# Design Method for Concrete Mix Proportion Based on Indices of Strength and Durability of Concrete Exposed to Chloride Attack

Authors: Jiaqi Zhao; Wanting Sun; Tuo Shi; Chenghui Qin
Journal: Advances in Materials Science and Engineering
 (2023)

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## Abstract

In the current design code, concrete mix proportions are not based on durability parameters that can quantitatively design the requirement of service life for chloride-exposed structures. Multifactor calculation models of durability index are developed, and the replacement of mineral admixtures in concrete is determined according to predetermined concrete strength and durability requirements. The chloride diffusion coefficient and aging factor of concrete are chosen as index parameters for the resistance of concrete to chloride salts. A multifactor model of the chloride diffusion coefficient of concrete is then established by regression analysis based on the experimental data, and explicit expressions for the water-to-binder ratio and replacement of admixture were determined by solving the multivariate nonlinear equation system. By calculating the average compressive strength and chloride diffusion coefficient from their specified values, the mix proportion of concrete is determined. The superiority of the proposed method is demonstrated by the comparison with the current code. Lastly, an orthogonal experiment was conducted to demonstrate the validity of the proposed method for designing concrete mix proportions by referencing engineering practices. The results show that major parameters of concrete mix proportion can be determined quantitatively from structural design parameters such as compressive strength, chloride diffusion coefficient, and aging factor, satisfying the requirements of ultimate bearing capacity and service life for concrete structures in chloride environments, as demonstrated in this paper.


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## Body

## 1. Introduction

In chloride environments, steel bars embedded in reinforced concrete structures suffer corrosion resulting in shorter service lives. Today, mineral admixtures, such as fly ash and ground granulated blast furnace slag (GGBFS), are widely used as cementreplacements in high-performance concrete (HPC) for their economical and durable properties. The superior durability performance of mineral admixtures is due to microaggregate filling and pozzolanic effects, which reduce overall concrete pore volume and modify pore size distribution [1].It is necessary to reasonably determine the appropriate content of mineral admixtures in HPC through its mix ratio design in order to achieve the necessary strength and durability. However, in most of the situation, the content of mineral admixtures in mix ratios is determined by experience or experiments [2–7].A multifactor model described the relationship between mix proportion parameters and concrete properties such as strength, workability, and frost durability was developed by Wang et al. [8], Lim et al. [9], and Lee et al. [10]. By using an optimization algorithm, the content of mineral admixtures can be solved quantitatively. Ji et al. [11], Khan [12], and Yeh [13] established a nonlinear relationship between strength, workability, and mineral admixtures, by using artificial neural networks. Note that the above quantitative analysis methods don’t take into account the influence of mineral admixtures on resistance to chloride ion penetration. Based on linear regression analysis, Ozbay et al. [14] established a multifactor model, which described the relation between mix proportion parameters and transmission performance index parameters including charge passed, water absorption and so on, and then used multiobjective optimization to determine the parameters of concrete mix ratio, including mineral admixtures replacement. However, due to the numerous parameters included in the previous models, this method cannot provide an explicit expression for determining mineral admixture replacement, so it is not suitable for engineering applications.This paper deduces explicit expression of mineral admixture content in the concrete mix based on the multifactor models of concrete strength and chlorine resistance performance index parameters in view of the extreme importance of concrete strength and chlorine resistance performance index to concrete structure in a chloride environment. Based on practical projects, 16 concrete mixtures were designed using the orthogonal experimental method and tested. The accuracy and applicability of the proposed method were validated.


## 2. Relation between Mineral Admixtures and Concrete Properties

Fly ash and GGBFS, which are widely used as mineral admixtures, have a significant influence on compressive strength [15, 16] and resistance to chloride penetration [17].


### 2.1. Computational Model for Concrete Compressive Strength

Standards of Ministry of Construction in China JGJ55-2011 proposed a model of 28-day compressive strength for concrete as follows:(1)fcu,0=αa⋅fb1RW/B−αb,where RW/B is the water-to-binder ratio, αa and αb are the empirical coefficients; for crushed limestone, the values of αa and αb are 0.53 and 0.20 respectively; for river gravel, the values of αa and αb are 0.49 and 0.13, respectively; fb is the actual strength of the cementitious matrix, which can be described as follows:(2)fb=γsγffce=γsγfγcfce,g,where fce,g is the strength grade value of cement, fce is the actual strength of cement, γc is the spare coefficient, which can be either determined by historical data or equal to 1.16 and 1.10, respectively for 42.5 and 52.5 cement grades, and γf and γs are the influence coefficients of fly ash and GGBFS, which can be described as follows:(3)γf=1−RFA,γs=1+0.14RSG−1.2RSG2,where RFA is the replacement of fly ash, and RSG is the replacement of GGBFS.


### 2.2. Computational Model for Aging Factor

Luping and Gulikers [18] investigated that the instantaneous chloride diffusion coefficient decreased with the age of concrete and can be represented by a power function as follows:(4)Dt′=D0t0t′n,where Dt′ is the instantaneous chloride diffusion coefficient at the concrete age t′, t0 is the age of concrete initially exposed to chloride environment, which is usually set to be 28 d, D0 is the instantaneous chloride diffusion coefficient (m2/s) at t0, which is usually called initial diffusion coefficient, and n is the aging factor of instantaneous chloride diffusion coefficient. Mangat and Molloy [19] investigated that the average chloride diffusion coefficient also decreased with the age of concrete and can be represented by a power function as follows:(5)Dat′=D0t0t′m,where Dat′ is the average chloride diffusion coefficient at the concrete age t′, and m is the aging factor of average chloride diffusion coefficient.According to the equations (4) and (5), it is observed that both instantaneous chloride diffusion coefficient and average chloride diffusion coefficient are closely associated with diffusion coefficient at t0 and aging factor. Hence, in this study, initial diffusion coefficient D0 and aging factor n are adopted as durability indices. Chalee et al. [20] proposed a model to describe the relationship between n and the water-to-binder ratio, as well as the replacement of fly ash, which was based on field test data in a marine tidal zone of the Gulf of Thailand. However, it ignores the influence of the replacement of GGBFS on n. The relationship can be expressed as follows:(6)n=−0.15RW/B+0.34⋅RFA−0.175RW/B+0.840.Jin and Jin [21] established a model for m in terms of water-to-binder ratio, replacement of fly ash, and replacement of GGBFS, which was based on indoor accelerated test data. The relationship can be expressed as follows:(7)n=0.80×RW/B+0.35×RFA0.50+RSG0.70,Atmosphericzone,0.80×RW/B−0.14+0.35×RFA0.50+RSG0.70,Dry−wetcyclingzone,0.80×RW/B−0.14,HPCinsubmergedzone,OPCinsubmergedzone.Nevertheless, it is not applicable for the long term field test, because the test period of the indoor accelerated test is too short to simulate the time dependency of chloride diffusivity in the long term field test. DuraCrete [22] provided the characteristic values of the aging factor for different types of mineral admixtures and environmental conditions. The characteristic values of the aging factor are listed in Table 1. However, due to the lack of reliable data on long-term exposure, some values of m are based on experience only, without taking into account the influence of replacement of mineral admixture.Table 1
The characteristic values of the aging factor.
Concrete typeExposure zoneCharacteristicOPCSubmerged0.30OPCTidal and splash0.37OPCAtmospheric0.65FASubmerged0.69FATidal and splash0.93FAAtmospheric0.66SGSubmerged0.71SGTidal and splash0.60SGAtmospheric0.85Life 365 [23] proposed a model (Life 365 model) to describe the relationship between m and the replacement of fly ash, as well as the replacement of GGBFS, which was based on long-term field test data from different regions in the marine environment. Their relationship can be expressed as follows:(8)m=0.2+0.8RFA+0.714RSG.It should be noted that the Life 365 model is superior to other models in the aspect of the accuracy and applicability. However, it is necessary to convertm in equation (8) into n, because n is adopted as one of the durability indices in this study. Frederiksen et al. [24] established the relationship between n and m, which can be described as follows:(9)n=m+ln1−m+mt0/t′lnt0/t′.Take the seaport wharf concrete structure as an example, the design service life is 50 a, and the age of concrete initially exposed to chloride environment is 0.077 a. Moreover, lett′ and t0 set to be 50 a and 0.077 a, respectively, equation (9) can be rewritten as follows:(10)n=m−0.154×ln1−0.998×m.It is inconvenient for engineering applications to use equation (10), since the functional form is complicated. Hence, the approximately linearized method is adopted to simplify equation (10) in this study.The solution procedure can be summarized as follows:(1)
Sample selection. The values ofm range from 0.20 to 0.60 in increments of 0.01, and then the corresponding values of n can be obtained according to equation (10).(2)
Based on the least square method, a linear regression model is established to describe the relationship betweenn and m as follows:(11)n=1.263m−0.023.The coefficient of determination (R2) is used for judging the goodness of fit of the linear regression model, and R2 is 0.9996.As shown in Figure1, the curves simulated with equations (10) and (11) agree well with each other when the value of m ranges from 0.2 to 0.6. The obtained results show that the approximately linearized method is effective.Figure 1
Relation betweenn and m.By substituting equation (9) into equation (12), one gets(12)n=0.229+1.011RFA+0.722RSG.Finally, equation (12) is adopted as the computational model for the aging factor.


