# Simplified Analysis of Thermal Cracks in Low-Heat Portland Cement Concrete

Authors: Longlong Liu; Shangchuan Zhao; Jianda Xin; Zhenhong Wang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

This article introduces some experimental data from the compression test, the tension test, and the temperature stress test machine (TSTM) test for low-heat Portland (LHP) cement concrete with different fly ash (FA) content. With the proposed equations, the thermal cracks can be predicted with the TSTM test along with the mechanical properties. The higher FA content exhibits a lower hydration degree and a smaller adiabatic temperature rise. Although the FA content decreases the hydration heat and the strength at an early age, there are few influences on the relational expression between compressive strength and tensile strength. The proposed equations can also be applied to estimate the thermal cracks risk in LHP cement concrete. The cracking stress of LHP cement concrete in the temperature stress test is not only related to the tensile stress history but also connected to the compressive stress history. Considering the strength history, an analytical method for evaluating the cracking temperature difference was proposed. With the proposed method, only compressive strength can be used to estimate the thermal cracks risk for LHP cement concrete at different ages. It greatly simplifies the evaluation method of thermal cracks in low-heat Portland cement concrete. Furthermore, this evaluation method can also be applied to other types of concrete.


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## Body

## 1. Introduction

The low-heat Portland (LHP) cement concrete is widely used in massive concrete structures due to its relatively small self-shrinkage and low adiabatic temperature rise, which has good crack resistance in the long run [1–5]. In the past few decades, LHP cement concrete has been used in massive constructions such as Baihetan, Shuangjiangkou, and Wudongde dams. Besides, there will be more low-heat cement used in massive constructions in China [5]. Since the restraint stress has always been an important issue for cracks in early-age concrete [6], especially the tensile stress that exceeds the tensile strength, the cracking stress is always used as a comprehensive evaluation index to predicate the thermal cracks in massive concrete structures [7]. The thermal stress happens when the deformation caused by the temperature difference between the adiabatic temperature rise and the ambient temperature is restrained [8]. Therefore, the adiabatic temperature is the primary reason for the great tensile stress, while the ambient temperature is constant in general [9]. To reduce the tensile stress history in massive concrete structures, the hydration heat in the early-age cement concrete should be reduced. This is also the main reason that LHP cement concrete can be widely used around the world [6].To further reduce the adiabatic temperature rise, fly ash (FA) is applied to replace part of the cement [10]. The reduction of the hydration heat and the decrease of the autogenous shrinkage have been seen in the long run, which extends from the appearance of the ordinary cement with FA and LHP cement concrete to the application of LHP cement concrete with FA. Compared with ordinary cement, the adiabatic temperature rise can be reduced to 10°C if the LHP cement is mixed with 30 wt.% FA [5]. The FA content in LHP cement concrete not only reduces the adiabatic temperature rise but also influences the mechanical properties. The cement replaced by FA in the cement concrete does not participate in the hydration reaction between the binder and water. The strengths of the mortars formed in cement concrete are easily affected by the hydration degree, especially for tensile strength [11]. Those mechanical properties are the most used model parameters to estimate the cracking risk in the early-age cement concrete [12–20], especially for the cracking risk of LHP cement concrete containing FA [5, 21–24]. With the increment of the FA content, the compressive strength decreases relatively little, while the loss of tensile strength does not affect the cracking risk of the early-age cement concrete structure [7, 21]. Combined with the empirical equation, the tensile strength may be forecasted by the compressive strength ranging from 4 MPa to 120 MPa [12]. The tensile Young’s modulus predicted by the known compressive strength or the tensile strength is enough [25, 26]. Furthermore, the influence of FA on compressive strength may not be as great as the influence of FA on tensile strength. The empirical correlation between the tensile strength and the compressive strength may be affected by the FA content in the cement concrete [27]. Although there have been many studies of the influences of FA on material properties, various conclusions are obtained from different cement materials. The FA content mainly changes the strength development for most used materials, which has few influences on the final strength and the elastic modulus. The low-heat cement concrete itself has little hydration heat with a slow growth rate at an early age [21, 28]. If the FA is added to the low-heat cement concrete, the growth rate of strength will be reduced again. For LHP cement concrete, it is still unknown whether the relationship between the strength development will be changed after adding FA. These assumptions have not been verified in the previous research.Estimating the cracking risk of LHP cement concrete containing blending materials at an early age has been studied [29]. Several mechanical properties, such as the elastic modulus, relaxation modulus, creep coefficient, and tensile strength, are investigated [30–33]. Despite the fact that the tensile stress history affects the cracking resistance of cement concrete, the effect of the compressive stress history on the cracking resistance is always neglected [34]. In general, compared with the compressive strength at an early age, for the whole thermal stress history, the maximum compressive stress during the temperature rise is extremely little. For LHP cement concrete, whether the compressive stress history influences the cracking stress has not been determined. As mentioned, the growth rate of strength will be lower if the FA is added to the LHP cement concrete. Compared with the normal strength of concrete, it reaches the maximum temperature of about two days, although the hydration heat is low. The maximum compressive stress is reached two days earlier when the elastic modulus development has few changes. For early-age restrained concrete structures, the cracking risk is increased if the growth rate of strength is decreased. A small restrained compressive stress may be close to the compressive strength, which may lead to immeasurable strength damage to the early-age concrete. Therefore, it is necessary to determine the relationship between the restrained compressive stress and the compressive strength of LHP cement concrete after adding fly ash. Thus, the TSTM test is conducted to investigate the influence of the compressive stress history on cracking stress using the same temperature history at different ages.To get a simple method for estimating the crack risk of LHP cement concrete, the relationship between the compressive strength and other mechanical properties should be determined [25]. In this article, the mechanical properties of LHP cement concrete with different FA content were investigated. Then, both the tensile strength and the elastic modulus can be predicted by the known compressive strength. The goal here is to predict the thermal stress history directly if LHP cement concrete is applied in concrete structures [9]. Meanwhile, the restrained stress behaviors due to the same temperature history at different ages were investigated under sealed conditions to investigate the effect of the compressive stress development on the cracking stress of LHP cement concrete.


