A predictive model of the effective tensile and compressive strengths of concrete considering porosity and pore size

Construction and Building Materials 170 (2018) 520–526

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

A predictive model of the effective tensile and compressive strengths of concrete considering porosity and pore size Dongqi Li a, Zongli Li a,⇑, Congcong Lv b, Guohui Zhang c, Yueming Yin a a

College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, PR China College of Hydraulic & Environmental Engineering, China Three Gorges University, Yichang 443002, PR China c Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650051, PR China b

h i g h l i g h t s
A predictive model of the effective tensile and compressive strengths is established.
The distribution and influence of pore size are considered in the strength model.
Rationality and accuracy of the proposed expressions are assessed.
Porosity higher influences compressive strength than tensile strength.
Tensile and compressive strengths decrease with increases in porosity and pore size.

a r t i c l e

i n f o

Article history: Received 8 August 2017 Received in revised form 28 February 2018 Accepted 2 March 2018

Keywords: Concrete Tensile strength Compressive strength Porosity Pore size Predictive model

a b s t r a c t Pores, voids correspond to inevitable products in the formation of concrete and significantly affect its macroscopic effective strength. A lot of researches show that both porosity and pore structure have significant influence on the concrete strength and differences exist with respect to those influences on the tensile and compressive strengths. In this paper, first, the relationships between the effective tensile and compressive strengths of concrete and porosity are expressed respectively from the proposed simplified center pore model. Second, in order to further consider the influence of the pore structure, the pore size is chosen as the representative index of pore structure and an influence function related to the pore size is proposed based on the correlation analysis. The total influence coefficient of pore size is obtained by combining the pore size distribution function and influence function. Taking the total influence coefficient into account in the relation between the effective strength and porosity, explicit formulae of porous concrete effective tensile and compressive strengths are finally deduced. Through comparing the present approach with a few classical analytical solutions and experimental observations, the results indicate the reliability and accuracy of the proposed approach and the derived equations are simple and convenient to use. Analysis and discussion suggest that the effect of porosity on the compressive strength of concrete exceeds that of the tensile strength and that the effective concrete strength with the same porosity can be improved by decreasing the pore size. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction As widely-known, pores, voids and other defects govern the most important mechanical properties of cement-based materials and especially in terms of strength and permeability [1–3]. The strength of cement-based materials depends on the formation of cementitious material. However, the defects are considered as factors that reduce the effective cementitious area and further

⇑ Corresponding author. E-mail address: [email protected] (Z. Li). https://doi.org/10.1016/j.conbuildmat.2018.03.028 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

weaken the cement-based materials strength. In order to understand this influence, several related theories were proposed to predict the effective strength. Initially, many researchers select the porosity as the only factor to establish strength models of cement-based materials. Powers proposed a gel space ratio theory to examine the compressive strength of Portland cement [4]. Hansen [5] assumed that defects were focused into a spherical void and that strength was proportional to the gel area on the maximum cross section of the void. Furthermore, many typical semi-empirical equations were also proposed to reflect the relationship between strength and porosity

