Composite modeling to predict shrinkage of concretes containing supplementary cementitious materials from paste volumes

Construction and Building Materials 43 (2013) 139–155

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Composite modeling to predict shrinkage of concretes containing supplementary cementitious materials from paste volumes Arkamitra Kar a, Indrajit Ray b,⇑, Avinash Unnikrishnan a, Julio F. Davalos c a

Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26506-6103, USA Visiting Faculty in Civil Engineering, Department of Mechanical Engineering, Purdue University Calumet, Hammond, IN 46323, USA c Department of Civil Engineering, The City College of New York, New York, NY 10031, USA b

h i g h l i g h t s " Developed two-step composite models to predict shrinkage from pastes containing SCMs. " Compared the accuracy of the results from lab-scale results. " Validated the models with external practical sources of shrinkage data. " Compared the proposed models with different existing code-type models.

a r t i c l e

i n f o

Article history: Received 2 July 2012 Received in revised form 28 December 2012 Accepted 2 January 2013 Available online 19 March 2013 Keywords: Cement paste Composite modeling Concrete Prediction models Shrinkage Supplementary cementitious materials

a b s t r a c t The inclusion of supplementary cementitious materials (SCMs) such as slag, fly ash and silica fume into ordinary concrete diverts SCMs from landfills, reduces CO2 emissions, and produces durable and sustainable concrete. Concrete shrinkage is a significant parameter for durability. There is no systematic research on predicting the shrinkage of concrete mixed with various SCMs starting from paste properties. In this study, a two-step composite model is developed to predict concrete shrinkage from the paste level to the specimen level for 14 concrete mixtures containing only-slag, only-fly ash, only-silica fume, and slag plus silica fume, fly ash plus silica fume, and slag plus fly ash at two different water-to-cementitious materials ratios (w/cm). The composite model is used to find the shrinkage of mortar from paste properties in the first step, and then the range of the concrete shrinkages is determined using the shrinkage of the mortar and the Hashin–Shtrikman bounds [16] with the corresponding elastic modulus. The accuracy of the proposed models is evaluated using shrinkage data obtained through lab-scale experiment. The prediction models are found to agree well with the experimental data. Also, data from external sources are used to compare with the proposed models. Finally, these proposed models are compared with existing shrinkage models such as: ACI 209, CEB MC 90, and GL 2000. It is observed that the proposed model agrees more closely with the experimental results from the present study than the existing shrinkage prediction models. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The inclusion of supplementary cementitious materials (SCMs) such as slag, fly ash, and silica fume into concrete has several benefits. The use of SCMs as partial replacement of portland cement divert these SCMs from landfills, reduces CO2 emissions, and produces highly durable concrete leading to development of sustainable and green construction materials. Since the substitution of cement with SCMs alters the microstructure, constituents, and ⇑ Corresponding author. E-mail addresses: [email protected] (A. Kar), [email protected] (I. Ray), [email protected] (A. Unnikrishnan), jdavalos@ccny. cuny.edu (J.F. Davalos). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.01.002

properties of concrete, there is a critical need for better understanding of the effects of SCMs on concrete shrinkage, which is an important durability parameter. Several models have been proposed in the recent years to estimate the drying shrinkage of concrete by different researchers over the years. The most popular models for estimation of hardened concrete shrinkage are: (i) the ACI 209 model [2]; (ii) the Bazant–Baweja B3 model [6,7]; (iii) the CEB – MC90 model [33,10,11,12,1]; and, (iv) the GL 2000 model [18]. These prediction models were primarily based on pure portland cement concrete; no systematic inclusion effect of SCM such as slag, fly ash, silica fume, and other natural pozzolans was considered in these models. ACI 209R-92 committee [1] also suggested that there is a strong

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Fig. 1. Flowchart describing the multi-scale methodology to obtained the shrinkage of concrete from the shrinkage of cementitious pastes.

need to study the effect of chemical and mineral admixtures on shrinkage of concrete. Moreover, from the previous studies by Brooks and Neville [9], Khatri et al. [24], and ACI 209.1R-05 [1], it is evident that the addition of SCM has significant influence on the shrinkage properties of concrete, but very limited information is available to quantitatively predict the shrinkage of concrete modified with SCM. Considering the strong predicted growth of use of various by-products (or SCMs) as replacement of cement in the coming years, it will be prudent to develop model to predict the shrinkage of concrete containing different types, percentage, and combinations of SCM. The shrinkage of concrete at the specimen level is primarily a function of the shrinkage of the cementitious pastes at the microstructural level. Some researchers [28,29] studied the behavior of portland cements with various compositions at micro, meso and macro levels with the help of the DuCOM software platform to predict different mechanical properties. But, they did not consider the effect of the SCMs at the different levels. Eguchi and Teranishi [16] proposed a two-step composite model to study the drying shrinkage strain and Young’s modulus of elasticity for ordinary concrete and mortar. They assumed a twophase material at each step and modeled on the basis of equations given by Kishitani and Baba [27]. Their study was undertaken at mesoscale or millimeter level, but they did not include the effect of SCM. Also, no other systematic studies are available on the prediction of shrinkage from the paste level to the concrete specimen level for mixtures with different types, quantities and combinations of SCM. Thus, the present study adopts a simplified composite modeling approach to predict shrinkage of concretes containing SCM, starting from the corresponding paste shrinkages. At the microscale (micrometer level), the role of cementitious pastes is most important. The variations and amounts of C-S-H and pore structures formed due to changes in constituents of cementitious materials with SCM and water–cementitious material ratios (w/ cm) are considered to predict the mechanisms. The mesoscale con-

siders the material at millimeter level to consider the composite behavior of the cement paste and the aggregates including interfacial transition zone. The macroscale focuses on the coupon or member level of the concrete. The external parameters like size and shape of the specimens, relative humidity of the surroundings are incorporated at this stage. The present study makes two significant contributions: (1) predict model for shrinkage of concrete containing SCMs and (2) adopt a simplified composite modeling approach, beginning from the paste level and continuing to the concrete specimen level to predict the shrinkage. In brief, the present study develops a model to predict the shrinkage of concrete containing SCM at macroscale (or lab-scale), by using the paste shrinkage data obtained from microscale. The different scales used for the present study are illustrated in Fig. 2. The details of the materials and the mix proportions used in this study are described in Section 2 of this paper; the basis of the composite model and the corresponding assumptions are described in Section 3; the formulation of the proposed model is described in Section 4; and, the validation of the proposed model and its comparison with other existing shrinkage prediction models are described in Section 5 followed by subsequent conclusions based on the present study.

2. Materials and experimental program 2.1. Materials Commercially available Type I portland cement conforming to ASTM C150 (Standard Specification for Portland Cement) is used in this study. The oxide compositions and physical properties are provided in Table 1. The Bogue’s composition of portland cement is provided in Table 2. Ground granulated blast furnace slag or slag conforming to Grade 100 of ASTM C989 (Standard Specification for Slag Cement for Use in Concrete and Mortars) obtained from local steel plant and commercially available silica fume conforming to ASTM C1240 (Standard Specification for Silica Fume Used in Cementitious Mixtures) are used in this study (Table 1). Class F fly ash used in this study conforming to ASTM C618 (Standard Specification for Coal

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Macroscale (Length scale – meter); Millimeter phases assumed homogeneous

Mesoscale (Length scale – millimeter); Micrometer phases assumed homogeneous

Microscale (Length scale – micrometer); Nanometer phases assumed homogeneous

Fig. 2. Schematic diagram showing three-phase hierarchical model of concrete.

Table 2 Compound compositions of the portland cement based on Bouge’s formula and corresponding volume fractions.

