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The relationship between porosity and strength for porous concrete

C.Lian a ,?,Y.Zhuge b ,S.Beecham a

a School of Natural and Built Environments,University of South Australia,Adelaide,South Australia,Australia b

Faculty of Engineering and Surveying,University of Southern Queensland,Brisbane,Queensland,Australia

a r t i c l e i n f o Article history:

Received 19November 2010

Received in revised form 5April 2011Accepted 23May 2011

Available online 14June 2011Keywords:

Porous concrete

Compressive strength Porosity

a b s t r a c t

As for many porous media,the strength of porous concrete is signi?cantly affected by the porosity of its internal structure.This paper describes the development of a mathematical model to characterize the relationship between compressive strength and porosity for porous concrete by analyzing empirical results and theoretical derivations.The suitability of existing equations for porous concrete is assessed and a new model is then proposed.The new model,which was derived from Grif?th’s theory,presents a better agreement with the experimental data for porous concrete.It is demonstrated that the proposed model could provide a better prediction of porous concrete compressive strength based on the material porosity.

1.Introduction

It is well known that the mechanical behavior of a building material is predominately dependent on its composited structure.The presence of pores can adversely affect the material’s mechan-ical properties such as failure strength,elasticity and creep strains [1].Porous concrete,which differs from conventional concrete,has a large volume of air voids.Currently it is mainly utilized in perme-able pavements and in?ltration beds [2].For maximizing the ben-e?t of its permeability,several studies have been conducted to reveal the relationship between pore features and the hydraulic or acoustic conductivity of porous concrete [3,4].But as a construc-tion material,porous concrete also needs to be able to withstand traf?c loads.It is also important to determine how its mechanical performance is affected by the presence of pores.

In a previous experimental investigation [5],the compressive strength of porous concrete has been tested.This testing could be used as an index to characterize the mechanical capacity of por-ous concrete in this study.On the other hand,the pore structure of a porous material can be characterized by a number of parameters including pore size,pore connectivity,pore surface roughness and pore volume fraction (porosity).Of these,the porosity is regarded as the primary parameter of porous material microstructures [6].Normally the strength of a porous material is in?uenced by poros-ity,the other parameters listed above having less in?uence.Thus,in this study the porosity is chosen as an independent variable to relate to the material strength.The objective of this study is to establish a quantitative relationship between porosity and com-pressive strength of porous concrete.

The in?uence of porosity on the strength of cement paste has al-ready been investigated [7–9].In these studies,hydrated cement paste was considered the main source of pores within conventional concrete.However,porous concrete contains a higher fraction of large macroscopic pores which are required to achieve suf?cient hydraulic conductivity.Therefore,the suitability of the existing relationships between porosity and strength developed for normal concrete need to be examined and extended for porous concrete.This is the purpose of the work described in this paper.

2.Experimental study 2.1.Mix compositions

The compositions used to prepare porous concrete in this study consisted of coarse aggregates,ordinary Portland cement and water.However,admixtures such as quarry sand,silica fume and superplasticizer were also used in some of the mixes to produce a strength variation.Two groups of samples were developed.The mix-ture proportions for each were summarized in Table 1.The ?rst group was pro-duced with only coarse aggregate,cement and water.Quartzite,limestone and dolomite were used as coarse aggregates and three gradings were selected (G1:13.2–4.75mm;G2:9.5–6.7mm;G3:9.5–4.75mm).The second group was made with additives including 7%silica fume and 0.8%superplasticizer by weight of ce-ment and some quarry sands as ?ne aggregates.For this second group,dolomite was used as the coarse aggregate.In this second group,the water to cement ratio was incrementally changed from 0.30to 0.38.In this way,samples of different strength and porosity were obtained.The preparation and mixing procedures are discussed in [5]https://www.wendangku.net/doc/b5942770.html,pressive strength

The compressive strength of porous concrete was determined through sample tests according to AS1012.9-1999.The samples were cylinders of 100mm diameter and 200mm height.After 24h the samples were removed from the steel moulds and moist cured until the day of testing.The curing condition complied with AS1012.8.1-2000.Prior to testing,the samples were weighed to determine the den-

Corresponding author.Tel.:+61883029941;fax:+61883025721.