### 2.3. Computational Model for Chloride Diffusion Coefficient

Bogas and Gomes [25], van Noort et al. [26], Yu and Ye. [27], Real et al. [28], Lollini et al. [29], Zhao et al. [30], Liu et al. [31], and Özbay et al. [32] found that the initial diffusion coefficient is affected by the water-to-binder ratio and replacement of mineral admixtures. Also, the initial diffusion coefficient increases linearly with the water-to-binder ratio and decreases linearly with the replacement of mineral admixtures. Breugel and Polder [33] proposed a model for the initial diffusion coefficient as a function of the water-to-binder ratio. However, it ignores the influence of the replacement of mineral admixtures on the initial diffusion coefficient. The relationship can be expressed as follows:(13)D0=A⋅Rw/b+B,where A and B are the constants for particular cement types.Persson [34] proposed a model for the chloride diffusion coefficient by taking into account the water-to-binder ratio, content of mineral admixtures, content of water, and curing time. The relationship can be expressed as follows:(14)D=0.0055lnt−0.2122⋅ceq−3.5lnt+104⋅4RW/B−1.20.4,ceq=c+0.2⋅cFA+1.6⋅cSF+1.0⋅cSG,where c is the cement content (kg/m3), ceq is the equivalent cement content (kg/m3), t is the curing time (month), cFA is the content of fly ash (kg/m3), cSF is the content of silica fume (kg/m3), and cSG is the content of GGBFS (kg/m3).The initial diffusion coefficient can be obtained when the curing time is equal to 28 d. However, it ignores the coupling effect of the water-to-binder ratio and the content of mineral admixtures. Hence, an improved computational model was proposed by taking into account the influence of the water-to-binder ratio, replacement of mineral admixtures, and the coupling effect:(15)D0X,a=a0RW/B+a1a2RFA+a3FA,a0RW/B+a1a2RSG+a3SG,a0RW/B+a1RFA+a2RSG+a3FA&SG,(16)a=a0,a1,a2,a3T,X=RW/B,RFA,RSGT,where a is the vector of the fitting coefficient and X is the vector of the materials parameter.The individual laboratory results for the chloride diffusion coefficient test are inevitably affected by environmental conditions, material properties, and experimental equipment. In order to get around these limitations, 161 experimental data of initial diffusion coefficients from different laboratories in different regions have been collected [35], and Table 2 lists the basic information of these data.Table 2
General situation of RCM test database.
Concrete typeAmount of samplesRW/BRFARSGD0FA610.25∼0.620.10∼0.4002.20∼15.50SG410.29∼0.5500.10∼0.702.40∼14.60FA&SG590.27∼0.510.05∼0.400.05∼0.551.60∼8.50Data collection should follow the following principles: (1) tests were performed using the RCM test, described in NT BUILD 492 or DuraCrete, (2) concrete was made with Ordinary Portland Cement, Class F fly ash, and Grade 95 GGBFS, and (3) all specimens were stored in a standard curing room until 28 days after pouring. Li et al. [36] found that the tested value from variable voltage RCM is 1.3 times the tested value from constant voltage RCM. Based on this relationship, all the tested values from variable voltage RCM should be transformed into tested values from constant voltage RCM.Moreover, the tested values of RCM can be described as follows:(17)D0=D0,1,D0,2,…,D0,nT,where D0,i is the ith tested value of the initial diffusion coefficient for the specimen.Xi is the ith vector of the materials parameter, and D0(Xi, a) is the ith predicted value of initial diffusion coefficient for specimen. We expand the D0(Xi, a) in a first-order Taylor series about a0 as follows:(18)D0a=f0+J0a−a0+e,where J0 is Jacobian matrix, which is the first partial derivatives of the function D0 with respect to a at a0; e is the error vector, which consists of modeling error and truncation error.The sum of squared residuals of all data can be defined as follows:(19)Π=∑i=1nD0Xi,a−D0,i2.According to the method of least squares, the optimal estimatesa can be obtained by minimizing the sum of squared residuals П, as listed in Table 3.Table 3
The optimal estimation fora.
Concrete typeAFA−41.105,−0.186,1,−1.104TSG−46.753,−0.189,1,−1.003TFA&SG−9.350,−0.113,3.641,−2.931TSubstituting the value ofa into equation (17), one gets the computational model for initial diffusion coefficient by taking into account the water-to-binder ratio, replacement of fly ash, and replacement of GGBFS as follows:(20)D0=−41.105RW/B−0.186RFA−1.104,FA,−46.753RW/B−0.189RSG−1.003,SG,−9.350RW/B−0.113RFA+3.641RSG−2.931,FA&SG.Several computational models of the initial diffusion coefficient were selected to validate the accuracy and applicability of the proposed model.As shown in Figure2, the data were collected in order to compare the accuracy and applicability among the reference model [33], the reference model [34], and the proposed model. It is clear that a large number of predicted values obtained by the two models [33, 34] are located previously the equality line, which indicates that predicted values are significantly larger than tested values. Furthermore, the predicted values obtained by the proposed model are not only much closer to the equality line than those of the other two models. In addition, they are uniformly distributed on both sides of the equality line, which indicates that the accuracy of the proposed model is satisfactory. The ratio of predicted value to tested value, as well as the mean value and standard deviation, can be defined as follows:(21)Ri=D0XiD0,i,μR=1n∑i=1nRi,σR=1n−1∑i=1nRi−μR.Figure 2
Comparison of proposed models with experimental data.The values forμR and σR are listed in Table 4. Clearly, for the two models [33, 34], the values for μR are much larger than 1, and the values for σR are larger than the proposed model, which indicates that the predicted values are far away from tested values, and the dispersal of the two models are much larger than the proposed model. For the proposed model, the values for μR are much closer to 1, which indicates that the predicted values are much closer to tested values, and the dispersal of the proposed is the smallest, thus demonstrating the high accuracy of the proposed model. This is because the model [33] only takes into account the water-to-binder ratio and ignores the significant influence of the replacement of mineral admixtures, whereas the model [34] ignores the importance of coupling the relationship between the water-to-binder ratio and replacement of mineral admixtures. In contrast, the proposed model takes into account the water-to-binder ratio, replacement of mineral admixtures, and the coupling relationship between water-to-binder ratio and replacement of mineral admixtures. Consequently, the proposed model is of higher accuracy.Table 4
Comparison of the precision of models.
ModelμRσRProposed model1.041.22Reference [33] model1.293.63Reference [34] model2.6560.21


## 2.1. Computational Model for Concrete Compressive Strength

Standards of Ministry of Construction in China JGJ55-2011 proposed a model of 28-day compressive strength for concrete as follows:(1)fcu,0=αa⋅fb1RW/B−αb,where RW/B is the water-to-binder ratio, αa and αb are the empirical coefficients; for crushed limestone, the values of αa and αb are 0.53 and 0.20 respectively; for river gravel, the values of αa and αb are 0.49 and 0.13, respectively; fb is the actual strength of the cementitious matrix, which can be described as follows:(2)fb=γsγffce=γsγfγcfce,g,where fce,g is the strength grade value of cement, fce is the actual strength of cement, γc is the spare coefficient, which can be either determined by historical data or equal to 1.16 and 1.10, respectively for 42.5 and 52.5 cement grades, and γf and γs are the influence coefficients of fly ash and GGBFS, which can be described as follows:(3)γf=1−RFA,γs=1+0.14RSG−1.2RSG2,where RFA is the replacement of fly ash, and RSG is the replacement of GGBFS.


