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 revid 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 significantly 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 assd and a new model is then propod.The new model,which was derived from Griffith’s theory,prents a better agreement with the experimental data for porous concrete.It is demonstrate
d that the propod model could provide a better prediction of porous concrete compressive strength bad on the material porosity.
Ó2011Elvier Ltd.All rights rerved.
1.Introduction
It is well known that the mechanical behavior of a building material is predominately dependent on its composited structure.The prence of pores can adverly 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 infiltration beds [2].For maximizing the ben-efit of its permeability,veral 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 traffic loads.It is also important to determine how its mechanical performance is affected by the prence of pores.
In a previous experimental investigation [5],the compressive strength of porous concrete has been tested.This testing could be ud as an index to characterize the mechanical capacity of por-ous con
crete 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 the,the porosity is regarded as the primary parameter of porous material microstructures [6].Normally the strength of a porous material is influenced by poros-ity,the other parameters listed above having less influence.Thus,in this study the porosity is chon 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 influence of porosity on the strength of cement paste has al-ready been investigated [7–9].In the 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 sufficient 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 purpo of the work described in this paper.
2.Experimental study 2.1.Mix compositions
The compositions ud to prepare porous concrete in this study consisted of coar aggregates,ordi
nary Portland cement and water.However,admixtures such as quarry sand,silica fume and superplasticizer were also ud 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 first group was pro-duced with only coar aggregate,cement and water.Quartzite,limestone and dolomite were ud as coar aggregates and three gradings were lected (G1:13.2–4.75mm;G2:9.5–6.7mm;G3:9.5–4.75mm).The cond group was made with additives including 7%silica fume and 0.8%superplasticizer by weight of ce-ment and some quarry sands as fine aggregates.For this cond group,dolomite was ud as the coar aggregate.In this cond 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 discusd in [5].2.2.Compressive 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-
0950-0618/$-e front matter Ó2011Elvier Ltd.All rights rerved.doi:10.buildmat.2011.05.005
Corresponding author.Tel.:+61883029941;fax:+61883025721.
E-mail address:chunqi.lian@unisa.edu.au (C.Lian).
sity.The samples were then sulphur capped and the unconfined compressive strengths were tested at 7and 28days.The 28-day results were taken as repren-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 coar aggregates and they distinguished be-tween porosity and air void content.In their rearch,the fraction of measureable voids migrated by fluids 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 defined differently.In this study,for clarity,the measureable voids are de-fined as the effective porosity since this relates to permeability and the overall air content is accordingly defined 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 firstly oven dried at 110°C and then immerd 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 expresd as a percentage as an effective porosity percentage:p e =(V b ÀV d )/V b Â100%.
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 prent from the nano-scale to the macro-scale.The difficulty of accurately testing the total porosity of porous concrete aris from its unique microstructure.Compared with the pores within ce-ment paste,the interconnected voids between coar aggregate are larger by v-eral millimetres.Although it is well known that the method of mercury intrusion porosimetry (MIP)is effective for obrving the pore configuration in normal con-crete,the large amount of connected voids within porous concrete will cau drip-ping and leakage of mercury if pressure is applied.Thus,the method of MIP is not feasible for porous concrete.Vacuum aling apparatus is more appropriate to test a relatively accurate porosity for porous concrete in lab
oratory rearch [11].How-ever,in practice,tting 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 ud the Hoff equation [13]to estimate the total porosity of foam concrete.Similarly,Zheng [14]has prented 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 ¼100þP c þ0:25P c
100
q
þP c
q c
þð0:25P c Â0:75Þ
Âq w ;ð1Þ
where q t is the theoretical density,P c is the cement to aggregate ratio by weight,q c
is the specific gravity of cement,q w is the unit weight of water and q is the aggregate apparent density.It can be en 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 ¼1À
q b
q t
;ð2Þ
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 conquently 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 fitted regression line for the data is given by:
p ¼1:28p e À18:11:ð3Þ
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 coefficient 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 influence of porosity on strength
has been investigated for veral engineering materials,such as ceramics,metals,plaster and rocks [1].The rearch pre-nted in this paper focus 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-bad 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)propod by Ryshkevitch [17]was valid for describing porous concrete.Fig.2shows how an exponential regression equa-tion can be fitted to the data from the prent study.