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such as power, index, linear, and logarithmic equations [6–9]. The equations are widely used to predict the strength of porous materials [10,11]. However, the experimental results show that these strength models are not accurate enough for strength prediction, because not only the porosity but also the pore structure, such as pore size and pore-connectivity, has vital influence on the strength. Therefore, various researchers investigated the effect of pores structure on the strength. Based on the Hanshin model, Huang et al. [12] introduced the concept of relative specific surface area of the pores structure into the strength equation and proposed an expression of compressive strength. Deo and Neithalath [13] explored the relationship between pore structure and concrete compressive strength by examining the response of pervious concrete. Gao et al. [14] studied the relationship between pore structure and concrete strength by examining samples involving the defects which are obtained the addition of air entraining agents. Jin et al [15] expressed the relation between strength and pore structure of hardened mortar by introducing the ratio parameter of fractal dimension to capillary pore volume. Although the fractal theory characterized the pore size distribution accurately, the influence of pore structure on the effective strength was not further explained. Griffith theory was adopted to evaluate the strength of porous materials and the influence of pore size was also considered [16–18]. However, many researchers suggested that differences exist with respect to the impacts of pore structure on the tensile and compressive strengths [2,19,20] and the model based on Griffith theory is unable to explain this phenomenon. The relationships between the tensile and compressive strengths and pore structure are required to be established separately. Recently, various scholars attempted to use meso-mechanics theory of composite materials to examine elastic parameters of porous materials, such as the self-consistent method, MoriTanaka method, and three-phase sphere model [21]. However, the effective strength of porous materials was not obtained by meso-mechanics theory. Given the fore-mentioned reasons, the objective of this paper is to establish a mathematical model of concrete effective strength which not only considers the influence of pore structure on the strength but also expresses the different relationships between the tensile and compressive strengths and pore structure. The rationality and accuracy of the developed model are verified by comparison with three classical analytical solutions and experimental observations. Finally, the influences of porosity and pore structure on the tensile and compressive strengths are analyzed and discussed.

2. Effective strength of porous concrete 2.1. A simplified central pore model Various researchers concentrated on the pore in the center of the model to investigate deformation characteristics and the strength of porous material. Hill [22] and Christensen and Lo [23] investigated elastic parameters of composite material based on a three phase sphere model. Du et al. [24] developed a hollow sphere model to study the effective strength of porous concrete. In view of these findings, this paper assumes that the concrete with defects is composed of a central round pore inclusion embedded in nonporous concrete matrix. The equivalent process of the porous concrete is illustrated in Fig. 1. First, with respect to the meso-scale, concrete is considered as a four-phase composite material composed of aggregates, cement paste, an interface transition region (ITZ), and initial defects as shown in Fig. 1(a). Second, aggregates, cement paste, and ITZ are considered as the matrix, and the defects in the cement paste and ITZ are concentrated in the center of the

matrix as the inclusion phase to establish the central pore model as shown in Fig. 1(b). The central pore model is subject to an externally applied uniaxial tensile load, and the locations are defined as M, N, O, and P respectively when variable q corresponds to the inside radius a and the variable u corresponds to 0; p2 ; p and 32p. The defects including cracks and voids expand, propagate and collapse under external load, which is the main factor leading to the nonlinear elastic behavior of concrete. Thus the concrete matrix without defects can be assumed as isotropic, homogeneous, and elastic, which is consistent with the assumption in literature [24]. According to Wong and Chau research [25] and Chen et al. research [26], the defects are randomly distributed in the concrete, and the cross section porosity in the concrete volume fluctuates within a small range and is considered to be equal. Besides, the cross section porosity is generally adopted to represent the concrete porosity in the research of concrete mechanical properties [14,25–27]. Therefore, in this paper the representative cross section with average porosity is selected to research the relation between the concrete strength and porosity. To aid in the convenience of calculation, the model presented in Fig. 1(b) is changed into a polar coordinate system model as shown in Fig. 1(c), and the corresponding original stress boundary is transformed based on the elastic theory [28] as follows:





rq

 q¼b

squ





 q q q þ cos 2u; rq q¼a ¼ 0; squ q¼b ¼  sin 2u; 2 2 2 ¼0

¼

q¼a

ð1Þ

where q denotes externally applied uniaxial tensile load (MPa), u denotes the angle (°) shown in Fig. 1(b) and (c), and subscripts q and u denote radial direction and circumferential direction, respectively. Based on the superposition principle of elastic mechanics, the boundary conditions are divided into two parts. The first and second parts of the stress boundary condition are as follows:



rIq



rIIq



q¼b





sIIqu

q¼b



¼

    q  I  ; squ ¼ 0; rIq ¼ 0; sIqu ¼0 q¼b q¼a q¼a 2

¼

    q q ¼  sin 2u; rIIq ¼ 0; cos 2u; sIIqu q¼b q¼a 2 2

q¼a

ð2Þ

¼0

ð3Þ

Based on the lame solution of elastic mechanics [28], the radial, circumferential, and tangential stresses of the equivalent ring applied to the first part of the stress boundary condition are obtained. With respect to the second part of the stress boundary condition, the stress components are obtained by a semi-inverse method of elastic mechanics [29]. Subsequently, the matrix stresses that satisfy the two-part boundary conditions are superimposed, and the total radial, circumferential, and tangential stresses are obtained as follows:

rq ¼

" 2 2 q q2  cb q 4c2 þ c þ 1 b  2ðc3 þ c2 þ cÞ 2  3 2 2 q ð1  cÞ ðc  1Þ 2 q # 3 2 4 3ðc þ c Þb cos 2u þ 2q4

ru ¼

ð4Þ

" # 2 q q2 þ cb q q2 4c2 þ c þ 1 3ðc3 þ c2 Þb4 þ þ 6c þ cos 2u 2 2 q2 ð1  cÞ ðc  1Þ3 2 2 q4 b ð5Þ

squ ¼

q ðc  1Þ3

" 3c

q2 4c2 þ c þ 1 b

2

þ

2

þ ðc3 þ c2 þ cÞ

b

2

q

 2

3ðc3 þ c2 Þb 2q4

4

# sin 2u ð6Þ

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D. Li et al. / Construction and Building Materials 170 (2018) 520–526

Fig. 1. Process of equivalence: (a) stress state of a representative micro-unit; (b) a simplified central pore model; (c) equivalent polar coordinate model of (b).

where q denotes the distance from the round center, parameters a and b denote the equivalent inside radius and outer radius, respec2

tively, and parameter c denotes the porosity of concrete, i.e., c ¼ ab2 . When the equivalent outer radius b approaches infinity, the situation is equivalent to an infinite space containing a circular hole, and the above Eqs. (4)–(6) are identical to the Kirsch solution of elastic mechanics [28]. 2.2. Effective tensile strength of porous concrete The effective strength of porous concrete represents the homogenization strength of representative elementary volume and is determined by pore structure. A portion of pores start to yield under externally applied load. When all pores are damaged, the concrete reaches the ultimate strength. Since the pores are concentrated in the center of the matrix in this research, the ultimate strength of concrete is determined by the center pore of the matrix. In this section, the porosity is selected as an independent variable that is related to the concrete strength without considering the influence of pore size. The concrete matrix is assumed as linear elastic up to its tensile strength. The maximum tensile stress criterion is selected as the failure criterion. Thus, the ultimately effective strength of porous concrete is determined by using the maximum principle stress at a certain point of the matrix. Based on the elastic theory [28], the maximum value of the maximum principle stress appears at points N and P when the tensile stress is applied, and both of the stresses are identical.

Therefore, only point N is considered for the analysis. According to Eqs. (4)–(6), the radial and tangential stresses of point N correspond to zero, and the maximum principal stress is the circumferential stress that is expressed as follows:

rN1 ¼

q 2ð2c þ 1Þq 3ðc þ 1Þq þ ¼ 1c ð1  cÞ2 ð1  cÞ2

ð7Þ

where rN1 denotes the maximum principal stress of matrix point N when the tensile load is applied (MPa). It is assumed that the tensile strength of matrix point N corresponds to rNt . The above analysis indicates that point N reaches its strength rNt when an external applied load corresponds to the macroscopic effective tensile strength St . Therefore, the effective tensile strength St of the concrete with a porosity of c is obtained based on Eq. (7) as follows:

St ¼

ð1  cÞ2 rNt 3ðc þ 1Þ

ð8Þ

When the porosity of concrete approaches zero, the effective tensile strength St0 of non-porous concrete is obtained based on Eq. (7) as follows:

St0 ¼

rNt 3

ð9Þ

Therefore, based on Eqs. (8) and (9), the quantitative relationship between the effective tensile strength and porosity is expressed as follows

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St ð1  cÞ2 ¼ St0 cþ1

ð10Þ

2.3. Effective compressive strength of porous concrete With respect to the application of the compressive stress on the boundary, the maximum value of the maximum principle stress appears at points M and O based on the elastic theory [28], and similarly only point M is considered for the analysis. According to Eqs. (4)–(6), the radial and tangential stresses of point M correspond to zero, and the maximum principal stress is the circumferential stress as follows:

rM1 ¼

q 2ð2c þ 1Þq ð5c þ 1Þq ¼  1c ð1  cÞ2 ð1  cÞ2

ð11Þ

where rM 1 denotes the maximum principal stress of point M when the compressive load is applied (MPa). It is assumed that the tensile strength of matrix point M corresponds to rMt . Point M is assumed to reach strength rMt when an external applied load corresponds to the macroscopic effective compressive strength Sc . Therefore, the effective compressive strength Sc of the concrete with a porosity of c is obtained based on Eq. (11) as follows:

ð1  cÞ2 Sc ¼  rMt ð5c þ 1Þ

ð12Þ

When the porosity of concrete approaches zero, the effective compressive strength Sc0 of non-porous concrete is obtained according to Eq. (11) as follows:

Sc0 ¼ rMt

ð13Þ

According to Eqs. (12) and (13), the quantitative relationship between the effective compressive strength and porosity is as follows: 2

Sc ð1  cÞ ¼ Sc0 5c þ 1

ð14Þ

2.4. Influence of pore structure on the effective tensile and compressive strengths Experiments conducted by Duan et al. [30] indicate that concrete with a higher fine porosity ratio and a reasonable pore size distribution exhibits a correspondingly higher compressive strength. Several studies indicated that the concrete pores are divided into C-S-H layer pores, micros-pores, and macros-pores and that C-S-H layer pores do not adversely affect the strength of concrete [4,31]. The strength of concrete is mainly determined by micro-pores and macro-pores, and the influence of large pores is evident. The pore structure of concrete is complex and the pore size reflects the leading characteristic of the pore structure. Accordingly, pore size is chosen as the representative index of pore structure for strength model. By considering the distribution and the contribution of the pore size, the total influence coefficient determined by the pore size is obtained as follows. First, the real pore size distribution of concrete is complex. Due to fact that the mechanical properties of concrete are close to rock mass, the pore size distribution function of concrete can be replaced by that of rock mass for convenience. A previous study [32] indicated that the pore size of the rock mass is consistent with an exponential function distribution as follows:

  d pðdÞ ¼ E exp  F

ð15Þ

where d denotes the pore size (mm); E and F denote the parameters of the exponential function. The value of parameters E and F are mainly determined by the water cement ratio, the cement sand ratio, aggregate content and type in the concrete mix proportion [33], and can be solved by fitting the test results of the pore structure. Second, based on the grey correlation analysis between pore size and concrete strength in literature [14], the values of the correlation grade have a power function to the pore size, and the large pores in the concrete leads to an increased weakened effect on the strength. The grey correlation analysis is a good measure to reflect the correlation among the factors [34], and thus it can be adopted to research the influence of pore size on the strength. Besides, the influence coefficient corresponds to zero when the pore size is equal to zero. Therefore, the influence function is described as follows: H

kðdÞ ¼ Gd

ð16Þ

where k denotes the influence coefficient related to the concrete pore size; G and H denote parameters of the influence function. The value of parameters G and H can be determined by fitting the test results of concrete strength and the grey correlation analysis. The exponential function and the influence function are combined, and the total influence coefficient is obtained as follows:

Z

dmax

kðdÞpðdÞdd K¼

dmin

Z

ð17Þ

dmax

pðdÞdd dmin

where K denotes the total influence coefficient determined by the pore size of concrete; dmin and dmax are the minimum radius and maximum radius of adverse pores (mm), respectively. Due to the complicated integration, Eq. (17) is replaced by Eq. (18) for simplicity:

Pn

K¼

i¼1 c i ki

ð18Þ

c

where ci and ki denote porosity and influence coefficient of the i-th interval when the range of pore size is divided into n intervals. ci is Rd dependent on the pore size distribution, i.e., ci ¼ diiþ1 pðdÞdd, and ki takes the influence coefficient of the pore size median of the i-th P interval. c is the total porosity, thus c ¼ ni¼1 ci . When the interval is smaller, the Eq. (18) is higher close to Eq. (17). According to Eqs. (10) and (14), the damage values of the concrete tensile and compressive strengths caused by the porosity 2

2

can be defined as Dt ¼ 1  ð1cÞ and Dc ¼ 1  ð1cÞ . The influence cþ1 5cþ1 of pore size on the effective strength is taken into account by multiplying the damage value by the total influence coefficient K; and the effective tensile and compressive strengths of concrete are expressed as follows:

" # St ð1  cÞ2 ¼ 1  KDt ¼ 1  K 1  St0 cþ1

" # Sc ð1  cÞ2 ¼ 1  KDc ¼ 1  K 1  Sc0 5c þ 1

ð19Þ

ð20Þ

3. Model verification 3.1. Comparisons of the proposed model with other models Concrete porosity approximately ranges from 9% to 21% as indicated by the Yaman’s experimental observations [35]. Therefore, the maximum porosity of 30% is considered in the present study.

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D. Li et al. / Construction and Building Materials 170 (2018) 520–526 2

ð1cÞ fitting test results corresponds to Sc ¼ 42:616 ð5cþ1Þ with an R2 value

Fig. 2. Comparisons of the present method and three classical solutions.

The quantitative relationship between porosity and the effective tensile strength is obtained based on Eq. (19). In order to verify the rationality of the proposed model, the magnitude of the total influence coefficient is considered 1, and the present approach and three classical results, namely Hansen’s model [5], Wischers’ equation and Du’s model [24] are compared as shown in Fig. 2. It is observed that the relationship between the effective tensile strength of concrete and porosity as obtained by the present approach is in good agreement with those models.

of 0.934 and an absolute error mean value of 4.691%. Second, the effect of pore size is considered on the strength of concrete, and the pore size range is divided into six intervals based on the experimental observations listed in Table 1. By fitting the test results and correlation grade between the pore size and the compressive strength of concrete, the regressive value of the compressive strength of non-porous concrete corresponds to 44.650 MPa, and the regressive values of parameters G and H in Eq. (16) correspond to 1.365 and 0.156 respectively. The influence coefficients of each interval are considered as that of the pore size median, and thus, based on Eq. (16), the parameters k1 , k2 , k3 , k4 , k5 and k6 correspond to 0.868, 1.015, 1.158, 1.276, 1.365, and 1.438, respectively. The predicted values of the effective compressive strength with considering the effect of pore size are shown in Table 1. It is obtained that the errors of the predicted values are less than 10.500% and that the mean absolute error corresponds to 3.474%. The predicted results that consider the effect of pore size were preferred over those that do not consider the effect of pore size because the results that consider the effect of pore size lead to a better prediction of the effective compressive strength with a lower value of the mean absolute error (3.474% when compared to 4.691%). Therefore, the effect of pore size is necessarily considered on the effective strength. The fitting results indicate that the proposed model is reliable and accurate for predicting the effective compressive strength of porous concrete.