Table 1 Properties of the materials used. Materials

Portland cementa

Slagb

Fly ash

silica fume

Specific gravity Specific surface (m2/kg) Loss on ignition (%) SiO2 (%) Al2O3 (%) CaO (%) MgO (%) SO3 (%) Na2O + 0.685 K2O (%) Fe2O3 (%) Others (%)

3.15 320 (Blaine) 1.2

2.88 580 (Blaine) 0.0

2.47 490 (Blaine) 3.00

2.18 21400 (nitrogen absorption) 1.28

20.7 5.5 63.6 0.9 2.7 0.5

36.0 12.0 42.0 6.0 0.2 0.74

49.34 22.73 3.09 1.06 0.97 2.75

98.0 – – – 0.15 0.57

3.6 1.3

1.8 1.2

16.01 1.05

– –

a The initial and final setting times for the portland cement are 90 min and 260 min, respectively. b The pH value (in water) for the slag is in the range of 10.5  12.7.

Compounds

Compound formula

% by mass

% by volumea

Tricalcium silicate Dicalcium silicate Tricalcium aluminate Tetracalciumaluminoferrite Gypsum Free lime Free magnesia Others

C3S C2S C3A C4AF CSH2 CaO MgO –

49.0 25.0 12.0 8.0 2.8 0.8 1.4 2.0

50.6 25.2 13.1 7.0 4.0 0.01 0.01 –

a

Volume fractions were calculated based on density values provided in Table 4

[6].

The mix proportions, as shown in Table 3, are selected in order to compare their influences on the microstructural properties of the cementitious systems for different combinations of portland cement with slag, fly ash and silica fume, at different w/cm as well as at different ages. A total of 14 mixes are produced using different combinations of pure portland cement, slag, fly ash, and silica fume, as shown in Table 4. 2.2. Mix Proportions

Fly Ash and Raw or Calcined Natural Pozzolan for Use in Concrete), was obtained from a local coal power plant. The specific gravity, specific surface area and oxide composition are listed in Table 1. The coarse aggregate is 12.5 mm (½ in.) graded and crushed limestone conforming to ASTM C33 (Standard Specification for Concrete Aggregates). The saturated surface dry (SSD) bulk specific gravity is 2.68. Locally available 4.75 mm (0.187 in.) graded river sand conforming to ASTM C33 is used for this study. The fineness modulus and the SSD bulk specific gravity of sand are 2.79 and 2.59, respectively. A commercially available high-range water reducing admixture (HRWRA), conforming to ASTM C494 Type F (Specification for Chemical Admixtures for Concrete) [5], and air entraining agent (AEA), conforming to ASTM C260 (Specification for Air-Entraining Admixtures for Concrete) [4], are used in this study.

2.2.1. Experimental Program The materials used in this study, as described in Section 2, were mixed in the laboratory according to ASTM C192/192M (Standard Practice for Making and Curing Concrete Test Specimens in the Laboratory). All the specimens for shrinkage and compressive strength were cured in the molds under wet burlap for 24 h at room temperature. They were then demolded; the shrinkage specimens were kept in the environmental (humidity-controlled) chamber with an RH of 50% and temperature of 23 ± 2 °C. After removing them from molds after 24 h, the compressive strength cylinders were cured under water at 23 ± 2 °C until they were tested. For shrinkage tests, the prisms specimens of dimensions 76 mm  76 mm  254 mm were cast for all the concrete mixes. Length change measurements were conducted according to ASTM C157/C157M (Standard Test Method

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Table 3 Mix proportions.

a b

Mixtures

No. of mixesa

Short names

w/cm

Cement (%)

Slag (%)b

Fly ash (%)b

Silica fume (%)b

100% Cement 35% Slag 25% Fly ash 10% Silica fume 35% Slag + 10% silica fume 25% Fly ash + 10% silica fume 35% Slag + 15% fly ash

2 2 2 2 2 2 2

CC SG 35 FA 25 SF 10 SG 35 + SF 10 FA 25 + SF 10 SG 35 + FA 15

0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3,

100 65 75 90 55 65 50

– 35 – – 35

– – 25 – – 25 15

– – – 10 10 10 –

0.4 0.4 0.4 0.4 0.4 0.4 0.4

35

No. of mixes represent one mix each corresponding to w/cm of 0.3 and 0.4 (total 2) for each of the mixtures. All replacements are by weight of cement.

for Length Change of Hardened Hydraulic-Cement Mortar and Concrete). All the specimens were cured for 24 h under wet burlap and were placed in an environmental chamber maintained at 23 ± 2 °C and 50 ± 2% RH, immediately after the demolding. Length change of three replicate specimens were measured using vibratory strain gage for a period of 90 days at an interval of 2 s each. The presence of vibratory strain gauges ensures the accurate collection of the early age shrinkage data. From the values of length change, the free shrinkage strains were computed in terms of microstrain for each individual mix. The data for the length change at ages of 1-day, 3-day, 7-day, 28-day, 56-day and 90-day were compared with the predicted values from the shrinkage model for validation. Compressive strengths of 101.6 diameter and 203.2 mm long cylinder specimens were measured in accordance with ASTM C39 (Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens) [3]. Each specimen was cured under water at 23 ± 2 °C for 24 h. Tests were conducted at 1, 7, 28, and 90 days after casting. Average values of 28-day compressive strength of all three replicate specimens are shown in Table 4.

3. Principles of composite modeling approach to predict concrete shrinkage from paste level Fig. 1 provides a flowchart which describes the step by step methods adopted in this study. At the microscale level, the volumes of the hydrated cementitious phases including the contribution by SCMs are obtained from the microanalysis data using the volume stoichiometries of the hydration reactions, as mentioned in Section 3.1.1. Subsequently, the shrinkage of the cementitious pastes is determined from the volumes of their hydration products, as described in Section 3.1.2. Then, the shrinkage of concrete is evaluated from the shrinkage of the cementitious paste using the concept of composite modeling in two steps. At the mesoscale, the mortar shrinkage is predicted using the paste shrinkage; and, at the macroscale, the concrete shrinkage is predicted using the mortar shrinkage. The present study also incorporates the effect of the interfacial transition zone (ITZ) between the mortar and the coarse aggregate. The influence of the different SCM on the shrinkage of concrete is also included in the model.

3.1. Shrinkage of paste from micro-scale studies 3.1.1. Analysis of test data using volume stoichiometry and statistical optimization The addition of SCMs to the cementitious system produces a secondary calcium silicate hydrate (C-S-H(S)), in addition to the primary calcium silicate hydrate (C-S-H(P)). This complicates the hydration process of cement because of the different Ca/Si ratios derived from C-S-H(P) and C-S-H(S). It is necessary to distinguish between the two kinds of C-S-H, as they have distinct effects on the total volume of hydrated phases produced by cement hydration. But, it is very difficult to do so by ordinary chemical methods. Due to this a technique combining microanalysis results with optimization was proposed by the present authors for quantitative estimation of C-S-H(P) and C-S-H(S) in a previous study [25,26] which provided fairly accurate results in case of cementitious pastes containing various combinations of fly ash and silica fume

as well as slag and silica fume. A brief description of the methods is as follows. The 2-D model concept for portland cement hydration model, developed by the researchers at NIST [8], is used as the basis of the stoichiometry equations to quantify and obtain optimized results for the volumes of the hydration products of cementitious pastes. The oxide composition of the portland cement is used to find the Bogue’s compound compositions as mass fraction. By dividing the mass fractions with the molar densities, the corresponding volume fractions of C3S, C2S, C3A, C4AF, gypsum are determined as: 50.55%, 25.24%, 13.11%, 7.1%, and, 3.99%, respectively. The microanalysis data are plotted on 3D axes to validate the results experimentally. The volume stoichiometry of the hydration reactions is used to estimate the quantities of the primary and secondary calcium silicate hydrate (C-S-H) and the calcium hydroxide produced by these reactions. The 3D plots of Si/Ca, Al/ Ca and S/Ca atomic ratios obtained from the microanalyses are compared with the volumes of C-S-H from the prediction equation, based on the optimized Ca/Si ratio of the cementitious system to successfully determine the Ca/Si ratio of the cementitious systems at four different ages using a constrained nonlinear least squares optimization formulation. All the formulations are coded in the General Algebraic Modeling System (GAMS) and solved using the built-in CONOPT solver [37]. Further experimental validation was conducted through comparison of predicted and observed quantities of calcium hydroxide from TGA studies. The results from the above methods are used to obtain the shrinkage of cementitious pastes containing SCM, and from which the mortar and finally concrete shrinkages are determined through step-by-step approach. 3.1.2. Chemical shrinkage of cementitious paste For the present study, the volume shrinkage of the cementitious paste is calculated using the volume stoichiometries obtained from the above method (Section 3.1.1). The initial and final volumes of the reactants and the products are computed based on weight and density. The hydration processes for the cement clinker phases are all exothermic and result in a decreased volume of the reaction products. This volume reduction, or chemical shrinkage, begins immediately as the cement is mixed with water and the rate of reaction is at its peak at early stage. Using Eq. (1), the volume shrinkage of the cementitious paste was computed in terms of the total volume of hydrated products present in the system at any given instant of time and the corresponding degree of hydration and expressed as:

DV aðV ci þ V wi Þ  V hcp;a ¼ V ðV ci þ V wi Þ

ð1aÞ

where DVV is the volumetric or chemical shrinkage of cementitious paste, a is the degree of hydration, V ci is the volume of cement before mixing, Vhcp,a is the volume of hydrated cementitious paste at a degree of hydration, and V wi is the volume of water before mixing. The degree of hydration in Eq. (1a) is then expressed as a hyperbolic function of time, in order to transform Eq.1a into the same

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A. Kar et al. / Construction and Building Materials 43 (2013) 139–155 Table 4 Mixture proportions for concrete mixes and basic concrete properties. Mixture

CC SG 35 FA 25 SF 10 SG 35 + SF 10 FA 25 + SF 10 SG 35 + FA 15

Cement

Slag

Fly ash

Silica fume

Coarse aggregate

Fine aggregate

HRWRA

AEA

Slump

28 days Compressive strength (MPa)

kg/m3

kg/ m3

kg/m3

kg/ m3

kg/m3

kg/m3

mL/m3

mL/ m3

mm

w/cm 0.3

w/cm 0.4

870 1350 870 870 870 870 870

200 190 200 190 190 200 200

45 51 66 56 76 90 84

38 31 34 47 70 65 58

396 253 293 350 209 245 190

136 98 133 133

94 57

39 38 38

1038 1038 1038 1038 1038 1038 1038

699 699 699 699 699 699 699

w/ cm 0.3

w/ cm 0.4

500 1950 1200 1200 2200 700 1200

100 700 300 300 1200 200 300

Notes: For all the mixtures, the air contents were 5.5–6%; 1 MPa = 145 psi; 1 kg/m3 = 1.6855 lb/yd3; 1 in. = 25.4 mm; 1 fl oz/yd3 = 39 mL/m3.

format as the ACI shrinkage prediction equation. The degree of hydration at different ages is determined using the non-evaporable water content of the cementitious pastes as suggested by Powers [35]. The calculated values of the degree of hydration, a are plotted against time, t, and the equation of a as a function of time is obtained in the form of a rectangular hyperbola, as shown below:

a¼

t A þ Bt

ð1bÞ

where A and B are constants which vary depending on the w/cm ratio. Also, a is expressed as decimal fraction and t is in days. The values are provided in Table 5. Using Eq. (1b), the volumetric shrinkage from Eq. (1a) is finally expressed in terms of the hyperbolic time-function as t  ðV ci þ V wi Þ  V hcp;a DV AþBt ¼ V ðV ci þ V wi Þ

ð1cÞ

A previous study by Neubauer et al. [34] showed that the shrinkage of the cementitious paste is governed mostly by the shrinking characteristic of the C-S-H. The unreacted tricalcium silicate (C3S), and the dicalcium silicate (C2S) in the system were assumed to act as restraints to the volume shrinkage. As the pores in the cementitious system have zero static moduli, it may be assumed that the contribution of the pores to the shrinkage was negligible. So, for the present study, the chemical shrinkage of the cementitious paste is solely considered to be due to the volume deformation accompanying the formation of C-S-H from C3S and C2S hydrations. The shrinkage of paste determined using the above formula considers the volume deformation due to the formation of C-S-H from the hydration reaction; but it does not take into account, the restraining characteristics of the unreacted C3S and C2S. So, the paste shrinkage obtained as above is in fact an over estimation of the actual deformation that takes place in the paste. This paste shrinkage is then modified by multiplying it with a factor considering the respective volume fractions of the C-S-H and the silicates (C3S and C2S) and their corresponding elastic moduli. Thus, the final expression for the shrinkage of the cementitious paste, ep, is given as,

ep;chemical ¼

v f ;CSH  ECSH DV  V ðv f ;C3S  EC3S þ v f ;C2S  EC2S Þ

ð2aÞ

where vf,CSH, vf,C3S, vf,C2S refer to the volume fractions of C-S-H, C3S, and C2S and ECSH, EC3S, EC2S denote the elastic moduli of C-S-H, C3S, and C2S, respectively. The multiplicative term in Eq. (2a) is a modification factor by which the chemical shrinkage expression must be multiplied in order to obtain the actual chemical shrinkage of the cementitious paste. However, the numerator of the modification factor is a function of only the C-S-H. The CH is not considered due to its shrinkage restraining property. The effect of this modification factor is significant at early age (about 1 day) only. At later ages (>24 h), the factor has a value of about 1.0 and is not taken into account for the corresponding calculations.

3.1.3. Expression of chemical shrinkage as function of mixture proportions The expression of chemical shrinkage in Eq. (1c) is a function of the volume of the hydrated cementitious paste at the corresponding degree of hydration. In order to make the model suitable for practical use, it is preferable to express the chemical shrinkage solely as a function of the percentage of each material in the mixture so that user can use the percentages of cement and SCM as input and obtain the chemical shrinkage as output. The initial volumes of cementitious materials and water are also expressed in terms of the mixture proportions and the corresponding specific gravity of each material. The chemical shrinkage equation is now expressed as:

DV ¼ V

t AþBt

  100 þ



w=cm104 Q P R S qC þqSG þqFA þqSF

 100 þ



t Pba Q c a s Rda s S aAþBt SG SF SF FA FA 100



w=cm104

ð2bÞ

Q P R S qC þqSG þqFA þqSF

where P, Q, R, and S are the respective proportions in percentage of cement, slag, fly ash and silica fume in the mixture; a, b, c, and d are coefficients corresponding to cement, slag, fly ash and silica fume which denote the contribution of the respective materials to the hyt drated cementitious paste. The expression AþBt denotes the degree of hydration of cement at time t. aSG, aFA, and aSF denote the degrees of reaction for slag, fly ash and silica fume respectively. The coefficients sFA and sSF denote the silica content in fly ash and silica fume respectively. The coefficients qC, qSG, qFA, and qSF denote the specific gravity values for cement, slag, fly ash and silica fume respectively. The values for a, b, c, and d are calculated for each of the mixtures in the current study. They are shown in Table 6 for each of the different combinations of cement and SCM, i.e. corresponding to each P, Q, R, and S, as well as different w/cm. The values of a, b, c, or d for any other mixture proportion can be interpolated by the user from the tabulated values. The coefficients are assumed to be valid only if its corresponding material is present in the mix proportion. For example, if a mixture contains cement, slag and silica fume (i.e. R = 0 for this mix), the coefficient, c, corresponding to fly ash is assumed to be inapplicable. Using Eq. (2b), the expression for chemical shrinkage can be written as

ep;chemical ¼

t AþBt

  100 þ







w=cm104 Q P R S qC þqSG þqFA þqSF

100 þ



t Pba Q c a s Rda s S aAþBt SG SF SF FA FA 100

w=cm104 Q P R S qC þqSG þqFA þqSF



v f ;CSH :ECSH  v f ;C3S :EC3S þ v f ;C2S :EC2S ð2cÞ

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A. Kar et al. / Construction and Building Materials 43 (2013) 139–155 Table 5 Constants for degree of hydration expression as function of time.