E-mail address:chunqi.lian@https://www.wendangku.net/doc/b5942770.html,.au (C.Lian).

sity.The samples were then sulphur capped and the uncon?ned compressive strengths were tested at 7and 28days.The 28-day results were taken as represen-tative values of compressive strengths of porous concrete and the tested average values for each batch are listed in Table 1.2.3.Porosity

Ghafoori and Dutta [10]stated that the majority of pores in porous concrete are formed by the spaces left between coarse aggregates and they distinguished be-tween porosity and air void content.In their research,the fraction of measureable voids migrated by ?uids in their experiments was termed porosity and the sum of measureable voids between aggregates plus entrained or entrapped air in the ce-ment paste was termed air content.In other words,the porosity of porous concrete could be de?ned differently.In this study,for clarity,the measureable voids are de-?ned as the effective porosity since this relates to permeability and the overall air content is accordingly de?ned as total porosity.

2.3.1.Effective porosity

The effective porosity was determined by testing the volume of water displaced by samples.The sample was ?rstly oven dried at 110°C and then immersed in water for up to 24h.By measuring the difference in the water level before and after immersing the sample,the volume of water repelled by the sample (V d )can be readily determined.Subtracting V d from the sample bulk volume (V b )yields the vol-ume of open pores.This volume was then expressed as a percentage as an effective porosity percentage:p e =(V b àV d )/V b ?100%.

2.3.2.Total porosity

The strength of concrete is affected by the volume of its overall voids [9].In the complex microstructure of concrete,the pores can be present from the nano-scale to the macro-scale.The dif?culty of accurately testing the total porosity of porous concrete arises from its unique https://www.wendangku.net/doc/b5942770.html,pared with the pores within ce-ment paste,the interconnected voids between coarse aggregate are larger by sev-eral millimetres.Although it is well known that the method of mercury intrusion porosimetry (MIP)is effective for observing the pore con?guration in normal con-crete,the large amount of connected voids within porous concrete will cause drip-ping and leakage of mercury if pressure is applied.Thus,the method of MIP is not feasible for porous concrete.Vacuum sealing apparatus is more appropriate to test a relatively accurate porosity for porous concrete in laboratory research [11].How-ever,in practice,setting up such a delicate apparatus is challenging for concrete manufacturers and a simpler method is preferred.In the literature,Kearsley and Wainwright [12]have successfully used the Hoff equation [13]to estimate the total porosity of foam concrete.Similarly,Zheng [14]has presented an equation to esti-mate the total porosity of porous concrete,which was analogous to the Hoff equa-tion,but incorporated the aggregate proportions for porous concrete.This is shown in the following equation:

q t ?100tP c t0:25P c

100

tP c

q c

te0:25P c ?0:75T

?q w ;e1T

where q t is the theoretical density,P c is the cement to aggregate ratio by weight,q c

is the speci?c gravity of cement,q w is the unit weight of water and q is the aggregate apparent density.It can be seen that this equation was derived by understanding the

cement hydration process:0.25is taken as the ratio of hydration water to cement by weight,so the non-evaporable water mass is 0.25times the anhydrous cement mass P c ;and the volume of this water reduces to approximately 0.75of the original vol-ume after chemically hydrating the cement [9].Thus,the total porosity can be cal-culated as:

p ?1à

q b

q t

;e2T

where p is the total porosity and q b is the bulk density of the sample.

2.3.3.Relationship between effective porosity and total porosity

In this study,silica fume was employed in Group 2to improve the strength of porous concrete and consequently the gravity of q c was slightly adjusted.For calcu-lating q c ,the relative gravity for cement and silica fume was taken as 3.15and 2.2respectively.Hence,q c =(1+0.07)/(1/3.15+0.07/2.2)=3.063for Group 2.The mea-sured effective porosity and estimated total porosity for each mix are shown in Fig.1.The relationship is approximately linear.