## 2. Experiment

### 2.1. Cement Type and Mix Proportions of Concrete

In this study, one type of LHP (P⋅LH 42.5) was used. FA was partly mixed with the LHP cement to investigate the effect of FA content on the mechanical properties of the LHP cement concrete. The chemical and physical composition of LHP cement and FA adopted are stated in Table1, and the mechanical properties of the LHP cement are stated in Table 2. The setting time was tested according to the standard of GB/T 1346–2001 [35], and the strength characteristics of cement were determined by the standard of GB/T 17671–2021 [36].Table 1
Chemical and physical composition of LHP cement and FA (%).
ItemCaOSiO2Al2O3Fe2O3MgOSO3Na2OK2OSpecific surface (m2/kg)Specific gravity (kg/m3)LHP58.722.84.34.34.23.00.10.33323210FA3.25921.69.61.20.10.530.453862320Table 2
Mechanical strength properties of cement.
ItemSetting time (min)Compressive strength (MPa)Flexural strength (MPa)InitialFinal3 days7 days28 days3 days7 days28 daysCement23334314.923.644.54.35.28.2


### 2.2. Mix Proportion Design

The mix proportions of concrete are given in Table3. The mechanical test was conducted with a 0.5 w/b mixture along with different fly ash (FA) content. The FA contents in the present study were 15%, 25%, and 35% of the LHP cement, referred to as the FA content in [10, 25]. The LHP cement concrete containing 35% FA is adopted in the TSTM test.Table 3
Mix proportion of concrete (kg/m3).
ItemWaterCementFAFineCoarseSuperplasticizerAir entraining agentFA-151662825011188441.9920.053FA-251662498311188441.9920.053FA-3516621611611188441.9920.053


### 2.3. Method

The general view of the experiment device is shown in Figures1 and 2. The elastic modulus and tensile strength at 14, 28, 90, and 180 days were measured by the axial tensile specimen 150 × 150 × 550 mm, and the compressive strength at 14, 28, 90, and 180 days was measured by cubes 100 × 100 × 100 mm in the compression test [37]. In addition to the temperature stress test machine, as shown in Figure 2, the direct tensile strength can also be tested in this device. It is more convenient to measure the elastic modulus of concrete at any arbitrary time during the TSTM test. However, to prevent the early failure of concrete specimens, new specimens were tested in the traditional mechanics' experiments. Specific TSTM test details are given in 2.3.Figure 1
Mechanical property test: (a) compression test; (b) splitting tension test.
(a)(b)Figure 2
Sketch of a temperature stress test machine (TSTM) and removed concrete specimen. (a) Sketch of TSTM; (b) removed concrete specimen.
(a)(b)


### 2.4. Test Procedure

Figure3 shows the dimensions of the concrete specimen, whereby the embedded bars are used for fixing the measuring equipment and the embedded copper pipe is used for placing the temperature sensor. The application method of the embedded equipment is identical to that used in [34, 38]. Concrete was poured immediately into the mold after mixing the blend for 3–5 minutes. All specimens except those for the temperature stress test were cured and tested at a constant temperature of 20 ± 1°C. For this reason, the maturity of specimens is neglected in this article since the concrete specimen has the same temperature history. In the TSTM test, the complete restraint is controlled by displacement, while the concrete specimen will be pushed or pulled back to the original length by the load cell if the length of the concrete specimen exceeds a threshold value of 2 μm/m [34, 38], as seen in Figure 4. By continuously compensating for the free shrinkage strain of concrete, the TSTM test finished while the tensile stress decreased rapidly and cracks appeared in the middle of the concrete specimen.Figure 3
Dimensions of concrete specimen.Figure 4
The TSTM test [34].The mechanical properties of LHP cement concrete were tested before the TSTM test was finished. With the same external conditions, the mechanical properties are invariably stable. Therefore, there may be a relational expression between the test results. Both the tensile strength and the elastic modulus can be expressed by the compressive strength. Finally, these proposed expressions can be used to simplify the evaluation method of cracking risk of LHP cement concrete.


## 2.1. Cement Type and Mix Proportions of Concrete

In this study, one type of LHP (P⋅LH 42.5) was used. FA was partly mixed with the LHP cement to investigate the effect of FA content on the mechanical properties of the LHP cement concrete. The chemical and physical composition of LHP cement and FA adopted are stated in Table1, and the mechanical properties of the LHP cement are stated in Table 2. The setting time was tested according to the standard of GB/T 1346–2001 [35], and the strength characteristics of cement were determined by the standard of GB/T 17671–2021 [36].Table 1
Chemical and physical composition of LHP cement and FA (%).
ItemCaOSiO2Al2O3Fe2O3MgOSO3Na2OK2OSpecific surface (m2/kg)Specific gravity (kg/m3)LHP58.722.84.34.34.23.00.10.33323210FA3.25921.69.61.20.10.530.453862320Table 2
Mechanical strength properties of cement.
ItemSetting time (min)Compressive strength (MPa)Flexural strength (MPa)InitialFinal3 days7 days28 days3 days7 days28 daysCement23334314.923.644.54.35.28.2


## 2.2. Mix Proportion Design

The mix proportions of concrete are given in Table3. The mechanical test was conducted with a 0.5 w/b mixture along with different fly ash (FA) content. The FA contents in the present study were 15%, 25%, and 35% of the LHP cement, referred to as the FA content in [10, 25]. The LHP cement concrete containing 35% FA is adopted in the TSTM test.Table 3
Mix proportion of concrete (kg/m3).
ItemWaterCementFAFineCoarseSuperplasticizerAir entraining agentFA-151662825011188441.9920.053FA-251662498311188441.9920.053FA-3516621611611188441.9920.053


## 2.3. Method

The general view of the experiment device is shown in Figures1 and 2. The elastic modulus and tensile strength at 14, 28, 90, and 180 days were measured by the axial tensile specimen 150 × 150 × 550 mm, and the compressive strength at 14, 28, 90, and 180 days was measured by cubes 100 × 100 × 100 mm in the compression test [37]. In addition to the temperature stress test machine, as shown in Figure 2, the direct tensile strength can also be tested in this device. It is more convenient to measure the elastic modulus of concrete at any arbitrary time during the TSTM test. However, to prevent the early failure of concrete specimens, new specimens were tested in the traditional mechanics' experiments. Specific TSTM test details are given in 2.3.Figure 1
Mechanical property test: (a) compression test; (b) splitting tension test.
(a)(b)Figure 2
Sketch of a temperature stress test machine (TSTM) and removed concrete specimen. (a) Sketch of TSTM; (b) removed concrete specimen.
(a)(b)


## 2.4. Test Procedure

Figure3 shows the dimensions of the concrete specimen, whereby the embedded bars are used for fixing the measuring equipment and the embedded copper pipe is used for placing the temperature sensor. The application method of the embedded equipment is identical to that used in [34, 38]. Concrete was poured immediately into the mold after mixing the blend for 3–5 minutes. All specimens except those for the temperature stress test were cured and tested at a constant temperature of 20 ± 1°C. For this reason, the maturity of specimens is neglected in this article since the concrete specimen has the same temperature history. In the TSTM test, the complete restraint is controlled by displacement, while the concrete specimen will be pushed or pulled back to the original length by the load cell if the length of the concrete specimen exceeds a threshold value of 2 μm/m [34, 38], as seen in Figure 4. By continuously compensating for the free shrinkage strain of concrete, the TSTM test finished while the tensile stress decreased rapidly and cracks appeared in the middle of the concrete specimen.Figure 3
Dimensions of concrete specimen.Figure 4
The TSTM test [34].The mechanical properties of LHP cement concrete were tested before the TSTM test was finished. With the same external conditions, the mechanical properties are invariably stable. Therefore, there may be a relational expression between the test results. Both the tensile strength and the elastic modulus can be expressed by the compressive strength. Finally, these proposed expressions can be used to simplify the evaluation method of cracking risk of LHP cement concrete.