Table 6 Coefficients to determine the volume of hydrated cementitious paste depending on the mix proportions.

w/cm

A

B

P

Q

R

S

w/cm

a

b

c

d

0.3 0.4

1.919 1.164

1.561 1.328

100 65 55 75 65 90 55 65 50

0 35 45 0 0 0 35 0 35

0 0 0 25 35 0 0 25 15

0 0 0 0 0 10 10 10 0

0.4

140.396 140.396 140.396 140.396 140.396 140.396 142.696 142.696 142.696

– 198.280 198.288 – – – 198.28 – 198.28

– – – 1.286 1.286 – – 1.2854 1.2854

– – – – – 1.2847 1.2847 1.2854 –

100 65 55 75 65 90 55 65 50

0 35 45 0 0 0 35 0 35

0 0 0 25 35 0 0 25 15

0 0 0 0 0 10 10 10 0

0.3

137.307 137.307 137.307 137.307 137.307 137.307 139.556 139.556 139.556

– 198.28 198.288 – – – 198.28 – 198.28

– – – 1.286 1.286 – – 1.2847 1.2854

– – – – – 1.2847 1.2854 1.2847 –

3.1.4. Total shrinkage of cementitious paste The shrinkage strain in Eq. (2c), accounts for only the chemical shrinkage (a material property) of the paste and not its drying component. The determination of the total shrinkage of the cementitious paste is discussed below. According to Holt [22], the total shrinkage should be taken as the sum of each individual volume change due to the different modes of shrinkage – carbonation, thermal expansion, drying and chemical deformations. In 1995, Tazawa and Miyazawa observed that with decreasing w/cm, chemical shrinkage has significantly greater contribution to total shrinkage, compared to drying shrinkage [22].The contribution of chemical shrinkage to the total cementitious paste shrinkage is shown in Table 6 [22]. Thus, at low w/cm the total shrinkage may be taken as nearly equal to the chemical shrinkage. Moreover, it is accepted by researchers that autogenous shrinkage cannot be avoided but in the majority of cases it can be assumed to be negligible. The exact contribution of autogenous and drying shrinkages to the total shrinkage is still unknown in most cases – particularly at early ages and in high-strength or high-performance concrete [22]. Hence, for the sake of simplicity and keeping the safety issue in mind, it is assumed that the chemical shrinkage fully contributes to the autogenous shrinkage, even for matured cementitious pastes, i.e. the chemical shrinkage is equal to the autogenous shrinkage at all ages. Table 7 shows that at a w/ cm of 0.40 the chemical contributes 40% of the total shrinkage magnitude, while at a w/cm of 0.23 the contribution increases to about 80% of the total shrinkage [22]. So, the shrinkages of the cementitious paste computed from Eq. (2) are converted to the corresponding total shrinkage using this concept. For example, the paste shrinkage is divided by 0.4 in case of w/cm of 0.4 whereas it is divided by 0.5 in case of w/cm of 0.3. The coefficients are taken from the plots provided by Tazawa and Miyazawa in 1995 as mentioned by Holt [22]. The contribution of autogenous shrinkage to the total shrinkage varies with the w/cm as shown in Table 7. This variation can be represented by a linear regression equation as

b ¼ 2:6792ðw=cmÞ þ 1:4118 ½R2 ¼ 0:93

ð3aÞ

where b denotes the fractional contribution of chemical shrinkage to total shrinkage. Incorporating the effect of the w/cm into the chemical shrinkage expression (2c), the expression of the total paste shrinkage, ep,tot, is obtained as shown in Eq. (3b).The equation of shrinkage for the cementitious paste is finally expressed as:

ep;tot ¼ or;

ep;chemical b

ep;tot ¼

ep;chemical

ð3bÞ

2:6792ðw=cmÞ þ 1:4118

where b = 2.6792(w/cm) + 1.4118 denotes the function of the w/ cm obtained from Eq. (3a). It is essential to note that the volume fractions of the C-S-H for different mixtures depend on the chemical compositions of the cement and the SCM comprising those [20,21]. Hence, ep;tot is also a function of the chemical compositions of the mixtures.

3.2. Concept of Composite Model applied to Determine Concrete Shrinkage This section explains the use of composite modeling concept to formulate mortar shrinkage from paste shrinkage and then, the concrete shrinkage from the mortar shrinkage, with the relevant assumptions and their bases. According to Neubauer et al. [34], mortar and concrete are both composite materials. The arrangement and characteristics of each component of the microstructure influence the overall properties. The shrinkage of mortar and concrete must be described, based on a three-phase component model, comprising the bulk cement paste, the aggregate and the interfacial transition zone. The properties of the interfacial transition zone (ITZ) between the paste and the aggregate have significant influence on the mechanical properties of the concrete. Based on these criteria, concrete is modeled as a three-phase material in each of two separate steps in the present study. In the first step, concrete is assumed to be composed of three distinct phases – (i) cementitious paste; (ii) fine aggregate, and (iii) the (ITZ) between the paste and the fine aggregate (Fig. 2). At this step, it is assumed that the shrinkage of mortar is mostly caused by the shrinkage of the cementitious paste and the fine aggregates offer a restraining effect. The contribution of the ITZ at this step has been incorporated using the concept of the Hashin–Shrtikman bounds, as explained in Section 3.4. In the next step, concrete is again assumed to be composed of three distinct phases – (i) mortar; (ii) coarse aggregate, and (iii) the ITZ between the mortar and the coarse aggregate (Fig. 2). At this step, it is assumed that the shrinkage of concrete is mostly caused by the shrinkage of the mortar and the aggregates offer a restraining effect. The influence of the ITZ has a gradient that decreases from the paste to the aggregate [34]. It is also assumed that there is no direct contact between the aggregates, because ACI 209R-92 [1] recommends that shrinkage predictions models must be simple and easy to use for practical purposes. If direct contact between the aggregates were taken into account, the shrinkage prediction model becomes too complex due to incorporation of concepts of viscoelasticity [29]. So, effectively, the three-phase system is converted to an equivalent two-phase system. The mechanical properties of the interfacial zone were incorporated into the two-phase paste-aggregate model using the concept of the Hashin bounds. The concept of the step-wise composite model is inspired by the work of Kishitani and Baba [27], which was also used later on by Eguchi and Teranishi [16].

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1 Vp Va 2 6 ¼ þ þ 0:4V a tr þ GðÞ Gp Ga K i þ 4Gi =3 Gi

Table 7 Contribution of chemical to total shrinkage at varying w/cm (Courtesy: [22]). w/cm

Fractional contribution of chemical shrinkage to total shrinkage (b)

0.17 0.23 0.30 0.40

1.0 0.8 0.5 0.4

GðþÞ ¼ V p Gp þ



V a Ga 2:5Ga t r 1 þ ðK i þ4G i =3Þþ2Gi

ð5aÞ

ð5bÞ

The concept of two-step modeling adopted by Eguchi and Teranishi, based on the formulation suggested by Kishitani and Baba [22], is taken as the basis for the approach used in the current study. The above mentioned model has the limitation of not being able to incorporate the ITZ. This model is modified in the present study so as to take the influence of the ITZ in account, using the concept proposed by Hashin and Shtrikman [21]. They provided the range of elastic moduli for concrete, based on the bulk and shear moduli and gave the corresponding upper and lower limits known as Hashin and Shtrikman (H–S) bounds. This concept is modified for the present study, which assumes the following two-step composite model incorporating all the three phases present in concrete at two different stages.

