# A Prediction Method of Tensile Young's Modulus of Concrete at Early Age

Authors: Isamu Yoshitake; Farshad Rajabipour; Yoichi Mimura; Andrew Scanlon
Journal: Advances in Civil Engineering
 (2012)

---

## Abstract

Knowledge of the tensile Young's modulus of concrete at early ages is important for estimating the risk of cracking due to restrained shrinkage and thermal contraction. However, most often, the tensile modulus is considered equal to the compressive modulus and is estimated empirically based on the measurements of compressive strength. To evaluate the validity of this approach, the tensile Young's moduli of 6 concrete and mortar mixtures are measured using a direct tension test. The results show that the tensile moduli are approximately 1.0–1.3-times larger than the compressive moduli within the material's first week of age. To enable a direct estimation of the tensile modulus of concrete, a simple three-phase composite model is developed based on random distributions of coarse aggregate, mortar, and air void phases. The model predictions show good agreement with experimental measurements of tensile modulus at early age.


---

## Body

## 1. Introduction

An accurate estimation of the Young’s modulus is important for proper structural design of concrete members, and ensuring their serviceability, such as controlling deflections and crack widths. In particular, the time-dependent development of the tensile Young’s modulus at early ages is needed for estimation of the tensile stresses that are generated due to restrained thermal and hygral shrinkage. These tensile stresses may lead to premature cracking of concrete members. Currently, the tensile modulus is assumed to be equal in value to the compressive modulus and is estimated using empirical correlations based on the compressive strength of concrete [1, 2]. The Architectural Institute of Japan (AIJ) [3] points out that employing the tensile modulus is more appropriate for estimation of the risk of early-age cracking; however, the specification indicates that the compressive modulus may be used instead of the tensile modulus because investigations dealing with the tensile modulus are currently insufficient. Since the tensile behavior of concrete is more significantly affected by the presence of flaws (e.g., microcracks or large capillary pores common in early-age concrete), it is important to develop tools to predict or measure the tensile properties more accurately.Direct tension tests have been conducted in earlier studies to investigate the tensile strength and the tensile strain capacity of concrete. Although the tensile moduli can be obtained from the linear portion of the stress-strain diagram in these reports, the focus of these earlier studies has been primarily on mature concrete. As such, little information is available on the early-age (i.e., less than 28 days) tensile modulus and its development with time. In addition, a reliable model to aid design engineers in estimating the tensile modulus based on concrete’s proportions and age does not currently exist.Xie and Liu [4] conducted a direct tension test using small and large specimens of mature concrete with various aggregate sizes, and measured the tensile strength, strain capacity, and Young’s modulus. They observed that increasing the maximum aggregate size does not have a proportional impact on tensile strength and tensile modulus of concrete. Oluokun et al. [5] researched the compressive Young’s modulus and Poisson’s ratio of early-age concrete. They concluded that the compressive modulus is proportional to the 0.5 power of the compressive strength, and the ACI 318 formula for estimation of the compressive modulus is valid after the age of 12 hours. Hagihara et al. [6] investigated the tensile creep of high-strength concrete at early age, and reported that the tensile Young’s moduli are approximately 15% higher than the compressive moduli. This is an important conclusion and should be evaluated for other concretes with normal strength. Swaddiwudhipong et al. [7] investigated the mechanical properties of concrete containing ground granulated blast furnace slag and pulverized fuel ash. They reported that the tensile moduli of all concretes tested had correlated well with the tensile strength; although no predictive formula is presented for the tensile modulus. Aoki et al. [2] also conducted direct tension test in order to obtain tensile strength and Young’s modulus of mature concrete and found that the tensile modulus is 9–12% higher than the compressive modulus for these concretes. Bissonnette et al. [8] researched the tensile creep of concrete at early age, and presented some measurements of the tensile Young’s modulus at the age of 7 and 28 days. In an earlier work [9], the authors investigated the tensile Young’s moduli by using a direct tension test, and presented a composite model derived from the Hirsh model [10] to predict the tensile modulus. The model showed good agreement with the experimental data as well as other composite models offered by Counto [11] and Hashin [12]. Recently, Mihashi and Leite [13] presented a state-of-the-art report on early age cracking of concrete and its mitigation techniques that includes some information on the mechanical properties of concrete at early ages.As described above, the investigations focusing on the tensile Young’s modulus of early-age concrete are few, and more experimental data will be needed to establish reliable predictive correlations for estimation of the tensile modulus. The present paper reports laboratory measurements of the tensile modulus within the first 7 days of hydration. Three concrete mixtures (with differentW/C) were tested. In addition, to evaluate the effect of aggregate size, duplicate concrete mixtures were prepared and sieved before setting using a 5 mm mesh sieve. The resulting mortars were tested to determine their tensile Young’s modulus. Using the measurements results, a composite model was developed and calibrated which can serve as a simple method for estimating the tensile Young’s modulus of concrete.