## 3. Test Results and Analysis

### 3.1. Mechanical Properties of LHP Cement Concrete

#### 3.1.1. Compressive Strength

Combined with the early-age development law of the strength characteristics, a modified expression for the compressive strengthfct in CEB-FIP [39] can be described as follows:(1)fct=fc,28⋅exps1⋅1−28t−ts0.5, where fc,28 is the compressive strength at 28 days, t is the age of concrete from pouring, s1 is the model parameter, and ts is the apparent setting time. The model parameter s1 can be obtained by fitting the compressive test data, as seen in Figure 5.Figure 5
The compressive strength development at varying ages.The regression coefficientss1 of FA-15, FA-25, and FA-35 were obtained as 1.38, 1.48, and 2.03, respectively. The apparent setting time ts was obtained from the penetration resistance test. The model parameters for the mechanical property of LHP cement concrete are given in Table 4.Table 4
Model parameters for the mechanical properties of LHP cement concrete.
Items1s2sefc,28 (MPa)ft,28 (MPa)Ec,28 (GPa)FA-151.381.210.4732.93.3537.5FA-251.481.350.530.93.1137.2FA-352.031.650.5624.92.8636.8


#### 3.1.2. Tensile Strength

Combined with (2) mentioned in CEB-FIP [39], the tensile strength ftt is fitted with the test results, which can be applied to predict the tensile strength. Figure 6 shows the tensile strength for FA-15, FA-25, and FA-35 concrete from 14 to 180 days. The test data suggest that the tensile strength reduced with the increment of the FA content before 90 days. However, this decreasing tendency was reserved for the curing age when there is little variation in 180 days. This shows that LHP cement concrete containing the FA content obtains lower early-age tensile strength but nearly equivalent later-age tensile strength.(2)ftt=ft,28⋅exps2⋅1−28t−ts0.5,where ft,28 is the tensile strength at 28 days and s2 is the model parameter. The regression coefficients s2 were obtained as 1.21, 1.35, and 1.65.Figure 6
The tensile strength development at varying ages.


#### 3.1.3. Elastic Modulus

The elastic modulus can be predicted by (2), which is shown in CEB-FIP 1990 [40].(3)Ecte=Ec,28×expse×1−28te0.5,where Ec,28 is the elastic modulus at 28 days, se is a model parameter, and te is the equivalent age considering the temperature.A modified E-modulus was made by Kanstad [41], which considers the apparent setting time ts. The “ts-concept“ means that the concrete starts to achieve mechanical properties.(4)Ect=Ec,28⋅expse⋅1−28t−ts0.5.To calculate the development of the elastic modulus accurately,ts∗ was introduced in (4), which was explained in [42]:(5)Ect=Ec,28⋅expse⋅1−28−ts∗t−tsnE.As the model parameter is a bit complicated after consideringts∗, it is difficult to get from the limited test data. Thus, (4) is commonly used to calculate the elastic modulus; it is shown in CEB-FIP 2010 [39]. As seen in Figure 7, The model parameter se can be obtained as 0.47, 0.5, and 0.56 from fitting test results, respectively.Figure 7
Measured elastic modulus and calculated development by means of equation (4).


### 3.2. Estimation of the Tensile Strength

The tensile strength is always used for estimating the cracking risk of early-age cement concrete. However, the direct tensile test is difficult to conduct, which is generally evaluated by splitting tensile and expressed by the compressive strength [12, 13, 17, 43]. There have been a few formulas to express the tensile strength in terms of compressive strength, as shown in Figure 8. Every equation has a difference in the tensile strength with the same compressive strength [39, 44, 45].Figure 8
Comparison of the tensile models and the fitting results.JSCE [44]:(6)ft=0.23⋅fc0.67.ACI-318-08 [45]:(7)ft=0.53⋅fc0.5.CEB-FIP [39]:(8)ft=0.3⋅fck2/3for concrete grade≤C50,ft=2.12⋅ln1+0.1⋅fck+Δffor concrete grade<C50.fck is the specific characteristic compressive strength. The measured compressive strength can be used. Furthermore, the compressive strength of LHP cement concrete in this article is smaller than 50 MPa, so equation (8) is adopted, Δf=8 MPa.Nevertheless, the splitting tensile strength can be easily obtained from the test when the direct tensile strength is extremely difficult to obtain from the direct tension test. Even though the relation formula between the tensile strength and the compressive strength is given, it is always varying with the strength development or the test conditions [13, 46]. By virtue of the advantage of the TSTM, the tensile strength can be expressed as the compressive strength [34]. Figure 8 illuminates that the compressive strength can be applied to predict tensile strength development. However, the theoretical values from equations (7) and (8) underestimate the tensile strength development from 30 MPa to 50 MPa. Equation (8) from CEB-FIP overestimates the tensile strength of LHP cement concrete when the compressive strength is less than 20 MPa. These model parameters just need to be determined again according to the characteristics of the model. The relationship is exponential growth if the tensile strength can be expressed as the compressive strength, which is determined as follows:(9)ft=α1⋅fcα2,where ft is the tensile strength, MPa. α1andα2 are model parameters, which can be determined by fitting from the test results.α1 and α2 can be determined from fitting the test data with (9), which is shown in Table 5. The fitting curve of different FA content was drawn in Figure 8.Table 5
Model parameters for the estimation of the tensile strength and the elastic modulus.
Itemα1α2β1β2FA-150.150.927.10.3FA-250.140.9127.00.3FA-350.130.9426.50.33By summarizing the effect of FA content on the mechanical properties, these results suggest that the LHP cement concrete with different FA content attains a lower early-age speed of hydration rate but a higher later-age speed of hydration rate. With the increment of the FA content, the ratio between the tensile strength and the compressive strength shows smaller early-age values but larger later-age values.Only the tensile strength may be overestimated when the compressive strength is extremely small at an early age. This may be the reason why the compressive strength value is extremely small when the great ratio happens. To obtain a more accurate equation, more compression and tension tests at an earlier age should be carried out.