4. Results and discussion 4.1. Influence of porosity on effective tensile and compressive strengths

3.2. Experimental verification of the proposed model The concrete samples with different porosity and pore size were obtained from Gao’s tests [14] by adding different kinds and amounts of air entraining agents. Due to the water-cement ratio and aggregate gradation of the samples unchanged, the test can reflect the relationship between the porosity and pore structure and the strength more reasonably. Therefore, the experimental observations obtained from Gao’s tests are adopted in order to verify the rationality and accuracy of the proposed model. First, a comparison between the theoretical results without considering the influence of pore size and the experimental observations are shown in Fig. 3. It is obtained based on Eq. (14) that the

The effects of porosity on the effective tensile and compressive strengths are different. Therefore, the same total influence coefficient K is maintained, and the relationships between the tensile and compressive strengths of concrete and porosity are analyzed as shown in Fig. 4. It is observed in Fig. 4 that with respect to the condition involving the same total influence coefficient of 1.2, the effective tensile and compressive strengths reduce with increases in porosity and the effective compressive strength is lower than the effective tensile strength. This illustrates that the effect of porosity on the effective compressive strength exceeds that on the effective tensile strength, and this is in good agreement with Chen’s conclusion [20]. The reason for the phenomenon is that the stress around the pore increases faster with porosity when the compressive load is applied. As shown in Fig. 2, the classical models, which not distinguish the relationships between the concrete tensile and compressive strengths and porosity, may lead to the decrease of accuracy in predicting the concrete compressive strength. 4.2. Influence of pore size on the effective tensile and compressive strengths The

same 0:156

Fig. 3. Comparison between predicted strength without considering pore size and Gao’s experimental observations.

porosity

of

5%

and

influence

function

kðdÞ ¼ 1:365d are maintained, and the effects of pore size are examined on the total influence coefficient and the effective tensile and compressive strengths. According to Eqs. (15), (17), (19) and (20), pore size range, the total influence coefficient and the effective tensile and compressive strengths are calculated as shown in Table 2. It is observed in Table 2 that with respect to decreases of maximum pore size for the same porosity, the total influence coefficient decreases and the effective tensile and compressive strengths of concrete increase. This illustrates that the effect of pore size on the effective strength of concrete should not be

525

D. Li et al. / Construction and Building Materials 170 (2018) 520–526 Table 1 A comparison of predicted strength and Gao’s experimental observations. Total porosity (%)

Grading porosity (%) >0.01–0.1 (mm)

>0.1–0.2 (mm)

1.05 0.1 0.11 1.81 0.36 0.45 2.62 0.38 0.61 3.31 0.37 0.79 3.16 0.62 0.97 4.12 0.61 1.39 4.86 0.72 1.75 8.21 1.06 2.56 2.51 0.44 0.64 4.62 0.46 0.94 1.95 0.28 0.34 2.11 0.31 0.39 2.29 0.32 0.4 3.54 0.55 0.9 5.78 0.5 1.55 6.07 0.56 1.76 8.28 0.81 2.22 2.6 0.28 0.62 5.87 0.31 1.51 1.66 0.2 0.29 2.47 0.21 0.23 6.74 0.57 1.6 6.81 0.48 2 7.09 0.63 1.85 8.29 0.9 2.38 6.7 0.23 1.55 7.18 0.52 2.07 7.33 0.55 1.87 5.35 0.49 1.31 8.34 0.92 2.41 8.68 0.82 2.36 The average of absolute error: 3.474%

>0.2–0.5 (mm)

>0.5–0.8 (mm)

>0.8–1.2 (mm)

>1.2–1.6 (mm)

0.18 0.41 0.73 1.04 0.9 1.31 1.67 2.74 0.57 1.44 0.5 0.55 0.62 1.11 1.87 1.89 2.82 0.75 2.36 0.5 0.84 2.81 2.71 2.51 2.96 2.86 2.44 2.7 1.79 2.94 3.07