## 2. Experimental Program

### 2.1. Materials and Mix Proportions of Concrete

This study employed ordinary Portland cement with a density of3.14 g/cm3. Tables 1 and 2 provide the details of the cement and aggregate used. Proportions of the six concrete and mortar mixtures tested in this study are given in Table 3. The proportions of the concrete were designed by referring to mixture proportions used in a ready mixed concrete plant in Japan. Mortars were obtained by sieving plastic concrete mixtures as discussed above. This was done to duplicate the mechanical properties (i.e., tensile modulus) of the mortar portion of the concretes as closely as possible. These results are needed to develop the composite model as discussed in Section 3. In all mixtures, proper dosages of air entraining and water reducing admixtures were used to ensure consistency and workability of the concrete (Table 3).Table 1
Physical and chemical compositions of cement.
Ordinary Portland cementDensity3.14 g/cm3Blaine fineness3340cm2/gSetting time start-end2 h 26 m–3 h 34 mComp. strength at 3 days30.8 MPaat 7 days46.3 MPaat 28 days63.6 MPaChemical compositionsCaO64.5%SiO220.5%Al2O35.7%Fe2O32.9%MgO1.27%SO32.15%Cl-0.009%Loss of ignition1.89%Table 2
Properties of aggregate.
Fine agg.SCrushed rockGMaterialsSea sandAndesiteDensity2.56 g/cm32.73 g/cm3Fineness modulus3.366.66Absorption1.3%1.3%Size (max. – min.)5 mm20–5 mmTable 3
Mixture proportions of concrete.
IDW/C(%)Water (kg/m3)Cement (kg/m3)S(kg/m3)G(kg/m3)WRA (kg/m3)Air (%)O575716529081210302.94.5O57m#1572654661304—4.7—#2O39391694347909334.33.8O39m#1392576591200—6.5—#2O25251706806948186.83.6O25m#125243971991—9.7—#2#1Mortar is made from wet screening of concrete (maximum mesh size: 5 mm).#2Air content of mortar was not measured because it is difficult to obtain wet-screened mortar volume required for the test.


### 2.2. Test Methods and Specimens

Figure1 shows the direct tension apparatus used in this study. Figure 2 shows the geometry of the dog-bone specimens tested. To reduce bending moment during test, the ends of the dog-bone specimens were not fully fixed but were allowed rotational freedom. The direct tension apparatus manually provides a tensile force to a specimen using a lever. Overall, an approximately constant strain rate of 2 to 3×10-6/sec was applied. To measure tensile strain of specimens, an embedded strain sensor was used which includes an electrical resistance wire strain gage of 60 mm long coated by epoxy resin with tensile modulus of 2.8 kPa as shown in Figure 2. The overall sensor size was 120×10×3mm; as such the area ratio of the sensor to concrete was smaller than 1.4% to ensure that the sensor has little influence on the behavior of specimens when subjected to tensile force. While this setup was used to measure the tensile modulus, it may not be suitable for measurement of the tensile strength since a number of breaks occurred within the end zones of the dog-bone specimens (Figure 3).Figure 1
Direct tension test using a dog-bone-shaped specimen.Figure 2
A dog-bone-shaped specimen and an embedded sensor.Figure 3
Typical failure within the end zone of the dog-bone-shaped specimen.In addition to the tensile modulus, the compression and indirect tension tests were conducted using cylindrical specimens. The compressive modulus of concrete was obtained using an extensometer equipped with 2 displacement gages, and the modulus of mortar was measured using 2 wire strain gages 30 mm long. The cylindrical specimens (diameter × height) tested were100×200mm for concrete, and 50×100mm for mortar. Three cylindrical specimens were used for each test per each mixture, and the average of the three measurements was used. The tests were performed at ages 1, 2, 3, and 7 days.


### 2.3. Determination of the Tensile Young’s Modulus

To measure the tensile modulus, the tensile force is applied to the specimen at strain intervals of10×10-6. In order to prevent failure of specimen, the maximum strain during the test is set at 60×10-6. Since the plastic strain of concrete at early age may comprise a high percentage of the total strain measured, the tensile modulus is obtained from the unloading branches of the stress-strain relation as shown in Figure 4. The slope of the unloading branches after the specimen was loaded to 10×10-6, 20×10-6, and so forth, is determined and averaged to obtain the tensile modulus of the specimen. The force is measured twice at each strain level using a load cell with an accuracy of 0.1 kN and capacity of 200 kN. As will be discussed later, the maximum residual strain after unloading of each specimen was measured as 3×10-6. In each case, concrete and mortar specimens were made from the same batch using the sieving procedure mentioned above.Figure 4
Evaluation method for tensile Young’s modulus.