### 3.3. Estimation of the Elastic Modulus

Elastic modulus is invariably estimated from the measured compressive strength [17], especially in the design codes for predicting elastic modulus in terms of compressive strength. Models for estimating the elastic modulus in codes ACI 318–08 and CEB-FIP 2010 [39] were commonly used. Four typical models were collected as follows:Model developed by Kim [17]:(10)Ec=5.25⋅fc0.46.Model developed in JSCE [44]:(11)Ec=4.7⋅fc.Model developed in ACI-318-08 [45]:(12)Ec=22⋅fc100.3.Model developed in CEB-FIP 2010 [39]:(13)Ec=21.5×fc+Δf101/3.Ec is the elastic modulus (kN/mm2) and fc is the compressive strength (MPa), Δf=8 MPa.Figure9 shows the elastic modulus from the test data and the theoretical values from equations (11) to (14). The developed models in [17, 39, 44] all underestimate the elastic modulus. Compared with normal strength concrete, LHP cement concrete has a lower hydration degree and smaller early-age strength [21]. Although the compressive strength of LHP cement concrete may have a lower growth rate in the early stage, it may surpass the ultimate strength of normal strength concrete. Therefore, the elastic modulus may be expressed by the compressive strength:(14)Ec=β1⋅fc10β2,where β1 and β2 are model parameters, which can be determined by regression analysis from test data.Figure 9
Comparison of the elastic modulus models and fitting results.Referring to the previous models from equations (11) to (14), as shown in Figure 9, β1 and β2 can be determined by fitting the test data with (14), which is shown in Table 5. The fitting curve of different FA content was drawn in Figure 9. The FA content of 15% and 25% has a similar curve when the FA content of 35% attains a lower early-age elastic modulus but higher later-age elastic modulus.Compared with the models in [17, 39, 44], Kim [17] only investigated the effect of one kind of cement concrete containing 15% FA on the mechanical properties. The results from Kim [17] demonstrate that adding fly ash reduces strength growth rate and has little effect on the elastic modulus. The measured compressive strengths used in the Kim [17] and JSCE [44] models are based on the experimental data, while the specified compressive strength is applied in ACI 318–08 and the characteristic compressive strength is used in CEB-FIP 2010 [39]. All four developed theoretical models underestimate the elastic modulus.


### 3.4. Verification for an Estimation of Tensile Strength and Elastic Modulus

#### 3.4.1. Tensile Strength

To verify the proposed equation of the tensile strength, (1) and (9) are combined to estimate the tensile strength for FA-35 concrete; see Figure 10. The maximum difference in the different regression analysis is less than 5%. This suggests that the proposed (1) and (9) can be applied to predict the tensile strength of LHP cement concrete. In addition, the tensile strength development can be predicted from a measured or a given compressive strength at 28 days.Figure 10
Comparison of the proposed model and test results in tensile strength.


#### 3.4.2. Elastic Modulus

Figure11 shows two prediction methods for the elastic modulus. Excepted for the test method combined with (1) and (14), it can also predict the elastic modulus.Figure 11
Comparison of the proposed model and test results in elastic modulus.Compared with the test results and the calculation value of (1) and (14), the difference in elastic modulus is less than 3%. Therefore, it is rational to assume that the compressive strength can be applied to predict the elastic modulus. If the compressive strength development can be accurately predicted, the estimation of the elastic modulus is greatly simplified.


### 3.5. Thermal Cracks Analysis in LHP Cement Concrete

#### 3.5.1. Temperature History

The temperature history is shown in Figure12. The maximum temperature of LHP cement concrete for FA-35 is 32°C. To investigate the cracking resistance of LHP cement concrete, thermal stresses at different ages with the same temperature development were measured. As seen in Figure 12, T0 presents the pouring temperature or initial temperature, T1 is the temperature at the maximum compressive stress, ΔT1 is the temperature difference from T0 to T1, T2 is the maximum temperature through the whole temperature, Tc is the cracking temperature while the concrete specimen fails, and ΔT2 is the temperature difference from Tc to T2.Figure 12
Temperature history under semiadiabatic conditions.


#### 3.5.2. Thermal Stresses at Different Ages

Figure13 shows the thermal stress development at ages 0.417, 3, 7, and 20 days with FA-35. All specimens were cured at 20°C before the TSTM test. The same temperature history is applied to the concrete specimen at different ages. The test results show that the maximum compressive stress at the temperature rise stage is −0.33, −1.00, −1.07, and −1.91 MPa when the loading time Δt of LHP cement concrete is 1.95, 1.96, 2.07, and 1.97 days, respectively, as shown in Table 6. The maximum compressive stresses have reached before the maximum temperature throughout the whole temperature history. The cracking stress is 0.65, 0.92, 1.06, and 1.04 MPa at loading time Δt = 5.86, 7.56, 7.97, and 7.10 days, respectively. This reveals that the cracking resistance ability of the FA-35 specimen increases with the increment of loading age from 0.417 to 7 days and starts to reduce from 7 to 20 days. The tensile strength from (2) is 1.06, 1.66, 2.08, and 2.80 MPa when the elastic modulus from (4) is 26.3, 30.57, 33.07, and 36.5 GPa at loading time Δt = 5.86, 7.56, 7.97, and 7.10 days, respectively. Compared with the first specimen at a loading age of 0.417 days, the tensile strength of the later three specimens increases 56.6%, 96.2%, and 164.2% when the elastic modulus increases 16.23%, 25.74%, and 38.78%, respectively. This reveals that the growth rate of elastic modulus could not catch up with that of tensile strength 7 days later. In other words, the thermal stress increases very small at the same temperature difference, but the tensile strength increases surely when the curing age exceeds 7 days. This suggests that the later cooling behaves better with cracking resistance.Figure 13
Restrained stress development at different ages.Table 6
Test data for thermal cracks analysis in LHP cement concrete.
Loading ages (day)Δt (day)T1 (°C)Tc (°C)ΔT1 (°C)ΔT2 (°C)Maximum compressive stress (MPa)Cracking stress (MPa)Elastic modulus (GPa)0.4171.9531.3411.34−0.3316.825.8618.5013.500.6526.331.9631.3811.38−1.0024.37.5613.2618.740.9230.5772.0731.8111.81−1.0729.47.9712.1419.861.0633.02201.9731.4311.43−1.9133.387.1014.5817.421.0436.5


#### 3.5.3. Effect of Maximum Compressive Stress on the Cracking Stress

To investigate the effect of the maximum compressive stresses on the cracking stress, the maximum compressive stresses in Figure14 are connected and drawn as a curve. The cracking stress ratio (the cracking stress/the tensile strength) and the compressive stress ratio (the maximum compressive stress/the compressive strength) are also seen in Figure 14. The test data suggest that the compressive stress has an extreme effect on the cracking stresses in the LHP cement concrete. The cracking stress ratio is 0.61, 0.55, 0.50, and 0.37 when the maximum compressive stress is −0.33, −1.00, −1.07, and −1.91 MPa, respectively. The compressive stress ratio is 0.33, 0.19, 0.13, and 0.09 when the maximum compressive stress is compared with the compressive strength at the corresponding time. Obviously, the cracking stress ratio decreased with the increment of the maximum compressive stress, but the compressive stress ratio was so small that the compressive stress was always neglected in the past.Figure 14
Cracking stress ratios at different ages.Although there is a zero-stress point between the maximum compressive stress and the cracking stress, the whole temperature history is continuous. If the whole stress development is divided into two parts, one is the compression stage while the other is the tension stage. The influence of compressive stress on tensile stress will be ignored. From the test results analysis, it is found that the maximum compressive stress has an influence on the cracking stress. Therefore, except that the cooling rate should be controlled, the heating rate should also be controlled to avoid small cracking stress due to the great compressive stress.