The best ?tted regression line for the data is given by:

p ?1:28p e à18:11:e3T

The R 2value of 0.947shows a good correlation between measured effective porosity and estimated total porosity.A similar relationship was also determined by Ghafoori and Dutta [10]but with different coef?cient values.This could be attributed to the different mix proportions and compaction energy applied in mak-ing the samples.

3.Existing models for cementitious materials

Before creating a quantitative model to characterize the rela-tionship between porosity and compressive strength for porous concrete,it is worth noting that the in?uence of porosity on strength has been investigated for several engineering materials,such as ceramics,metals,plaster and rocks [1].The research pre-sented in this paper focuses developing a mathematical model be-tween total porosity and compressive strength of porous concrete.Historically,four general types of model have been developed [19,20]for cement-based materials,as summarized in Table 2.In Eqs.(a)–(c),the porosity (p )and the corresponding strength (f )of a porous material are related through a parameter,r 0,which is the material strength when porosity is zero.In Eq.(d),p 0is the porosity when the material has zero strength.Chindaprasirt et al.[21,22]have demonstrated that the exponential relationship (type c in Table 2)proposed by Ryshkevitch [17]was valid for describing porous concrete.Fig.2shows how an exponential regression equa-tion can be ?tted to the data from the present study.

From Fig.2,the ?tted exponential curve yields the equation:r =231exp (à0.09p ),with an R 2value of 0.90,which is lower than the value of 0.96obtained by Chindaprasirt et al.[22].This may be

Table 1

28-Day mix proportioning and compressive strength of porous concrete.Sample number

Mix proportions Compressive strength (MPa)

Coarse aggregate

W /C ratio Percentage of sand Group 1(no additives)1–1Q-G20.36011.8

1–2Q-G30.36015.51–3L-G20.36015.51–4L-G30.36014.01–5D-G10.36015.51–6D-G20.36015.81–7D-G3

0.36

19.0

Group 2(7%silica fume and 0.8%plasticizer by weight of cement)2–1D-G30.381823.32–2D-G30.361833.22–3D-G30.341846.22–4D-G30.321840.52–5D-G30.301840.32–6D-G30.301543.02–7D-G30.281824.3

Note :(1)Q –quartzite;L –limestone;D –dolomite;(2)the percentage of sand was based on the weight of coarse

aggregate.

C.Lian et al./Construction and Building Materials 25(2011)4294–4298

4295

ity.Grif?th found that the critical stress incurs crack propagation within a brittle material and can be expressed by:

????????

2E c

p a

;e4T

where r is the stress at the fracture(Pa),E is the elasticity modulus (Pa),c is the fracture surface energy(J/m2)and a is the half length of an internal crack(m).

When considering this criterion for a porous material,the effec-tive value of E and r need to be determined.This is because the presence of pores affects both elasticity and fracture energy.The elasticity and fracture energy are both reduced compared to the pore-free solid material.After mixing well and compacting the por-ous concrete,the cement paste wraps around the aggregate and behaves as one unit with air voids.Therefore all the pores,includ-ing both the large interconnected voids between aggregates and the small ones in the paste,are taken as defects that can lead to fracture of the porous concrete.The failure stress of the paste ma-trix can be determined by Eq.(4).

Various equations have been developed to describe the in?u-ence of pore content on Young’s Modulus and surface energy for different materials[24,25].In the present study,two different methods were considered when choosing the appropriate empiri-cal equations for E and c to use in Eq.(4).Firstly,Rice[26]observed the reduction of Young’s modulus as E=E0exp(àtp),where E0is the elastic modulus of the material at zero porosity with t as a con-stant.He also determined the fracture energy of pores as:c=c0 exp(àqp),where c0is the fracture energy at zero porosity and q is a constant.This showed the variation of the fracture energy would be the same as that of the Young’s modulus in terms of porosity.If these empirical relationships are assumed for porous concrete,the following relationship can be derived from Eq.(4): r?