## 2.1. Materials and Mix Proportions of Concrete

This study employed ordinary Portland cement with a density of3.14 g/cm3. Tables 1 and 2 provide the details of the cement and aggregate used. Proportions of the six concrete and mortar mixtures tested in this study are given in Table 3. The proportions of the concrete were designed by referring to mixture proportions used in a ready mixed concrete plant in Japan. Mortars were obtained by sieving plastic concrete mixtures as discussed above. This was done to duplicate the mechanical properties (i.e., tensile modulus) of the mortar portion of the concretes as closely as possible. These results are needed to develop the composite model as discussed in Section 3. In all mixtures, proper dosages of air entraining and water reducing admixtures were used to ensure consistency and workability of the concrete (Table 3).Table 1
Physical and chemical compositions of cement.
Ordinary Portland cementDensity3.14 g/cm3Blaine fineness3340cm2/gSetting time start-end2 h 26 m–3 h 34 mComp. strength at 3 days30.8 MPaat 7 days46.3 MPaat 28 days63.6 MPaChemical compositionsCaO64.5%SiO220.5%Al2O35.7%Fe2O32.9%MgO1.27%SO32.15%Cl-0.009%Loss of ignition1.89%Table 2
Properties of aggregate.
Fine agg.SCrushed rockGMaterialsSea sandAndesiteDensity2.56 g/cm32.73 g/cm3Fineness modulus3.366.66Absorption1.3%1.3%Size (max. – min.)5 mm20–5 mmTable 3
Mixture proportions of concrete.
IDW/C(%)Water (kg/m3)Cement (kg/m3)S(kg/m3)G(kg/m3)WRA (kg/m3)Air (%)O575716529081210302.94.5O57m#1572654661304—4.7—#2O39391694347909334.33.8O39m#1392576591200—6.5—#2O25251706806948186.83.6O25m#125243971991—9.7—#2#1Mortar is made from wet screening of concrete (maximum mesh size: 5 mm).#2Air content of mortar was not measured because it is difficult to obtain wet-screened mortar volume required for the test.


## 2.2. Test Methods and Specimens

Figure1 shows the direct tension apparatus used in this study. Figure 2 shows the geometry of the dog-bone specimens tested. To reduce bending moment during test, the ends of the dog-bone specimens were not fully fixed but were allowed rotational freedom. The direct tension apparatus manually provides a tensile force to a specimen using a lever. Overall, an approximately constant strain rate of 2 to 3×10-6/sec was applied. To measure tensile strain of specimens, an embedded strain sensor was used which includes an electrical resistance wire strain gage of 60 mm long coated by epoxy resin with tensile modulus of 2.8 kPa as shown in Figure 2. The overall sensor size was 120×10×3mm; as such the area ratio of the sensor to concrete was smaller than 1.4% to ensure that the sensor has little influence on the behavior of specimens when subjected to tensile force. While this setup was used to measure the tensile modulus, it may not be suitable for measurement of the tensile strength since a number of breaks occurred within the end zones of the dog-bone specimens (Figure 3).Figure 1
Direct tension test using a dog-bone-shaped specimen.Figure 2
A dog-bone-shaped specimen and an embedded sensor.Figure 3
Typical failure within the end zone of the dog-bone-shaped specimen.In addition to the tensile modulus, the compression and indirect tension tests were conducted using cylindrical specimens. The compressive modulus of concrete was obtained using an extensometer equipped with 2 displacement gages, and the modulus of mortar was measured using 2 wire strain gages 30 mm long. The cylindrical specimens (diameter × height) tested were100×200mm for concrete, and 50×100mm for mortar. Three cylindrical specimens were used for each test per each mixture, and the average of the three measurements was used. The tests were performed at ages 1, 2, 3, and 7 days.


## 2.3. Determination of the Tensile Young’s Modulus

To measure the tensile modulus, the tensile force is applied to the specimen at strain intervals of10×10-6. In order to prevent failure of specimen, the maximum strain during the test is set at 60×10-6. Since the plastic strain of concrete at early age may comprise a high percentage of the total strain measured, the tensile modulus is obtained from the unloading branches of the stress-strain relation as shown in Figure 4. The slope of the unloading branches after the specimen was loaded to 10×10-6, 20×10-6, and so forth, is determined and averaged to obtain the tensile modulus of the specimen. The force is measured twice at each strain level using a load cell with an accuracy of 0.1 kN and capacity of 200 kN. As will be discussed later, the maximum residual strain after unloading of each specimen was measured as 3×10-6. In each case, concrete and mortar specimens were made from the same batch using the sieving procedure mentioned above.Figure 4
Evaluation method for tensile Young’s modulus.