### 3.6. A Simple Method for Predicting the Cracking Temperature Difference

The thermal stress can be approximately calculated as follows:(15)σt=RKpEctαΔT,where s is the thermal stress, R is the restraint coefficient, Kp is the stress relaxation coefficient, a is the coefficient of linear expansion when it is 7.5×10−6με/°C for the LHP cement concrete, and ΔT is the temperature difference.As the cracking stress ratio is 0.61, 0.55, 0.50, and 0.37 in the TSTM test, to prevent the thermal cracks of concrete, the thermal stress ratio should not exceed the average value of 0.5. The thermal stress should not exceed the allowable tensile stress [47].(16)σt≤0.5⋅ftt.As the concrete specimen is fully restrained,R = 1, the stress relaxation coefficient can be simplified as 0.2, while the elastic modulus and the tensile strength can be estimated by (9) and (14) with the model parameters in Table 5. The cracking temperature difference can be predicted as follows:(17)ΔT≤0.026⋅fct0.61α.The cracking temperature differences of the FA-35 concrete specimens at 6.27, 10.56, 14.97, and 27.10 days were 11.8, 16.4, 19.4, and 24.3°C, calculated by (17). The test results were 13.50, 18.7, 19.86, and 17.42°C. The deviations between results obtained from (17) and test data were 1.7, 2.3, 0.46, and 6.88°C, respectively. Compared with the test results, the deviations were small. Furthermore, the cracking risk of concrete for long periods can also be expressed by the compressive strength. For instance, the values of permissible temperature differences at 180, 360, 720, and 1000 days could be obtained from (17) were 35.8, 38.5, 40.5, and 41.3°C. Compared with the value of permissible temperature difference at 180 days, the permissible temperature differences at 360, 720, and 1000 days increased by 2.7, 4.7, and 5.5°C, respectively. Therefore, (17) could be utilized to predict the cracking risk of LHP cement concrete. However, the influence of compressive stress on the cracking risk was not considered in the present study. Further investigations of the influence of compressive stress, especially larger than 2 MPa, on the cracking risk at early age are necessary. Although the proposed model for the tensile strength, the elastic modulus, and the prediction equations of cracking risk were arbitrary due to the limited data, the cracking risk of concrete structures demonstrated by the test data could be illustrated directly.


## 3.1. Mechanical Properties of LHP Cement Concrete

### 3.1.1. Compressive Strength

Combined with the early-age development law of the strength characteristics, a modified expression for the compressive strengthfct in CEB-FIP [39] can be described as follows:(1)fct=fc,28⋅exps1⋅1−28t−ts0.5, where fc,28 is the compressive strength at 28 days, t is the age of concrete from pouring, s1 is the model parameter, and ts is the apparent setting time. The model parameter s1 can be obtained by fitting the compressive test data, as seen in Figure 5.Figure 5
The compressive strength development at varying ages.The regression coefficientss1 of FA-15, FA-25, and FA-35 were obtained as 1.38, 1.48, and 2.03, respectively. The apparent setting time ts was obtained from the penetration resistance test. The model parameters for the mechanical property of LHP cement concrete are given in Table 4.Table 4
Model parameters for the mechanical properties of LHP cement concrete.
Items1s2sefc,28 (MPa)ft,28 (MPa)Ec,28 (GPa)FA-151.381.210.4732.93.3537.5FA-251.481.350.530.93.1137.2FA-352.031.650.5624.92.8636.8


### 3.1.2. Tensile Strength

Combined with (2) mentioned in CEB-FIP [39], the tensile strength ftt is fitted with the test results, which can be applied to predict the tensile strength. Figure 6 shows the tensile strength for FA-15, FA-25, and FA-35 concrete from 14 to 180 days. The test data suggest that the tensile strength reduced with the increment of the FA content before 90 days. However, this decreasing tendency was reserved for the curing age when there is little variation in 180 days. This shows that LHP cement concrete containing the FA content obtains lower early-age tensile strength but nearly equivalent later-age tensile strength.(2)ftt=ft,28⋅exps2⋅1−28t−ts0.5,where ft,28 is the tensile strength at 28 days and s2 is the model parameter. The regression coefficients s2 were obtained as 1.21, 1.35, and 1.65.Figure 6
The tensile strength development at varying ages.


### 3.1.3. Elastic Modulus

The elastic modulus can be predicted by (2), which is shown in CEB-FIP 1990 [40].(3)Ecte=Ec,28×expse×1−28te0.5,where Ec,28 is the elastic modulus at 28 days, se is a model parameter, and te is the equivalent age considering the temperature.A modified E-modulus was made by Kanstad [41], which considers the apparent setting time ts. The “ts-concept“ means that the concrete starts to achieve mechanical properties.(4)Ect=Ec,28⋅expse⋅1−28t−ts0.5.To calculate the development of the elastic modulus accurately,ts∗ was introduced in (4), which was explained in [42]:(5)Ect=Ec,28⋅expse⋅1−28−ts∗t−tsnE.As the model parameter is a bit complicated after consideringts∗, it is difficult to get from the limited test data. Thus, (4) is commonly used to calculate the elastic modulus; it is shown in CEB-FIP 2010 [39]. As seen in Figure 7, The model parameter se can be obtained as 0.47, 0.5, and 0.56 from fitting test results, respectively.Figure 7
Measured elastic modulus and calculated development by means of equation (4).


## 3.1.1. Compressive Strength

Combined with the early-age development law of the strength characteristics, a modified expression for the compressive strengthfct in CEB-FIP [39] can be described as follows:(1)fct=fc,28⋅exps1⋅1−28t−ts0.5, where fc,28 is the compressive strength at 28 days, t is the age of concrete from pouring, s1 is the model parameter, and ts is the apparent setting time. The model parameter s1 can be obtained by fitting the compressive test data, as seen in Figure 5.Figure 5
The compressive strength development at varying ages.The regression coefficientss1 of FA-15, FA-25, and FA-35 were obtained as 1.38, 1.48, and 2.03, respectively. The apparent setting time ts was obtained from the penetration resistance test. The model parameters for the mechanical property of LHP cement concrete are given in Table 4.Table 4
Model parameters for the mechanical properties of LHP cement concrete.
Items1s2sefc,28 (MPa)ft,28 (MPa)Ec,28 (GPa)FA-151.381.210.4732.93.3537.5FA-251.481.350.530.93.1137.2FA-352.031.650.5624.92.8636.8


## 3.1.2. Tensile Strength

Combined with (2) mentioned in CEB-FIP [39], the tensile strength ftt is fitted with the test results, which can be applied to predict the tensile strength. Figure 6 shows the tensile strength for FA-15, FA-25, and FA-35 concrete from 14 to 180 days. The test data suggest that the tensile strength reduced with the increment of the FA content before 90 days. However, this decreasing tendency was reserved for the curing age when there is little variation in 180 days. This shows that LHP cement concrete containing the FA content obtains lower early-age tensile strength but nearly equivalent later-age tensile strength.(2)ftt=ft,28⋅exps2⋅1−28t−ts0.5,where ft,28 is the tensile strength at 28 days and s2 is the model parameter. The regression coefficients s2 were obtained as 1.21, 1.35, and 1.65.Figure 6
The tensile strength development at varying ages.