where Va is the volume fraction of aggregate in the concrete (however, it denotes the volume fraction of sand in mortar), Vp is the volume fraction of paste in the concrete (however it denotes volume fraction cementitious paste in mortar), K is the bulk modulus of the concrete (in the meter scale) and the mortar (in the millimeter scale), respectively, G is the shear modulus of the concrete (in the meter scale) and the mortar (in the millimeter scale), respectively, Ki is the bulk modulus of the ITZ between mortar and concrete (at meter scale) and of the ITZ between paste and fine aggregate (at millimeter scale), Gi is the shear modulus of the ITZ between mortar and concrete (at meter scale) and of the ITZ between paste and fine aggregate (at millimeter scale), tr is the ratio of the thickness of the ITZ to the radius of the spherical inclusions = 0.1. For the millimeter scale, the average radius of the sand particles was taken as 200 lm and the thickness of the ITZ between the sand particles and the cementitious paste as 20 lm. For the meter scale, the average radius of the coarse aggregate particles was taken as 12 mm and the thickness of the ITZ between the coarse aggregate particles and the mortar as 1.2 mm. Subsequently, the H–S bounds for the elastic modulus of the mortar were computed as:

3.4. Hashin–Shtrikman bounds

Lower Bound :

EðÞ ¼

9K ðÞ GðÞ 3K ðÞ þ GðÞ

ð6aÞ

Upper bound :

EðþÞ ¼

9K ðþÞ GðþÞ 3K ðþÞ þ GðþÞ

ð6bÞ

3.3. Two-step composite modeling

According to Hashin and Shtrikman [21], the measured values of elastic modulus for some concretes differ from the predictions by the available two-phase models due to the presence of the ITZ. By using the three-phase model proposed by Hashin and Shtrikman [21], the upper and lower bounds (H–S Bounds) for the elastic modulus of the mortar can be determined from the knowledge of the corresponding bulk and shear moduli. The bulk modulus and the shear modulus of the ITZ and the ratio of the thickness of the ITZ to the equivalent radius of the spherical inclusions (aggregate particles), must be known in order to use this model. The bulk modulus of the ITZ is taken as half of that of the cementitious paste. The shear modulus of the ITZ is taken as 0.4 times that of the paste, based on the findings of Neubauer et al. [34]. The upper and lower bounds represent the two possible extreme cases – (a) the mortar phase in parallel with the coarse aggregate phase at the meter scale (or the cementitious paste matrix in parallel with the sand particles at the millimeter scale), together with the ITZ at the respective length scale and (b) the mortar phase in series with the coarse aggregate phase at the meter scale (or the cementitious paste matrix in parallel with the sand particles at the millimeter scale), together with the ITZ at the respective length scale (Fig. 3). The heterogeneity of the ITZ and the uncertainty regarding its orientation in the system at both the scales makes it difficult to specify one particular value for the elastic modulus. Thus, the concepts of composite modeling for two-phase materials are extended to the three-phase material and Hashin and Shkritman [21] came up with the upper and lower bounds for the elastic moduli as shown below: Hashin–Shtrikman bounds:

1 Vp Va 3V a tr ¼ þ þ K ðÞ K p K a K i þ 4Gi =3 K ðþÞ ¼ V p K p þ

V aKa a tr 1 þ K 3K þ4G =3 i

i

The different phases present in the hydrated cementitious pastes have different effects on the elastic moduli due to difference in stiffness of each phase. This aspect is incorporated in the present study and explained in the following section. 3.4.1. Contribution of different phases to stiffness The three phases that play the major role in the elastic moduli of the cement paste are the clinker materials, calcium hydroxide and the C-S-H due to their volumetric abundance in the cementitious system [17]. Ettringite is another phase that appears in a lesser quantity in the paste and thus has a minor effect on the elastic modulus. The researchers at NIST used cement paste samples at later ages to identify the influence of each phase on the elastic modulus with the help of grayscale analyses of 3D images of the microstructure, which were simulated using the CEMHYD3D software. The values of bulk and shear moduli for the standard clinker materials and the hydrated phases must be known apriori. This requires the help of nanoindentation techniques. The effective bulk modulus and the effective shear modulus of the cementitious pastes were computed by considering the bulk and shear moduli of each individual phase and using the expressions provided by Garboczi [17], as shown in Eqs. (7) and (8). The coefficients of the bulk and shear moduli terms in Eqs. (7) and (8) denote the contribution of the corresponding phase to the bulk and shear moduli of the cementitious paste. For w/cm of 0.3,

K p ¼ ð0:32ÞK CSH þ ð0:27ÞK cement þ ð0:22ÞK CH ð4aÞ

þ ð0:005ÞK pores þ ð0:15ÞK others Gp ¼ ð0:32ÞGCSH þ ð0:27ÞGcement þ ð0:2ÞGCH þ ð0:17ÞGothers

ð4bÞ

For w/cm of 0.4,

ð7aÞ ð7bÞ

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A. Kar et al. / Construction and Building Materials 43 (2013) 139–155

(a)

(b)

Fig. 3. Orientation of the ITZ between mortar and aggregate in two extreme cases: (a) Mortar phase in parallel with the coarse aggregate phase at the meter scale (or the cementitious paste matrix in parallel with the sand particles at the millimeter scale) and (b) Mortar phase in series with the coarse aggregate phase at the meter scale (or the cementitious paste matrix in parallel with the sand particles at the millimeter scale).

K p ¼ ð0:4ÞK CSH þ ð0:14ÞK cement þ ð0:25ÞK CH þ ð0:01ÞK pores þ ð0:18ÞK others Gp ¼ ð0:43ÞGCSH þ ð0:14ÞGcement þ ð0:23ÞGCH þ ð0:18ÞGothers

ð8aÞ ð8bÞ

where Kp denotes the bulk modulus of the cementitious paste and Gp denotes the corresponding shear modulus. However, Eqs. (7) and (8) are valid for pure portland cements only. So, the values of Kp and Gp are modified in the case of the cementitious systems containing the SCMs. The effect of the C-SH(S) was incorporated such that its stiffness was 20% more than that of C-S-H(P). As the inclusion of SCM improves the mechanical properties of the concrete, it was assumed that the SCM had positive effect on the stiffness of the cementitious paste. Also, the contribution of the unhydrated cement clinker materials is modified, keeping in mind the replacement of cement by the SCMs. The modified expressions are shown in Eqs. (10) and (11). For w/cm of 0.3,

K p ¼ ð0:16ÞK CSH þ ð0:16Þð1:2ÞK CSH þ ð0:27Þ  ðV cement K cement þ V slag K slag þ V flyash K flyash þ V silicafume K silicafume Þ þ ð0:22ÞK CH þ ð0:005ÞK pores þ ð0:15ÞK others

ð9aÞ

Gp ¼ ð0:175ÞGCSH þ ð0:175Þð1:2ÞGCSH þ ð0:27Þ  ðV cement Gcement þ V slag Gslag þ V flyash Gflyash þ V silicafume Gsilicafume Þ þ ð0:2ÞGCH þ ð0:18ÞGothers

ð9bÞ

For w/cm of 0.4,

K p ¼ ð0:2ÞK CSH þ ð0:2Þð1:2ÞK CSH þ ð0:27ÞðV cement K cement þ V slag K slag þ V flyash K flyash þ V silicafume K silicafume Þ þ ð0:22ÞK CH þ ð0:005ÞK pores þ ð0:18ÞK others

ð10aÞ

Gp ¼ ð0:215ÞGCSH þ ð0:215Þð1:2ÞGCSH þ ð0:27Þ  ðV cement Gcement þ V slag Gslag þ V flyash Gflyash þ V silicafume Gsilicafume Þ þ ð0:2ÞGCH þ ð0:17ÞGothers

Taking all the above factors into consideration, the two-step composite model for concrete shrinkage are formulated as described in Section 4.

ð10bÞ

The values of the bulk and shear moduli for the SCMs are taken from available research data [36,30].