## 3. Prediction of the Young’s Modulus Using a Composite Model

As mentioned earlier, the tensile Young’s modulus of concrete is often assumed to be equal in value to the compressive modulus. In addition, the compressive modulus of concrete is frequently estimated based on empirical correlations with concrete compressive strength. The compressive modulus of concrete has also been related to the volume and mechanical properties of concrete’s constituents (aggregates, paste, etc.) using some classical composite models such as those presented by Zhou et al. [14], Top00E7u [15], and Yoshitake et al. [9]. A brief overview of these models is provided below.The typical composite models for estimation of the elastic modulus are illustrated in Figure5. These include (a) the Parallel model, (b) the Series model, (c) the Counto model [11], and (d) the Hashin model [12]; as represented by the following:
(1)Parallelmodel:E=∑i=1nEi⋅Vi,(2)Seriesmodel:1E=∑i=1nViEi,(3)Countomodel:EEm=1+VgVg-Vg+Em/(Eg-Em),(4)Hashinmodel:EEm=VmEm+(1+Vg)Eg(1+Vg)Em+VmEg,
where Ei and Vi represent the Young’s modulus and the volume fraction of concrete constituents (e.g., mortar, coarse aggregate, etc.), n is the number of constituents, and the subscripts m and g refer to mortar and coarse aggregate, respectively. The Counto and Hashin models are based on a 2-phase composite (mortar and aggregate). While these models are generally more accurate than simple parallel and series models, they may estimate the Young’s modulus inappropriately for concretes containing high aggregate volumes or high air content.Typical composite models for predicting Young’s modulus: (a) Parallel model, (b) Series model, (c) Counto model, and (d) Hashin model.
(a)(b)(c)(d)In the present work a new triphase model is proposed based on random distribution of elements within a 2-dimensional80×80 grid (Figure 6). Each element in the model is composed of mortar, coarse aggregate, or air. The number elements corresponding to each phase is proportional to the volume fraction of that phase in concrete. Elements are placed randomly in the model using a Monte Carlo procedure. To determine the tensile modulus of the grid, simple micromechanical calculations are performed based on the series and parallel models. First the tensile modulus of each row of elements is determined using the parallel model and then the modulus of the grid is determined by combining all rows using the series model. Alternatively, the modulus of each column can first be determined using the series model and then the columns are combined using the parallel model.Figure 6
An example of a simple composite model using parallel and series models.


## 4. Experimental Results and Discussion

### 4.1. Evaluation of the Reliability of the Embedded Strain Sensor

It is important to ensure proper measurements of the tensile strain using the embedded strain sensor. For this purpose, a dog-bone concrete specimen is tested in tension and the tensile strain is measured by both the embedded gage as well as 2 wire strain gages mounted on the surface of the concrete specimen. Figure7 presents the results showing that the internally measured strain is practically equal to the surface measured strains. This implies that the tensile stress is applied uniformly to the specimen and the embedded sensor can be used to monitor concrete’s tensile strain. Based on this conclusion, further measurements in this study employ only the embedded sensor.Figure 7
Comparison of inside strain and surface strain.


### 4.2. Tensile Stress-Strain Responses

Figure8 presents examples of the tensile stress-strain responses of each concrete. As shown, the slope of each stress-strain regression line (i.e., the tensile modulus) develops with increasing age and reducing the water-cement ratio (W/C) of concrete. Figure 8(a) shows the stress-strain response at 1 day; the results indicate that the residual strain after unloading of specimens is zero (i.e., plastic strain at age of 1 day when specimen are loaded to 60 × 10−6 is negligible). Note that the response of mixture O57 (W/C= 57%) at 1 day could not be obtained because the concrete was too weak to allow performing the direct tension test. Figure 8(b) presents the stress-strain responses at 7 days; the maximum residual strain after loading specimens to 60×10-6 is 3×10-6 corresponding to the mixture O39. For this mix, had the loading branches of the stress-strain response been used to determine the modulus, the tensile modulus would be estimated as 31.2 GPa, comparing with 33.0 GPa obtained from using the unloading stress-strain branches. The ratio of the modulus obtained by the two methods is approximately 0.95.Tensile stress-strain responses: (a) age of 1 day and (b) age of 7 days.
(a)(b)