## 3.1.3. Elastic Modulus

The elastic modulus can be predicted by (2), which is shown in CEB-FIP 1990 [40].(3)Ecte=Ec,28×expse×1−28te0.5,where Ec,28 is the elastic modulus at 28 days, se is a model parameter, and te is the equivalent age considering the temperature.A modified E-modulus was made by Kanstad [41], which considers the apparent setting time ts. The “ts-concept“ means that the concrete starts to achieve mechanical properties.(4)Ect=Ec,28⋅expse⋅1−28t−ts0.5.To calculate the development of the elastic modulus accurately,ts∗ was introduced in (4), which was explained in [42]:(5)Ect=Ec,28⋅expse⋅1−28−ts∗t−tsnE.As the model parameter is a bit complicated after consideringts∗, it is difficult to get from the limited test data. Thus, (4) is commonly used to calculate the elastic modulus; it is shown in CEB-FIP 2010 [39]. As seen in Figure 7, The model parameter se can be obtained as 0.47, 0.5, and 0.56 from fitting test results, respectively.Figure 7
Measured elastic modulus and calculated development by means of equation (4).


## 3.2. Estimation of the Tensile Strength

The tensile strength is always used for estimating the cracking risk of early-age cement concrete. However, the direct tensile test is difficult to conduct, which is generally evaluated by splitting tensile and expressed by the compressive strength [12, 13, 17, 43]. There have been a few formulas to express the tensile strength in terms of compressive strength, as shown in Figure 8. Every equation has a difference in the tensile strength with the same compressive strength [39, 44, 45].Figure 8
Comparison of the tensile models and the fitting results.JSCE [44]:(6)ft=0.23⋅fc0.67.ACI-318-08 [45]:(7)ft=0.53⋅fc0.5.CEB-FIP [39]:(8)ft=0.3⋅fck2/3for concrete grade≤C50,ft=2.12⋅ln1+0.1⋅fck+Δffor concrete grade<C50.fck is the specific characteristic compressive strength. The measured compressive strength can be used. Furthermore, the compressive strength of LHP cement concrete in this article is smaller than 50 MPa, so equation (8) is adopted, Δf=8 MPa.Nevertheless, the splitting tensile strength can be easily obtained from the test when the direct tensile strength is extremely difficult to obtain from the direct tension test. Even though the relation formula between the tensile strength and the compressive strength is given, it is always varying with the strength development or the test conditions [13, 46]. By virtue of the advantage of the TSTM, the tensile strength can be expressed as the compressive strength [34]. Figure 8 illuminates that the compressive strength can be applied to predict tensile strength development. However, the theoretical values from equations (7) and (8) underestimate the tensile strength development from 30 MPa to 50 MPa. Equation (8) from CEB-FIP overestimates the tensile strength of LHP cement concrete when the compressive strength is less than 20 MPa. These model parameters just need to be determined again according to the characteristics of the model. The relationship is exponential growth if the tensile strength can be expressed as the compressive strength, which is determined as follows:(9)ft=α1⋅fcα2,where ft is the tensile strength, MPa. α1andα2 are model parameters, which can be determined by fitting from the test results.α1 and α2 can be determined from fitting the test data with (9), which is shown in Table 5. The fitting curve of different FA content was drawn in Figure 8.Table 5
Model parameters for the estimation of the tensile strength and the elastic modulus.
Itemα1α2β1β2FA-150.150.927.10.3FA-250.140.9127.00.3FA-350.130.9426.50.33By summarizing the effect of FA content on the mechanical properties, these results suggest that the LHP cement concrete with different FA content attains a lower early-age speed of hydration rate but a higher later-age speed of hydration rate. With the increment of the FA content, the ratio between the tensile strength and the compressive strength shows smaller early-age values but larger later-age values.Only the tensile strength may be overestimated when the compressive strength is extremely small at an early age. This may be the reason why the compressive strength value is extremely small when the great ratio happens. To obtain a more accurate equation, more compression and tension tests at an earlier age should be carried out.


## 3.3. Estimation of the Elastic Modulus

Elastic modulus is invariably estimated from the measured compressive strength [17], especially in the design codes for predicting elastic modulus in terms of compressive strength. Models for estimating the elastic modulus in codes ACI 318–08 and CEB-FIP 2010 [39] were commonly used. Four typical models were collected as follows:Model developed by Kim [17]:(10)Ec=5.25⋅fc0.46.Model developed in JSCE [44]:(11)Ec=4.7⋅fc.Model developed in ACI-318-08 [45]:(12)Ec=22⋅fc100.3.Model developed in CEB-FIP 2010 [39]:(13)Ec=21.5×fc+Δf101/3.Ec is the elastic modulus (kN/mm2) and fc is the compressive strength (MPa), Δf=8 MPa.Figure9 shows the elastic modulus from the test data and the theoretical values from equations (11) to (14). The developed models in [17, 39, 44] all underestimate the elastic modulus. Compared with normal strength concrete, LHP cement concrete has a lower hydration degree and smaller early-age strength [21]. Although the compressive strength of LHP cement concrete may have a lower growth rate in the early stage, it may surpass the ultimate strength of normal strength concrete. Therefore, the elastic modulus may be expressed by the compressive strength:(14)Ec=β1⋅fc10β2,where β1 and β2 are model parameters, which can be determined by regression analysis from test data.Figure 9
Comparison of the elastic modulus models and fitting results.Referring to the previous models from equations (11) to (14), as shown in Figure 9, β1 and β2 can be determined by fitting the test data with (14), which is shown in Table 5. The fitting curve of different FA content was drawn in Figure 9. The FA content of 15% and 25% has a similar curve when the FA content of 35% attains a lower early-age elastic modulus but higher later-age elastic modulus.Compared with the models in [17, 39, 44], Kim [17] only investigated the effect of one kind of cement concrete containing 15% FA on the mechanical properties. The results from Kim [17] demonstrate that adding fly ash reduces strength growth rate and has little effect on the elastic modulus. The measured compressive strengths used in the Kim [17] and JSCE [44] models are based on the experimental data, while the specified compressive strength is applied in ACI 318–08 and the characteristic compressive strength is used in CEB-FIP 2010 [39]. All four developed theoretical models underestimate the elastic modulus.


## 3.4. Verification for an Estimation of Tensile Strength and Elastic Modulus

### 3.4.1. Tensile Strength

To verify the proposed equation of the tensile strength, (1) and (9) are combined to estimate the tensile strength for FA-35 concrete; see Figure 10. The maximum difference in the different regression analysis is less than 5%. This suggests that the proposed (1) and (9) can be applied to predict the tensile strength of LHP cement concrete. In addition, the tensile strength development can be predicted from a measured or a given compressive strength at 28 days.Figure 10
Comparison of the proposed model and test results in tensile strength.


