lightweight high-strength concrete with a w/b of 0.37. Table 3 lists the mixture propor-tions of these two mixture designs.

Compressive strengths tests for Concrete Mixture Designs HSC8-3A and LTHSC 3B were conducted on 100 x 200 mm (4 x 8 in.) cylindrical specimens according to ASTM C39. Each measurement was the mean of two tests.

Shrinkage specimens for Concrete Mixture Design HSC8-3A were placed in the control room with a temper-ature of 23.0°C ± 1.7°C (73.4°F ± 35.06°F) and an RH of 50 ± 4%, and those for Concrete Mixture Design LTHSC 3B were placed in the control room with a temperature of 23.0°C ± 1.7°C (73.4°F ± 35.06°F) and an RH of 45 ± 4%. Shrinkage measurements were taken on the schedule set forth in ASTM C512 using a Whittemore gauge to measure changes in length between the gauge points over time. Descriptions of the materials, fabrication, curing, and testing procedures are given in detail in References 23 and 24.

RESULTS AND DISCUSSIONThe experimental results of the compressive strengths are

listed in Table 4. The values of this table were used to predict the shrinkage by the investigated models. Figures 4 to 9 show the experimental and predicted shrinkage results for the investigated mixtures. Figures 4 to 7 show the results of total shrinkage for Concrete Mixture Designs C-C, C-SF, C-S,

was replaced by SF. In C-S, 25% of cement weight was replaced by GGBS, whereas in Concrete Mixture Design C-SFS, 32.5% of cement was substituted by both GGBS (25% of cement weight) and SF (7.5% of cement weight).

Specimen fabrication and testingCompressive specimens were fabricated for each concrete

mixture in accordance with ASTM C192 and then tested at 28 days according to ASTM C39. Each measurement was the mean of three tests.

Fig. 2—Comparator used to measure length change of specimens.

Fig. 3—Overview of specimens in position ready for testing.

Table 3—Mixture proportions of Concrete Mixture Designs HSC8-3A and LTHSC 3B

Concrete

Mixture proportioning Workability

Cement, kg/m3 (lb/ft3)

Slag, kg/m3 (lb/ft3)

Fine aggregate (lightweight), kg/m3 (lb/ft3)

Fine aggregate (normalweight),

kg/m3 (lb/ft3)

Coarse aggregate (lightweight), kg/m3 (lb/ft3)

Coarse aggregate (normalweight),

kg/m3 (lb/ft3) w/bSlump,

mm (in.)

HSC8-3A 303 (18.9) 202 (12.6) — 586 (36.6) — 1157 (72.2) 0.3 216 (8.5)

LTHSC 3B 268 (16.7) 179 (11.2) 231 (14.4) 321 (20) 413 (25.8) 359 (22.4) 0.37 140 (5.5)

Table 4—Compressive strength of concretes at 28 daysMixture C-C C-SF C-S C-SFS HSC8-3A LTHSC 3B

Compressive strength, MPa (psi) 51 (7400) 59 (8560) 58 (8410) 70 (10,200) 90.3 (13,300) 43 (6290)


and C-SFS before and after applying IA for 28- and 60-day measurements of shrinkage. Figures 8 and 9 show the results of total shrinkage for Concrete Mixture Designs HSC8-3A

and LTHSC 3B before and after applying IA for 7-day measurements of shrinkage. For all mixtures, each experi-mental curve represents an average of three specimens.

Fig. 4—Total shrinkage of Concrete Mixture Design C-C: (a) before IA; (b) after IA for 28-day measurements; (c) after IA for 60-day measure-ments; and (d) after averaging.

Fig. 5—Total shrinkage of Concrete Mixture Design C-SF: (a) before IA; (b) after IA for 28-day measurements; (c) after IA for 60-day measure-ments; and (d) after averaging.


Fig. 6—Total shrinkage of Concrete Mixture Design C-S: (a) before IA; (b) after IA for 28-day measurements; (c) after IA for 60-day measure-ments; and (d) after averaging.

Fig. 7—Total shrinkage of Concrete Mixture Design C-SFS: (a) before IA; (b) after IA for 28-day measurements; (c) after IA for 60-day measurements; and (d) after averaging.

The values of the coefficients C1 and C2 obtained by the IA method are given in Table 5. The curves presented in these figures show that, before applying IA, some of the models were capable of computing shrinkage strains with good

agreement for a given mixture, but none of them was suit-able for all the investigated mixtures. For example, the model proposed by Tadros et al.8 was the best for Concrete Mixture Design C-SF (Fig. 5) but failed for the other mixtures. In


Fig. 8—Total shrinkage of Concrete Mixture Design HSC8-3A: (a) before IA; (b) after IA for 7-day measurements; and (c) after averaging.

Fig. 9—Total shrinkage of Concrete Mixture Design LTHSC 3B: (a) before IA; (b) after IA for 7-day measurements; and (c) after averaging.

addition, the figures show that there is a large dispersion of the results obtained by the investigated models.