### 4.3. Time Dependent Development of the Mechanical Properties

The results of the compressive and splitting tensile strength measurements of the three concrete mixtures are presented in Figure9. At 7 days, the concrete mixtures have compressive strengths in the range of 20 to 45 MPa, and splitting tensile strengths in the range of 2 to 3.3 MPa. The time-dependent compressive and tensile Young’s moduli are presented in Figure 10. The compressive moduli in the graph demonstrate the secant moduli under 33% of the maximum stress. Based on Figure 10, the tensile Young’s moduli are approximately 1.0–1.3-times larger than the compressive moduli. The result may be caused by different response to stress applied to each specimen, such that the tensile stresses were less than 10% of the compressive stresses. This difference can be especially significant at early ages when the large macropores dictate the tensile response of concrete.Figure 9
Strengths versus age of concrete.Figure 10
Young’s moduli versus age of concrete.A comparison between the tensile modulus of concrete and the corresponding mortar specimens are provided in Figure11. The results indicate that the concrete tensile modulus is always higher than the mortar modulus due to the stiffness provided by the coarse aggregates in concrete. However, the difference is narrowed as concrete ages due to hydration of cement which results in an increased stiffness of the mortar.Figure 11
Tensile Young’s modulus predicted by employing grids of different fineness.


## 4.1. Evaluation of the Reliability of the Embedded Strain Sensor

It is important to ensure proper measurements of the tensile strain using the embedded strain sensor. For this purpose, a dog-bone concrete specimen is tested in tension and the tensile strain is measured by both the embedded gage as well as 2 wire strain gages mounted on the surface of the concrete specimen. Figure7 presents the results showing that the internally measured strain is practically equal to the surface measured strains. This implies that the tensile stress is applied uniformly to the specimen and the embedded sensor can be used to monitor concrete’s tensile strain. Based on this conclusion, further measurements in this study employ only the embedded sensor.Figure 7
Comparison of inside strain and surface strain.


## 4.2. Tensile Stress-Strain Responses

Figure8 presents examples of the tensile stress-strain responses of each concrete. As shown, the slope of each stress-strain regression line (i.e., the tensile modulus) develops with increasing age and reducing the water-cement ratio (W/C) of concrete. Figure 8(a) shows the stress-strain response at 1 day; the results indicate that the residual strain after unloading of specimens is zero (i.e., plastic strain at age of 1 day when specimen are loaded to 60 × 10−6 is negligible). Note that the response of mixture O57 (W/C= 57%) at 1 day could not be obtained because the concrete was too weak to allow performing the direct tension test. Figure 8(b) presents the stress-strain responses at 7 days; the maximum residual strain after loading specimens to 60×10-6 is 3×10-6 corresponding to the mixture O39. For this mix, had the loading branches of the stress-strain response been used to determine the modulus, the tensile modulus would be estimated as 31.2 GPa, comparing with 33.0 GPa obtained from using the unloading stress-strain branches. The ratio of the modulus obtained by the two methods is approximately 0.95.Tensile stress-strain responses: (a) age of 1 day and (b) age of 7 days.
(a)(b)


## 4.3. Time Dependent Development of the Mechanical Properties

The results of the compressive and splitting tensile strength measurements of the three concrete mixtures are presented in Figure9. At 7 days, the concrete mixtures have compressive strengths in the range of 20 to 45 MPa, and splitting tensile strengths in the range of 2 to 3.3 MPa. The time-dependent compressive and tensile Young’s moduli are presented in Figure 10. The compressive moduli in the graph demonstrate the secant moduli under 33% of the maximum stress. Based on Figure 10, the tensile Young’s moduli are approximately 1.0–1.3-times larger than the compressive moduli. The result may be caused by different response to stress applied to each specimen, such that the tensile stresses were less than 10% of the compressive stresses. This difference can be especially significant at early ages when the large macropores dictate the tensile response of concrete.Figure 9
Strengths versus age of concrete.Figure 10
Young’s moduli versus age of concrete.A comparison between the tensile modulus of concrete and the corresponding mortar specimens are provided in Figure11. The results indicate that the concrete tensile modulus is always higher than the mortar modulus due to the stiffness provided by the coarse aggregates in concrete. However, the difference is narrowed as concrete ages due to hydration of cement which results in an increased stiffness of the mortar.Figure 11
Tensile Young’s modulus predicted by employing grids of different fineness.