### 3.4.2. Elastic Modulus

Figure11 shows two prediction methods for the elastic modulus. Excepted for the test method combined with (1) and (14), it can also predict the elastic modulus.Figure 11
Comparison of the proposed model and test results in elastic modulus.Compared with the test results and the calculation value of (1) and (14), the difference in elastic modulus is less than 3%. Therefore, it is rational to assume that the compressive strength can be applied to predict the elastic modulus. If the compressive strength development can be accurately predicted, the estimation of the elastic modulus is greatly simplified.


## 3.4.1. Tensile Strength

To verify the proposed equation of the tensile strength, (1) and (9) are combined to estimate the tensile strength for FA-35 concrete; see Figure 10. The maximum difference in the different regression analysis is less than 5%. This suggests that the proposed (1) and (9) can be applied to predict the tensile strength of LHP cement concrete. In addition, the tensile strength development can be predicted from a measured or a given compressive strength at 28 days.Figure 10
Comparison of the proposed model and test results in tensile strength.


## 3.4.2. Elastic Modulus

Figure11 shows two prediction methods for the elastic modulus. Excepted for the test method combined with (1) and (14), it can also predict the elastic modulus.Figure 11
Comparison of the proposed model and test results in elastic modulus.Compared with the test results and the calculation value of (1) and (14), the difference in elastic modulus is less than 3%. Therefore, it is rational to assume that the compressive strength can be applied to predict the elastic modulus. If the compressive strength development can be accurately predicted, the estimation of the elastic modulus is greatly simplified.


## 3.5. Thermal Cracks Analysis in LHP Cement Concrete

### 3.5.1. Temperature History

The temperature history is shown in Figure12. The maximum temperature of LHP cement concrete for FA-35 is 32°C. To investigate the cracking resistance of LHP cement concrete, thermal stresses at different ages with the same temperature development were measured. As seen in Figure 12, T0 presents the pouring temperature or initial temperature, T1 is the temperature at the maximum compressive stress, ΔT1 is the temperature difference from T0 to T1, T2 is the maximum temperature through the whole temperature, Tc is the cracking temperature while the concrete specimen fails, and ΔT2 is the temperature difference from Tc to T2.Figure 12
Temperature history under semiadiabatic conditions.


### 3.5.2. Thermal Stresses at Different Ages

Figure13 shows the thermal stress development at ages 0.417, 3, 7, and 20 days with FA-35. All specimens were cured at 20°C before the TSTM test. The same temperature history is applied to the concrete specimen at different ages. The test results show that the maximum compressive stress at the temperature rise stage is −0.33, −1.00, −1.07, and −1.91 MPa when the loading time Δt of LHP cement concrete is 1.95, 1.96, 2.07, and 1.97 days, respectively, as shown in Table 6. The maximum compressive stresses have reached before the maximum temperature throughout the whole temperature history. The cracking stress is 0.65, 0.92, 1.06, and 1.04 MPa at loading time Δt = 5.86, 7.56, 7.97, and 7.10 days, respectively. This reveals that the cracking resistance ability of the FA-35 specimen increases with the increment of loading age from 0.417 to 7 days and starts to reduce from 7 to 20 days. The tensile strength from (2) is 1.06, 1.66, 2.08, and 2.80 MPa when the elastic modulus from (4) is 26.3, 30.57, 33.07, and 36.5 GPa at loading time Δt = 5.86, 7.56, 7.97, and 7.10 days, respectively. Compared with the first specimen at a loading age of 0.417 days, the tensile strength of the later three specimens increases 56.6%, 96.2%, and 164.2% when the elastic modulus increases 16.23%, 25.74%, and 38.78%, respectively. This reveals that the growth rate of elastic modulus could not catch up with that of tensile strength 7 days later. In other words, the thermal stress increases very small at the same temperature difference, but the tensile strength increases surely when the curing age exceeds 7 days. This suggests that the later cooling behaves better with cracking resistance.Figure 13
Restrained stress development at different ages.Table 6
Test data for thermal cracks analysis in LHP cement concrete.
Loading ages (day)Δt (day)T1 (°C)Tc (°C)ΔT1 (°C)ΔT2 (°C)Maximum compressive stress (MPa)Cracking stress (MPa)Elastic modulus (GPa)0.4171.9531.3411.34−0.3316.825.8618.5013.500.6526.331.9631.3811.38−1.0024.37.5613.2618.740.9230.5772.0731.8111.81−1.0729.47.9712.1419.861.0633.02201.9731.4311.43−1.9133.387.1014.5817.421.0436.5


### 3.5.3. Effect of Maximum Compressive Stress on the Cracking Stress

To investigate the effect of the maximum compressive stresses on the cracking stress, the maximum compressive stresses in Figure14 are connected and drawn as a curve. The cracking stress ratio (the cracking stress/the tensile strength) and the compressive stress ratio (the maximum compressive stress/the compressive strength) are also seen in Figure 14. The test data suggest that the compressive stress has an extreme effect on the cracking stresses in the LHP cement concrete. The cracking stress ratio is 0.61, 0.55, 0.50, and 0.37 when the maximum compressive stress is −0.33, −1.00, −1.07, and −1.91 MPa, respectively. The compressive stress ratio is 0.33, 0.19, 0.13, and 0.09 when the maximum compressive stress is compared with the compressive strength at the corresponding time. Obviously, the cracking stress ratio decreased with the increment of the maximum compressive stress, but the compressive stress ratio was so small that the compressive stress was always neglected in the past.Figure 14
Cracking stress ratios at different ages.Although there is a zero-stress point between the maximum compressive stress and the cracking stress, the whole temperature history is continuous. If the whole stress development is divided into two parts, one is the compression stage while the other is the tension stage. The influence of compressive stress on tensile stress will be ignored. From the test results analysis, it is found that the maximum compressive stress has an influence on the cracking stress. Therefore, except that the cooling rate should be controlled, the heating rate should also be controlled to avoid small cracking stress due to the great compressive stress.


## 3.5.1. Temperature History

The temperature history is shown in Figure12. The maximum temperature of LHP cement concrete for FA-35 is 32°C. To investigate the cracking resistance of LHP cement concrete, thermal stresses at different ages with the same temperature development were measured. As seen in Figure 12, T0 presents the pouring temperature or initial temperature, T1 is the temperature at the maximum compressive stress, ΔT1 is the temperature difference from T0 to T1, T2 is the maximum temperature through the whole temperature, Tc is the cracking temperature while the concrete specimen fails, and ΔT2 is the temperature difference from Tc to T2.Figure 12
Temperature history under semiadiabatic conditions.