0.24 0.3 0.35 0.55 0.33 0.43 0.42 1.07 0.41 0.89 0.36 0.37 0.45 0.57 0.89 0.93 1.21 0.43 0.96 0.32 0.45 1.05 0.93 1.09 1.15 1.07 1.02 1.14 0.82 1.09 1.26

0.2 0.15 0.35 0.36 0.23 0.25 0.16 0.5 0.21 0.56 0.25 0.31 0.23 0.29 0.65 0.55 0.75 0.28 0.49 0.18 0.44 0.53 0.47 0.55 0.59 0.59 0.63 0.68 0.52 0.57 0.7

0.21 0.14 0.2 0.2 0.1 0.13 0.15 0.28 0.24 0.32 0.24 0.17 0.26 0.12 0.32 0.39 0.46 0.23 0.24 0.17 0.29 0.19 0.22 0.45 0.31 0.4 0.49 0.39 0.41 0.41 0.47

Test strength (MPa)

K

Sc (MPa)

Error (%)

42 40.7 38 32.9 35.1 33.4 31.4 27.5 39.7 30.3 38.8 37.5 36.4 32.1 28 27.8 24.2 37.8 29.8 37.2 34.4 27.6 27.5 26.1 25 24.2 26.2 23.7 29.5 24 21.9

1.227 1.123 1.147 1.151 1.090 1.101 1.092 1.114 1.134 1.165 1.186 1.157 1.163 1.122 1.152 1.147 1.142 1.155 1.154 1.172 1.207 1.144 1.135 1.145 1.127 1.169 1.148 1.152 1.154 1.128 1.141

40.831 38.839 36.373 34.486 35.397 32.967 31.402 24.629 36.773 31.077 38.082 37.768 37.204 34.155 28.644 28.107 24.005 36.376 28.419 39.049 36.388 26.800 26.802 26.102 24.248 26.491 25.883 25.534 29.542 24.145 23.335

2.784 4.574 4.281 4.820 0.845 1.295 0.006 10.440 7.372 2.565 1.852 0.714 2.209 6.403 2.300 1.104 0.805 3.767 4.635 4.970 5.779 2.898 2.539 0.007 3.007 9.466 1.209 7.737 0.144 0.604 6.555

neglected and that the effective strength of concrete can be improved by decreasing the pore size.

5. Conclusion In this study, considering not only the different relations between the effective tensile and compressive strengths and porosity but also the influence of pore size on the strength, explicit formulas of the macroscopic effective tensile and compressive strengths of porous concrete are deduced. The developed analytical solution is compared with the classical solutions and experimental observations. The results show that the derived equations are rational and accurate to research the porous concrete strength. The concrete strength decreases with increase of porosity, and the decrease in the compressive strength is higher than that in the tensile strength. The effective strength of concrete with the same porosity can be improved by decreasing the pore size. Fig. 4. The effect of porosity on the effective tensile and compressive strengths of concrete.

Acknowledgment Table 2 The effect of pore size on the effective tensile and compressive strengths of concrete. Total porosity (%)

Parameter E

Parameter F

Pore size range (mm)

K

5 5 5 5 5

0.05 0.06 0.07 0.08 0.09

6 8 10 12 14

0.1–1.22 0.1–1 0.1–0.85 0.1–0.75 0.1–0.67

1.253 1.228 1.194 1.179 1.156

St/St0

The authors gratefully acknowledge the financial support of National Science Foundation of China (Grant No. 51379178).

Sc/Sc0

References 0.824 0.828 0.832 0.834 0.838

0.652 0.659 0.668 0.672 0.679

[1] J.J. Beaudoin, R.F. Feldman, P.J. Tumidaiski, Pore structure of hardened Portland cement pastes and its influence on properties, Adv. Cem. Based Mater. 1 (5) (1994) 224–236. [2] M. Röbler, I. Odler, Investigations on the relationship between porosity, structure and strength of hydrated Portland cement pastes I. effect of porosity, Cem. Concr. Res. 15 (2) (1985) 320–330.

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