As seen in the figures, the dispersion of the results is reduced after applying IA. This is mainly due to the effect

of modification in the shrinkage prediction models. It may also be deduced that the predicted results were improved by using 60-day measurements of shrinkage instead of 28-day data. Therefore, it can be concluded that the longer


Table 5—Coefficients C1 and C2 obtained in shrinkage models’ fitting

Mixture Age, days Coefficients ACI3 CEB4 B35 Huo et al.1 Tadros et al.8 Mazloom9

C-C

28C1 0.9275 0.1177 2.7379 0.9897 0.8856 1.0478

C2 0.7665 1.4356 1.2410 0.8239 0.7898 1.0543

60C1 0.9657 0.6790 2.3499 1.1039 0.9382 1.2324

C2 0.7930 1.1229 1.1205 0.8860 0.8309 1.2588

C-SF

28C1 0.7647 4.6780 1.9532 0.9473 0.8364 1.0028

C2 0.4295 0.8001 0.7190 0.4697 0.4558 0.5366

60C1 0.7907 2.7119 1.8503 1.0016 0.8785 1.0848

C2 0.4490 0.8992 0.6793 0.5096 0.4888 0.6094

C-S

28C1 0.8977 0.0993 2.8989 1.0641 1.0025 1.0604

C2 0.9105 1.4440 1.4333 1.0441 1.0085 1.3149

60C1 0.7097 0.2090 1.8238 0.8783 0.8367 0.9646

C2 0.7422 1.2990 1.0520 0.8891 0.8500 1.1852

C-SFS

28C1 0.8126 0.0660 2.4786 0.9739 0.9244 0.9615

C2 0.8735 1.5035 1.3837 1.0495 1.0369 1.1682

60C1 0.7374 0.1130 1.8915 0.9593 0.8932 0.9570

C2 0.8050 1.4014 1.1621 1.0185 1.0033 1.1627

HSC8-3A 7C1 0.3348 0.3500 1.0114 0.4389 0.4221 0.4118

C2 0.5805 1.1025 1.0844 0.6679 0.7244 0.6642

LTHSC 3B 7C1 0.8932 0.4374 2.3547 0.9636 0.7064 0.8919

C2 0.6088 1.1700 1.1423 0.6156 0.5650 0.6994

the period of time of shrinkage measurements, the better the results achieved by the IA method.

Figures 4 to 9 also show that after applying IA, one or more modified models seem to be outliers—in which case, they should be disregarded in the rest of the analysis. To identify the possible outlier modified model(s), Chauvenet’s crite-rion22 was used. It can be concluded from this criterion that in almost all cases, the B3 model5 is an outlier model. Unsuit-ability regarding the B3 model5 can be seen throughout the experiments on different mixtures. This does not imply that this model cannot be used to predict the long-term shrinkage of concrete; it only shows that the results obtained from the modified form of this model in this study do not seem to be in harmony with the modified form of other models. It is possible to explain this by focusing on the equation type used for each model. The form of the equation used to predict shrinkage by the B3 model5 is notably different from the equations used in other models. The B3 model5 uses a hyperbolic function not seen in other equations, whereas most of the other models use a homographic function. There-fore, by including the modified form of the B3 model5 in itself would disrupt the results and create a significant differ-ence and error in the final prediction compared to when the B3 model5 is not considered. However, more research into this aspect is required to fully explain the difference in the results for the B3 model5 compared to other models, and additional experiments on other mixtures are also necessary to prove the usefulness of the B3 model.5

Even though it seems that the results of the B35 and Mazloom9 models differ greatly from others for Concrete Mixture Design LTHSC 3B (Fig. 9), the results of Chauvenet’s criterion22 indicate that both the B35 and Mazloom9 models should either be omitted from the results

or should be considered simultaneously. Taking these models into account does not cause any significant errors in the averaged calculations. The results of the B35 model are overestimated and the results of the Mazloom9 model are underestimated; therefore, they neutralize the effects of each other to some extent. It should be noted that the results were assessed using these models and then were assessed after omitting them. Little difference was observed between the results of these two cases.

After applying IA and omitting the outlier model(s), it is not possible to list the modified models in order of importance or dependability because no evidence could be observed to show that a model(s) worked better for all mixtures compared to other models. For example, the modi-fied form of the Huo et al.1 model was the best for Concrete Mixture Design C-SFS at 28 days but failed for the other mixtures (Fig. 7). This randomness is acceptable because each model was proposed only on the basis of results from a handful of mixtures. Also, in predicting shrinkage, different parameters were considered for each model, so some would provide better predictions for particular mixtures, whereas they may make an unacceptable prediction in other situations. The difference between the modified form of the B35 and Mazloom9 models with other modified models in Fig. 9 also results from this issue. It only seems rational to take the average from the rest of the modified models to represent a more dependable prediction. By averaging the results from all the modified models mentioned, in fact, the effects of all the parameters and mixtures used for these models can be considered.