## 3.5.2. Thermal Stresses at Different Ages

Figure13 shows the thermal stress development at ages 0.417, 3, 7, and 20 days with FA-35. All specimens were cured at 20°C before the TSTM test. The same temperature history is applied to the concrete specimen at different ages. The test results show that the maximum compressive stress at the temperature rise stage is −0.33, −1.00, −1.07, and −1.91 MPa when the loading time Δt of LHP cement concrete is 1.95, 1.96, 2.07, and 1.97 days, respectively, as shown in Table 6. The maximum compressive stresses have reached before the maximum temperature throughout the whole temperature history. The cracking stress is 0.65, 0.92, 1.06, and 1.04 MPa at loading time Δt = 5.86, 7.56, 7.97, and 7.10 days, respectively. This reveals that the cracking resistance ability of the FA-35 specimen increases with the increment of loading age from 0.417 to 7 days and starts to reduce from 7 to 20 days. The tensile strength from (2) is 1.06, 1.66, 2.08, and 2.80 MPa when the elastic modulus from (4) is 26.3, 30.57, 33.07, and 36.5 GPa at loading time Δt = 5.86, 7.56, 7.97, and 7.10 days, respectively. Compared with the first specimen at a loading age of 0.417 days, the tensile strength of the later three specimens increases 56.6%, 96.2%, and 164.2% when the elastic modulus increases 16.23%, 25.74%, and 38.78%, respectively. This reveals that the growth rate of elastic modulus could not catch up with that of tensile strength 7 days later. In other words, the thermal stress increases very small at the same temperature difference, but the tensile strength increases surely when the curing age exceeds 7 days. This suggests that the later cooling behaves better with cracking resistance.Figure 13
Restrained stress development at different ages.Table 6
Test data for thermal cracks analysis in LHP cement concrete.
Loading ages (day)Δt (day)T1 (°C)Tc (°C)ΔT1 (°C)ΔT2 (°C)Maximum compressive stress (MPa)Cracking stress (MPa)Elastic modulus (GPa)0.4171.9531.3411.34−0.3316.825.8618.5013.500.6526.331.9631.3811.38−1.0024.37.5613.2618.740.9230.5772.0731.8111.81−1.0729.47.9712.1419.861.0633.02201.9731.4311.43−1.9133.387.1014.5817.421.0436.5


## 3.5.3. Effect of Maximum Compressive Stress on the Cracking Stress

To investigate the effect of the maximum compressive stresses on the cracking stress, the maximum compressive stresses in Figure14 are connected and drawn as a curve. The cracking stress ratio (the cracking stress/the tensile strength) and the compressive stress ratio (the maximum compressive stress/the compressive strength) are also seen in Figure 14. The test data suggest that the compressive stress has an extreme effect on the cracking stresses in the LHP cement concrete. The cracking stress ratio is 0.61, 0.55, 0.50, and 0.37 when the maximum compressive stress is −0.33, −1.00, −1.07, and −1.91 MPa, respectively. The compressive stress ratio is 0.33, 0.19, 0.13, and 0.09 when the maximum compressive stress is compared with the compressive strength at the corresponding time. Obviously, the cracking stress ratio decreased with the increment of the maximum compressive stress, but the compressive stress ratio was so small that the compressive stress was always neglected in the past.Figure 14
Cracking stress ratios at different ages.Although there is a zero-stress point between the maximum compressive stress and the cracking stress, the whole temperature history is continuous. If the whole stress development is divided into two parts, one is the compression stage while the other is the tension stage. The influence of compressive stress on tensile stress will be ignored. From the test results analysis, it is found that the maximum compressive stress has an influence on the cracking stress. Therefore, except that the cooling rate should be controlled, the heating rate should also be controlled to avoid small cracking stress due to the great compressive stress.


## 3.6. A Simple Method for Predicting the Cracking Temperature Difference

The thermal stress can be approximately calculated as follows:(15)σt=RKpEctαΔT,where s is the thermal stress, R is the restraint coefficient, Kp is the stress relaxation coefficient, a is the coefficient of linear expansion when it is 7.5×10−6με/°C for the LHP cement concrete, and ΔT is the temperature difference.As the cracking stress ratio is 0.61, 0.55, 0.50, and 0.37 in the TSTM test, to prevent the thermal cracks of concrete, the thermal stress ratio should not exceed the average value of 0.5. The thermal stress should not exceed the allowable tensile stress [47].(16)σt≤0.5⋅ftt.As the concrete specimen is fully restrained,R = 1, the stress relaxation coefficient can be simplified as 0.2, while the elastic modulus and the tensile strength can be estimated by (9) and (14) with the model parameters in Table 5. The cracking temperature difference can be predicted as follows:(17)ΔT≤0.026⋅fct0.61α.The cracking temperature differences of the FA-35 concrete specimens at 6.27, 10.56, 14.97, and 27.10 days were 11.8, 16.4, 19.4, and 24.3°C, calculated by (17). The test results were 13.50, 18.7, 19.86, and 17.42°C. The deviations between results obtained from (17) and test data were 1.7, 2.3, 0.46, and 6.88°C, respectively. Compared with the test results, the deviations were small. Furthermore, the cracking risk of concrete for long periods can also be expressed by the compressive strength. For instance, the values of permissible temperature differences at 180, 360, 720, and 1000 days could be obtained from (17) were 35.8, 38.5, 40.5, and 41.3°C. Compared with the value of permissible temperature difference at 180 days, the permissible temperature differences at 360, 720, and 1000 days increased by 2.7, 4.7, and 5.5°C, respectively. Therefore, (17) could be utilized to predict the cracking risk of LHP cement concrete. However, the influence of compressive stress on the cracking risk was not considered in the present study. Further investigations of the influence of compressive stress, especially larger than 2 MPa, on the cracking risk at early age are necessary. Although the proposed model for the tensile strength, the elastic modulus, and the prediction equations of cracking risk were arbitrary due to the limited data, the cracking risk of concrete structures demonstrated by the test data could be illustrated directly.


## 4. Conclusions

This article mainly studied the mechanical properties of one concrete mix with three different FA contents. Both the tensile strength and the elastic modulus expressed by the compressive strength and the cracking resistance of the LHP cement concrete predicted by the proposed equations were investigated. The following conclusions can be drawn as follows:(1)
The test results of compressive strength, tensile strength, and elastic modulus at 28 days show that the FA content has negative influences on the compressive strength and the tensile strength, and the elastic modulus decreases slightly with the increment of FA content.(2)
The prediction formulas for the compressive strength, tensile strength, and elastic modulus were established according to the test results. Furthermore, both the tensile strength and the elastic modulus can be indicated by the compressive strength, respectively.(3)
In the TSTM test, the maximum compressive stresses and cracking stresses increase with the increment of loading age, but the increment of the cracking stress is so small that the cracking stress ratios decrease with the increment of the maximum compressive stress. Therefore, too rapid increment of temperature should be avoided in the temperature control design.(4)
Considering the equations of tensile strength and elastic modulus predicted by compressive strength, a model for predicting the cracking temperature difference of LHP cement concrete was proposed.Notably, only 35% FA replacement was used to predict the cracking risk of the LHP cement concrete in this article. Further research is needed to investigate the influence of the amount of FA replacement on the cracking risk of the LHP cement concrete. Moreover, the formula parameters were obtained from limited test data. Therefore, further experiments should be done to verify the proposed model of the LHP cement concrete after 28 days.



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*Source: 7630568-2022-04-16.xml*