4. Formulation of two-step composite model The purpose of the two-step composite model for our study is to find the shrinkage of mortar from that of the paste and then the range of the concrete shrinkage using the shrinkage of the mortar and the H-S bounds for the corresponding elastic moduli. The aggregates are assumed to have negligible shrinkage strain values as they mainly restrain the shrinkage of the concrete. The composite model is based on the formulae provided by Eguchi and Teranishi [16] from the works of Kishitani and Baba [27]. As mentioned in Section 3.3, Eguchi’s formula had two steps so that he could incorporate the composite model into the prediction of the shrinkage of concrete. For our study, the two-step equation can be re-written as

em ½1  ð1  m1 n1 ÞV s ½n1 þ 1  ðn1  1ÞV s  ¼ n1 þ 1 þ ðn1  1ÞV s ep

ð11aÞ

ec ½1  ð1  m2 n2 ÞV a ½n2 þ 1  ðn2  1ÞV a  ¼ n2 þ 1 þ ðn2  1ÞV a em

ð11bÞ

where ep is the shrinkage of paste (for the present study we will use ep,tot to denote total shrinkage of paste), em is the shrinkage of mortar, ec is the shrinkage of concrete, m1 is the ratio of shrinkage strain of sand to that of paste, m2 is the ratio of shrinkage strain of coarse aggregate to that of mortar, n1 is the ratio of elastic modulus of sand to that of cementitious paste, n2 is the ratio of elastic modulus of coarse aggregate to that of mortar, Vs is the volume fraction of sand in the mortar, Va is the volume fraction of coarse aggregate in the concrete. Since we assume that the fine aggregate (sand) and coarse aggregate to resist shrinkage, we set m1 ¼ 0 and m2 ¼ 0 in Eqs. (11a) and (11b). For the above equations, the values of the elastic modulus of the paste, Ep. are calculated in terms of the corresponding compressive strength of the paste. The compressive strength values of the cementitious pastes are determined experimentally. The tests for compressive strength are conducted according to ASTM C109/

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A. Kar et al. / Construction and Building Materials 43 (2013) 139–155

The final shrinkage equation is modified by incorporating the effects of the size and shape of the specimen and other environmental factors. According to the ACI 209R-92 [1], the following factors have an influence on the shrinkage of concrete: (i) Volume-to-surface ratio (csh,vs); (ii) Curing time (csh,tc); (iii) Relative humidity (csh,RH); (iv) Slump factor (csh,s); (v) Fine aggregate factor (csh,w); (vi) Cement content factor (csh,c); and, (vii) Air-content factor (csh,a). The symbols within the parentheses represent the modification coefficient corresponding to each factor, by which the shrinkage term has to be multiplied in order to incorporate the effects of these factors. The equation for concrete shrinkage proposed in Section 4 does not take into account the influence of the factors mentioned above. These factors are incorporated into the concrete shrinkage equation (Eq. (13b)) by multiplication with the coefficients suggested by ACI-209R92 [1].

C109M (Standard Test Method for Compressive Strength of Hydraulic Cement Mortars (Using 2-in. or [50-mm] Cube Specimens)). For the purpose of the present study, the cubes are used to test the cementitious pastes instead of mortar. According to Maekawa et al. [24], the elastic modulus of the paste varies with its compressive strength, ðfc0 Þp , as: 1

Ep ¼ 8500  ðfc0 Þpð3Þ 1

Ep ¼ 8:5  ðfc0 Þðp3Þ

ðin MPa; 1MPa ¼ 145 psiÞ

ð12aÞ

ðin GPa; 1GPa ¼ 145 ksiÞ

ð12bÞ

The elastic modulus for the coarse aggregate is obtained from Garboczi [17]. The elastic modulus for the fine aggregate is obtained from a publication by the researchers at Texas A&M University (web link provided in references) [38].



2 t  AþBt

100þ

6

2 þ1ðn2 1ÞV a  1 þ1ðn1 1ÞV s csh  ½1Vna2½n  ½1Vn1s ½n 6 þ1þðn2 1ÞV a þ1þðn1 1ÞV s 4

ec ¼

w=cm104 Q P R S qC þqSG þqFA þqSF







w=cm104 100þ Q P R S qC þqSG þqFA þqSF

½1  V a ½n2 þ 1  ðn2  1ÞV a  ¼ n2 þ 1 þ ðn2  1ÞV a 

½1  V s ½n1 þ 1  ðn1  1ÞV s  ep;tot n1 þ 1 þ ðn1  1ÞV s 

ð13aÞ

Using the expression for the shrinkage of cementitious paste from Eq. (2b), the concrete shrinkage strain can be written as

2



t  AþBt

6 ½1V a ½n2 þ1ðn2 1ÞV a  1 þ1ðn1 1ÞV s  ½1Vn1s ½n 6 þ1þðn1 1ÞV s n2 þ1þðn2 1ÞV a 4

ec ¼



 ðv

v f ;CSH ECSH v

f ;C3S EC3S þ f ;C2S EC2S Þ

7 7 5 ð14Þ

2:6792ðw=cmÞ þ 1:4118

For the second step of the model, the elastic modulus of the mortar, Em, is found out considering the mortar to be composed of the cementitious paste, the fine aggregate (sand) and the ITZ. The bulk modulus of the ITZ is taken as half of that of the cementitious paste. The shear modulus of the ITZ is taken as 0.4 times that of the paste, based on the findings of Neubauer et al. [34]. The formulation mentioned above is followed to determine the range of the values using the concept of the H–S bounds. Thus, when these ranges are plugged into the second step of the composite model, the range of shrinkage strains are predicted for any given concrete at each age. The equation for the shrinkage strain of the concrete can thus be expressed as

ec

3

a t PbaSG QcaFA sFA RdaSF sSF S AþBt 100

w=cm104 Q P R S qC þqSG þqFA þqSF

100þ



100þ

P

 

where csh = (csh,tc), (csh,RH), (csh,vs), (csh,s), (csh,w), (csh,c), (csh,a) denotes the combined effect of all the factors mentioned above. The final expression is written in a compact manner as follows:

ec ¼ csh  ca  cs  ep;tot where ca ¼

cs ¼

½1  V a ½n2 þ 1  ðn2  1ÞV a  n2 þ 1 þ ðn2  1ÞV a

½1  V s ½n1 þ 1  ðn1  1ÞV s n1 þ 1 þ ðn1  1ÞV s

and csh and ep,tot are as defined earlier in Eq. (14) and Eq. (3b) respectively. The shrinkage of concrete as predicted by the above expression is a function of the chemical composition of the cementitious materials in the mixtures, following the explanation provided in Section 3.1.4. The details on these factors are provided in ACI 209R-92 [1]. In brief, this model has been developed on cementitious systems containing varying proportions of SCM with w/cm of 0.3 and

3

a t PbaSG QcaFA sFA RdaSF sSF S AþBt 100

w=cm104 Q R S

qC þqSG þqFA þqSF

2:6792ðw=cmÞ þ 1:4118

This equation can be used to predict the range of the concrete shrinkage strains. The upper and lower values were obtained by plugging in the respective upper bound and lower bound for the elastic modulus of the mortar, which were computed using the modified H–S bounds, as described earlier.

ð14aÞ



 ðv

v f ;CSH ECSH v

f ;C3S EC3S þ f ;C2S EC2S Þ

7 7 5 ð13bÞ

0.4. After studying all the models the ACI model is chosen as a baseline and modifications are made to incorporate the effects of SCM on drying shrinkage. The above model is then tested for accuracy by comparing with experimental data, as described in Section 5.

A. Kar et al. / Construction and Building Materials 43 (2013) 139–155

350

1000 900 800 700 600 500 400 300 200 100 0

shrinkage (microstrain)

Shrinkage (microstrain)

148

exp data

300 250 200 150 100 50

exp data

0 0

50

100

0

20

40

60

Age (days)

Age (days)

(a) w/cm = 0.4

(b) w/cm = 0.3

80

100

700

400

600

350

Shrinkage (microstrain)

Shrinkage (microstrain)

Fig. 4. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of 100% cement mixes at two different w/cm.

500 400 300 200 exp data

100

300 250 200 150 100 exp data

50 0

0 0

20

40

60

80

100

0

20

40

60

Age (days)

Age (days)

(a) w/cm = 0.4

(b) w/cm = 0.3

80

100

Fig. 5. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of mixes containing replacement of cement by 35% slag at two different w/cm.