## 2.2. Computational Model for Aging Factor

Luping and Gulikers [18] investigated that the instantaneous chloride diffusion coefficient decreased with the age of concrete and can be represented by a power function as follows:(4)Dt′=D0t0t′n,where Dt′ is the instantaneous chloride diffusion coefficient at the concrete age t′, t0 is the age of concrete initially exposed to chloride environment, which is usually set to be 28 d, D0 is the instantaneous chloride diffusion coefficient (m2/s) at t0, which is usually called initial diffusion coefficient, and n is the aging factor of instantaneous chloride diffusion coefficient. Mangat and Molloy [19] investigated that the average chloride diffusion coefficient also decreased with the age of concrete and can be represented by a power function as follows:(5)Dat′=D0t0t′m,where Dat′ is the average chloride diffusion coefficient at the concrete age t′, and m is the aging factor of average chloride diffusion coefficient.According to the equations (4) and (5), it is observed that both instantaneous chloride diffusion coefficient and average chloride diffusion coefficient are closely associated with diffusion coefficient at t0 and aging factor. Hence, in this study, initial diffusion coefficient D0 and aging factor n are adopted as durability indices. Chalee et al. [20] proposed a model to describe the relationship between n and the water-to-binder ratio, as well as the replacement of fly ash, which was based on field test data in a marine tidal zone of the Gulf of Thailand. However, it ignores the influence of the replacement of GGBFS on n. The relationship can be expressed as follows:(6)n=−0.15RW/B+0.34⋅RFA−0.175RW/B+0.840.Jin and Jin [21] established a model for m in terms of water-to-binder ratio, replacement of fly ash, and replacement of GGBFS, which was based on indoor accelerated test data. The relationship can be expressed as follows:(7)n=0.80×RW/B+0.35×RFA0.50+RSG0.70,Atmosphericzone,0.80×RW/B−0.14+0.35×RFA0.50+RSG0.70,Dry−wetcyclingzone,0.80×RW/B−0.14,HPCinsubmergedzone,OPCinsubmergedzone.Nevertheless, it is not applicable for the long term field test, because the test period of the indoor accelerated test is too short to simulate the time dependency of chloride diffusivity in the long term field test. DuraCrete [22] provided the characteristic values of the aging factor for different types of mineral admixtures and environmental conditions. The characteristic values of the aging factor are listed in Table 1. However, due to the lack of reliable data on long-term exposure, some values of m are based on experience only, without taking into account the influence of replacement of mineral admixture.Table 1
The characteristic values of the aging factor.
Concrete typeExposure zoneCharacteristicOPCSubmerged0.30OPCTidal and splash0.37OPCAtmospheric0.65FASubmerged0.69FATidal and splash0.93FAAtmospheric0.66SGSubmerged0.71SGTidal and splash0.60SGAtmospheric0.85Life 365 [23] proposed a model (Life 365 model) to describe the relationship between m and the replacement of fly ash, as well as the replacement of GGBFS, which was based on long-term field test data from different regions in the marine environment. Their relationship can be expressed as follows:(8)m=0.2+0.8RFA+0.714RSG.It should be noted that the Life 365 model is superior to other models in the aspect of the accuracy and applicability. However, it is necessary to convertm in equation (8) into n, because n is adopted as one of the durability indices in this study. Frederiksen et al. [24] established the relationship between n and m, which can be described as follows:(9)n=m+ln1−m+mt0/t′lnt0/t′.Take the seaport wharf concrete structure as an example, the design service life is 50 a, and the age of concrete initially exposed to chloride environment is 0.077 a. Moreover, lett′ and t0 set to be 50 a and 0.077 a, respectively, equation (9) can be rewritten as follows:(10)n=m−0.154×ln1−0.998×m.It is inconvenient for engineering applications to use equation (10), since the functional form is complicated. Hence, the approximately linearized method is adopted to simplify equation (10) in this study.The solution procedure can be summarized as follows:(1)
Sample selection. The values ofm range from 0.20 to 0.60 in increments of 0.01, and then the corresponding values of n can be obtained according to equation (10).(2)
Based on the least square method, a linear regression model is established to describe the relationship betweenn and m as follows:(11)n=1.263m−0.023.The coefficient of determination (R2) is used for judging the goodness of fit of the linear regression model, and R2 is 0.9996.As shown in Figure1, the curves simulated with equations (10) and (11) agree well with each other when the value of m ranges from 0.2 to 0.6. The obtained results show that the approximately linearized method is effective.Figure 1
Relation betweenn and m.By substituting equation (9) into equation (12), one gets(12)n=0.229+1.011RFA+0.722RSG.Finally, equation (12) is adopted as the computational model for the aging factor.