?????????????????????????????

2E0eàqp c0eàtp

p a

?????????????

2E0c0

p a

eàmp?keàmp;e5Twhere k and m are constants.

This equation is similar to the empirical one which was ob-served by Chindaprasirt et al.[22].Kendall et al.[27]proposed an alternative method.They applied a different function of Young’s modulus:E=E0(1àp)3with fracture energy c=c0exp(àtp)into a fracture criterion formula and the predicted failure stress was compared with the test results from their study of concrete made Fig.3.The proposed model for compressive strength versus porosity.

with polymers.In light of this,a combined functional model based on this approach is proposed for porous concrete as shown in the following equation:

???????????????????????????????????????

2E0e1àpTm c0eànp

p a

;e6T

where m and n are new material constants for porous concrete. 4.2.Examination of the proposed model

To assess the validity of the proposed model,a regression anal-ysis was performed on Eq.(6)based on the available experimental data.The proposed model was shown in Fig.3.It can be seen from the?gure that Eq.(6)is complicated as a combined format of power and exponential relations.Thus,in order to utilize a linear least-squares regression technique,Eq.(6)has to be rearranged:

First of all,2E0c0

p a ?A is assumed,regardless of the possible differ-

ent pore sizes formed in different samples.Then:

??????????????????????????????

Ae1àpTm eànp

Squaring both sides and taking the natural logarithm of each

side:

2ln r?m lne1àpTànptln A;e7T

which now can be regarded as a linear equation of the form:

Y?mx1tnx2tc;e8T

with Y?2ln r;x1?lne1àpT;x2?p and c?ln A:e9T

The values of Y,x1and x2are calculated in Table3.

4.3.Results and discussion

The multiple linear regression run by least square method gen-

erates the best?tted plane and parameters,as shown in Fig.4.The

regression results are:m=5.96and n=à10.01when c=10.61for

Eq.(8),which is:Y=5.96x1à10.01x2+10.61.The coef?cient of

determination R2for this equation is estimated to be0.99and

the standard error of estimated Y is0.306.This indicates that the

model could describe the correlation between compressive

strength and porosity for porous concrete with acceptable

Table3

Analytical and experimental data for modeling.

Sample number Cement to aggregate

ratio

Speci?c gravity of

binder

Aggregate density(?103

kg/m3)

Sample

density

Total porosity

(%)

Compressive strength

(MPa)