## 5. Prediction of the Tensile Young’s Modulus at Early Age

### 5.1. Input Data for Development of the Simplified Composite Model

The tensile Young’s moduli of coarse aggregates and mortar are needed as input parameters for use in the composite models. While the moduli of mortar is directly measured (Figure11), it is difficult to directly obtain the tensile modulus of the aggregate due the size and number of specimens required for tensile testing, difficulty in using a proper tensile grip method of the rock specimen, as well as the variability in the material properties due to layering and impurities of the rock. Thus, the tensile Young’s modulus of the coarse aggregate is indirectly calculated by employing the Counto and Hashin models in the present study. The study estimates the tensile Young’s modulus (Eg) of 36.0 GPa from the experimental result of O25 at age of 7 days, for which the modulus of mortar was almost equal to the modulus of concrete. That is, the estimated value to the modulus is little affected by the models used for obtaining the modulus of the aggregate. Similar values are obtained based on testing O25 at earlier ages or by testing the other two mixtures.Table4 provides the input data for the composite model. The volume fractions of each component (mortar, coarse aggregates, and air) are obtained from the mix proportions given in Table 3. To determine the appropriate number of elements in the model, the study estimates the tensile Young’s moduli of concrete by employing models of various mesh sizes. Figure 12 presents the resulting tensile moduli predicted as a function of mesh fineness. The graph shows the maximum, minimum, and average moduli obtained from 10 consecutive simulations in each model. Based on these results, a 80×80 model was chosen for the remaining simulations in this work.Table 4
Input data for the composite model.
Mix. IDVmVgVaEt of mortar (GPa) shown in Figure 11Eg (GPa)1 day2 days3 days7 daysO5757.7%37.7%4.5%N/A19.022.425.636.0O3962.0%34.2%3.8%20.024.227.030.036.0O2566.4%30.0%3.6%23.729.132.335.336.0Figure 12
Comparison of experiment and predicted tensile Young’s modulus.


### 5.2. Quality of the Model Predictions

Figure13 shows the tensile Young’s moduli of all concrete specimens predicted by using the proposed composite model. Each bar presents the average of 10 simulations for each concrete mixture and at each age. The bar graph indicates that the predicted Young’s moduli are in good agreement with the experimental values; the ratio of the two is in range of 0.90 to 1.07. This implies that the tensile modulus of concrete can be predicted appropriately by employing the composite model when the volume fraction of constituents is known.Figure 13
Tensile Young’s modulus predicted by employing grids of different fineness.


### 5.3. Empirical Formula for the Tensile Modulus of Mortars in This Study

Considering that the Young’s modulus of coarse aggregate and air content of concrete are age-independent, the Young’s modulus of concrete at early age may be predicted if the modulus of mortar can be estimated appropriately. For the mortar studied in this work, empirical correlations between the experimental measurement modulus, theW/C, and the age of mortars are established as presented in Figure 14. The figure shows a linear correlation between the inverse water to cement ratio (C/W) and the tensile moduli of mortar (Et) at ages of 1, 2, 3, and 7 days. Interestingly, the slope ki of all regression lines in the graph is approximately 4.3 GPa:
(5)RelationofEt-CW:Et=ki⋅CW+Ci,
where Et (GPa) is the tensile modulus of mortar, and Ci (GPa) is an age-dependent parameter in each regression line. The change in Ci with age of mortar is shown in Figure 15. A Goral curve [16], which is often used for estimating concrete strength development with age, is fit to the data points resulting in the following:
(6)CoefficientCiwithage:Ci=21TT+1.5,
where T is mortar age in days. Combining (5) and (6) results in(7)TensileYoung’smodulusofmortar:Et(T)=4.3CW+21TT+1.5,
where Et is the estimated tensile Young’s modulus of mortar (GPa).Figure 14
Relations between cement-water ratio and tensile Young’s moduli of mortar.Figure 15
CoefficientCi of the regression line versus age of mortar.It must be noted that similar to modulus of concrete, the tensile modulus of mortar is a function of the volume fraction and stiffness of sand, volume fraction and modulus of cement paste (itself a function of age andW/C), and the air content of the mortar. By accounting for age and W/C, (7) can provide an estimate for the tensile modulus of mortars with similar volume fraction and stiffness of the fine aggregates. Young’s modulus of coarse aggregate can be considered as an influencing factor to the tensile modulus of concrete, so this equation including the effect of fine aggregate may be useful for normal concrete using sea sand when an appropriate value of the modulus for coarse aggregate is provided. To be applicable to mortars other than those used here, the most significant remaining parameter is the volume fraction of sand which must be taken into consideration.