## 2.3. Computational Model for Chloride Diffusion Coefficient

Bogas and Gomes [25], van Noort et al. [26], Yu and Ye. [27], Real et al. [28], Lollini et al. [29], Zhao et al. [30], Liu et al. [31], and Özbay et al. [32] found that the initial diffusion coefficient is affected by the water-to-binder ratio and replacement of mineral admixtures. Also, the initial diffusion coefficient increases linearly with the water-to-binder ratio and decreases linearly with the replacement of mineral admixtures. Breugel and Polder [33] proposed a model for the initial diffusion coefficient as a function of the water-to-binder ratio. However, it ignores the influence of the replacement of mineral admixtures on the initial diffusion coefficient. The relationship can be expressed as follows:(13)D0=A⋅Rw/b+B,where A and B are the constants for particular cement types.Persson [34] proposed a model for the chloride diffusion coefficient by taking into account the water-to-binder ratio, content of mineral admixtures, content of water, and curing time. The relationship can be expressed as follows:(14)D=0.0055lnt−0.2122⋅ceq−3.5lnt+104⋅4RW/B−1.20.4,ceq=c+0.2⋅cFA+1.6⋅cSF+1.0⋅cSG,where c is the cement content (kg/m3), ceq is the equivalent cement content (kg/m3), t is the curing time (month), cFA is the content of fly ash (kg/m3), cSF is the content of silica fume (kg/m3), and cSG is the content of GGBFS (kg/m3).The initial diffusion coefficient can be obtained when the curing time is equal to 28 d. However, it ignores the coupling effect of the water-to-binder ratio and the content of mineral admixtures. Hence, an improved computational model was proposed by taking into account the influence of the water-to-binder ratio, replacement of mineral admixtures, and the coupling effect:(15)D0X,a=a0RW/B+a1a2RFA+a3FA,a0RW/B+a1a2RSG+a3SG,a0RW/B+a1RFA+a2RSG+a3FA&SG,(16)a=a0,a1,a2,a3T,X=RW/B,RFA,RSGT,where a is the vector of the fitting coefficient and X is the vector of the materials parameter.The individual laboratory results for the chloride diffusion coefficient test are inevitably affected by environmental conditions, material properties, and experimental equipment. In order to get around these limitations, 161 experimental data of initial diffusion coefficients from different laboratories in different regions have been collected [35], and Table 2 lists the basic information of these data.Table 2
General situation of RCM test database.
Concrete typeAmount of samplesRW/BRFARSGD0FA610.25∼0.620.10∼0.4002.20∼15.50SG410.29∼0.5500.10∼0.702.40∼14.60FA&SG590.27∼0.510.05∼0.400.05∼0.551.60∼8.50Data collection should follow the following principles: (1) tests were performed using the RCM test, described in NT BUILD 492 or DuraCrete, (2) concrete was made with Ordinary Portland Cement, Class F fly ash, and Grade 95 GGBFS, and (3) all specimens were stored in a standard curing room until 28 days after pouring. Li et al. [36] found that the tested value from variable voltage RCM is 1.3 times the tested value from constant voltage RCM. Based on this relationship, all the tested values from variable voltage RCM should be transformed into tested values from constant voltage RCM.Moreover, the tested values of RCM can be described as follows:(17)D0=D0,1,D0,2,…,D0,nT,where D0,i is the ith tested value of the initial diffusion coefficient for the specimen.Xi is the ith vector of the materials parameter, and D0(Xi, a) is the ith predicted value of initial diffusion coefficient for specimen. We expand the D0(Xi, a) in a first-order Taylor series about a0 as follows:(18)D0a=f0+J0a−a0+e,where J0 is Jacobian matrix, which is the first partial derivatives of the function D0 with respect to a at a0; e is the error vector, which consists of modeling error and truncation error.The sum of squared residuals of all data can be defined as follows:(19)Π=∑i=1nD0Xi,a−D0,i2.According to the method of least squares, the optimal estimatesa can be obtained by minimizing the sum of squared residuals П, as listed in Table 3.Table 3
The optimal estimation fora.
Concrete typeAFA−41.105,−0.186,1,−1.104TSG−46.753,−0.189,1,−1.003TFA&SG−9.350,−0.113,3.641,−2.931TSubstituting the value ofa into equation (17), one gets the computational model for initial diffusion coefficient by taking into account the water-to-binder ratio, replacement of fly ash, and replacement of GGBFS as follows:(20)D0=−41.105RW/B−0.186RFA−1.104,FA,−46.753RW/B−0.189RSG−1.003,SG,−9.350RW/B−0.113RFA+3.641RSG−2.931,FA&SG.Several computational models of the initial diffusion coefficient were selected to validate the accuracy and applicability of the proposed model.As shown in Figure2, the data were collected in order to compare the accuracy and applicability among the reference model [33], the reference model [34], and the proposed model. It is clear that a large number of predicted values obtained by the two models [33, 34] are located previously the equality line, which indicates that predicted values are significantly larger than tested values. Furthermore, the predicted values obtained by the proposed model are not only much closer to the equality line than those of the other two models. In addition, they are uniformly distributed on both sides of the equality line, which indicates that the accuracy of the proposed model is satisfactory. The ratio of predicted value to tested value, as well as the mean value and standard deviation, can be defined as follows:(21)Ri=D0XiD0,i,μR=1n∑i=1nRi,σR=1n−1∑i=1nRi−μR.Figure 2
Comparison of proposed models with experimental data.The values forμR and σR are listed in Table 4. Clearly, for the two models [33, 34], the values for μR are much larger than 1, and the values for σR are larger than the proposed model, which indicates that the predicted values are far away from tested values, and the dispersal of the two models are much larger than the proposed model. For the proposed model, the values for μR are much closer to 1, which indicates that the predicted values are much closer to tested values, and the dispersal of the proposed is the smallest, thus demonstrating the high accuracy of the proposed model. This is because the model [33] only takes into account the water-to-binder ratio and ignores the significant influence of the replacement of mineral admixtures, whereas the model [34] ignores the importance of coupling the relationship between the water-to-binder ratio and replacement of mineral admixtures. In contrast, the proposed model takes into account the water-to-binder ratio, replacement of mineral admixtures, and the coupling relationship between water-to-binder ratio and replacement of mineral admixtures. Consequently, the proposed model is of higher accuracy.Table 4
Comparison of the precision of models.
ModelμRσRProposed model1.041.22Reference [33] model1.293.63Reference [34] model2.6560.21


## 3. Determination of Replacement of Mineral Admixtures

Solving the set of nonlinear equations where equations (1) and (20) are not linear and can be shown to be equivalent to solving the equation of higher degree. Therefore, the values of the water-to-binder ratio and replacement of mineral admixtures can be determined by solving the equation of higher degree.


### 3.1. Algorithm for Solving the Set of Nonlinear Equations

For concrete with ternary blends of cement, fly ash, and GGBFS, we get the quartic equation with regard toRFA by combining the equations (1), (12) and (20) as follows:(22)RFA4+e1RFA3+e2RFA2+e3RFA+e4=0.Solving equation (22) by Ferrari’s method, we obtain(23)RFA=−2p4−p1+−2p4−p1+2p2/2p4−p18−e14.According to equation (12), we have(24)RSG=−0.318+1.386n−1.401RFA.According to equation (20), we have(25)RW/B=D0−9.350RFA+3.641RSG−2.931+0.113,where(26)e1=4.7391−28.4107n+3.2104αbn−0.2302D0αb−0.5355αb/8.8496+αb,e2=−11.9795+1.3537αb+0.1084αbD0+0.4569αbnD0+0.3662n+0.0414αbn8.8496+αb+30.2313n2−3.4162αbn28.8496+αb,e3=0.2061D0αb+0.5357αb−3.0748αbn−28.6346n−0.3343D0αbn−15.7859n2+1.7838αbn28.8496+αb+−0.2263D0αbn2−10.6703n3+1.2058αbn3−6.03118.8496+αb−0.4252fcu,08.8496fceαa+fceαaαb,e4=3.1319−0.3539αb−0.0820D0αb+0.8345n−0.0943αbn−0.1229D0αbn+1.6324αbn23.0659+αb+−14.4454n2+10.6703n3+0.2263D0αbn2−1.2058αbn38.8496+αb+0.5235fcu,0n−0.4243fcu,0−0.0982fcu,0D08.8496fceαa+fceαaαb,p1=−6e12+16e2,p2=8e13−32e1e2+64e3,p3=−3e14+16e12e2−64e1e3+256e4,p4=16p1+43p3+p129⋅cos13arccosp2216−p1p36+p132163612p3+p123.


### 3.2. Concrete Performance Parameters

The values of concrete performance parameters are random and discrete since concrete is a heterogeneous composite material composed of mortar, randomly distributed aggregates, and microcracks. The specified values must be achieved with a predetermined probability in order to guarantee high-quality concrete. The specified compressive strength depends on the bearing capacity of the concrete structure. The specified chloride diffusion coefficient and its aging factor depend on the durability of concrete structures.When the specified compressive strengthfcu,k is less than 60 MPa, the required average compressive strength fcu,0 can be described as follows:(27)fcu,0=fcu,k+tσf,where t is a probability, σf is the standard deviation of compressive strength, fcu,k is the specified compressive strength, and fcu,0 is the required average compressive strength. When the confidence level is 95%, the value of t is equal to 1.645. When fcu,k≤20 MPa, the value of σf is equal to 4 MPa. When 25≤fcu,k≤45 MPa, the value of σf is equal to 5 MPa. When 50≤fcu,k≤55 MPa, the value of σf is equal to 6 MPa.When the specified compressive strength is not less than 60 MPa, the required average compressive strength can be described as follows:(28)fcu,0=1.15fcu,k.Similarly, the required average chloride diffusion coefficientD0 can be described as follows:(29)D0=D0,k−tσD,where σD is the standard deviation of diffusion coefficient the value of σD ranges from 0.15D0 to 0.30D0, which can be also determined by reference to historical data [37–39]. In this paper, the value of σD is taken as 0.30D0.