From Fig.2,the fitted 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)
Coar 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 bad on the weight of coar
aggregate.
C.Lian et al./Construction and Building Materials 25(2011)4294–4298
4295
ity.Griffith found that the critical stress incurs crack propagation within a brittle material and can be expresd by:
r¼
ffiffiffiffiffiffiffiffi
2E c
p a
r
;ð4Þ
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).
莲藕汤When considering this criterion for a porous material,the effec-tive value of E and r need to be determined.This is becau the prence 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 influ-ence of pore content on Young’s Modulus and surface energy for different materials[24,25].In the prent study,two different methods were considered when choosing the appropriate empiri-cal equations for E and c to u in Eq.(4).Firstly,Rice[26]obrved 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 the empirical relationships are assumed for porous concrete,the following relationship can be derived from Eq.(4): r¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E0eÀqp c0eÀtp
p a
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2E0c0
p a
r
eÀmp¼keÀmp;ð5Þwhere k and m are constants.
This equation is similar to the empirical one which was ob-rved by Chindaprasirt et al.[22].Kendall et al.[27]propod 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 propod model for compressive strength versus porosity.
with polymers.In light of this,a combined functional model bad on this approach is propod for porous concrete as shown in the following equation:
r¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E0ð1ÀpÞm c0eÀnp
p a
日出的诗句
s
;ð6Þ
where m and n are new material constants for porous concrete. 4.2.Examination of the propod model
To asss the validity of the propod model,a regression anal-ysis was performed on Eq.(6)bad on the available experimental data.The propod model was shown in Fig.3.It can be en from thefigure 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 ¼A is assumed,regardless of the possible differ-
ent pore sizes formed in different samples.Then:
r
颜的词语
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Að1ÀpÞm eÀnp
q
:
Squaring both sides and taking the natural logarithm of each
side:
小学生家长寄语
2ln r¼m lnð1ÀpÞÀnpþln A;ð7Þ
which now can be regarded as a linear equation of the form:
Y¼mx1þnx2þc;ð8Þ
with Y¼2ln r;x1¼lnð1ÀpÞ;x2¼p and c¼ln A:ð9Þ
The values of Y,x1and x2are calculated in Table3.
4.3.Results and discussion
The multiple linear regression run by least square method gen-
修改病句的口诀erates the bestfitted 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 coefficient 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
手机号查物流strength and porosity for porous concrete with acceptable
Table3
Analytical and experimental data for modeling.
Sample number Cement to aggregate
ratio
Specific gravity of
binder
Aggregate density(Â103
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
机械工程及自动化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 as 4361.7,with an extremely small probability of 1.857E À42,indicating that the ob-rved relationship did not occur by chance.This means that the propod model is reliable for predicting the failure compressive strength of porous concrete.
In addition,the propod 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 n when the porosity is clo to 1.Therefore it offers a wider range for appli-cation.On the other hand,it is noticed that while the current pro-pod model is inclusive of all three aggregate types ud in this study,some factors such as aggregate shape and absorption were not accounted for parately.Another constant B is suggested to be employed in the propod Eq.(6)for future work to account for the additional factors.In this ca,a general format for this future model could be:
r ¼B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1Àp Þm e Ànp q ;
where B could be determined when the additional factors are tested and quantified.5.Conclusions
The dependence of compressive strength on porosity for porous concrete was analyd 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 ud when total porosity testing apparatus is not available.
(2)Existing equations relating compressive strength and poros-ity for cement-bad materials were prented and a poten-tial equation for porous concrete has been appraid by fitting 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 coefficient.
(3)With a large t of data on porosity and tested compressive
strength,a new model using Griffith’s fracture theory has been propod.It has been shown that the propod model provides a stronger relationship between the compressive strength and the porosity of porous concrete,with a model regression statistic R 2of up to 0.99.This reprents a signif-icant improvement over the simple exponential equation.Other statistics also verified that this mi-empirical model could predict the compressive strength of porous concrete bad 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.References
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Fig.4.Relationship between Y and X 1,X 2.
Materials 25(2011)4294–4298