700

shrinkage (microstrain)

shrinkagle (microstrain)

800

600 500 400 300 200 100

exp data

0 0

20

40

60

Age (days)

80

100

500 450 400 350 300 250 200 150 100 50 0

exp data 0

20

40

60

80

100

Age (days)

Fig. 6. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of mixes containing replacement of cement by 25% fly ash at two different w/cm.

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A. Kar et al. / Construction and Building Materials 43 (2013) 139–155

600

shrinkage (microstrain)

shrinkage (microstrain)

700 600 500 400 300 200

500 400 300 200 exp data 100

100

exp data

0

0 0

20

40

60

80

100

0

20

40

60

80

100

Age (days)

Age (days)

Fig. 7. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of mixes containing replacement of cement by 10% silica fume at two different w/cm.

500 450

600

shrinkage (microstrain)

shrinkage (microstrain)

700

500 400 300 200 100

exp data

400 350 300 250 200 150 100

exp data

50

0

0 0

20

40

60

80

100

0

20

40

Age (days)

60

80

100

Age (days)

900

700

800

600

shrinkage (microstrain)

shrinkage (microstrain)

Fig. 8. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of mixes containing replacement of cement by 35% slag and 10% silica fume at two different w/cm.

700 600 500 400 300 200

exp data

100 0

500 400 300 200 exp data 100 0

0

20

40

60

Age (days)

80

100

0

20

40

60

80

100

Age (days)

Fig. 9. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of mixes containing replacement of cement by 25% fly ash at two different w/cm.

A. Kar et al. / Construction and Building Materials 43 (2013) 139–155

800

600

700

500

shrinkage (microstrain)

shrinkage (microstrain)

150

600 500 400 300 200

exp data

100

400 300 200 exp data

100

0

0 0

20

40

60

80

100

0

20

40

60

80

100

Age (days)

Age (days)

Fig. 10. Comparison of concrete shrinkage experimental data with the upper and lower bounds predicted by the proposed model in case of mixes containing replacement of cement by 35% slag and 15% fly ash at two different w/cm.

Table 8a Observations based on the residuals (w/cm = 04). w/cm = 0.4

Age (Days) 3

7

28

90

Mixture

l

r

Max

Min

l

r

Max

Min

l

r

Max

Min

l

r

Max

Min

CC SG 35 FA 25 SF 10 SG 35 + SF 10 FA 25 + SF 10 SG 35 + FA 15

64 44 50 45 50 64 93

20 29 23 19 25 18 54

79 65 67 58 68 77 132

50 23 34 32 32 51 55

73 34 41 35 44 54 70

18 20 18 16 16 16 43

95 65 67 58 68 77 132

50 23 25 23 32 42 30

43 19 40 38 31 39 45

33 19 14 11 17 19 35

95 65 67 58 68 77 132

4 3 17 23 8 17 25

35 17 45 47 20 45 42

25 13 12 11 16 22 30

95 65 67 59 68 89 132

0 3 17 23 2 17 1

Table 8b Observations based on the residuals (w/cm = 0.3). w/cm = 0.3

Age (Days) 3

7

28

90

Mixture

l

r

Max

Min

l

r

Max

Min

l

r

Max

Min

l

r

Max

Min

CC SG 35 FA 25 SF 10 SG 35 + SF 10 FA 25 + SF 10 SG 35 + FA 15

15 20 28 23 28 34 19

8 14 12 25 12 19 17

21 30 37 40 37 47 31

10 10 20 5 20 21 7

8 11 21 13 21 19 20

9 13 12 19 11 21 11

21 30 37 40 37 47 20

0 2 9 0 12 0 31

5 10 16 11 22 23 19

6 8 9 12 8 15 7

21 30 37 40 37 47 31

0 2 8 0 12 0 7

10 15 10 12 27 24 18

7 7 8 9 7 16 10

23 30 37 40 37 47 32

0 2 1 0 12 0 0

Note: l = mean value of the cumulative residuals up to each corresponding age; r = standard deviation of the cumulative residuals up to each corresponding age; Max = maximum value of the residuals the cumulative residuals up to each corresponding age; Min = minimum value of the residuals the cumulative residuals up to each corresponding age.

5. Comparisons of models with the shrinkage data 5.1. Accuracy of the proposed model and comparison with experimental data The accuracy of the proposed model is first checked by plotting the experimental data and the predicted upper and lower bounds of the proposed model along the same axes, as shown in Figs. 4– 10. This is done for all the mixes and all the cases showed that the experimental data lied between the predicted limits. But in certain cases, there was a distinct randomness regarding how the experimental data lies within the predicted values, i.e. in some case, it is very close to the upper bound whereas in other cases, it is very close to the lower bound. This randomness is further

investigated by computing the means and standard deviations of the residuals (difference between predicted and experimental values) for each mix at different ages, as explained in this section. The results obtained from the shrinkage tests mentioned are compared with the range of the shrinkage predicted by the proposed equation. The results are tested to find out if they lied within the lower and upper bounds of the predicted model and the residuals are computed by subtracting the mean of the predicted values from the experimental data. The plots of the experimental data with the ranges given by the proposed model are furnished in Figs. 4–10. The experimental data in all of the cases are found to lie within the upper and the lower bounds predicted by the proposed model. This shows the effectiveness of the model in providing an estimate of the shrinkage strains

151

Shrinkage (microstrain)

A. Kar et al. / Construction and Building Materials 43 (2013) 139–155

1000 800 600 400 predicted lower bound predicted upper bound WVDOH 40th Street

200 0 0

10

20

30

40

50

60

70

80

90

100

Age (days)

Shrinkage (microstrain)

Fig. 11a. Comparison of predicted shrinkage strains from proposed model with shrinkage strain data from other sources in case of CC mixes at w/cm of 0.4.

800 700 600 500 400 300 200 100 0

WVDOH_Morgantown WVDOH_Martinsburg WVDOH_Summersville

0

20

40

60

80

100

Age (Days)

Shrinkage (microstrain)

Fig. 11b. Comparison of predicted shrinkage strains from proposed model with shrinkage strain data from other sources in case of FA 20 + SF 5 mixes at w/cm of 0.4.

The residuals show that the mean differences of the experimental data from the mean of the predicted values are fairly close for most of the mixtures. For w/cm of 0.4, the mean deviations were generally about 40–50 microstrains, with a maximum of 93 microstrains for the SG 35 + FA 15 mixture (Table 8a). The values of the mean differences in case of w/cm of 0.3 are generally around 15– 25 microstrains, with the highest difference being 34 microstrains in the case of FA 25 + SF 10 mixture. Also, the largest value of standard deviation observed is 54 microstrains (for SG 35 + FA 15 with w/cm = 0.4). So, it can be concluded that the predicted values of shrinkage strains from the proposed model were in fairly close agreement with the experimental data. Based on the above findings, it can be concluded that the residual analyses corroborate the accuracy of the proposed model for predicting concrete shrinkage. The accuracy of the proposed model was further examined by comparison with the existing shrinkage prediction models, as shown in the next section.

700 600 5.2. Evaluating the accuracy of the proposed model by comparison with available data from other sources

500 400 300 200 100

WVDOH_Morgantown

WVDOH_Martinsburg

Predicted upper bound

Predicted lower bound

0 0

20

40

60

80

100

Age (Days) Fig. 11c. Comparison of predicted shrinkage strains from proposed model with shrinkage strain data from other sources in case of SG 30 + SF 5 mixes at w/cm of 0.4.

of concrete containing SCM of varied types and proportions. In some cases, the experimental data are found to lie very close to the predicted upper bound (Figs. 4b, 5b, 6a, 7a, 8b and 10b), whereas, in a few other cases (Figs. 7b, 9a, and 10a), the experimental data approach the predicted lower bound values. In order to analyze these random occurrences more closely, residual analyses are carried out. A residual plot can be a good measure of accuracy of a prediction model [31]. If the residuals are randomly distributed with respect to the datum (usually the expected values) then the model under consideration can be taken as fairly accurate. The residuals are obtained by subtracting the mean value of the predicted upper and lower bounds, and then examined in detail, in terms of the mean, standard deviation, and maximum and minimum values, as shown in Tables 8a and b.