Y x1x2

1–10.22 3.15 2.65183130.9012.00 4.97à0.370.31

0.22 3.15 2.65173434.5712.00 4.97à0.420.35

0.22 3.15 2.65176333.4811.50 4.89à0.410.33

1–20.22 3.15 2.65188029.0617.50 5.72à0.340.29

0.22 3.15 2.65180032.0814.50 5.35à0.390.32

0.22 3.15 2.65184030.5714.50 5.35à0.360.31

1–30.22 3.15 2.74198527.5515.50 5.49à0.320.28

0.22 3.15 2.74210523.1719.50 5.94à0.260.23

0.22 3.15 2.74194728.9411.50 4.88à0.340.29

1–40.22 3.15 2.74196028.4615.50 5.48à0.340.28

0.22 3.15 2.74192029.9213.00 5.13à0.360.30

0.22 3.15 2.74190030.6513.50 5.21à0.370.31

1–50.22 3.15 2.70194028.1517.00 5.67à0.330.28

0.22 3.15 2.70194028.1516.50 5.61à0.330.28

0.22 3.15 2.70190029.6313.00 5.13à0.350.30

1–60.22 3.15 2.70186331.0015.00 5.42à0.370.31

0.22 3.15 2.70189529.8117.00 5.67à0.350.30

0.22 3.15 2.70188030.3715.50 5.48à0.360.30

1–70.22 3.15 2.70192028.8917.00 5.67à0.340.29

0.22 3.15 2.70198026.6722.50 6.23à0.310.27

0.22 3.15 2.70192028.8917.50 5.72à0.340.29

2–10.25 3.06 2.70208022.9630.50 6.84à0.260.23

0.25 3.06 2.70206023.7031.50 6.90à0.270.24

0.25 3.06 2.70204024.4428.00 6.66à0.280.24

2–20.25 3.06 2.70212021.4734.507.08à0.240.21

0.25 3.06 2.70212021.4732.00 6.93à0.240.21

0.25 3.06 2.70214020.7333.00 6.99à0.230.21

2–30.25 3.06 2.70224017.0249.007.78à0.190.17

0.25 3.06 2.70224017.0246.507.68à0.190.17

0.25 3.06 2.70224017.0243.007.52à0.190.17

2–40.25 3.06 2.70216019.9839.507.35à0.220.20

0.25 3.06 2.70220018.5042.007.48à0.200.19

0.25 3.06 2.70214020.7240.007.38à0.230.21

2–50.25 3.06 2.70218019.2441.007.43à0.210.19

0.25 3.06 2.70218019.2441.007.43à0.210.19

0.25 3.06 2.70214020.7239.007.33à0.230.21

2–60.25 3.06 2.70214020.7242.007.48à0.230.21

0.25 3.06 2.70214020.7244.007.57à0.230.21

0.25 3.06 2.70220018.5043.007.52à0.200.19

2–70.25 3.06 2.70196027.3923.00 6.27à0.320.27

0.25 3.06 2.70196027.3926.50 6.55à0.320.27

0.25 3.06 2.70192028.8723.50 6.31à0.340.29

C.Lian et al./Construction and Building Materials25(2011)4294–42984297

accuracy.Moreover,the F statistic is calculated as4361.7,with an extremely small probability of1.857Eà42,indicating that the ob-served relationship did not occur by chance.This means that the proposed model is reliable for predicting the failure compressive strength of porous concrete.

In addition,the proposed model predicts a zero strength when the material is assumed to be fully porous,conquering the limita-tion of the exponential function which cannot make sense when the porosity is close to1.Therefore it offers a wider range for appli-cation.On the other hand,it is noticed that while the current pro-posed model is inclusive of all three aggregate types used in this study,some factors such as aggregate shape and absorption were not accounted for separately.Another constant B is suggested to be employed in the proposed Eq.(6)for future work to account for these additional factors.In this case,a general format for this future model could be:

r?B ???????????????????????????e1àpTm eànp

;

where B could be determined when these additional factors are tested and quanti?ed.

5.Conclusions

The dependence of compressive strength on porosity for porous concrete was analysed empirically and theoretically in this paper. The following conclusions can be drawn:

(1)The effective porosity of porous concrete has been mea-

sured.However,since the non-intrusive pores weaken the strength of concrete,the total porosity was estimated and then compared with the effective porosity.It has been dem-onstrated that the estimated total porosity has a good corre-lation with the measured effective porosity.This estimation method could be used when total porosity testing apparatus is not available.

(2)Existing equations relating compressive strength and poros-

ity for cement-based materials were presented and a poten-tial equation for porous concrete has been appraised by ?tting to the experimental data.It has been shown that without extra knowledge of paste strength,the exponential function derived using experimental data resulted in a rela-tively low correlation coef?cient.

(3)With a large set of data on porosity and tested compressive

strength,a new model using Grif?th’s fracture theory has been proposed.It has been shown that the proposed model provides a stronger relationship between the compressive strength and the porosity of porous concrete,with a model regression statistic R2of up to0.99.This represents a signif-icant improvement over the simple exponential equation.

Other statistics also veri?ed that this semi-empirical model could predict the compressive strength of porous concrete based on the material porosity.