### 5.4. Prediction of the Tensile Modulus of Concrete by the Composite Model

By combining (7) with the composite model (Figure 6), the age-dependent tensile moduli of concrete can be estimated from its mixture proportions (i.e., volume fraction of constituents) and the aggregate modulus. Figure 16 presents a comparison between the model predictions and the experimental data from this study as well as those from previous investigations [2, 4, 7]. Herein, these predicted data were obtained with the assumption that the aggregate properties in other studies are equal to the values employed in this study, because the modulus of coarse aggregate used is not reported in the previous investigations. The figure shows that the proposed method can predict the tensile modulus of concrete, with reasonable accuracy, solely based on the mixture proportions.Figure 16
Comparison of experiments and predictions.Figure17(a) describes characteristics of tensile moduli of concrete with W/C= 55% and coarse aggregates of different volume fractions. The vertical axis in the graph presents the tensile Young’s moduli ratio of concrete to coarse aggregate. Herein, the model results are obtained from the assumptions shown in Table 5. The figure demonstrates that the tensile modulus of concrete increases with increasing the volume of coarse aggregates since the aggregates have a higher Young’s modulus than the mortars. It is also noted that tensile moduli of concrete having more coarse aggregates (i.e., larger vol. fractions) than the solid volume content of coarse aggregates [17] are unavailable. Figure 17(b) presents a similar set of curves corresponding to concrete with W/C= 30%. Both figures describe that the tensile modulus of concrete develops rapidly at early age and gradually plateaus at later ages.Table 5
Conditions for simulation of tensile Young’s modulus.
Composite modelSee Figure6Elements80 × 80Air content4.5%Tensile Young’s modulus of mortarsee (7)Tensile Young’s modulus of coarse aggregate36 GPaSolid volume content of coarse aggregate60%Tensile Young’s modulus of concreteAverage of 10 simulationsPrediction results for tensile Young’s modulus of concrete: (a)W/C = 55% and (b) W/C = 30%.
(a)(b)


## 5.1. Input Data for Development of the Simplified Composite Model

The tensile Young’s moduli of coarse aggregates and mortar are needed as input parameters for use in the composite models. While the moduli of mortar is directly measured (Figure11), it is difficult to directly obtain the tensile modulus of the aggregate due the size and number of specimens required for tensile testing, difficulty in using a proper tensile grip method of the rock specimen, as well as the variability in the material properties due to layering and impurities of the rock. Thus, the tensile Young’s modulus of the coarse aggregate is indirectly calculated by employing the Counto and Hashin models in the present study. The study estimates the tensile Young’s modulus (Eg) of 36.0 GPa from the experimental result of O25 at age of 7 days, for which the modulus of mortar was almost equal to the modulus of concrete. That is, the estimated value to the modulus is little affected by the models used for obtaining the modulus of the aggregate. Similar values are obtained based on testing O25 at earlier ages or by testing the other two mixtures.Table4 provides the input data for the composite model. The volume fractions of each component (mortar, coarse aggregates, and air) are obtained from the mix proportions given in Table 3. To determine the appropriate number of elements in the model, the study estimates the tensile Young’s moduli of concrete by employing models of various mesh sizes. Figure 12 presents the resulting tensile moduli predicted as a function of mesh fineness. The graph shows the maximum, minimum, and average moduli obtained from 10 consecutive simulations in each model. Based on these results, a 80×80 model was chosen for the remaining simulations in this work.Table 4
Input data for the composite model.
Mix. IDVmVgVaEt of mortar (GPa) shown in Figure 11Eg (GPa)1 day2 days3 days7 daysO5757.7%37.7%4.5%N/A19.022.425.636.0O3962.0%34.2%3.8%20.024.227.030.036.0O2566.4%30.0%3.6%23.729.132.335.336.0Figure 12
Comparison of experiment and predicted tensile Young’s modulus.


## 5.2. Quality of the Model Predictions

Figure13 shows the tensile Young’s moduli of all concrete specimens predicted by using the proposed composite model. Each bar presents the average of 10 simulations for each concrete mixture and at each age. The bar graph indicates that the predicted Young’s moduli are in good agreement with the experimental values; the ratio of the two is in range of 0.90 to 1.07. This implies that the tensile modulus of concrete can be predicted appropriately by employing the composite model when the volume fraction of constituents is known.Figure 13
Tensile Young’s modulus predicted by employing grids of different fineness.