Shrinkage data obtained from work done by different researchers using w/cm of 0.4, similar aggregate content, cementitious material and environmental conditions were used for validation of the model. Some of the researchers used different conditions of curing and relative humidity. For those cases, the shrinkage strain values are modified by multiplying them with appropriate coefficients from ACI-209R [1] to convert them to the equivalent conditions of the current study. Depending on the percentages of cementitious materials in the mixtures, the values of a, b, c, and d are found either directly or through interpolation from Table 6. The data for validating the model for the 100% Portland cement case are obtained from laboratory testing done by the WVDOH for the I-64 Dunbar bridge and 40th street projects [14]. Some other results were also obtained from the work done by Collins and Sanjayan [13], Goel et al. [20], and Ghodousi et al. [19]. However, limited results are available for the cementitious systems containing SCM. The sources used for the validation are Morris [32], and Davalos et al. [14,15] for three different sites in West Virginia, namely Morgantown, Martinsburg and Summersville. The plots showing the comparison between the available shrinkage strain data and the corresponding predictions made by the proposed model are shown in Figs. 11a–c. The data from the external sources lie within the range of the upper and lower bounds of the proposed model for the majority of the cases, as shown in Figs. 11a–c. The exceptions are the values at 7 days and 14 days for the WVDOH_Martinsburg, WVDOH_ Summersville, and Morris [32]. In these cases, the experimental data lie just below the predicted lower bound. But the deviation

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Fig. 14. Comparison of the ACI (and the modified ACI by [23] with the proposed model and experimental data in case of SG 35 + SF 10 and FA 25 + SF 10 mixes at w/cm of 0.4.

is of the order of 50 microstrains. So, the proposed model is fairly accurate in predicting the strains for the external sources also. 5.3. Comparison with available models This section discusses the accuracy of the proposed model, as compared to the existing shrinkage prediction models. The existing shrinkage prediction models such as ACI (and modified ACI by [23], CEB MC 90, and GL 2000 are used to predict the shrinkage strains for the mix proportions, mentioned in Table 4. The results from these predictions are then compared with the experimental values and the predicted results from the proposed model.

5.3.1. Comparisons with ACI and modified ACI (by [23] The comparisons of the proposed model with the ACI model (and modified ACI by Huo et al. [23]) are shown in this section. The ACI model does not take into account the effect of the compressive strength for high-performance concretes. This issue has been addressed in the modified ACI model proposed by Huo et al. [23], which introduced a function of the compressive strength in the prediction equation. So, the modified ACI model is also considered for comparison in the present study. The general trend of the comparison of the predicted model with the ACI (and the modified ACI by [23] model shows a tendency of the ACI model to comply well with the predictions from

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SG 35 + FA 15 Fig. 15. Comparison of the ACI (and the modified ACI by [23] with the proposed model and experimental data in case of SG 35 + FA 15 mix at w/cm of 0.4.

the proposed model for w/cm of 0.4 (Figs. 12–15). The ACI model (and the modified ACI by [23] underestimates the shrinkages for the ages below 28 days. However, both the models overestimates the predicted shrinkage strains at low w/cm case (w/cm = 0.3) in all the cases (figures not shown here for brevity). The largest value of the mean deviation of the ACI model observed with respect to the experimental data is 221 microstrains in case of the FA

25 + SF 10 (w/cm 0.4) (Fig. 14 b). This value significantly exceeds the maximum mean deviation of 132 microstrains exhibited by the proposed model (Table 8a). In general, the mean deviation of the ACI predictions from the experimental values are in the range of 60–170 microstrains. The possible reason behind the observation deviations is that the prediction equation of the ACI model does not contain any factor which takes into account the influence of the w/cm. The ACI does not even take into account the effect of the compressive strength for high-performance concretes. However, the modified ACI model by Huo et al. [23] introduced a function of the compressive strength in the prediction equation and the predictions in this case were much closer to those of the proposed model as compared to the ACI model. The largest value of the mean deviation of the ACI model observed with respect to the experimental data is 217 microstrains in case of the SG 35 (w/cm 0.3). This value significantly exceeds the maximum mean deviation of 132 microstrains exhibited by the proposed model (Table 8a). In general, the mean deviation of the ACI predictions from the experimental values are in the range of 50  150 microstrains. The possible reason behind the observation deviations is that the prediction equation of the Huo et al. [23] model does not contain any factors which explicitly consider the influence of the w/cm and also the presence of SCM in the system. For concretes with high compressive strengths, the shrinkage strains are overestimated by the modified ACI model. The high

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strength cases included mostly concretes containing replacement of portland cement by SCM at low w/cm (w/cm = 0.3). Overall, it can be concluded that, the experimental data agrees much more with the model proposed in the present study than the ACI (and the modified ACI by [23] model.

5.3.2. CEB MC 90 model The CEB MC 90 model agrees fairly well with the proposed model for the mixtures with w/cm 0.4 (Figs. 16–19). The largest value of the mean deviation of the ACI model observed with respect to the experimental data is only 128 microstrains in case of the FA 25 + SF 10 (w/cm 0.4) (Fig. 18b). This value is just less than the maximum mean deviation of 132 microstrains exhibited by the proposed model (Table 8a). In general, the mean deviation of the ACI predictions from the experimental values are in the range of 40–90 microstrains for the w/cm 0.4 cases. The possible reason is that the CEB MC 90 model considers the total shrinkage to be the sum of the autogenous shrinkage and the drying shrinkage for the high performance concretes. However, the CEB MC 90 model overestimates the concrete shrinkage strains in all the low w/cm cases (figures not shown here for brevity). The largest value of the mean deviation of the ACI model observed with respect to the experimental data is only 219 microstrains in case of the CC mixture (w/cm 0.3) which significantly exceeds the maximum deviation of 132 microstrains exhibited the proposed model. These discrepancies are observed as the CEB MC 90 model takes into ac-

count the effects of only the compressive strength at a given age as the governing factor of shrinkage and not the influence of the w/cm ratio or the presence of SCM in the system. Thus, the proposed model is evidently more effective as compared to the proposed model in case of concretes containing replacement of cement by SCM. The overall comparison with the available models shows that the proposed model in the predicted study is able to predict the shrinkage strains of the concrete mixes with the best accuracy. So, it can be concluded that the proposed model for predicting concreting shrinkage is quite capable of incorporating the effect of the SCM, as well as the multi-scale approach. 6. Conclusions The two-step composite models are developed to predict the range of shrinkage strain of concrete containing SCM using the shrinkage strains of the cementitious paste and mortar in successive steps. In this model the quantities and types of SCMs can be used as input parameters to obtain the predicted shrinkage. The influence of the ITZ is incorporated through the concept of the Hashin–Shtrikman bounds. The statistical analysis shows that the proposed composite models is in good agreement with the experimental shrinkage values measured in this study. The proposed model also complied well with the experimental data from the external sources. The general trend of the comparison of the predicted model with the ACI shows a tendency of the ACI model to be more or less in agreement with the lower predicted bound for the w/cm 0.4 cases. The modified ACI model [23] agrees with the proposed model to a greater extent. The predictions from the modified ACI model are close to the mean of the upper and lower bounds obtained from the proposed model. However, the ACI model, as well as the modified ACI model overestimates the predicted shrinkage strains at low w/cm (w/cm = 0.3). The CEM MC 90 model shows very good agreement with the proposed model for the 0.4 w/cm cases. However, the CEB MC 90 model overestimates the shrinkage strains for the low w/cm cases. The overall comparison with the available models shows the proposed model in the predicted study to have the closest correlation with the experimental data. Acknowledgements The authors would like to gratefully acknowledge the financial support received from WV Higher Education Policy Commission/

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