### 3.3. Concrete Performance Parameters

According to equations (23)–(25), the major mix proportion parameters, including water-to-binder ratio, replacement of fly ash, and GGBFS, can be calculated. The solution procedure can be summarized as follows:(1)
Based on the methodology of quantitative durability design for concrete structures [40, 41], first determine the chloride exposure class, the action of load and the intended service life, the specified compressive strength is determined by the ultimate bearing capacity requirement of concrete structures, and then the specified chloride diffusion coefficient and its aging factor are determined by durability design of concrete structures.(2)
According to the situation of local raw materials, the type of coarse aggregate and size gradation, types of mineral admixtures and their combinations, and chemical admixtures, are determined.(3)
According to equations (27)–(29), the required average compressive strength and the required average chloride diffusion coefficient can be calculated.(4)
According to equations (23)–(25), water-to-binder ratio and replacement of mineral admixtures can be calculated.The previous procedure is a quantitative analysis method for replacing mineral admixtures. Moreover, then the mixing water content and sand ratio are determined by the requirement of workability. Finally, the concrete mix proportion can be determined. The concrete prepared by using this method meets the requirements for strength and durability indices.


## 3.1. Algorithm for Solving the Set of Nonlinear Equations

For concrete with ternary blends of cement, fly ash, and GGBFS, we get the quartic equation with regard toRFA by combining the equations (1), (12) and (20) as follows:(22)RFA4+e1RFA3+e2RFA2+e3RFA+e4=0.Solving equation (22) by Ferrari’s method, we obtain(23)RFA=−2p4−p1+−2p4−p1+2p2/2p4−p18−e14.According to equation (12), we have(24)RSG=−0.318+1.386n−1.401RFA.According to equation (20), we have(25)RW/B=D0−9.350RFA+3.641RSG−2.931+0.113,where(26)e1=4.7391−28.4107n+3.2104αbn−0.2302D0αb−0.5355αb/8.8496+αb,e2=−11.9795+1.3537αb+0.1084αbD0+0.4569αbnD0+0.3662n+0.0414αbn8.8496+αb+30.2313n2−3.4162αbn28.8496+αb,e3=0.2061D0αb+0.5357αb−3.0748αbn−28.6346n−0.3343D0αbn−15.7859n2+1.7838αbn28.8496+αb+−0.2263D0αbn2−10.6703n3+1.2058αbn3−6.03118.8496+αb−0.4252fcu,08.8496fceαa+fceαaαb,e4=3.1319−0.3539αb−0.0820D0αb+0.8345n−0.0943αbn−0.1229D0αbn+1.6324αbn23.0659+αb+−14.4454n2+10.6703n3+0.2263D0αbn2−1.2058αbn38.8496+αb+0.5235fcu,0n−0.4243fcu,0−0.0982fcu,0D08.8496fceαa+fceαaαb,p1=−6e12+16e2,p2=8e13−32e1e2+64e3,p3=−3e14+16e12e2−64e1e3+256e4,p4=16p1+43p3+p129⋅cos13arccosp2216−p1p36+p132163612p3+p123.


## 3.2. Concrete Performance Parameters

The values of concrete performance parameters are random and discrete since concrete is a heterogeneous composite material composed of mortar, randomly distributed aggregates, and microcracks. The specified values must be achieved with a predetermined probability in order to guarantee high-quality concrete. The specified compressive strength depends on the bearing capacity of the concrete structure. The specified chloride diffusion coefficient and its aging factor depend on the durability of concrete structures.When the specified compressive strengthfcu,k is less than 60 MPa, the required average compressive strength fcu,0 can be described as follows:(27)fcu,0=fcu,k+tσf,where t is a probability, σf is the standard deviation of compressive strength, fcu,k is the specified compressive strength, and fcu,0 is the required average compressive strength. When the confidence level is 95%, the value of t is equal to 1.645. When fcu,k≤20 MPa, the value of σf is equal to 4 MPa. When 25≤fcu,k≤45 MPa, the value of σf is equal to 5 MPa. When 50≤fcu,k≤55 MPa, the value of σf is equal to 6 MPa.When the specified compressive strength is not less than 60 MPa, the required average compressive strength can be described as follows:(28)fcu,0=1.15fcu,k.Similarly, the required average chloride diffusion coefficientD0 can be described as follows:(29)D0=D0,k−tσD,where σD is the standard deviation of diffusion coefficient the value of σD ranges from 0.15D0 to 0.30D0, which can be also determined by reference to historical data [37–39]. In this paper, the value of σD is taken as 0.30D0.


## 3.3. Concrete Performance Parameters

According to equations (23)–(25), the major mix proportion parameters, including water-to-binder ratio, replacement of fly ash, and GGBFS, can be calculated. The solution procedure can be summarized as follows:(1)
Based on the methodology of quantitative durability design for concrete structures [40, 41], first determine the chloride exposure class, the action of load and the intended service life, the specified compressive strength is determined by the ultimate bearing capacity requirement of concrete structures, and then the specified chloride diffusion coefficient and its aging factor are determined by durability design of concrete structures.(2)
According to the situation of local raw materials, the type of coarse aggregate and size gradation, types of mineral admixtures and their combinations, and chemical admixtures, are determined.(3)
According to equations (27)–(29), the required average compressive strength and the required average chloride diffusion coefficient can be calculated.(4)
According to equations (23)–(25), water-to-binder ratio and replacement of mineral admixtures can be calculated.The previous procedure is a quantitative analysis method for replacing mineral admixtures. Moreover, then the mixing water content and sand ratio are determined by the requirement of workability. Finally, the concrete mix proportion can be determined. The concrete prepared by using this method meets the requirements for strength and durability indices.


## 4. Application Examples and Experimental Verification

According to the requirement of the ultimate bearing capacity and durability of concrete structures in a certain chloride environment, the mineral admixtures can be quantitatively determined.


### 4.1. An Example for Calculation of the Replacement of Mineral Admixtures

Concrete structures designed for urban railway traffic have a design life of 100 years. The total length of this project is 32.1 km, which is in a complex geological environment. In the previous survey, it was found that groundwater along the line of the project contained chloride ions and other aggressive substances. Concrete structures of subway stations and shield tunnel segments subjected to chloride attack. Therefore, the replacement of mineral admixtures must be calculated according to the strength and durability of the concrete.Firstly, the concrete slump is determined by the construction condition and the requirement of workability. The strength grade of concrete, the specified chloride diffusion coefficient, and its aging factor can be calculated according to the ultimate bearing capacity and durability requirements of concrete structures at different locations, as shown in Table5.Table 5
The indices of compressive strength, durability, and workability of concrete.
Concrete structuresStrength gradeD0,k (×10−12 m2/s)nSlump (mm)1DC3010.500.491802DC355.640.64180Shield segmentC507.270.40180“1D” and “2D” denote one-dimensional diffusion zone and two-dimensional diffusion zone, respectively. The shield segment denotes circular section diffusion zone. After that, CEM I Portland cement is used, depending on the raw material situation in the area. Crushed limestone with a size of 5–20 mm is used as coarse aggregate. The mineral admixtures are fly ash and GGBFS. A polycarboxylate superplasticizer is used and its water reduction rate is 20%. According to equations (27)–(29), the required average compressive strength and the required average chloride diffusion coefficient can be calculated, as listed in Table 6.Table 6
The required average compressive strength and chloride diffusion coefficient of concrete.
Concrete structuresfcu,0 (MPa)D0 (×10−12 m2/s)1D38.237.902D43.234.24Shield segment59.875.47Then, according to equations (23)–(25), water-to-binder ratio, replacement of fly ash, and replacement of GGBFS can be calculated, as shown in Table 7.Table 7
Mix proportion of concrete.
Concrete structuresRW/BRFARSG1D0.470.200.102D0.390.200.30Shield segment0.350.100.10Lastly, the sand ratio was determined by means of laboratory trials, the value of which is equal to 0.37. At the same time, segregation and bleeding did not occur, and the slump satisfied the requirement of workability. Based on the absolute volume method, the proportion of component materials of a concrete mixture were determined, as listed in Table8.Table 8
The mix proportion of concrete (kg/m3).
Concrete structuresWaterCementFly ashGGBFSSandGravel1D160238683470311972D160205821236771153Shield segment16036646466601123