Acknowledgment

The authors would like to express their special thanks to Mr. David Carver for his technical assistance during the experimental work.

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Useful Expressions: Words and phrases Friends and communication: solidify/ strengthen/ enhance/ promote communication / connection with mutual understanding relationship network/circle of f r i e n d s cultivate/develop friendship with s b . keep steady relationship with sb. establish interpersonal networksac build up the social circle spur message transmission Knowledge and experience widen one’s outlook broaden one’s vision/horizon acquire knowledge and skills comprehensive/overall quality

expand/enlarge one’s scope of knowledge knowledge reserve/base/storage theoretical knowledge practical skills social experience broaden one’s knowledge base promote one’s overall/ comprehensive competence accumulate experiences learn lessons from past experiences Work and experience the scarcity of employment o p p o r t u n i t i lay the foundations for career p r o s p e r i t y

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Who—My most favourite singers include Adele, James Blunt, Avril Lavigne, and so forth. ( 喜欢的歌手) Why—I am fond of music mainly because it can cheer me up greatly when I feel down/low/ blue/bored/tired/depressed. ( 心情不好的时候让我高兴起来)?Besides, I also believe that music is an indispensable part of culture and tradition, through which I could have a better understanding of different cultures around the world, including cowboy culture, African-American street culture, the three main reli- gions and so on. ( 有助于理解不同的文化) Do you like watching movies?? A—Speaking of movies, yes, I am a big fan of all types of movies, such as...?What—comedy, action, romance, sci-fi, manga, vampire, zombie, animation...

英语中冠词的用法以及练习题

英语冠词的用法一、冠词的基本概念冠词，是用来表示名词特性的一种词，它表示名词可数不可数、单数复数、任何一个还是特定的一个等。也就是说，冠词与名词是紧密连在一起的。如果用了a /an /the，不管后面是什么词，都变成了名词。所以，冠词就是放在名词前面的一种词。使用冠词有三种基本情况： 1、用a / an a / an 用在单数可数名词前面，表示这个人或事物是泛指的不确定的一个，相当于中文的“一个”。注意：可数的单数名词前，必须要用a / an 或the. 我们不能说boy, desk, dog，而必须说 a boy / the boy. a 用在读音为辅音开头的名词之前, 而an 用在读音为元音开头的名词之前。注意：这里指的是“读音”，而不仅仅指字母。例如： a university 一所大学(虽然u 是元音字母，但不读元音。) an hour 一个小时(虽然h 不是元音，但单词读音是元音开头) 另外，a / an 也可用在不可数名词前面，这时，这个不可数名词就变成了可数的单数名词，意义上也有变化。例如： glass 玻璃, a glass 一只玻璃杯/ wood 木头, a wood 一片树林 power威力, a power大国/ beauty 美丽, a beauty美人, 美的事物 2、用the the 相当于this/that 或these / those，总的用法是表示特定的人或事物。the 的用法最广，不管是可数还是不可数，不管是单数还是复数，都可以用the. 具体用法见下面的讲解。 3、不用冠词有的时候，名词前面不用任何冠词，没有a / an / the. 有的书上称为零冠词。不可数名词前可以不用冠词，可数的复数名词也可以不用，还有一些习惯用法上不需要用。下面，详细讲一个冠词的用法。二、不定冠词的用法: 1、表示某一类人或某事物中的任何一个, 通常在第一次提到某人或某物时用a / an，以表示与其他事物的区别。 I gave him a book yesterday. 我昨天给了他一本书。 I am reading an interesting story. 我在读一本有趣的故事书。 I have got a ticket. 我有一张票。 There is a tree in front of my house. 我的屋前有一棵树。 2、用其中的任何一个，代表他们所属种类的特性。 A horse is useful to mankind. 马对人类有用。 A bird can fly. 鸟会飞。 A steel worker makes steel. 炼钢工人炼钢。 3、用在事物的度量单位前, 如时间、速度、价格等, 表示每一个。 1