To evaluate the accuracy of IA in predicting the long-term shrinkage of concretes, the COV is calculated for all mixtures and the results are given in Fig. 10, which shows


that the COV is lower than 0.1 for all mixtures except for Concrete Mixture Design C-S in the 28-day measure-ments. The difference between the results of IA and the experimental values can be due to several reasons. The first reason is the insufficient measurements of shrinkage in the laboratory. For Concrete Mixture Designs C-C, C-SF, C-S, and C-SFS, the number of shrinkage measurements at 7 days was low because at the time of the experiment, the main aim was to measure the long-term shrinkage to investigate the dimensional stability of such mixtures as a repair concrete with substrate. Therefore, in this study, the results of 28 and 60 days were used as the short-term measurements. For Concrete Mixture Designs HSC8-3A and LTHSC 3B, however, there were seven measurements of shrinkage for each day after the curing period and the results of these 7 days were used to predict the long-term shrinkage. For determining an appropriate period of time for shrinkage measurements in the laboratory, it is better to assess measurements step by step using the IA method. When after a certain step and disregarding the outlier model(s), the dispersion of the predicted long-term results becomes low, the measurement of shrinkage can be stopped in the laboratory and the obtained experimental results can be used as short-term values in the IA method.

The accuracy of the shrinkage measurements in the laboratory is also important. This issue is critical when

the early-age measurements (such as the first-day hourly measurements of shrinkage) are used in IA for the long-term prediction of shrinkage. In this case, the determination of modifying coefficients (C1 and C2) based on these early-age measurements may lead to large errors in the long-term predicted values.

Table 6 shows the comparison between the experimental measurements of total shrinkage and the predicted values by IA for all mixtures where a good agreement between the predicted values and experimental measurements can be observed. This indicates that the IA method is capable of predicting the long-term shrinkage of concretes based on the short-term measurements.

CONCLUSIONSIn this study, the IA method was applied to six existing

shrinkage models to predict long-term shrinkage from short-term measurements. The results showed that by using the existing models for different concrete mixtures, one model can be more appropriate for a particular mixture than the others, and none of the existing models was suitable for all the investigated mixtures. It was concluded that an accu-rate prediction of the long-term concrete shrinkage can be achieved by applying the IA method for different concrete mixtures. This method can be used for various mixtures with a wide range of additives. To reduce the differences between the experimental values and the predicted results, however, two points should be considered:

1. The number of measured values and the time period of measurement should be sufficient to reduce the dispersion of results after applying IA.

2. Providing an accurate measurement of short-term shrinkage is necessary, especially when early-age data are used in the prediction.

Prediction of the long-term free shrinkage from short-term measurements was investigated as well. The proposed method in this study may be applied to predict autogenous shrinkage and creep. Moreover, to evaluate the capability of this method, another study may be conducted to investigate the predicted results of each case by early-age measurements (only 1-day hourly measurements).

ACKNOWLEDGMENTSThe authors would like to thank H. Layssi from McGill University for his

assistance in carrying out the experimental program. Also, the authors are grateful for the technical support provided by the Construction Materials Institute (CMI).

REFERENCES1. Huo, X. S.; Al-Omaishi, N.; and Tadros, M. K., “Creep, Shrinkage and

Modulus of Elasticity of High-Strength Concrete,” ACI Materials Journal, V. 98, No. 6, Nov.-Dec. 2001, pp. 440-449.

2. Decter, M. H., and Keeley, C., “Durable Concrete Repair—Importance of Compatibility and Low Shrinkage,” Construction & Building Materials, V. 11, No. 5-6, 1997, pp. 267-273.

3. ACI Committee 209, “Guide for Modeling and Calculating Shrinkage and Creep in Hardened Concrete (ACI 209.2R-08),” American Concrete Institute, Farmington Hills, MI, 2008, 45 pp.

4. CEB-FIP Bulletin 1, Structural Concrete: Textbook on Behaviour, Design and Performance: Updated Knowledge of the CEB/FIP Model Code 1990, The International Federation for Structural Concrete, Lausanne, Swit-zerland, 1999, pp. 37-52.

5. Bažant, Z. P., “Creep and Shrinkage Prediction Model for Analysis and Design of Concrete Structures—Model B3,” Materials and Structures, V. 28, No. 180, 1995, pp. 357-365.

6. Gardner, N. J., and Lockman, M. J., “Design Provisions for Drying Shrinkage and Creep of Normal-Strength Concrete,” ACI Materials Journal, V. 98, No. 2, Mar.-Apr. 2001, pp. 159-167.

Table 6—Comparison between results of IA method and experimental values

Mixture

Age of short-term values used in IA,

days

Experimental results of ultimate

shrinkage, meIA results of ultimate

shrinkage, me

C-C28 620 (in 500 days) 646

60 620 (in 500 days) 668

C-SF28 650 (in 500 days) 558

60 650 (in 500 days) 614

C-S28 529 (in 500 days) 620

60 529 (in 500 days) 523

C-SFS28 502 (in 500 days) 539

60 502 (in 500 days) 514

HSC8-3A 7 252 (in 98 days) 218

LTHSC 3B 7 648 (in 329 days) 626

Fig. 10—COV between experimental results and predicted values.


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Inverse Analysis Method for Concrete Shrinkage Prediction from Short-Term Tests