### 4.2. Comparison with Code

According to Table6, the total replacement of fly ash and GGBFS is less than 0.50, and the replacement of fly ash and GGBFS is less than 0.25 and 0.50, respectively, which complies with Standards of Ministry of Construction in China JGJ 55-2011 and ACI 211.According to the Chinese National Standard GB/T 50476-2008, the exposure class is IV-C, the maximum water-to-binder ratio is 0.40, and the minimum strength grade of concrete is C45. According to ACI 318, the maximum water-to-binder ratio is 0.40, and the minimum specified compressive strength is 5000 psi, which is approximately equivalent to C45. Therefore, the durability requirements of GB/T 50476-2008 is the same as ACI 318. For 1D and 2D concrete structures, the minimum strength grade of concrete are C30 and C35 respectively, which are below C45. For GB/T 50476-2008 and ACI 318, and it is apparent that quantitative control of the durability of concrete structures can be achieved by increased strength grade of concrete.Owning to the replacement of mineral admixtures is determined based on the experience, the values according to JGJ 55-2011 and ACI 211 were set to the same value as the proposed approach. Based on the relationship between the compressive strength and the water-to-binder ratio, the water-to-binder ratio can be calculated. Then, we selected the smaller value between the calculated water-to-binder ratio and the specified maximum water-to-binder ratio as the final value. The major mix proportion parameters are listed in Table9.Table 9
Comparison of calculation for JGJ 55-2011 and ACI 211.
Concrete structuresResults by JGJ 55-2011Result by ACI 211RW/BRFARSGRW/BRFARSG1D0.360.200.100.400.200.102D0.360.200.300.400.200.30Shield segment0.360.100.100.360.100.10As shown in Tables7 and 9, for shield tunnel segment concrete, the water-to-binder ratio determined by the proposed method is consistent with JGJ 55-2011 and ACI 211, to some extent, which demonstrates the validity of the proposed approach. However, for 1D and 2D concrete structures, the value of the water-to-binder ratio calculated by JGJ 55-2011 or ACI 211 is less than the value calculated by the proposed method, this is due to the fact that the value calculated by the proposed method is based on the durability and strength indices, whereas the value calculated by JGJ 55-2011 or ACI 211 is based on experience. It is obvious that the water-to-binder ratio determined by the previous codes cannot meet the quantitative requirements of durability. Since the smaller value of the water-to-binder ratio will result in greater durability and strength than intended, which leads to excess material performance.Yang [42] investigated that the correlation coefficient between 28 d compressive strength of concrete and 28 d chloride diffusion coefficient of concrete is less than 0.85, which is much lower than the statistically strong correlation coefficient of 0.95. Meanwhile, previous studies [43] shown that the porosity of pores of 10–20 nm is closely correlated to the 28 d compressive strength of concrete, and the porosity of pores of 20–50 nm contributes significantly to the 28 d chloride diffusion coefficient of concrete. Therefore, variating the strength grade of concrete is not the only way to satisfy durability requirements. For one-dimensional diffusion zones with lower durability requirements, instead of increasing concrete strength to achieve durability requirements, adjusting the replacement of mineral admixtures provides a design with better workability and cost-effectiveness. Currently, concrete mix proportions are primarily determined by strength alone in order to meet durability requirements, so the advantages of mineral admixtures have not been fully exploited.


### 4.3. Experimental Verification

#### 4.3.1. Design of Experiments

The orthogonal experimental design method, which has the advantage that there are fewer experiments, was used to verify the validity of the proposed approach, the accuracy of the computational model for the initial diffusion coefficient, and the rationality of the mix proportion for urban railway traffic. The influencing factors included water-to-binder ratio, replacement of fly ash, and replacement of GGBFS. Each factor had four different levels. According to the orthogonal experimental design, a total of 16 mixes were tested, as listed in Table10.Table 10
The contents of raw materials of concrete in 1 m3 concrete (kg/m3).
No.RW/BRFARSGStrength gradeD0,k(10−12 m2/s)WaterCementFly ashGGBFSSandAggregate10.3500C608.4516045700660112320.350.100.10C506.381603664646660112330.350.200.20C457.731602749191660112340.350.300.30C359.09160183137137660112350.3900.10C5011.26160369041677115360.390.100C4511.07160369410677115370.390.200.30C357.8116020582123677115380.390.300.20C3011.1216020512382677115390.4300.20C4512.221602980746911177100.430.100.30C356.41160223371126911177110.430.200C3512.921602987406911177120.430.300.10C3013.35160223112376911177130.4700.30C3510.9516023801027031197140.470.100.20C358.5316023834687031197150.470.200.10C3012.1616023868347031197160.470.300C2514.8716023810207031197It should be noted that some mixtures were inconsistent with the habits of engineering practice. This is because sample points in orthogonal experimental design were distributed in a particular place in the sample space. However, there are enough mixes to choose from for engineering practice. The mix proportions in Tables7 and 8 were selected from nos. 15, 7, and 2 in Table 10.


#### 4.3.2. Materials and Specimen Preparation

The cement used was Type I Portland cement provided by Huarun Cement Co. Ltd. Mineral admixtures including class F fly ash and grade 100 GGBFS were used for the replacement of Portland cement. The chemical composition of cement is shown in Table11.Table 11
Chemical composition of Portland cement (%).
CompositionSiO2Al2O3Fe2O3CaOMgOSO3LossContent19.895.353.8563.550.672.273.72The physical properties of fly ash and GGBFS are shown in Table12.Table 12
Physical properties and chemical compositions of FA and GGBFS.
FAGGBFSIndexParameterIndexParameterFiness18.97-day activity index (%)69Water requirement ratio8228 d activity index (%)95Ignition loss (%)5.92Specific surface (m2/kg)424CaO (%)3.13Fluidity ratio (%)243f-CaO0.21Ignition loss (%)0.46SO3 (%)0.12SO3 (%)0.40Cl−1 (%)0.002Cl−1 (%)0.004The coarse aggregate used in this study was crushed limestone with a size of 5–20 mm and an apparent density of 2590 kg/m3. The fine aggregate used was river sand with a fineness modulus of 2.7 and an apparent density of 2560 kg/m3. A polycarboxylate superplasticizer is used and its water reduction rate is 20%.The concrete cube specimens for each mix proportion with the size of 150 × 150 × 150 mm were demolded 24 hours after casting, then cured in a standard curing room for 28 days. The concrete cylinder specimens with the size of Φ100 × 200 mm were demolded 24 hours after casting, then cured in a standard curing room for 21 days. Each specimen was cut into three test specimens with a concrete saw. The size of each test specimen is Φ100 × 50 mm. After cutting, the test specimens were cured for another 7 days.


#### 4.3.3. Tests Methods and Results

Compressive strength tests were carried out on 150 mm cube specimens, according to Chinese Standard GB/T 50107-2010. A nonsteady state migration test was performed following the RCM method as described in DuraCrete. Comparison among tested, specified, and required average compressive strength is shown in Figure3.Figure 3
The compressive strength of concrete.Comparison among tested, specified, and required average chloride diffusion coefficient is shown in Figure4.Figure 4
Chloride diffusion coefficient of concrete.As shown in Figure3, there is a uniform distribution of compressive strength values on both sides of the reference line of required average values, and the values are significantly higher than the values specified. As shown in Figure 4, the tested values of the chloride diffusion coefficient are uniformly distributed on both sides of the reference line of required average values, and less than specified values. Hence, the specified values of compressive strength and chloride diffusion coefficient were attained with a predetermined probability of 95%, which demonstrates the validity of the proposed approach.In conclusion, a novel method is provided to quantitatively design the replacement of mineral admixtures based on durability requirements.