## 5.3. Empirical Formula for the Tensile Modulus of Mortars in This Study

Considering that the Young’s modulus of coarse aggregate and air content of concrete are age-independent, the Young’s modulus of concrete at early age may be predicted if the modulus of mortar can be estimated appropriately. For the mortar studied in this work, empirical correlations between the experimental measurement modulus, theW/C, and the age of mortars are established as presented in Figure 14. The figure shows a linear correlation between the inverse water to cement ratio (C/W) and the tensile moduli of mortar (Et) at ages of 1, 2, 3, and 7 days. Interestingly, the slope ki of all regression lines in the graph is approximately 4.3 GPa:
(5)RelationofEt-CW:Et=ki⋅CW+Ci,
where Et (GPa) is the tensile modulus of mortar, and Ci (GPa) is an age-dependent parameter in each regression line. The change in Ci with age of mortar is shown in Figure 15. A Goral curve [16], which is often used for estimating concrete strength development with age, is fit to the data points resulting in the following:
(6)CoefficientCiwithage:Ci=21TT+1.5,
where T is mortar age in days. Combining (5) and (6) results in(7)TensileYoung’smodulusofmortar:Et(T)=4.3CW+21TT+1.5,
where Et is the estimated tensile Young’s modulus of mortar (GPa).Figure 14
Relations between cement-water ratio and tensile Young’s moduli of mortar.Figure 15
CoefficientCi of the regression line versus age of mortar.It must be noted that similar to modulus of concrete, the tensile modulus of mortar is a function of the volume fraction and stiffness of sand, volume fraction and modulus of cement paste (itself a function of age andW/C), and the air content of the mortar. By accounting for age and W/C, (7) can provide an estimate for the tensile modulus of mortars with similar volume fraction and stiffness of the fine aggregates. Young’s modulus of coarse aggregate can be considered as an influencing factor to the tensile modulus of concrete, so this equation including the effect of fine aggregate may be useful for normal concrete using sea sand when an appropriate value of the modulus for coarse aggregate is provided. To be applicable to mortars other than those used here, the most significant remaining parameter is the volume fraction of sand which must be taken into consideration.


## 5.4. Prediction of the Tensile Modulus of Concrete by the Composite Model

By combining (7) with the composite model (Figure 6), the age-dependent tensile moduli of concrete can be estimated from its mixture proportions (i.e., volume fraction of constituents) and the aggregate modulus. Figure 16 presents a comparison between the model predictions and the experimental data from this study as well as those from previous investigations [2, 4, 7]. Herein, these predicted data were obtained with the assumption that the aggregate properties in other studies are equal to the values employed in this study, because the modulus of coarse aggregate used is not reported in the previous investigations. The figure shows that the proposed method can predict the tensile modulus of concrete, with reasonable accuracy, solely based on the mixture proportions.Figure 16
Comparison of experiments and predictions.Figure17(a) describes characteristics of tensile moduli of concrete with W/C= 55% and coarse aggregates of different volume fractions. The vertical axis in the graph presents the tensile Young’s moduli ratio of concrete to coarse aggregate. Herein, the model results are obtained from the assumptions shown in Table 5. The figure demonstrates that the tensile modulus of concrete increases with increasing the volume of coarse aggregates since the aggregates have a higher Young’s modulus than the mortars. It is also noted that tensile moduli of concrete having more coarse aggregates (i.e., larger vol. fractions) than the solid volume content of coarse aggregates [17] are unavailable. Figure 17(b) presents a similar set of curves corresponding to concrete with W/C= 30%. Both figures describe that the tensile modulus of concrete develops rapidly at early age and gradually plateaus at later ages.Table 5
Conditions for simulation of tensile Young’s modulus.
Composite modelSee Figure6Elements80 × 80Air content4.5%Tensile Young’s modulus of mortarsee (7)Tensile Young’s modulus of coarse aggregate36 GPaSolid volume content of coarse aggregate60%Tensile Young’s modulus of concreteAverage of 10 simulationsPrediction results for tensile Young’s modulus of concrete: (a)W/C = 55% and (b) W/C = 30%.
(a)(b)


## 6. Conclusions

This paper describes the experimental measurement of tensile Young’s modulus of concrete at early age using a direct tension setup. Moreover, a predictive composite model was developed to estimate the age-dependent tensile modulus of concrete using the volume fractions and properties of the constituents. The main conclusions are summarized as follows.(1)
The tensile stress-strain response of concrete was observed to be very linear even at early ages (e.g., 1 day old). The residual strains after repeated loading of specimens up to strains of60×10-6 were negligible. The tensile Young’s modulus obtained from the stress-strain response develops according to the water-cement ratio and the age of concrete.(2)
The tensile modulus of concrete is approximately 1.0–1.3-times larger than its compressive modulus within the material’s first week of age. As such, estimation of the tensile modulus based on empirical correlations with the compressive strength of concrete can be inaccurate.(3)
The age-dependent tensile moduli of concrete could be predicted appropriately by the proposed composite model as long as the volume fractions of coarse aggregates, mortar, and air, and the modulus of aggregates are known. In comparison with the experimental results, the model predictions showed accuracy better than±15%.



---

*Source: 391214-2012-01-11.xml*