## 4.1. An Example for Calculation of the Replacement of Mineral Admixtures

Concrete structures designed for urban railway traffic have a design life of 100 years. The total length of this project is 32.1 km, which is in a complex geological environment. In the previous survey, it was found that groundwater along the line of the project contained chloride ions and other aggressive substances. Concrete structures of subway stations and shield tunnel segments subjected to chloride attack. Therefore, the replacement of mineral admixtures must be calculated according to the strength and durability of the concrete.Firstly, the concrete slump is determined by the construction condition and the requirement of workability. The strength grade of concrete, the specified chloride diffusion coefficient, and its aging factor can be calculated according to the ultimate bearing capacity and durability requirements of concrete structures at different locations, as shown in Table5.Table 5
The indices of compressive strength, durability, and workability of concrete.
Concrete structuresStrength gradeD0,k (×10−12 m2/s)nSlump (mm)1DC3010.500.491802DC355.640.64180Shield segmentC507.270.40180“1D” and “2D” denote one-dimensional diffusion zone and two-dimensional diffusion zone, respectively. The shield segment denotes circular section diffusion zone. After that, CEM I Portland cement is used, depending on the raw material situation in the area. Crushed limestone with a size of 5–20 mm is used as coarse aggregate. The mineral admixtures are fly ash and GGBFS. A polycarboxylate superplasticizer is used and its water reduction rate is 20%. According to equations (27)–(29), the required average compressive strength and the required average chloride diffusion coefficient can be calculated, as listed in Table 6.Table 6
The required average compressive strength and chloride diffusion coefficient of concrete.
Concrete structuresfcu,0 (MPa)D0 (×10−12 m2/s)1D38.237.902D43.234.24Shield segment59.875.47Then, according to equations (23)–(25), water-to-binder ratio, replacement of fly ash, and replacement of GGBFS can be calculated, as shown in Table 7.Table 7
Mix proportion of concrete.
Concrete structuresRW/BRFARSG1D0.470.200.102D0.390.200.30Shield segment0.350.100.10Lastly, the sand ratio was determined by means of laboratory trials, the value of which is equal to 0.37. At the same time, segregation and bleeding did not occur, and the slump satisfied the requirement of workability. Based on the absolute volume method, the proportion of component materials of a concrete mixture were determined, as listed in Table8.Table 8
The mix proportion of concrete (kg/m3).
Concrete structuresWaterCementFly ashGGBFSSandGravel1D160238683470311972D160205821236771153Shield segment16036646466601123


## 4.2. Comparison with Code

According to Table6, the total replacement of fly ash and GGBFS is less than 0.50, and the replacement of fly ash and GGBFS is less than 0.25 and 0.50, respectively, which complies with Standards of Ministry of Construction in China JGJ 55-2011 and ACI 211.According to the Chinese National Standard GB/T 50476-2008, the exposure class is IV-C, the maximum water-to-binder ratio is 0.40, and the minimum strength grade of concrete is C45. According to ACI 318, the maximum water-to-binder ratio is 0.40, and the minimum specified compressive strength is 5000 psi, which is approximately equivalent to C45. Therefore, the durability requirements of GB/T 50476-2008 is the same as ACI 318. For 1D and 2D concrete structures, the minimum strength grade of concrete are C30 and C35 respectively, which are below C45. For GB/T 50476-2008 and ACI 318, and it is apparent that quantitative control of the durability of concrete structures can be achieved by increased strength grade of concrete.Owning to the replacement of mineral admixtures is determined based on the experience, the values according to JGJ 55-2011 and ACI 211 were set to the same value as the proposed approach. Based on the relationship between the compressive strength and the water-to-binder ratio, the water-to-binder ratio can be calculated. Then, we selected the smaller value between the calculated water-to-binder ratio and the specified maximum water-to-binder ratio as the final value. The major mix proportion parameters are listed in Table9.Table 9
Comparison of calculation for JGJ 55-2011 and ACI 211.
Concrete structuresResults by JGJ 55-2011Result by ACI 211RW/BRFARSGRW/BRFARSG1D0.360.200.100.400.200.102D0.360.200.300.400.200.30Shield segment0.360.100.100.360.100.10As shown in Tables7 and 9, for shield tunnel segment concrete, the water-to-binder ratio determined by the proposed method is consistent with JGJ 55-2011 and ACI 211, to some extent, which demonstrates the validity of the proposed approach. However, for 1D and 2D concrete structures, the value of the water-to-binder ratio calculated by JGJ 55-2011 or ACI 211 is less than the value calculated by the proposed method, this is due to the fact that the value calculated by the proposed method is based on the durability and strength indices, whereas the value calculated by JGJ 55-2011 or ACI 211 is based on experience. It is obvious that the water-to-binder ratio determined by the previous codes cannot meet the quantitative requirements of durability. Since the smaller value of the water-to-binder ratio will result in greater durability and strength than intended, which leads to excess material performance.Yang [42] investigated that the correlation coefficient between 28 d compressive strength of concrete and 28 d chloride diffusion coefficient of concrete is less than 0.85, which is much lower than the statistically strong correlation coefficient of 0.95. Meanwhile, previous studies [43] shown that the porosity of pores of 10–20 nm is closely correlated to the 28 d compressive strength of concrete, and the porosity of pores of 20–50 nm contributes significantly to the 28 d chloride diffusion coefficient of concrete. Therefore, variating the strength grade of concrete is not the only way to satisfy durability requirements. For one-dimensional diffusion zones with lower durability requirements, instead of increasing concrete strength to achieve durability requirements, adjusting the replacement of mineral admixtures provides a design with better workability and cost-effectiveness. Currently, concrete mix proportions are primarily determined by strength alone in order to meet durability requirements, so the advantages of mineral admixtures have not been fully exploited.


## 4.3. Experimental Verification

### 4.3.1. Design of Experiments

The orthogonal experimental design method, which has the advantage that there are fewer experiments, was used to verify the validity of the proposed approach, the accuracy of the computational model for the initial diffusion coefficient, and the rationality of the mix proportion for urban railway traffic. The influencing factors included water-to-binder ratio, replacement of fly ash, and replacement of GGBFS. Each factor had four different levels. According to the orthogonal experimental design, a total of 16 mixes were tested, as listed in Table10.Table 10
The contents of raw materials of concrete in 1 m3 concrete (kg/m3).
No.RW/BRFARSGStrength gradeD0,k(10−12 m2/s)WaterCementFly ashGGBFSSandAggregate10.3500C608.4516045700660112320.350.100.10C506.381603664646660112330.350.200.20C457.731602749191660112340.350.300.30C359.09160183137137660112350.3900.10C5011.26160369041677115360.390.100C4511.07160369410677115370.390.200.30C357.8116020582123677115380.390.300.20C3011.1216020512382677115390.4300.20C4512.221602980746911177100.430.100.30C356.41160223371126911177110.430.200C3512.921602987406911177120.430.300.10C3013.35160223112376911177130.4700.30C3510.9516023801027031197140.470.100.20C358.5316023834687031197150.470.200.10C3012.1616023868347031197160.470.300C2514.8716023810207031197It should be noted that some mixtures were inconsistent with the habits of engineering practice. This is because sample points in orthogonal experimental design were distributed in a particular place in the sample space. However, there are enough mixes to choose from for engineering practice. The mix proportions in Tables7 and 8 were selected from nos. 15, 7, and 2 in Table 10.


### 4.3.2. Materials and Specimen Preparation

The cement used was Type I Portland cement provided by Huarun Cement Co. Ltd. Mineral admixtures including class F fly ash and grade 100 GGBFS were used for the replacement of Portland cement. The chemical composition of cement is shown in Table11.Table 11
Chemical composition of Portland cement (%).
CompositionSiO2Al2O3Fe2O3CaOMgOSO3LossContent19.895.353.8563.550.672.273.72The physical properties of fly ash and GGBFS are shown in Table12.Table 12
Physical properties and chemical compositions of FA and GGBFS.
FAGGBFSIndexParameterIndexParameterFiness18.97-day activity index (%)69Water requirement ratio8228 d activity index (%)95Ignition loss (%)5.92Specific surface (m2/kg)424CaO (%)3.13Fluidity ratio (%)243f-CaO0.21Ignition loss (%)0.46SO3 (%)0.12SO3 (%)0.40Cl−1 (%)0.002Cl−1 (%)0.004The coarse aggregate used in this study was crushed limestone with a size of 5–20 mm and an apparent density of 2590 kg/m3. The fine aggregate used was river sand with a fineness modulus of 2.7 and an apparent density of 2560 kg/m3. A polycarboxylate superplasticizer is used and its water reduction rate is 20%.The concrete cube specimens for each mix proportion with the size of 150 × 150 × 150 mm were demolded 24 hours after casting, then cured in a standard curing room for 28 days. The concrete cylinder specimens with the size of Φ100 × 200 mm were demolded 24 hours after casting, then cured in a standard curing room for 21 days. Each specimen was cut into three test specimens with a concrete saw. The size of each test specimen is Φ100 × 50 mm. After cutting, the test specimens were cured for another 7 days.


### 4.3.3. Tests Methods and Results

Compressive strength tests were carried out on 150 mm cube specimens, according to Chinese Standard GB/T 50107-2010. A nonsteady state migration test was performed following the RCM method as described in DuraCrete. Comparison among tested, specified, and required average compressive strength is shown in Figure3.Figure 3
The compressive strength of concrete.Comparison among tested, specified, and required average chloride diffusion coefficient is shown in Figure4.Figure 4
Chloride diffusion coefficient of concrete.As shown in Figure3, there is a uniform distribution of compressive strength values on both sides of the reference line of required average values, and the values are significantly higher than the values specified. As shown in Figure 4, the tested values of the chloride diffusion coefficient are uniformly distributed on both sides of the reference line of required average values, and less than specified values. Hence, the specified values of compressive strength and chloride diffusion coefficient were attained with a predetermined probability of 95%, which demonstrates the validity of the proposed approach.In conclusion, a novel method is provided to quantitatively design the replacement of mineral admixtures based on durability requirements.


## 4.3.1. Design of Experiments

The orthogonal experimental design method, which has the advantage that there are fewer experiments, was used to verify the validity of the proposed approach, the accuracy of the computational model for the initial diffusion coefficient, and the rationality of the mix proportion for urban railway traffic. The influencing factors included water-to-binder ratio, replacement of fly ash, and replacement of GGBFS. Each factor had four different levels. According to the orthogonal experimental design, a total of 16 mixes were tested, as listed in Table10.Table 10
The contents of raw materials of concrete in 1 m3 concrete (kg/m3).
No.RW/BRFARSGStrength gradeD0,k(10−12 m2/s)WaterCementFly ashGGBFSSandAggregate10.3500C608.4516045700660112320.350.100.10C506.381603664646660112330.350.200.20C457.731602749191660112340.350.300.30C359.09160183137137660112350.3900.10C5011.26160369041677115360.390.100C4511.07160369410677115370.390.200.30C357.8116020582123677115380.390.300.20C3011.1216020512382677115390.4300.20C4512.221602980746911177100.430.100.30C356.41160223371126911177110.430.200C3512.921602987406911177120.430.300.10C3013.35160223112376911177130.4700.30C3510.9516023801027031197140.470.100.20C358.5316023834687031197150.470.200.10C3012.1616023868347031197160.470.300C2514.8716023810207031197It should be noted that some mixtures were inconsistent with the habits of engineering practice. This is because sample points in orthogonal experimental design were distributed in a particular place in the sample space. However, there are enough mixes to choose from for engineering practice. The mix proportions in Tables7 and 8 were selected from nos. 15, 7, and 2 in Table 10.


## 4.3.2. Materials and Specimen Preparation

The cement used was Type I Portland cement provided by Huarun Cement Co. Ltd. Mineral admixtures including class F fly ash and grade 100 GGBFS were used for the replacement of Portland cement. The chemical composition of cement is shown in Table11.Table 11
Chemical composition of Portland cement (%).
CompositionSiO2Al2O3Fe2O3CaOMgOSO3LossContent19.895.353.8563.550.672.273.72The physical properties of fly ash and GGBFS are shown in Table12.Table 12
Physical properties and chemical compositions of FA and GGBFS.
FAGGBFSIndexParameterIndexParameterFiness18.97-day activity index (%)69Water requirement ratio8228 d activity index (%)95Ignition loss (%)5.92Specific surface (m2/kg)424CaO (%)3.13Fluidity ratio (%)243f-CaO0.21Ignition loss (%)0.46SO3 (%)0.12SO3 (%)0.40Cl−1 (%)0.002Cl−1 (%)0.004The coarse aggregate used in this study was crushed limestone with a size of 5–20 mm and an apparent density of 2590 kg/m3. The fine aggregate used was river sand with a fineness modulus of 2.7 and an apparent density of 2560 kg/m3. A polycarboxylate superplasticizer is used and its water reduction rate is 20%.The concrete cube specimens for each mix proportion with the size of 150 × 150 × 150 mm were demolded 24 hours after casting, then cured in a standard curing room for 28 days. The concrete cylinder specimens with the size of Φ100 × 200 mm were demolded 24 hours after casting, then cured in a standard curing room for 21 days. Each specimen was cut into three test specimens with a concrete saw. The size of each test specimen is Φ100 × 50 mm. After cutting, the test specimens were cured for another 7 days.


## 4.3.3. Tests Methods and Results

Compressive strength tests were carried out on 150 mm cube specimens, according to Chinese Standard GB/T 50107-2010. A nonsteady state migration test was performed following the RCM method as described in DuraCrete. Comparison among tested, specified, and required average compressive strength is shown in Figure3.Figure 3
The compressive strength of concrete.Comparison among tested, specified, and required average chloride diffusion coefficient is shown in Figure4.Figure 4
Chloride diffusion coefficient of concrete.As shown in Figure3, there is a uniform distribution of compressive strength values on both sides of the reference line of required average values, and the values are significantly higher than the values specified. As shown in Figure 4, the tested values of the chloride diffusion coefficient are uniformly distributed on both sides of the reference line of required average values, and less than specified values. Hence, the specified values of compressive strength and chloride diffusion coefficient were attained with a predetermined probability of 95%, which demonstrates the validity of the proposed approach.In conclusion, a novel method is provided to quantitatively design the replacement of mineral admixtures based on durability requirements.


## 5. Conclusions

Multifactor models for durability indices were proposed to estimate quantitatively the influence of mineral admixtures on the properties of concrete. Based on this, a quantitative analysis method to determine the replacement of mineral admixtures was proposed. Comparing the proposed method to JGJ 55-2011and ACI 211, it became apparent that it has an inherent advantage of rational design for strength and durability. The accuracy of this method was validated by engineering examples and experimental data.In this paper, the quantitative durability design technique focuses mainly on the method of combining multifactor models to obtain explicit expressions for the major mix proportion parameters. It is not limited to the parameters and multifactor models mentioned in the article. The research will be conducted in the future to improve this method’s coverage of dependent variables and independent variables.



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*Source: 4134894-2023-02-10.xml*